Gate Two: An Analytic-Synthetic Analysis of Science

From the book A Presence and an Absence by Rabbi Michael Avraham. Translated from Hebrew using gpt-5.4 (reasoning_effort=high, batch API).

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An Analytic-Synthetic Analysis of Science: Characteristics and Methodology

Introduction

Science is regarded by many in the modern age as the principal anchor of rationality and progress. One indication of this is that various fields appropriate for themselves the aura of science, and use scientific—or pseudo-scientific—terminology in order to reinforce recognition of them as reliable bodies of knowledge.¹

At first glance, this is a highly puzzling phenomenon. The picture that emerged from the account in the first book was that our age is characterized, more than earlier ages, by skepticism, expressed in analytic and postmodern outlooks. This is the result of a historical development that began in mythical-synthetic ages and continued into the scientific-analytic present. We saw there that this process reached its peak in recent decades, when the myth of rationality itself also entered into crisis, and thus the analytic conception was actualized in full. As we saw, in the second half of the twentieth century, allegedly modernist conceptions were exposed in their postmodern nakedness. It suddenly became clear that modernism, for the most part, is nothing but a shaky covering, a fig leaf, for the postmodernism hidden within it.

What is strange in this situation is that the public attitude toward science does not seem to suffer from these symptoms.² The domain of scientific thought, which is patently synthetic—as will become clear below—enjoys an almost mythological aura of rationality and trust, and this even, and perhaps especially, in the skeptical-analytic-postmodern age in which we live. The claim that a certain assertion is “scientific” constitutes, not always justly, as we shall see below, a decisive argument against any opponent.

Beyond all this, the strangest phenomenon is that it is specifically those of analytic disposition, whose confidence in scientific truth ought to have been the most fragile, who often attack those holding synthetic positions—for example, religious or traditional people—by arguing that their positions do not accord with the results of scientific research. That is, in quite a few cases science is perceived as part of the analytic stance, and is used to attack synthetic positions, or various “myths,” on the grounds that these are “unscientific” domains.

In order to ground this trust, various thinkers developed analytic interpretations of scientific activity. As we shall see in the present gate, these are puzzling interpretations that do not withstand the test of common sense. We shall see here that confidence in science can rest only on a synthetic conception, and that there is no reasonable way to escape this conclusion.

The basic explanation for analytic trust in science is that in a skeptical age such as ours, the rational human being—who is synthetic at bottom—must create for himself a mythology of new concepts of rationality, as an alternative to the “dead God.” This is another example of the phenomenon that in the first book we called Bokononism,³ or the Copernican revolution. A person cannot live in a vacuum, and therefore adopts principles of thought and contents to which he grants the credit of rationality. As a result, they receive an almost mythological, or mystical, status and unconditional trust, often unjustifiably. Ironically, adherence to these principles is often stronger than the adherence of many religious people to their own synthetic, that is, mythological, principles. Science is, to a large extent, the myth of the modern world.⁴ In the next gate we shall discuss myth, including its expressions in the scientific world.

In the first book this process was presented on the philosophical plane and on the conceptual-ideological-value plane. Here we shall see it on the plane of rational thought itself. As we saw in that book, and as will also become clear from the discussion in the present one, it is precisely the synthetic stance, and not the analytic one, that can make coherent and rational use of scientific claims and of the scientific method.

It should be emphasized that those who hold various synthetic positions also believe in those same principles of scientific rationality, except that for them this belief is not a case of Bokononism. Those who hold a synthetic position have a principled justification for believing in these principles and forms of thought. Just as we saw in the first book with respect to various values, so too we shall see below with respect to scientific thought and its assumptions: it can be genuinely grounded only in synthetic ways. Of course, according to the synthetic approach these principles must be placed in proper proportion, and certainly must not be worshiped as though they were beyond challenge. As we shall see below, there are also synthetic critiques of the methodology of science and of its contents, but the synthetic critique is not skeptical or sweeping, like the analytic one; it is substantive. It points to the limits and proportions according to which scientific determinations should be examined, but it does not undermine the very possibility of advancing in the various fields of knowledge.

Bechler, in the above-mentioned book, discusses this point at length. Those who hold the analytic position—actualistic, in his terminology—seek certainty. They are unwilling to live with any doubt, at least in the sciences, and precisely for that reason they adopt analytic positions. In their view, only logical arguments are entitled to the degree of absolute certainty that science deserves, and therefore they construe science as absolute truth. For exactly this reason they are compelled to maintain that science operates on the logical-analytic plane and says nothing whatever about the world. Any claim about the world, being synthetic by its very nature, cannot be certain, and therefore, from the analytic point of view, it is expelled from the domain of science.⁵

Those who hold the synthetic position—informativist, in Bechler’s terminology—are prepared to accept uncertain truths, and therefore a theoretical opening is available to them for making synthetic claims, that is, statements about the world, despite the doubt inherently involved in their consequences. The holder of the synthetic position does not grant science absolute credit, although he certainly does adopt its conclusions, at least tentatively.⁶

In this gate I wish to examine the nature of the domain called science, and to see why, if at all, it may be regarded as a rational body of knowledge. In light of this, in the next gates we shall continue the discussion of the proper attitude toward myth. We shall also try to clarify whether science has the power to attack myth, which is generally perceived as a prominent representative of the synthetic mode.

These subjects have received, and continue to receive, broad treatment in the literature.⁷ Many of the basic points that follow—especially in the first chapter—are certainly not original, but a considerable part of the public, including many scientists, is unaware of them. I hope and believe that there will be something new here even for readers already familiar with these issues. In any event, this gate is a necessary introduction to the gates that follow.

Chapter One: The Naive Picture: Francis Bacon’s Inductivism and the Critiques of It

The Empiricism of Science

Science, as it is currently understood, is a rational and comprehensive reservoir of knowledge about the nature of the world. In antiquity, there prevailed a conception that saw science as the comprehensive formulation of a priori, or philosophical, intuitions about the world.⁸ Beginning in the modern era, it became increasingly clear that one of the important criteria for classifying some activity as scientific is that the theories advanced within it be subjected to experimental, that is, empirical, tests. In other words, scientific theories are required to fit reality.

We have already pointed out that the conception according to which the concepts and basic principles of science are drawn directly from experience is naive, and we presented Hume’s critique of this conception. In the first book we already proposed an alternative that allows science nevertheless to be seen as derived from experience, in a broader sense.⁹ Here we shall try to describe this process in its scientific context, in greater detail and breadth. At the end of the present gate the synthetic alternative will be presented, that is, the approach that relates to science as a synthetic domain, one that makes claims about the world.

The Methodology of Science According to Francis Bacon

The first systematic description of scientific work is generally attributed to Francis Bacon, one of the fathers of modern empiricism. According to Bacon, the process of scientific inquiry is composed of the following stages: first, the collection of facts. Then their classification and analysis, and finally the derivation from them of theoretical conclusions by means of generalization—induction. These conclusions are usually formulated as general laws of nature, from which the particular experimental facts, or the particular natural phenomena, are derived. After that, one identifies the experimental consequences—predictions—of the proposed theory, which can be empirically tested. Finally, experiments are conducted, and if in light of them the theory requires improvement or replacement, the whole process begins again. Thus, according to Bacon, we continually refine and improve the scientific knowledge in our possession so that it corresponds to more and more experimental facts.

Let us take Newton’s theory of gravitation as an example. According to this description, the scientist observes a collection of facts, namely, various bodies falling to the earth, and perhaps also additional facts concerning the heavenly bodies. He then infers from this a scientific generalization, which in our case is Newton’s theory, according to which every two masses attract one another with a force proportional to the product of the masses and inversely proportional to the square of the distance between them. The theory is then subjected to experimental testing. For example, two masses are placed in an area as free as possible of external influences, and their acceleration under the influence of the force acting on them is measured. Since, according to Newton’s second law of mechanics, acceleration is proportional to force, one can confirm or refute the scientific hypothesis proposed by Newton.

Clearly, a parallel account can be given in every field of science. In medicine as well, or in the social sciences, a hypothesis is proposed that describes a set of facts known to us by means of a general law, and then an experiment is devised to test the theory. In light of the experimental results, the theory in question is adopted, improved, or rejected.

Bacon, and after him John Stuart Mill, elaborated and refined the inductive method of science, and tried, as far as possible, to purify and distill it from the problems that accompany it, some of whose difficulties we already discussed in the first book. Bacon sought to neutralize as much as possible the unfounded assumptions that characterized ancient science, most of which were not drawn from experience but from various a priori considerations. Bacon saw the main failure of ancient science in its departure from pure empiricism. Therefore, in his eyes, the essential novelty of modern science was precisely its empiricism. Unlike Aristotle, for example, who based scientific principles on assumptions that seemed necessary to him for various a priori philosophical reasons, Bacon aspired to base all scientific knowledge and scientific method on experience, that is, on the inductive learning of comprehensive laws from collections of particular facts. Here began a central stage in the important transition from rationalist science in antiquity to empiricist science in the modern era.

Ancient Aristotelian science began with the subject, that is, with the human being and his philosophical-a priori assumptions, and ended with the object, that is, the world itself, in claims about real phenomena. Baconian science, by contrast, begins with the objective world, with the collection of experimental facts, passes through the subject, namely the scientist, who makes an inductive theoretical generalization, and finally returns to the objective world, establishing laws that make claims about the world itself.

According to Bacon, this yields a picture of scientific theory as a theory that can be empirically proved, or verified. As we shall see below, this is a naive, high-altitude description of scientific work. Once one enters a bit more deeply into the details of scientific methodology, one sees that this seemingly pastoral picture contains many minefields, and we now wish briefly to consider some of them.

Karl Popper on Verification and Falsification

The first problem we shall discuss is the problem of confirming a scientific theory, as presented by the philosopher of science Karl Popper.

A simple example of this problem may be seen if we consider the scientific theory that states: all ravens are black. Seemingly, in order to confirm this theory empirically one would have to observe all ravens now living—and perhaps also those that will be born in the future, as well as those that died in the past—and determine whether they are black or not. If we understand that this is indeed what would be required in order to confirm the theory, then a scientific theory would be valid only if it met the criteria that in the first book we called analytic. As we may recall, adherents of the analytic approach are prepared to recognize as correct or valid only a claim proved on the basis of logical tautologies or direct empirical observations. Accordingly, it is clear that according to their approach, in order to adopt the theory that all ravens are black, the scientist must observe all ravens and ascertain that they are indeed black.¹⁰

A full empirical examination of theories that are relevant to an infinite number of objects or events is, of course, impossible. More than that: even where the theory concerns a finite number of objects or events, the analytic principle of confirmation just described would be plainly irrelevant. Even if there were a finite number of ravens, and we could theoretically observe all of them and verify that they are black, in that case the scientific theory would have no value, since we would already know all the particulars that make up the generalization. The value of a scientific theory lies in a determination that emerges from a generalization and can teach us additional and new facts, including about events or objects that we have not observed. It is unreasonable to think that the collection of facts all of which we have directly observed is the heart of the scientific process. The heart of the scientific process is the generalization from observed facts and its application to a broader, perhaps infinite, set of phenomena. This application is expressed in the formulation of a comprehensive law of nature, and the predictions of the theory—those by means of which we shall subject it to empirical testing—will be derived from applying that comprehensive formulation to cases we have not yet observed.

This description depends deeply on the synthetic character of science. As stated above, synthetic thought—that is, thought built upon a generalization of already existing knowledge, and which therefore says things about the world beyond what is already known from direct observation—is the only kind that has the value of adding knowledge about the world, or, in Bechler’s term, informativeness. Observing all ravens in order to confirm such a theory is an analytic process, and therefore has no scientific value. In such a case, the theory adds nothing to our knowledge of the world beyond what is already known to us from direct observation. This is another aspect of the emptiness of the analytic, discussed in the previous gate.

From this brief discussion it follows that science is fundamentally a synthetic process of thought. In the example we saw, we observe some ravens, see that they are black, and determine by means of generalization—scientific induction—that all ravens are black. This determination has value as a scientific theory only so long as not all ravens have been observed. Once they all have been observed, it ceases to be a theory and becomes a collection of facts. Facts are something neutral. They are neither scientific nor unscientific; at most they provide an informational basis for science. Only generalizations, or general laws, can be regarded as scientific theories, and can therefore be scientific or unscientific. We shall return to this point at greater length below.

In light of all this, Popper asks: if the entire essence of science is speculative generalization beyond what is directly observed in experience, how can there be any process of confirming or proving a scientific theory, that is, a general law of nature? What justification do we have for doing so? Put differently: in what sense is the scientific process more rational than some speculative guess?¹¹

Popper continued to attack Baconian empiricism from another direction as well. The experimental testing of a scientific theory is itself highly problematic. If there is a theory T from which an experimental prediction B is derived under certain conditions, we test in an experiment in which those conditions obtain whether B indeed occurs. Let us now assume that B does in fact occur. Does that occurrence prove the theory? True, from accepting a theory T one may infer that B ought to occur, but the converse is not true: from the occurrence of B one cannot infer the truth of theory T, since the occurrence of B might arise even if it rests upon countless entirely different theories.

The logical expression of this Popperian critique is the following. In logical notation it is customary to denote implication by an arrow. The proposition A → B means: if A is true, then B is also true. From this proposition one cannot derive the proposition B → A. For example, if the first proposition expresses the sentence, “If one lights a match here, a fire will break out,” it is clear that one cannot derive from it the proposition that “If a fire breaks out here, then one has lit a match here.” The only equivalent proposition is ¬B → ¬A, where ¬A denotes the negation of A. In our example this means: if no fire broke out here, then no match was lit here. That proposition is equivalent to the first one. On the logical operation of implication, see also Note 4 below, on the raven paradox.

In the context of scientific theories, this logical consideration is applied in order to formulate Popper’s argument as follows: the first proposition is, “If theory T is true, B will occur.” From this it does not follow that if B occurs, then theory T is true. At most one may derive that if B does not occur, then theory T is not true. Below we shall formulate the meaning of this logical fact as follows: a scientific theory cannot be empirically proved; it can only be empirically refuted.

For example, there is no doubt that the theory of gravitation entails the fact that if we release a stone of some mass into the air, it will fall to the ground. But if we see that a stone released into the air does indeed fall to the ground, we cannot infer that the theory of gravitation is correct—that is, that there is some force exerted upon it by the earth. It may be that the stone fell to the ground because every stone strives to return to its quarry, as Aristotle held, or because it likes the smell of the plants down below. It is true, however, that if the stone does not fall, then the theory of gravitation has indeed been refuted.

Popper argued that there is, in principle, no way to define and construct an experiment that will test a given theory unambiguously. One cannot establish a one-to-one correspondence between a theory and any particular experimental observation.¹² Later on we shall see that for any given state of affairs in the world, infinitely many explanations can be found. This is an inherent problem in theoretical science, and we shall discuss it below.

Finally, it is important to note here that the two Popperian attacks presented here closely parallel David Hume’s two famous critiques: the critique of induction and the critique of causality.¹³

After Popper rejected the possibility of experimental verification of a scientific theory, he proposed a different account of what makes a theory scientific and of scientific methodology, an account based on the asymmetry between confirming a scientific theory and refuting it. As we saw above, one cannot fully confirm a scientific theory, unless it becomes trivial. By contrast, in order to refute the theory that all ravens are black, it is enough to observe a single raven that is not black. That is, although a scientific theory cannot be fully empirically proved, it can certainly be refuted by producing a counterexample.

In light of this asymmetry between proof and refutation, Popper argues that the definition of the scientific status of theories must be changed. The criterion he proposes for distinguishing a scientific theory from a non-scientific one is falsifiability, not provability. According to Popper, a theory that is not falsifiable is not scientific.

It should be noted that this criterion requires that a theory be, in principle, open to experimental testing, but it does not require that the theory successfully pass that test. A theory that is empirically testable is defined as scientific by virtue of the very fact that it is open to experimental examination. If the empirical test succeeds—that is, if the theory’s predictions do indeed accord with the facts observed in the experiment—then it is a scientific theory that has not been refuted, or perhaps has been corroborated, as we shall see below. And if the test fails, then it is a false scientific theory, not a non-scientific one.

According to this, the theory that above every blade of grass stands an angel telling it, “Grow,” is not a scientific theory.¹⁴

Above every blade of grass stands an angel and tells it: “Grow.”

It is not open to experimental testing, and therefore of course it is not open to falsification either. On the other hand, it is important to note that this does not mean that such a non-scientific theory is false.

Newtonian mechanics, by contrast, is a scientific theory, since it is open to experimental testing. Yet, as is now known after Einstein’s theory of relativity, it is a false scientific theory. In contrast to both of these, Einstein’s relativistic mechanics, according to the data currently known to us, is a scientific theory, since it is open to experimental falsification; in addition, it is also considered, as of today, a true scientific theory, since it has successfully withstood the experimental tests to which it has thus far been subjected.

As stated, no theory of any kind, scientific or non-scientific, is provable. According to Popper, the testability of a scientific theory is directed toward refuting the theory, not toward proving it. This is the criterion he proposes for assessing the scientific character of a theory.¹⁵ It is worth noting that, in Popper’s opinion, psychoanalysis and some branches of psychology are also not scientific fields; see below in Chapter Six.

Note 1: Testimony That Cannot Be Impeached by Alibi¹⁶

As we have seen, the Popperian criterion for the scientific status of a theory is its falsifiability.

There appears to be an element here of taking a risk. A theory that takes upon itself the risk of being refuted may also receive the “scientific” credit—below we shall see that perhaps such a theory may also be confirmed, and not only refuted. A theory that is not open to experimental examination, even if such examination can only refute it and not prove it, takes no risk and therefore receives less credit. In this note we shall briefly discuss a halakhic (Jewish law) example of a similar principle, one that appears in the context of the laws of testimony.

As a preliminary matter, halakha distinguishes between contradiction of witnesses and their impeachment by alibi. When there is one pair of two witnesses who testify that a certain person borrowed one hundred shekels from another person in Haifa on a certain date, a situation may arise in which another pair of two witnesses comes and testifies that no such event took place. In such a case, the second pair is regarded as merely contradicting the first pair. According to halakha, both pairs of witnesses retain their presumption of validity, since neither is preferable to the other.

By contrast, another situation is possible, in which the members of the second pair say that they do not know whether such a loan occurred, but they know for certain that the two witnesses of the first pair were not in Haifa on that day at all, but were with them in a completely different place, and could not even have reached Haifa. In such a case, the second pair is regarded as impeaching the first, and the members of the first pair are declared liars. Beyond that, they are also punished for their falsehood. The Torah states:

“You shall do to him as he schemed to do to his brother.”

That is, they receive the same punishment they schemed to impose upon the accused—in our example, the borrower.

Let us now take one more step. There is a halakhic rule that any testimony accepted in court must be open to such impeachment; in halakhic terminology, it must be “testimony that you can impeach by alibi.” See Babylonian Talmud, Makkot 5a, where this is derived from the verse in Parashat Shoftim: “And behold, the witness is a false witness; he has testified falsely against his brother.” Testimony that cannot be so impeached is not accepted. In other words: witnesses who, under certain circumstances, cannot be declared liars—witnesses who take no risk in their testimony—cannot have their testimony accepted.

This rule may be understood in several ways. One may understand it according to Popperian logic, namely: witnesses who take no risk upon themselves are not sufficiently reliable to convict someone else. In Popper’s language: every testimony must be falsifiable. But it may also be understood as a formal legal-halakhic rule, required for other reasons.

There are several halakhic questions whose answers depend on these two understandings. I shall present here, by way of example, two questions discussed in Kehillot Yaakov by Rabbi Yaakov Yisrael Kanievsky, on tractate Ketubot, section 24, and see also the same work on tractate Rosh Hashanah, section 20.¹⁷

There is a dispute as to whether the possibility of impeachment must necessarily include a situation in which the lie would also bring punishment, or whether the mere possibility of impeachment—even if the witnesses would not be punished as a result—satisfies the requirement and is sufficient to validate the testimony.¹⁸ At first glance, if this rule expresses a demand that the witnesses take a risk upon themselves, then falsifiability is intended to deter the witnesses. According to this view, it is reasonable to require that they also be liable to punishment, and not merely that their testimony could be rejected. Still, this is not necessary, since it may be that even the possibility that they will be declared liars, even if not punished, is sufficient for deterrence.

Another relevant halakhic question is this: must there be a possibility of impeaching the testimony even after the testimony has been completed—for example, at the time the verdict is rendered?¹⁹ If the problem is deterrence of the witnesses, then clearly this possibility need exist only at the time the testimony is given, and there is no point in requiring the possibility of impeachment also at the time of the verdict. The view that requires it even at the time of the verdict apparently understood the demand for impeachability as a formal legal-halakhic requirement.

To conclude, it should be noted that even if we understand the law of “testimony that can be impeached by alibi” as a legal requirement, and not as a requirement intended to deter witnesses from lying, this is not necessarily a different principle from the Popperian one.

It is certainly plausible that the requirement that a scientific theory be falsifiable is not intended to deter the person who proposes it. It may be understood as a methodological requirement, meant to filter among theories, so that time will not be wasted on theories that cannot be examined by scientific tools. Or perhaps this requirement is intended to ensure that the theory in question has empirical content, and does not amount merely to a heap of words devoid of empirical meaning.

In fact, it is not at all clear whether one can even find in this context a substantive, non-formal dimension and aspire for this requirement to serve as a criterion of the theory’s reliability. Can falsifiability be a criterion of a theory’s reliability? Why should a theory that cannot be subjected to empirical testing be less reliable? Seemingly, this too is only a formal requirement in the scientific context. This point will be examined below from additional angles.

Hempel on Verification and Confirmation

We have seen that Bacon argued for the possibility of verifying a scientific hypothesis or theory. Popper, by contrast, denied this because of problems such as the problem of induction—the impossibility of generalizing, that is, of arriving from a collection of particular facts at a general law—and therefore entirely gave up on any positive testing of scientific theories.

This approach is problematic, because there is a strong intuitive sense that science is a tool possessing some capacity to discover truths about the world. By contrast, according to Popper, science is not concerned with the question of what the natural world is like, but mainly with the question of what it is not like.

In the next note we shall discuss briefly a similar phenomenon that appears in the theology of many medieval thinkers.

Note 2: The Doctrine of Negative Attributes: Can Knowledge Be Accumulated Through Negation

In the twelfth gate of the first book, we dealt in detail with the meaning of the logical operation of negation. At the beginning of Chapter Four there we included a brief discussion of the divine attributes, and more precisely of the doctrine of negative attributes, which forms part of the theology of many medieval thinkers.

In somewhat oversimplified form, one may say that the doctrine of negative attributes maintains that every description of divinity can be understood only negatively. When we say that God is great, the meaning is that He is not small. When we say that He is wise, we mean that He is not foolish. That He exists means that He is not absent, and so forth. The assumption behind the doctrine of negative attributes is that a mortal human being cannot positively understand the concepts and attributes of divinity. Maimonides, for example, holds that the more we multiply positive descriptions of God, the farther we move from truly understanding Him. The only way to speak of knowledge of divinity is by means of negative attributes.

But here a difficult problem arises. Seemingly, through our study of the divine attributes we do not add knowledge about God, but at most about what He is not. The question is whether such study has value. Do we in fact learn anything about God in this way?

This is an example of passive learning, like what we saw in the mechanism of empirical testing of scientific theory. We only refute; we do not learn, or prove, in a positive way.

Maimonides is one of the fathers of this doctrine, and he discusses it extensively in his Guide of the Perplexed—see Part I, especially chapters 46-60.²⁰ Here I wish to cite a remark of his concerning the value of passive learning; see there, chapter 59:

For whenever you add attributes to that which is described, it becomes more specifically distinguished, and the one describing comes closer to grasping its true reality. So too, whenever you add negations concerning Him, exalted be He, you come closer to apprehension, and you will be nearer to Him than one who has not negated what you have proven must be negated… In this way one must approach apprehension of Him through inquiry and investigation until one knows the nullity of everything that is null with respect to Him… And the clearest thing said in this matter is what is stated in Psalms: “To You, silence is praise.”

The obstacles to a positive description of divinity are different from the obstacles to a positive testing of scientific theories. Yet the solutions may be very similar in both cases. Up to this point we have seen that Popper chooses the option of attending only to the refutation of theories. He sees no positive significance in a theory’s passing a test.

Maimonides’ remark cited here leads us to the position of Carnap and Hempel and their associates, to be presented now, which sees positive value—not merely the absence of failure—in a theory’s successful passage through empirical testing.

This is not the place to discuss the issues of the divine attributes in greater detail, nor the relation between them and the methodology of scientific inquiry. Here I have only hinted at initial directions of thought on these matters.

Because of this intuition concerning the informativeness of science, several philosophers of science argued that Popper’s position should be refined, and that one should try to discern some positive value as well in the empirical testing of scientific theories. Hempel, and also Carnap during a certain period, argued that although it is impossible to prove a scientific theory, there is certainly a way to confirm it. If we subject the theory to many tests of possible refutation, and in all of them it successfully withstands them—that is, no counterexample is found, or more generally, none of the predictions of the theory in question is negated in experiment—and it is not refuted, then it may be regarded as a confirmed theory.²¹

It should be noted that this description of scientific methodology stands between the Baconian description and the Popperian one. According to Bacon, a scientific theory can be experimentally verified. He almost completely ignores the problem of induction. According to Popper, who moved to the opposite pole precisely because of the problem of induction, a scientific theory cannot even be confirmed; it can only be refuted, or one can try to refute it. According to Popper, a theory’s surviving an experimental test says nothing at all. According to Hempel and Carnap, it is not true that a scientific theory can only be refuted when it fails in experience, as Popper thought. On the other hand, it is also not true that it can be verified when it survives experience, as Bacon thought. They argue that it can both be refuted, when it fails in experience, and confirmed, when it survives.

Note 3: Different Degrees of Confirmation: Halakhic and Scientific Approaches to Variety in Evidence

It seems that one of the indications that a theory’s surviving tests does indeed confirm it is that intuitively we regard a theory that has survived diverse tests as more firmly grounded than a theory that has survived only one kind of test, even if there were many such tests.

Variety in tests is indeed a standard criterion in the philosophy and methodology of science, but it certainly requires clarification and explanation. If Popper is correct in his arguments, then there is no possibility of drawing positive conclusions from the fact that some theory is not refuted by experiment. On the other hand, we saw in the previous note, in Maimonides, and we now encounter the same thing among modern philosophers of science, that one can learn something about natural reality by negating other possibilities.

Let us take, for example, the fact that we examined the color of one raven and found that it is indeed black. Popper asked: why does this confirm, or support, the theory that all ravens are black? We may now add another question concerning the significance of the number of empirical tests: is the situation after examining the colors of a thousand ravens any different? Can one now draw a positive conclusion about the color of ravens in general? Finally, let us add a question concerning the significance of variety in these tests: if we examine the colors of those same thousand ravens, but do so in Africa, Asia, Europe, and America, and find that all of them are black, are we now more convinced of the theory that all ravens are black?

As stated, intuition says that there is indeed significance in the very fact of withstanding empirical examination. We shall further see that the number of tests matters. And we shall further see that their variety also matters.

Now let us add that if the number of tests has a confirmatory significance, that is because at the end of the day we now know regarding more ravens that they are indeed black. But from that standpoint, it is clear that variety in the evidence has no significance at all.

It seems that variety plays another role in the scientific investigation of the theory in question. If we were to discover a raven in Africa whose color was white, we could say that the theory applies only to Europe, or to Asia. In Africa, because of special causes—climate and the like—the color of ravens is different. The purpose of variety is to negate alternative theories that would place qualifications upon the theoretical generalization we are testing, namely, that all ravens everywhere in the world are black.

But even here, it seems that the function of variety in testing is still negation. This time, not the negation of particular facts contrary to the theoretical determination, but the negation of theoretical qualifications that one might impose upon it.

Hence, the intuition that variety has significance does not necessarily indicate that empirical experiment also confirms, and not merely negates or refutes.

I shall say in advance that the significance of what we have seen so far is that within an analytic picture Popper is indeed correct. The position of Carnap and Hempel, if it is not accompanied by a fundamental change in perspective—and as far as I know it is not accompanied by such a change—is nothing more than a semantic trick. They do not offer a logical grounding for the trust, admittedly qualified, that we place in inductive scientific generalizations.

Below we shall see the synthetic explanation of science, and we shall realize that only according to it can one understand the claim that empirical tests confirm a theory, and do not merely “fail to refute” it. There it will also become clear that variety can have an important role in confirming scientific theories.

This is the place to note a Torah-halakhic aspect of this discussion.

In halakha too, weight is given to variety in the “empirical” evidence for a certain halakhic principle. Usually, when we encounter a halakhic principle in one context, we learn from it and extend it to all other contexts. If we see a source in a Torah verse teaching us that in the commandment of fringes one may attach threads of a different species from that of the garment—a combination that in every other context constitutes the prohibition of forbidden mixture—we infer a general principle, a halakhic theory, that says: “A positive commandment overrides a prohibition.” The meaning of this principle is that when there is a positive commandment—to place fringes on a garment—and it involves violating a prohibition—forbidden mixture—one may fulfill the positive commandment even at the cost of violating the prohibition, at least when there is no other choice.

Yet it turns out that if there were an additional verse teaching the same principle, the situation would be completely reversed. When there are two distinct sources for the same principle, we infer that this principle is valid only in the two contexts in which it appears. In halakhic formulation: two verses that come together do not teach a general rule. The reason is that if the Torah had intended to teach us a general principle, it could have sufficed with one of the contexts, and we would have learned everything from it alone. If the Torah expresses this principle in two contexts, it appears to mean that the principle is not generally correct, but only in those two contexts are we to act in accordance with it, for special reasons.

There is now a third situation, in which we have two different sources for a law, but from each of them one cannot infer the validity of the general principle throughout halakha. The reason one cannot infer the validity of the principle from each source is that each possesses a special characteristic, and therefore it may be that the principle appearing in it is unique only to contexts with a similar characteristic.

In such a case, if we try to learn the general principle for all of halakha from source A, an objection will arise, claiming that source A deals with a special context, and that perhaps only in such a context is the general principle valid. If such objections arise with respect to both sources, then the rule that two verses coming together do not teach no longer applies, since it is clear that there is a need for the principle to be written in both sources because of their uniqueness—one cannot learn the validity of the principle in context B from its validity in context A, because of the difference between them. Therefore we once again generalize this principle to all of halakha through a form of exposition called “the common denominator,” or “the shared element.” See the first book, Gate Thirteen, Chapter Three, for a discussion of another aspect of this form of exposition.²²

This form of exposition is called by some commentators binyan av, because it constructs a general principle—a parent principle—from the two particular examples that appear in the Torah.²³ It should be noted that the general principle is constructed only when those two examples differ from one another. Otherwise we would reject the generalization and say that the general principle is true only in those two contexts because of their uniqueness. But it is precisely the difference between the two contexts in which this principle appears in the Torah, the two sources, that persuades us that it is a general principle. This is the significance of variety in the evidence.²⁴

A complete logical process is described here, consisting of three stages:

1. Learning from one source and generalizing from it to all of halakha.
2. Finding an additional source that blocks the possibility of such generalization.
3. Solving the problem by determining that the additional source differs from the first. In talmudic terminology this stage is called tzrikhuta, because it proves that we need both sources. Therefore one may generalize from the two of them to all of halakha.

It should be noted that this process appears in very many places in the Talmud, and it is one of the building blocks of talmudic logic.

On the principle of variety in evidence, see also below in Note 8, on Ockham’s razor.

As noted, this mechanism of confirmation is seemingly highly problematic from the standpoint of pure logic. As far as I know, Carnap and Hempel do not address the problems raised by Popper, and therefore, at least in analytic contexts such as theirs, it appears to be nothing more than a semantic change. Popper pointed out that a theory cannot be verified, because the inductive method is problematic. As we saw, Popper also pointed to the difficulty of drawing positive conclusions from submitting a theory to empirical experiment. His conclusion was that there is no possibility of verifying a scientific theory, but at most of refuting it. The analytic thinkers now come and define non-refutation by the word confirmation, and in that they see a solution to the problem.

In the next note I shall briefly point to a certain difficulty—among others—in the mechanism of confirmation proposed here as a substitute for the naive Baconian mechanism of proof, or verification. One of the main problems in such a description of scientific activity is what is known in the philosophy of science as the raven paradox.

Note 4: The Raven Paradox and Material Implication²⁵

Let us again take the theory expressed in the proposition: all ravens are black. There are simple logical rules by means of which one can see that this proposition is equivalent to the proposition: everything that is not black is not a raven.²⁶ Seemingly, then, this is nothing more than a different formulation of the same theory.

Let us now try to subject the second formulation, which is also a general law of nature, to empirical testing. Seemingly, the way to do so is very simple. We take objects that are not black, and examine whether or not they are ravens. If they turn out to be ravens, then clearly the theory has been refuted, since we have found a non-black object that is a raven. In other words, we have found a raven that is not black.

The problem begins when we try to decipher the meaning of a successful experiment according to the second formulation. If we take an object that is not black, and ascertain that it is indeed not a raven, then seemingly the theory has been splendidly confirmed, since its predictions have successfully passed the empirical test.

But when we consider the meaning of the matter, we discover that it is very hard to understand why finding a white table, green grass, or a golden cloud—in all these cases, an object that is not black turns out also not to be a raven—constitutes confirmation of the theory that all ravens are black.

Against this background, it should be emphasized that if we were to complete the full set of possible experiments, that is, if we were to examine all objects in the world, the equivalence of the two formulations would once again emerge. Here too, the equivalence appears when one performs an analytic operation—proof or refutation. But confirmation has no meaning in an analytic world, and therefore it does not appear equivalently in the two formulations of the theory.

This paradox was raised by philosophers of science as a challenge to the meaning of the concept of confirmation. As we have seen, the direction of refutation—failure in the experimental test—is indeed equivalent in the two formulations. Refutation of one refutes the other as well. The direction of full proof is also equivalent for both. But the direction of confirmation, which as we noted above lies somewhere between proof and mere non-refutation, certainly does not appear equivalent. Confirmation of the theory in its first formulation does not have the same significance as confirmation in the second formulation, and vice versa.

The conclusion is that in an analytic mode of reflection, that is, one that examines matters only on the logical plane, confirmation has no real standing. The critique presented above was formulated in analytic terms, and at that level of reference it is indeed correct. Below we shall see what standing confirmation, and the critique of it, have within the synthetic picture.

We should further note here that, for one who accepts the concept of confirmation, confirmation of the contrapositive claim constitutes a weaker confirmation of the direct claim. The status of such weak confirmation is still better, and more relevant to the empirical testing of the theory, than finding a deep lake—which is not a raven, but whose depth is a property wholly unrelated to the color of the object.

There is another expression, from a different angle, of the raven paradox. This expression appears in the operation of logical implication. In logic it is customary to symbolize implication as A → B. The meaning of this notation in ordinary language is: if A, then B, or: A implies B.

What is the precise logical meaning of the relation of implication between two logical variables? In logic it is customary to define such a relation by a truth table. Such a table presents the status of the implication relation in terms of truth and falsity for every state of the two variables, that is, for cases in which A is true or false, and B is true or false.

The truth table of implication in its standard logical sense—where it is called material implication—is as follows:

A	B	A → B
F	F	T
F	T	T
T	F	F
T	T	T

The meaning of the material definition of implication is that we construe it in the most minimal way possible: it cannot happen that when the condition is satisfied, the conditioned result does not occur. For example, take the sentence: “If something is a raven, then it is black.” This is a formulation of the theory “All ravens are black” in terms of implication. The material definition of implication is that implication is always true unless it is plainly false.

If we assume that we are examining the implication statement, then four different situations are possible in the experiment: we find an object that is a raven or not a raven, and see that it is black or not black.

The first row in the table describes a situation in which something is not a raven and is also not black. Here the implication has not been refuted. But this very situation constitutes confirmation of the contrapositive formulation: whatever is not black is not a raven.

The second row in the table describes a situation in which something is not a raven, but is black. Even in such a case the implication has not been refuted. This is a neutral situation. It corresponds to the contrapositive of a claim opposite to the original implication: all ravens are not black, or, in contrapositive form, everything black is not a raven.

The third row represents a situation in which we have found a raven that is not black. This is of course an empirical refutation of the implication claim.

The fourth row represents a situation in which we have found a black raven. Here too the theory expressed by the implication has not been refuted. But here there is clearly more than that: it has been confirmed.

In the table above we see that states in which the theory has not been refuted are defined in the material sense as true. Only when we have directly refuted the implication is it regarded as false. In all other cases it receives, by the standard logical convention, the value “true.”

The material definition is so widespread in logic that it is already difficult for the logician to think in other terms. But this definition gives significance only to refutation as an operation with meaning and value. This is a clear expression of the analytic character of logic.

Seemingly, one might define a non-material implication in some other way. There we would regard implication as true only if it were verified, not merely if it were not refuted. But in that case we would obtain a trivial truth table, in which the entire implication column would be false.

The reason is that one cannot confirm an implication by any empirical means whatsoever. Even if there is a case in which we have found a black raven, or many such ravens, this does not mean that every raven is black. This is precisely Hume’s critique of the concept of cause. One cannot observe it, and therefore one cannot infer it from any concrete state of affairs. The relation of cause and effect is a conclusion that lies beyond the logical level.

True, finding a black raven certainly seems to confirm the implication claim, especially if it recurs many times. But as we have seen, confirmation cannot have logical standing, and therefore symbolic logic cannot express it in its formalism.

We thus come to see that logical definitions are subject to severe constraints because of their adherence to logical necessity and to the logical plane. Synthetic operations cannot receive expression in logic, because they lie outside it. But this does not mean that they add nothing to us. The operation of confirmation, as we shall also see below when we discuss the synthetic explanation of scientific methodology, certainly has scientific significance, even if not logical significance.

Just as Karl Popper defines scientific theory in terms of falsifiability, so logicians define logical implication—or, in fact, the relation of causality and conditionality—in a material way, that is, as false if refuted, not as true if verified.

We thus come to know that the logical perspective cannot distinguish the concept of confirmation. It is transparent to it. From the material logician’s standpoint, a state that confirms a theory is like a neutral state. In the table above, the material logician identifies row 4 with row 2.

Although we have not yet discussed the synthetic picture of science, I will already anticipate here that, because of the synthetic character of science, in order to describe scientific methodology correctly we must define a different implication, one that is not material. In scientific implication, and also in everyday implication, we shall define every state of confirmation as having positive standing, and unlike Popper we shall not identify a neutral state of mere absence of refutation with a state of confirmation. Row 2 in the table above is the absence of refutation, a neutral state, and it certainly has scientific standing, though not logical standing. That standing is different from the last row, which confirms the theory, and also from the first row, which confirms the contrapositive formulation—a weaker confirmation.

The truth table of synthetic, scientific implication—let us denote it by A ↝ B—will therefore be as follows:

A	B	A ↝ B
F	F	C1
F	T	N
T	F	F
T	T	C2

The values in the table are defined as follows:

• C1 — weak confirmation, that is, confirmation of the contrapositive formulation: what is not black is not a raven.
• N — a neutral state.
• F — logical falsehood, as in the material formulation.
• C2 — strong confirmation, that is, confirmation of the implication claim itself.

The value T, of course, does not exist in this picture, because it concerns the empirical plane. As we have seen, on the empirical plane there is no possibility of certain proof for any scientific theory. Empirically, a scientific theory can at most be confirmed, but certainly not proven.

To see why confirmation of the contrapositive claim, “Everything that is not black is not a raven,” is not equivalent to confirmation of the original claim, “All ravens are black,” one may examine the truth table of the contrapositive in the two tables above, and see whether one obtains the same logical table. In the case of material implication no difference will appear, for ¬B → ¬A. By contrast, the table of synthetic implication for the contrapositive will certainly not look the same—for example, C2 will switch places with C1.

Accordingly, the logical equivalence of the sentence under examination depends on the character of the implication that appears in it. For material implication, the contrapositive “Everything that is not black is not a raven” is equivalent. But for a sentence in which implication is synthetic, that is not the relevant equivalence.

Those of the analytic approach, who regard logic as all-encompassing, understand that the two propositions are equivalent, and therefore reject the concept of confirmation. By contrast, one who explains science through synthetic lenses will accept the concept of confirmation, and will not accept logical equivalence as the equivalence relevant to our issue.

Thus Carnap and his associates do not answer the problems raised by Popper, but merely propose calling the process of non-refutation confirmation. There is no real philosophical answer here to the substantive problems in scientific methodology as presented by Popper. That is, there is no explanation of why the concept of confirmation contains something positive beyond passive non-refutation. Put differently: when we say that the experiment confirms the theory, in what way have we progressed beyond the state we were in before conducting the experiment? Seemingly, what is offered here is only a proposal on the semantic plane.

Already here one can see the character of analytic solutions to problems in scientific methodology, and in philosophy generally. The analytic thinker solves every philosophical problem by means of an appropriate semantic change. In his view, language is the plane on which philosophical problems are located, and therefore it is also the plane on which they are solved. On this see the first book at greater length.

In the next chapter we shall discuss analytic explanations of science in greater detail, and there we shall also see Carnap’s approach brought to full expression.

The Relation Between Facts and Theory: What Is a Scientific Fact?

Let us now return to the simplistic Baconian description of the scientific process, as presented at the beginning of our discussion, and examine it from another angle.

Up to this point we have discussed the path from the facts placed before the scientist to the scientific theory that explains them. We saw that a scientific theory is not proved, but only confirmed, or alternatively, refuted. We now wish to discuss another problem present in Bacon’s empiricist account, namely, the problem of collecting facts. This problem has been discussed in several contexts which appear, at first glance, to be different, but whose root is one.

The historian E. H. Carr, in his book What Is History?,²⁷ describes the absurdity of the classical description of the historian’s work.²⁸ This description, parallel to the Baconian one above, presents a tranquil picture according to which the historian collects all the facts, then classifies and analyzes them, and finally proposes a theoretical explanation for them.

Carr argues against this description that there are infinitely many facts on every subject, and therefore one cannot simply contemplate “the facts.” First one must decide which facts are relevant, or in his language: what are historical facts.

For example, a historian coming to analyze the outcome of the Battle of Zama in the Punic Wars between Rome and Carthage would first be required to examine the facts. Is the fact that Scipio’s height at the battle of Zama was 1.70 meters relevant? Or should he examine what tents the armies had, or how many hours of sleep the soldiers got? Are the political situation and balance of power in Rome at that period, the stabbing of one soldier by his drunken comrade on the evening before the battle, the size of China at that time, the first letter in the name of the Persian idol, the number of inhabitants of Rome, the name of the artisan who made the weapons for each side, and so forth, relevant facts? The historian must decide which facts are historically relevant for him from the infinite set of facts, not all of which are, of course, even before him.

As stated, in every context there are many facts; indeed, their number is infinite. Only those among them that are relevant can be considered facts for the historian. Those facts with which the historian must concern himself are commonly called historical facts.

But the criterion of relevance of the facts, that is, what counts as historical facts, depends on the explanation being proposed. If the proposed explanation of the outcome of the battle of Zama is based on the morale of Scipio’s soldiers, then the stabbing of one of the soldiers by his comrade in a duel on the eve of the battle may be a highly relevant fact. If the explanation is the army’s confidence in the regime, then other facts relating to the government in Rome will be considered relevant. If the vulnerability of the commander is what matters, then Scipio’s physical height itself may be no less important, and so on.²⁹

But from this it follows that the collection of historical facts, those that stand at the base of the historical explanation, is derived from the explanation itself. The theoretical explanation determines which facts are relevant and which are not. On the other hand, the explanation itself is supposedly derived from the facts, since that is why the historian collects them. This is a circular process that admits of no linear logical description.

Note 5: The Circularity Underlying Analogy and Induction

We have seen a phenomenon of circularity in the relation between historical facts and the theory that explains them. A similar, more general phenomenon can be seen in processes of analogical and inductive inference.

When we draw an analogy between one raven and another, and decide that if one raven is black then all ravens are black, we have assumed that the property of being black is essential to ravens. After all, we do not draw an analogy between them and decide that, just like raven A, raven B also has a length of 35 centimeters. The reason is that the property of length is not an essential property of ravens.

This can also be formulated the other way around. The fact that the observed object is a raven is relevant to its black color. When we look for another object about which we can infer that its color too is black, we shall say this of a raven. Thus we are also assuming that being a raven is a relevant parameter of similarity. We shall not infer it of a sea turtle, even if its length is 35 centimeters, exactly like our black raven, that its color too must therefore be black. The reason is that length is not a similarity parameter relevant to the color of the object. Comparison on the basis of length is irrelevant for inferring color.

In both cases we assume a relevant parameter of similarity. Being a raven is a relevant parameter, whereas having a length of 35 centimeters is not. But seemingly the black color, as a property of ravens rather than of things of a certain length, is the result of the theory and not one of its assumptions. We discover that having a black color is a property of all ravens only as a result of the analogy and the generalization that follows it. Yet, on the other hand, as we have seen, this assumption also stands at the basis of the analogy between them, which leads to the generalization. We have once again discovered the same circle inherent in empirical research. Zoology and history are both nothing but different expressions of a basic logical problem in analogical comparison and inductive generalization.³⁰

It should be noted that this is not the problem of justifying analogy and induction, but a different problem. Just as the problem discussed here with respect to scientific theories is not the problem of their synthetic nature, but another one. The basic problem of the synthetic nature of scientific theories, and likewise of analogy and induction, is this: how do we justify the generalization on the basis of observations of particular facts?

Here we are dealing with a different problem: even if one can in principle infer generalizations from particular observations, how do we know from which observations to do so? What is the basis for the classification and sorting of the particular observations, based on similarity among them, that makes comparison and generalization possible?

One can formulate this paradox more sharply, from a slightly different angle. Suppose we are looking for essential properties that characterize the species of ravens. We are, of course, required to observe ravens in order to learn their properties from the observations. But how do we know what is a raven before we know its essential properties? Perhaps, in fact, we are observing a lion, and not a raven.

Once again, there is here a vicious circle between the properties and the definition of the kinds of things themselves.

Let me anticipate a bit here and say that there seems to be an analytic solution to this vicious circle. The analytic thinker will say that the theory according to which all ravens are black, and possess such-and-such additional properties, is not a synthetic claim at all. It is simply the definition of the species ravens, that is, an analytic proposition. We do not discover facts about ravens; rather, we classify animals according to their properties. All proponents of a certain cluster of properties—flying, black, having a beak of such-and-such a kind, and so on—will henceforth be called by us ravens. The analytic thinker needs such strange solutions only because he alone is liable to become entangled in the loop described above. Those who hold the synthetic position know, at least to some degree of probability, what a raven is even before its essential properties have been defined.³¹

For further detail, and for the continuation of the discussion, see below in the chapters dealing with the analytic and synthetic descriptions of scientific methodology.

Up to this point we have seen the logical loop present in the relation between theory and facts in the context of historical inquiry. But the same is true in all fields of science, and we shall illustrate it by an example discussed by the philosopher of science Carl Hempel, though in a somewhat different context.³²

The Hungarian physician Semmelweis, in the middle of the nineteenth century, desperately sought an explanation for the high mortality rate among women in his maternity ward from puerperal fever, as compared with a much lower rate in a parallel maternity ward. He of course did not know the cause, and therefore tried to collect and isolate various kinds of facts and build on their basis a theory that would explain the phenomenon.

Here too, as stated, the relevant facts are theory-dependent, and therefore, so long as Semmelweis did not know the causes of the disease, it was unclear where he ought to look. At that stage it was not at all clear to him which facts he should examine. He searched in various directions and tested different hypotheses by subjecting them to empirical examination.

He rejected the hypothesis concerning cosmic changes and atmospheric influences, because it does not explain the difference between two wards in the same hospital. He examined the crowding in the two wards and discovered that the “healthy” ward was actually the more crowded one—among other reasons, because women in labor tried to be admitted to the more successful ward.

A committee appointed to investigate the matter attributed the difference to rough treatment by medical students, and therefore recommended reducing the number of students. After the recommendation was adopted, there was a certain temporary decline in mortality, but immediately afterward there was a sharp rise, and mortality reached new peaks.

There were also psychological explanations. Some attributed the difference to the route taken by the priest when he came to administer the last rites to a dying patient. In front of the priest walked an attendant carrying a bell, and this could arouse anxiety among the women in labor. In the healthier ward the priest passed by a shorter route, whereas in the problematic ward he had to pass through five pavilions, and therefore there was concern that he aroused anxiety in more women. Semmelweis decided to test this theory, and changed the route taken by the priest and prevented the attendant from coming, but no effect on mortality was discovered.

After that, Semmelweis thought the difference lay in the different birthing positions used in the two wards, since in his ward the women gave birth lying on their backs, whereas in the healthier ward they gave birth lying on their sides. Semmelweis instituted side-lying births in his ward as well, but again to no avail.

This perplexity illustrates very well the problem we saw above in the historical context. So long as the theory explaining the facts is unknown, we have no way of knowing which facts are the relevant ones that we ought to examine.

What gave Semmelweis the clue to solving the mystery was a case in which one of his colleagues injured his finger by being pricked with the scalpel of a student who was performing an autopsy with him. The colleague died from the injury, and the process was accompanied by all the symptoms of puerperal fever as they appear in women in labor.

Although the role of microorganisms in infection was still not known, Semmelweis inferred that matter from the dead had entered his colleague’s bloodstream through the student’s scalpel, and that this was what caused both his illness and the death that followed it. Semmelweis noticed that on his own hands and on those of his colleagues there remained a substance after dissection in the autopsy room, and that he and his colleagues would then go to the maternity ward in order to examine the women there. It should be noted that in the second ward, care of the women in labor was carried out by midwives, whose training did not include the dissection of corpses.

Once again Semmelweis subjected his hypothesis to empirical testing, and ordered all the medical students to wash their hands in a solution of chlorinated lime before conducting a medical examination. The mortality rate from the fever began immediately to fall, and within a little over a year the mortality percentages in the two wards were identical.

This theory, of transmission of infection by matter from the bloodstream of the dead, was later verified through additional observations—for example, infants who contracted the disease were only the children of sick mothers—and it was extended, already by Semmelweis himself, to additional cases.

If we look at this fascinating case, we shall realize that there were infinitely many facts to which Semmelweis might have paid attention. Clearly, the names of the doctors and nurses differed in the two wards. It may be that the structure of the buildings was different, and so on. It is important to note that the transmission of matter from the dead, in an age before viruses and microbes were known, was no less mystical a principle than explanations hanging on the structure of the building or on the names of the doctors. So long as we are not equipped with some idea of a possible explanation, there is no meaning to examining the full set of facts.

If Semmelweis had noticed that the workers in his ward did not wash their hands whereas those in the other ward did, this would not even have aroused suspicion that the root of the problem lay precisely there. What connection is there between the handwashing of doctors and the mortality of women in labor? This is a connection that appears patently mystical so long as we are not equipped with a theory that supports it and gives it some meaning.

Before the connection between cleanliness, infection, and mortality became known, no one would have imagined examining whether the doctors and nurses wash their hands before treatment. That could not have been considered a more relevant fact than the first letter in the name of a Persian idol in antiquity, or the beverage that the doctors and students drink in the morning before they go to work.

Only after the theoretical explanation was found—infection as a result of lack of hygiene, namely the theory—can one know that among the relevant facts, the scientific facts, is, for example, the fact that the ward staff does not wash its hands before treatment.

Thus, the circularity built into the relation between facts and the theory that explains them appears in the empirical natural sciences just as it does in historical inquiry.

If we now examine the sciences of inanimate nature, physics and chemistry, the situation appears at first glance to be different. First, in the physical world there are no irrelevant facts. A complete picture of nature is supposed to describe all natural phenomena, and therefore, seemingly, all facts are relevant. If there is even one fact that does not conform to the proposed theory, the theory is refuted. Clearly, a complete physical picture is supposed to describe all the phenomena of the behavior of physical objects, and therefore it seems that we would not find here the problem described above.

However, this description is not exhaustive. The main problem in this description is the classification of facts. A given physical theory, such as gravitation, is supposed to describe a certain set of natural phenomena, not all of them. There are phenomena whose governing physical laws do not belong to the domain of gravitation, and for that reason there are several different domains even within physics itself. Heat phenomena are described by thermodynamics. Phenomena of the dynamics and statics of massive bodies are described by mechanics. Phenomena of electricity and magnetism are described by electromagnetic theory, and so on. Each such domain describes the phenomena within its own field of concern by means of a system of physical laws specific to it.

Here again a problem parallel to the one described above arises. The scientist must decide which set of phenomena belongs to a given natural law, or, in effect, how to divide physics and its laws into different classes. This is indeed a problem of classification, but in fact once again we stand before the problem of the relevant properties and characteristics of facts, or of occurrences.³³ In antiquity it was not clear whether the revolution of the heavenly bodies was a phenomenon belonging to mechanics, since some views held that these are living beings, possessing intellect and will, and not inanimate objects. Even in the modern age, before Newton, it was not clear that the motion of the heavenly bodies belonged to the domain of gravitation, since that domain still scarcely existed. It was known that every body falls toward the earth, but the general phenomenon according to which every two masses attract one another was not yet known. No one could have imagined that the phenomenon of free fall toward the earth was merely a particular case of the more general phenomenon of an attractive force between masses, later called gravitation, and that this same force also underlies the motion of the heavenly bodies.

Thus, before Newton established that every two masses attract one another, one could not think that the facts concerning the motion of the heavenly bodies were relevant to explaining the phenomenon of objects falling toward the earth. One might instead have thought that objects fall toward the earth because of their “striving” to return to their “natural” place or to their source, as ancient science, following Aristotle, held. And conversely, the motion of the heavenly bodies could be explained by their desires and aspirations, and not by submission to an inanimate natural force. In other words, these are phenomena belonging to different classes of science.³⁴

We thus come to see that even in the domain of the sciences of inanimate nature there is uncertainty regarding the relation between facts and the theory that explains them. The theory determines which facts will be explained, and the facts are the source from which the theory is generated. Therefore, from the standpoint of fundamental logic, there is no real difference between physics, and the natural sciences generally, on the one hand, and medicine and history on the other.

If we now return once more to medicine, then there too, before understanding the disease that caused the mortality, we must conduct research on the patients who died of this disease. Therefore we must determine which patients died of the same disease, that is, sort and classify the patients who died into classes of diseases. It follows that we must decide whether two patients died of the same disease before we have understood it, and perhaps before we even know how to identify it with certainty. This is the same problem as the classification of scientific facts in physics.

Two Types of Theories³⁵

The problem raised in the previous section may be formulated more generally. Scientific inquiry is often carried out in two stages. Earlier we spoke about the collection of facts as a first step, after which comes theoretical generalization. If we refine the definitions further, we may say that first one constructs a phenomenological, that is, descriptive, theory, and only afterward tries to find an essential theory. The relation between these two stages will come up in several places below as well, especially in the fourth gate, where we shall deal with semantics and syntax.

As an example, let us take a mathematical theory from plant biology. When we examine the number of petals in various flowers, we encounter flowers with three petals, others with five petals, and so forth. There is a theory according to which the number of petals of flowers belongs in many cases to a sequence of whole numbers known in mathematics as the Fibonacci sequence.³⁶ Each of the numbers in this sequence is the sum of the two preceding numbers. If we begin with 1 and 2, the next number is 3, then 5, then 8, 13, and so on.

A phenomenological theory is a theory that generalizes the facts we observe. For example, in our case, the theory that in all flowers, or at least in certain kinds of flowers, the number of petals belongs to the Fibonacci sequence is a phenomenological theory. It contains no explanation, but only a comprehensive hypothesis concerning the facts. It is a sophisticated form of sorting and classification. Once we have a phenomenological theory, we try to find an essential explanation. If we find one, then we possess an essential theory, as distinct from a phenomenology.³⁷

The more intricate the field with which we are dealing, the more we tend to engage in phenomenology, that is, description, and less in essence, that is, explanation. We shall return to this phenomenon later in the book.

In these terms one may say that the paradox concerning the relation between facts and theory is in fact the question of how one can construct a phenomenological theory before understanding the essential layer. We must decide how to classify the facts before we find a description for them, yet this classification is often conditioned by an essential understanding of the processes.³⁸

On the other hand, the essential theory can be built only on the basis of a defined phenomenology. We explain processes of a certain kind only after we know what it is that we must explain, and which processes belong to the same class of phenomena. This, in brief, is the paradoxical relation between facts and theory. The phenomenological theory is only a processed and coherent description of a collection of facts. The essential theory is a scientific explanation for that collection of facts.³⁹

Correlations

A large part of scientific discussion deals with correlations. The scientist’s basic datum, when he comes to build a theory, is a correlation between one fact and another. For example, as we saw in the case of Semmelweis, several correlations stood before him that might have been relevant. In one ward the priest passed along a longer route, or the caregivers were nurses rather than students. These facts could be linked to the fact of the difference in mortality rates from puerperal fever. Semmelweis tried to determine which of them was real and which illusory.

Another level on which correlations appear is the very decision of which facts belong to the same class. We must decide that two patients died of the same disease, or that two natural phenomena derive from the same law.

In the first book, in Gate Five, Chapter Four, and in Note 17 there, the problem of genuine and spurious correlations was discussed. We saw there that there are correlations that have no real basis, and there are correlations that do have such a basis.

Here we wish to discuss another aspect of the problem of correlations, one that exists also, and in fact only, in genuine correlations: determining the direction of the correlation. Even if Semmelweis finds a clear correlation between not washing hands and high mortality from puerperal fever, several principled possibilities of interpretation still remain before him regarding the direction of the correlation.

In general, there are three possibilities for explaining a genuine correlation between A and B:

1. A is the cause and B is the result.
2. B is the cause and A is the result.
3. There is one cause of both phenomena.⁴⁰

In Semmelweis’s case, it is not clear whether handwashing is the cause of low mortality, or whether low mortality is the cause of the doctors’ tendency not to wash their hands—for example, because the patients, most of whom remain alive, repel them less than in the parallel wards—and so forth.⁴¹

Another example would be a study that links cigarette smoking, or any other phenomenon, such as coffee drinking, with cancer. One possibility is to claim that smoking causes cancer. But it is also possible to say that the disease incubating in the body causes nervousness, and that this leads the person to smoke. And there is also a third possibility, according to which there is some other cause that generates both the disease and the tendency to drink coffee. It should be noted that if one of the latter two possibilities is correct, then there would be no point in recommending cessation of smoking in order to avoid cancer. The practical conclusions of the study depend to a very significant degree on the direction of the correlation.

There are ways of refining research and guiding investigators as to the correct direction of the correlation, but the problem exists in many studies. Sometimes it is impossible to decide unequivocally regarding the direction of a given correlation.

Some of the ways are based on continuing the experiment and subjecting the different hypotheses concerning the direction of the correlation themselves to empirical testing, in an attempt to isolate the correct one from among them.

A different, and sometimes parallel, way of trying to overcome this problem is to find the theoretical explanation that underlies the correlation. If we find a mechanistic explanation of how smoking causes cancer, then it is clear that the direction of the correlation is from smoking to the disease, and not from the disease to smoking. Finding the theoretical cause involves generalization and the formulation of a hypothesis about the mechanism that underlies the correlation, and then testing that hypothesis by empirical and theoretical means. Of course, such generalization is subject to all the difficulties we pointed out above. This is, in fact, an expression of the transition from phenomenology to essence, as described above. In any event, it is clear that the mere existence of a correlation cannot serve as an unambiguous indication of the explanation of any natural phenomenon.

The General Problematic Nature of Analogy and Induction

In the first gate we presented the conclusion of the discussion from the first book: that deductive inference is necessarily true precisely because it adds nothing, that is, because of its emptiness. Besides deduction, there are two other principal forms of inference: analogy and induction.

In the first book we became aware of the difficulties inherent in these two forms of inference. When we infer from the fact that one particular person is mortal that another person is also mortal, because he too is human, we have tacitly assumed that the fact that the two are human is what underlies their mortality. It would have been possible to interpret the matter differently, namely, that the first person is mortal because he was born in a certain country, or because God is angry with him on account of his sins. In such a case, of course, we could not immediately infer about the second person, who is also human, that he too is mortal. See also Note 5 above.

Thus, in analogical inference there is always a problem of identifying the relevant parameter of similarity. In many respects, this is the same problem as identifying a genuine correlation, or identifying the direction of the correlation, problems we discussed above.

This problem of identifying the relevant parameter of similarity also underlies the classification of scientific facts into their respective domains. Above we asked whether the motion of the heavenly bodies and the fall downward toward the earth of objects possessing mass are facts that belong to the same theoretical class, that is, whether the same laws stand behind them. Here too the question is the relevance of the classification parameters: does the fact that all these objects have mass suffice to infer that this is what lies at the basis of these phenomena, which appear so very different from one another? The similarity among these objects, by virtue of their all having mass, does not necessarily mean that their mode of motion will be similar. We must assume here the relevance of the parameter of analogy—in this case, the existence of mass—to the theory in question, and this even before the formulation of the theoretical explanation itself.

In the same way one may conclude that induction from a collection of particular facts to a comprehensive law suffers from a similar difficulty. Here too we must decide that this collection of facts belongs to one class, and only then can these phenomena be generalized into a general law.⁴²

Thus, one may view the problem of analogy and induction as a certain generalization of the types of problems we have discussed in the previous sections.

Theoretical Entities

We saw above that one of the characteristics of scientific theories, especially modern ones, is the fact that they use entities that are not open to direct empirical observation. The observed facts are explained by mechanisms that are not always themselves observable.

A simple example of this is those pieces of matter from the dead which, according to Semmelweis’s hypothesis, underlay the mortality from puerperal fever. Semmelweis could not see the microorganisms that we now know transmit diseases. He inferred their existence from circumstantial evidence.

Another example is Newton’s second law of mechanics. This law establishes a fixed relation between the force acting upon a body and the acceleration that it develops as a result of the force’s action upon it. The formal formula of the second law is:

F = m × a

Here F is the force acting on the body, a is the acceleration it develops, and m is the mass of the body.

We directly observe the acceleration of the body. We see that the body changes its velocity, and can even measure the rate at which it does so. The mass of the body is also a direct observational datum. One can weigh the body and determine its quantity of matter. By contrast, force, for example, is a theoretical concept. We cannot directly see that there is a force between two masses that we observe. We can only see that one of them accelerates after coming under the influence of the other mass. For this reason, before Newton explained the manner of motion of the heavenly bodies, we could not know that there was a force acting between them. All that we directly observed were their forms of motion.

Many may perhaps be surprised to hear that no one has yet directly seen the electron or the proton, and certainly not many other elementary particles. These are theoretical entities, whose existence we infer from facts available to us in direct observation. The same is true of force fields, the quantum wave function, and many other theoretical concepts in modern science.

It is therefore difficult to see theoretical scientific explanation as a simple generalization of particular facts. Science does not merely generalize particular facts into general laws, as Bacon thought; it also creates, as if from nothing, new and abstract systems of concepts, including theoretical entities, which stand behind the phenomena explained and cause them. These conceptual systems are not a direct result of observation, and it is very difficult to understand their emergence as mere induction and generalization from simple facts.

As a result of these problems, many doubt the existence of theoretical entities. They treat them as useful fictions, intended to organize the phenomena in an elegant and usable way. Below we shall discuss these approaches in greater detail, but already here one may point out that they fail to distinguish between a phenomenological theory and an essential theory. These claims are nothing but a reflection of a general analytic approach, as it is mirrored in the philosophy of science.

Note 6: The Ontological Status of a Physical Force Field

One of the basic entities investigated by physics is force. There are several fundamental forces in nature: gravitation, the electromagnetic force, the weak force, and the strong force. We saw above that some would interpret the concept of force as a fictional definition, which is nothing but a different description of the fact that a certain body is in acceleration.

Physics defines additional theoretical entities on the basis of the concept of force. The first of these is a force field. The second is a potential of a force, or simply a potential.

For each of the forces listed above there are also relevant charges. A charge is the property of objects that senses the force acting upon them and responds to it. There are also charges that generate the force, and these are usually charges of the same type, but that is not our concern here. For example, electric force is sensed by electric charges. A particle carrying electric charge, and located at a point where an electric field is acting, will feel a force acting upon it. A particle that is not electrically charged will not feel this force at all. The greater the particle’s charge, the greater the force it will feel. The same applies to the gravitational field, that is, the force of attraction. There the relevant charge is mass. A body with a greater mass will feel a stronger gravitational force—for example, a stronger attraction to the earth. That is why a fat person weighs more than a thin one. Since he has a greater mass, the force exerted upon him by the earth is greater.

The concept of a force field at a certain point in space is defined as the quantity of force that a unit charge would feel if located at that point. For example, if there is a certain point in space at which a particle carrying ten units of charge feels a force of intensity fifty units, then a particle with a charge of fifteen units will feel a force of seventy-five units. We define the force field at that point as having an intensity of five units. The meaning is that each unit of charge feels five units of force. Therefore ten units of charge feel fifty units of force, and fifteen units of charge feel seventy-five units of force.

At first glance, this seems to be a convenient fictional definition that helps us calculate the force that any particle will feel for any charge at that point. But one should note the following point. The fact that every charged particle standing at that point feels a force acting upon it indicates that even when no charged particle is located there to feel any force at all, it is still not correct to say that there is nothing there. There must be some entity there that is what causes a force to act upon every charged particle that reaches that point.

Thus it is clear that the force field is not merely a convenient fictional definition of force per unit of charge, but a description of the entity that is located at that point in space. This entity is not force, for force always acts upon some body, and every body that stands there will feel a force of different intensity. Rather, it is a certain entity that causes the operation of forces of varying intensity on particles with different charges. This entity is what we call a force field. We say that in that place there is a force field of intensity five units. This is not merely a method for calculating the forces that will act on every kind of particle found there, but a description of reality. Admittedly, without the presence of a charged particle at that point, we do not directly sense its existence.

As a result of this description, the analytic thinker will say that a force field is a fictional definition. It is a nonexistent entity, defined solely for the convenience of theoretical calculation. The holder of the synthetic position, by contrast, regards the force field as a description of objective reality. What truly exists is not force, but the force field. When there is a charge at that point in space, a force will act upon that charge, and the source of that force is the field. In fact, the force field is a more concrete entity than force itself. Force is only a kind of action of this entity, or an interaction between it and the charged particle, but it is not itself a real entity.

As we mentioned, physics defines a concept more abstract than the force field, namely potential. This concept is obtained by mathematical abstraction from the force field. If we return to the gravitational field, in which we know the phenomenon of potential in a more tangible form, the potential of the gravitational field at a point on a mountain is higher than the potential of the field at a lower point in the valley. Therefore, when a body with mass is located on the mountain, it will “tend” to descend to a lower point, to a point of lower potential in space, namely, to the valley. This is another description of the fact that on the mountain there is a force field such that if a particle with mass stands on the mountain, it will feel a downward gravitational force, tending to make it move toward the valley. In this picture, force is nothing but the difference between different values of potential.

My tentative inclination is that potential does not represent a real entity, but rather a different mathematical description of the force field. The reason is that we are not compelled to posit its existence by a causal consideration. There is no event caused by the potential, and therefore no event that forces us to assume that it exists. Every event occurring in the relevant space can be explained by means of the force field, and therefore potential probably does not belong to the ontology of science, but rather to the fictional part of scientific theory. Here even the holder of the synthetic position, who upholds the possibility—but not the necessity—of the existence of theoretical entities, will agree that this is a concept defined for the sake of theoretical convenience.

This claim is also connected to a further discussion of the concept of potential that will be conducted below, when we discuss teleology in science, that is, what it means to say that the particle “strives” to move downward.

Simplifying Generalizations

In the introduction to the second unit of the first book we dealt with simplifying generalizations. We saw there that when science tries to understand a complex phenomenon, it breaks it down into its components, and at the first stage tries to understand each component separately. After that, it attempts to reassemble all those components and examine how they fit together within the framework of the complex phenomenon.

The fundamental laws of nature do not deal with real phenomena, which are generally very complex, but precisely with pure, theoretical, hypothetical situations that can be described with great precision. For example, mechanics describes a system of laws of motion without friction, even though such motion never exists in empirical reality. It then characterizes the force of friction in itself, even though that too never appears in reality by itself, without other components. Finally, physics inserts these two components into the comprehensive equations that describe the real situation. Usually one can understand exactly what happens without friction, and how the force of friction itself is characterized. But the complex situation, in which all the components are present, is often not fully understandable. There we use various approximations and discuss limiting cases, such as a point particle, friction with a simple mathematical form, and so forth.

The conclusion, then, is that science seeks simplifying generalizations, and indeed is entirely built upon them. Simplification, as we saw there, is not a term of abuse in science. It is an existential necessity. Without simplifying generalizations, almost nothing can be understood.

Similar phenomena can also be seen in the social sciences. In that same introduction we saw the relation established between frustration and aggression. Clearly, this relation never appears alone. There are always additional factors that influence the reaction of the particular person we are observing. There are people who do not react violently when they are frustrated. There are people who react violently even though they are not frustrated. This does not mean that the law is false, but only that in addition to it other components are at work, and they too influence the person’s response. Simplifying generalizations are true with respect to theoretical situations, and as such they constitute entirely correct laws, not merely approximate ones. But the description of real behavior—whether of a physical object or of a human being—is composed of the influence of additional components, and these are governed by further laws.⁴³

Note 7: The Meaning of Restrictive Construals in the Talmud: The Purpose of Study⁴⁴

There is a common phenomenon in the Talmud which, at first glance—and even afterward—appears very puzzling. Sometimes the Gemara interprets the Mishnah in a way that does not accord with the language of the Mishnah itself. In such cases, the Gemara explains that the Mishnah is dealing with a very specific situation, only in which the law stated in the Mishnah is correct. This form of interpretation is called in the study hall an okimta, that is, a restrictive construal.

In such situations a difficulty arises: from where does the Gemara derive the right to construe the Mishnah in ways so far removed from its wording? And from the other side: could Rabbi Judah the Patriarch, the editor of the Mishnah, not have written explicitly that his intention was to discuss only a certain unique situation?

To sharpen the point, let us take an example from Babylonian Talmud, Gittin 21a. There the Gemara makes a restrictive construal of the words of an Amora, not of a Tanna in the Mishnah.

Rava said: If he wrote her a get and placed it in the hand of his slave, and then wrote her a deed of gift regarding the slave, she acquires him and is divorced through him.

The meaning is this: a man who wants to divorce his wife writes her a get, that is, a Jewish bill of divorce, and gives it into her hand. The Talmud here discusses a man who wrote a get for his wife, placed it in the hand of his slave, and afterward wrote her a deed of gift conveying the slave to her. The deed transferred the slave to her, and Rava’s innovation is that the act of transferring the slave, who is holding the get in his hand, is also considered a giving of the get itself to the woman.

The Gemara assumes that the get is acquired by the woman through the slave by means of a proprietary mode of acquisition known as acquisition through one’s domain. According to halakha, when an object is situated in a person’s domain, under certain conditions, it is acquired by him. For this purpose, the Gemara assumes that the slave is considered the woman’s domain, and therefore the get in his hand is acquired by her through that domain. The Gemara then raises an objection:

But why? It is a moving domain, and a moving domain does not acquire.

That is, the Gemara objects that a slave is a moving creature, and therefore is not considered a “domain” in the halakhic sense. Hence one cannot convey through it a get to the woman in the way one can do so through an ordinary courtyard.

The Gemara resolves this by means of a restrictive construal:

The law applies when he is bound.

That is, we are speaking of a situation in which the slave is tied up and cannot walk. In such a case he is considered an ordinary domain, one that is not moving, and it is possible to convey to the woman the get in his hand.

Already here the difficulty arises: how does the Gemara impose upon Rava’s statement such a strange restrictive construal? Could Rava not have told us that he was speaking specifically of a bound slave? The simple assumption is, of course, that when one says merely “a slave,” the intention is to an ordinary slave.

In Tosafot there, under the heading “ve-hilkheta,” an additional difficulty is added. The Tosafists object from Babylonian Talmud, Gittin 78a, from which it is evident that only if the slave is also asleep can one acquire the objects in his hand by acquiring the slave who is holding them. The conclusion of Tosafot is that Rava’s statement refers to a slave who is both bound and asleep. This is an additional restriction beyond the one made in the Gemara.

The difficulty, as stated, now becomes even stronger. There is no hint in Rava’s words that he is speaking of a slave in so unusual and uncommon a state. How do we permit ourselves interpretive freedom so broad with respect to the words of an Amora, or in other cases, with respect to the language of the Mishnah?

This difficulty has occupied many in the study hall. The common answer to it is that Rava did not really intend to discuss laws pertaining to a sleeping and bound slave. Rava’s intention was to teach us the law that transferring ownership of a slave who is holding a get in his hand is effective in transferring the get to the woman, and thereby divorcing her.

The problem is that in actual reality there are additional difficulties with this mode of transfer. A slave who is not tied up is not regarded as a domain, and therefore it is impossible to transfer a get that is in his hand. In addition, because of another constraint, if he is not asleep, it again cannot be done.

To this the Gemara responds that such a reality can be implemented in a case where the slave is bound and asleep. But Rava’s intention was not to teach us the law that applies only when the slave is bound and asleep. His intention was to teach us a general principle: a slave is like a domain, and transferring a get into it counts as giving it to the woman. If we are troubled by secondary constraints that arise on the practical plane, that is, constraints that do not allow the principle to be implemented in reality, then we shall find a purer situation, even one far removed from practical probability, simply in order to enable us to think concretely about the general principle. Rava teaches us a theoretical principle, and it makes no difference to him through what practical circumstances it can be implemented in reality.

According to this interpretation, the restrictive construal serves to set aside secondary objections. When Rava teaches us theoretical principles, there is no point in asking him how they can be implemented in reality. The importance lies in the very understanding of the theoretical principle. That principle can have practical consequences in entirely different areas.

Thus, the Amoraim, or the Tannaim in the Mishnah, sometimes speak in a general way in order to teach a theoretical principle. Later, when we examine how this theoretical principle can be implemented in practice in the case under discussion, we are compelled to add parameters to the description of the situation. Rava himself was not at all interested in the question whether the slave was bound and asleep or not. Therefore the question why he did not specify the situation is misplaced. Had he specified more, his concern would have been to teach practical law, and not a theoretical principle. When the issue is a theoretical principle, the details are unimportant.⁴⁵

It now appears that this is an approach very similar to the scientific abstraction discussed above. When a teacher teaches mechanics to his students, he always begins with a description of motion without friction. This is a theoretical situation, in which an object moves and encounters no frictional resistance at all—neither air resistance nor rough ground, and so on. Such a state is entirely hypothetical. A student now comes and asks the teacher: why is this important, since there is no such situation in the world as motion without friction?

The teacher will answer this student that the question is irrelevant. “We are now studying the theory of the mechanics of motion, and the problems of how to implement the theory in reality are not the subject of this lesson,” he will say. In other words, this lesson does not deal with engineering, but with theoretical physics.

Another, different in form but entirely equivalent, way of answering this student is by way of a restrictive construal: we are speaking here of motion that takes place far out in space, far from every star, so that there is no air at all and no friction resisting the motion.

One may ask why the teacher did not say in advance that he was speaking only of motion in space, since there was no hint of this in his words. How were the students supposed to understand that by themselves? The answer is that it did not interest him at all whether they would understand something about friction or about the absence of friction. Friction will be taught in another lesson. At present they are studying the theory of motion. The teacher did not teach the mechanics of motion in space, but principles in the theory of general mechanics. These principles are applicable on earth as well, except that in that context we must take additional factors into account.

Another interesting question that arises here is: what is the purpose of this study? In the halakhic context we ask: if it does not teach practical conduct, why is it important at all? And in the scientific context we ask: if it does not teach about phenomena in the real world, why is it important at all? And why is this science, rather than science fiction?

Let us begin with scientific study and research. Is the purpose of scientific inquiry to arrive, in a pedagogical and gradual way, at fuller knowledge, one capable of taking all factors into account? According to this, the ignoring of friction stems from pedagogical considerations of the order of learning. But one may also understand it differently: perhaps the teacher, or the researcher, does not intend at all to arrive at a description of real motion. There is importance in the very knowledge of the theoretical principles, and not only as an instrument for understanding the real world.

According to the second conception, the theoretical principles whose combination yields the description of complex and entangled empirical reality are not merely an instrument for understanding reality. The reverse is true: reality is an instrument for knowing the principles. The scientist’s purpose is not the knowledge of empirical reality, but the knowledge of the abstract laws of nature.

Within the framework of Torah study, an exactly analogous conception prevails in the yeshivot. The purpose of study is not only to know how to act in life in every situation—that is the task of halakhic ruling—but also to know what the will of God is. The will of the Holy One, blessed be He, is expressed in the theoretical principles themselves, and therefore the purpose of study is the knowledge of those theoretical principles. Afterward, in order to know how one should act in practice, we try to take into account the whole set of factors—both theoretical and practical, reality-based ones—in order to arrive at halakhic directives applicable to concrete realities. On this matter see also Note 9 below.

A final point that must be discussed here is the question whether these are true principles, that is, principles that correctly describe reality. Seemingly, motion without friction simply does not exist at all, even in deep space, where there is no perfect vacuum.

It appears that these principles do indeed describe reality. There is a correct and true principle according to which the characteristics of frictionless motion are such and such. In empirical reality there is no frictionless state, but if such a state did exist, the motion in it would look exactly like this. So too, the description of friction by itself is a description of the true force of friction, which is part of reality, but a part that never appears by itself. In the real situation one must combine the whole set of true and relevant rules in order to arrive at a full and correct description of empirical reality itself.

The matter is similar to the example from the social sciences discussed above.

One can also relate to simplifying generalizations as fictions. The analytic thinker may say that the situation of motion without friction, or of a person who responds solely according to the law linking frustration and aggression, are theoretical situations, and therefore those laws are not true. They describe no reality and therefore do not exist except in the scientist’s mind. This is another aspect of the fictitiousness of theoretical entities.

However, as we have seen, it is more plausible to understand that these laws are entirely true, except that they describe only part of the complex reality. See also the previous note on this point. We shall return to this below, when we discuss the synthetic explanation of science.

Simplicity and Elegance of a Scientific Theory

Suppose we are given a collection of facts for which we seek a scientific explanation. Let us set aside for the sake of discussion the simplistic assumption, which as we saw above is indeed simplistic, that we have already divided the facts into classes and selected the relevant set, and that we are now seeking for it a theoretical explanation. There is no doubt that most of the theories—the explanations—that we might imagine will not fit the given set of facts, and these of course will not be candidates for the scientific theory relevant to this set of facts.

But by the same token there is also no doubt that more than one theoretical explanation is possible for every given set of phenomena. In fact there are infinitely many possible explanations, and an infinity, at least equally large, of impossible explanations. Each explanation will have its own language and its own laws, but all of them will fit the given set of facts.

With respect to phenomenological theories, one can see this as follows. Essential theories will be discussed below, but the principle regarding them is very similar. Let us represent the set of known facts in some domain as a set of points on a coordinate system. For simplicity’s sake, let us say that these points represent the results of several experiments carried out in the laboratory in that domain. For example, let us draw a coordinate system describing experiments intended to examine the relation between the force acting on a body and the acceleration it develops. Suppose the results are those shown in the following illustration:

[Illustration: a graph of points with several possible ways of connecting them; see Hempel, p. 47.]

The general law we seek is the line connecting all these points. This is precisely the phenomenological theory relevant to this case. We seek a comprehensive description of all the experimental facts we have measured, one that is both generalizing and simple. The predictions of the general law, that is, of the theory, are the points that we have not yet measured on the graph. Therefore the empirical testing of the theory will be carried out by measuring an additional point on the graph, and comparing the experimental result with the result expected according to the theoretical graph.

But clearly, any given set of points on a coordinate system can be connected in infinitely many different ways: by a straight line between every two points, as in the illustration above, or by some variation in the path the line takes between each pair of points, or in other forms. Thus there are infinitely many theories capable of explaining, to the same degree of success, any given set of facts.⁴⁶

In such a situation, the scientist must decide on the form of the theoretical line that will connect the points, that is, on the outcome of the experiment, or on the true law of nature as it emerges from the experiment. Usually he will choose the simplest form. If the points can be connected by a straight line, that will be the theoretical law adopted.

Here we encounter a strange criterion for the adoption of a scientific theory: simplicity and elegance.

The concepts of simplicity and elegance are themselves very unclear. Why, in the previous case, is the simple description specifically a straight line? Why not some other line? Usually the answer to this is that we choose the line with the smallest number of mathematical parameters that can fit the experimental results. A straight line is the kind of line with the smallest number of mathematical parameters among the other possible line-shapes.

But now another problem rises to the surface, one more fundamental, concerning the assumptions of simplicity and elegance. Even if it were clear to us what simplicity and elegance are, it is not clear what justifies our assumption that nature always chooses the simplest and most elegant mode of behavior. Why should the selection among theories by means of a criterion of simplicity and elegance lead to knowledge about natural reality? Who said that natural reality conducts itself according to those same simple and elegant criteria? It should be noted that we are dealing here with a set of theories, or general laws, all of which fit the experimental results in our possession, and therefore there is no way to decide among them by means of laboratory experiment. If we perform yet another experiment and thereby eliminate many possible lines—every experiment eliminates infinitely many lines—we shall still be left with a new coordinate system that includes the point representing the additional experimental result, and the previous argument will once again lead us to the conclusion that there are infinitely many possible lines to describe the general law, even for the new situation.

Thus, the meaning of simplicity and elegance, and the scientific justification for using them as a criterion that selects among possible scientific theories, all of which fit the facts, is a difficult problem for which no evident solution appears.⁴⁷

It is important to emphasize here that we illustrated the problem by means of a discussion of the possibility of drawing a line connecting points that represent experimental results. This is a problem of finding a general phenomenological law for a given set of facts. The problem of selecting among possible essential theories is far more difficult. After finding the general phenomenological law, we usually seek an explanation for this law, that is, an essential theory for the phenomenological theory we have found. Why is the line connecting force and acceleration specifically of this shape and not of some other shape? In such a situation we introduce into the picture abstract theoretical concepts, and every general law we have found can be explained by infinitely many systems of theoretical concepts, each of which uses a different system of entities and theoretical laws.

The selection among the different explanatory systems for the phenomenological law that has been obtained likewise cannot be made empirically. The assumption is that all the relevant systems of essential explanation will explain the general phenomenological law arrived at in analysis of the experiment, and that there are infinitely many such systems. Thus, when we come to choosing an essential theory, the number of candidates is vastly greater than in the selection among phenomenological theories, since for each phenomenological theory that fits the experimental results there are countless essential theories that explain it. Hence, at the level of choosing a theoretical language, the problem becomes doubly complicated.

In sum, the problem of elegance and simplicity in scientific theories is one of the pressing problems in the philosophy of science, and seemingly it is one of the strongest supports—or strongest motivations—for seeking analytic solutions, of the sort to be discussed below.

Note 8: Ockham’s Razor

Here is a well-known example of the criterion of simplicity. This criterion was first formulated by William of Ockham, a fourteenth-century Franciscan monk, and it is called Ockham’s razor. This principle states that one should not multiply assumptions concerning the existence of objects beyond what is required. The common formulation is an expansion of this principle to assumptions of any kind, not necessarily ontological ones. The expanded principle of Ockham’s razor states that one should not multiply assumptions beyond necessity. In fact, in the modern age the term Ockham’s razor is used to describe the criterion of simplicity and elegance in general.

According to this principle, if we can explain the collection of scientific facts in a certain field by means of two theories, one containing N assumptions and the other containing M assumptions, we shall tend to prefer the theory with the smaller number of assumptions.

At first glance, this intuitive principle serves various domains of thought, and not science alone. For example, if a person enters his house and sees that it has burned, he will not assume that one fire broke out and burned the living room, and another fire, which broke out later, burned the bedroom. The simpler assumption will be that it was one fire that burned the whole house. Another example: if we notice that classified secrets have disappeared from the Ministry of Defense, we shall assume that there is one spy there, and after we have found him we shall not continue to search for others, certainly not for others unconnected to his network. And another example from medicine: if a certain person suffers from a variety of symptoms, and we know that there are two possible explanations for the medical picture before us—either he has some disease that can explain them, or two other diseases, each of which causes part of the symptoms—we shall tend to adopt the simpler theory.

It is worth noting a common aspect to all of these examples. In all these situations we are dealing with events whose probability is low. Disease is a pathological phenomenon, and is therefore less probable than a healthy state. A fire is a pathological state, and usually most houses at most times do not burn. The same is true of diseases. If so, here it appears clearly that a state in which two improbable events occur has a lower probability than a state in which one improbable event occurs. Here, then, the principle of thought that assumes simplicity is an entirely straightforward statistical principle.

When we discuss Ockham’s razor, it seems that we are speaking about an entirely different kind of claim. For example, if plane geometry could have been derived on the basis of a system of six axioms rather than four, as in the standard formulation, then we would have had two possible ways to formulate that mathematical system. In this context there is no simple statistical consideration that leads us to prefer the system with fewer axioms. There is nothing implausible about the axioms themselves, and therefore a system with more axioms is no less probable, in the statistical sense, than one with fewer axioms. Here the principle known as Ockham’s razor enters and tells us that even here we should adopt the system with fewer axioms. It is quite obvious that this is a principle of simplicity and elegance, not a statistical principle.

This becomes even sharper in logical systems. We know of quite a few ways to present the theorems of symbolic logic: some employ one assumption, others three assumptions, or more. Every logician understands that a system with fewer axioms is of greater value. One of the goals of logicians is to reduce the number of assumptions in every axiomatic system they encounter, and there are even mathematical algorithms whose purpose is to do exactly that.⁴⁸

This principle serves us mainly in scientific or mathematical thought, because it guides intellectual activity. Its appearances in everyday thought are rarer, and there one usually encounters its statistical version, which we discussed above, though not always.

At first glance this principle too is very intuitive, yet it does not seem to have a simple explanation, statistical or otherwise. I will return to this below.

It should be noted that, in a certain sense, Ockham’s razor actually lies at the basis of the demand for diversity of evidence as well (see Note 3 above). We saw that if one can bring more varied evidence for a given theory, this confirms it more strongly than bringing a larger number of pieces of evidence of the same kind. On the one hand, we tend to assume that diverse facts have a single theory that explains them all; this is Ockham’s razor. On the other hand, we assume that if there is diverse evidence for a particular theoretical explanation, it is probably correct; this is the principle of diversity in evidence. Put differently: it is preferable to assume that all the contexts in which the phenomenon appears have one explanation, rather than to assume that each has a different explanation. More than that, as we saw, diversity strengthens the plausibility of the explanation in comparison with an explanation supported by the same quantity but a less rich variety of evidence.

To conclude, I will cite several halakhic (Jewish legal) discussions of this principle.

1. In Note 3 we saw the hermeneutical form known as mah ha-tzad or binyan av—an inference from a shared principle—and its relation to the principle of diversity in evidence. From the side of Ockham’s razor we would say that if two laws can be grounded in a single explanation, we should adopt that explanation rather than an alternative that proposes a different explanation for each of them. This is precisely the basis of the interpretive form known as mah ha-tzad.⁴⁹

2. There is a common rule in the study hall regarding the interpretation of disputes among the tannaitic and amoraic sages: we do not multiply disputes unnecessarily.⁵⁰ This rule says that one should not attribute more points of disagreement to disputing sages than is necessary. In Talmudic give-and-take, down to our own day, an explanation that yields two disagreements between the parties is rejected in favor of an explanation that grounds the disagreement in a single principle.

According to this rule, one may derive halakha even from the words of one of the disputants from whom the final ruling does not follow. The assumption is that if the principle being derived is not connected to the explicit disagreement between them, then the other disputant also agrees with that point. Anyone wishing to reject such an inference must show that the conclusion really depends on the disputed principle between the views—in other words, that it is not independent of that dispute. In halakhic discourse one cannot reject such an inference merely on the ground that it is based on the words of the sage whose opinion was not accepted as normative law.

As one example among hundreds and thousands, the Ran, in his commentary on the Rif in tractate Yoma (4b in the Rif pagination, under the heading Ve-garsinan), cites a responsum of the Ra’avad, one of the great sages of Provence and the author of the glosses on Maimonides, regarding a sick person on Shabbat for whom a physician determined that he must eat meat, but the only meat available is carrion, which is not kosher. The question is whether one should slaughter an animal for him on Shabbat—which is an extremely severe prohibition—or feed him carrion, which is a lighter prohibition. The medieval authorities disagreed on this question, and some wrote that he should be fed carrion because it is the lighter prohibition. The Ran argues that one should slaughter a live animal for him, because although slaughtering on Shabbat is indeed more severe, it is only one prohibition. By contrast, in eating carrion there is a prohibition for every olive-sized portion the patient consumes.

Later authorities derive from the Ran the principle that eating several legal measures of forbidden food entails several punishments, one for each measure. This is so even though the Shulchan Arukh did not rule in accordance with the Ran.⁵¹

As stated, the reason is that the disagreement between the Ran and the other medieval authorities did not revolve around this question, or at least so the commentators understood, but around the question whether quantity overrides quality. If so, with regard to any point not tied to their fundamental dispute, one may understand that there is agreement. Therefore one may derive halakhic conclusions from it even from the words of someone whose opinion was not accepted in practice.⁵²

In effect, this anticipates a certain version of Ockham’s razor, since we assume that the parties have no more disputes than those stated explicitly.⁵³ Still, here we are dealing with a guiding rule, usually implicit, for interpreting tannaitic and amoraic disputes, not with a comprehensive formulation of a principle of thought.

3. A similar principle is known in the name of Rabbi Chaim Soloveitchik of Brisk, Lithuania, in the late nineteenth and early twentieth centuries. Rabbi Chaim’s formulation is already closer to that of a comprehensive principle of thought. I found a written source for it in the book Torat Chaim, which transmits teachings from the Brisk school. It writes there:⁶⁶

Rabbi Chaim, of blessed memory, used to say that when there is one question in a Talmudic discussion, one answers with one answer; when there are two questions, one answers with two answers; but when there are already three questions, one no longer gives three answers. Rather, one must say a single answer, namely: we are not learning the discussion correctly. For the multiplicity of difficulties proves that something basic in the discussion is not understood by us, and at that point further answers will not help; rather, one must learn the discussion again from the beginning.

He brought proof for this from what appears in the Babylonian Talmud, Hagigah 3b:

“The Sages taught: Who is a fool? One who goes out alone at night, spends the night in a cemetery, and tears his clothing…”

The case is always one in which he does these things in a foolish manner. But if he spends the night in a cemetery, perhaps he does so in order that a spirit of impurity rest upon him; if so, he is not necessarily a fool. And if he goes out alone at night, perhaps he is gripped by anxiety, or perhaps he is overheated; again, he is not necessarily a fool. And if he tears his clothing, perhaps he is preoccupied, and in his distraction he tore it; again, he is not necessarily a fool. But once he has done all of them, he is like an ox that has gored an ox, a donkey, and a camel and thereby becomes established as dangerous for everything—that is, he is evidently a fool.

We thus see from the Talmud that the first and second times one offers an explanation for why he acted as he did, but by the third time there is no longer any such explanation. For in place of offering three separate answers, it is preferable to offer one answer: this person must necessarily be different in essence from others; he is a fool.⁵⁴

A similar line of thought may also be found in Rabbi Abraham Isaac Kook’s Orot, in the section called “Zeraonim,” under the subsection “For the War of Opinions and Beliefs.”⁵⁵

4. It seems that one may add here the principles that “it is enough for what is derived by inference to be like the source from which it is derived” (see Babylonian Talmud, Bava Kamma 24a and parallels), and “if you grasp too much, you have grasped nothing” (see Babylonian Talmud, Yoma 80a).⁵⁶

An interesting study in its own right would be to examine the relationship among these principles, their common elements and their differences, but that is not the place for it here.

Summary and Conclusions

In this chapter we have seen that the Baconian description of scientific work as the collection of facts followed by their analysis and the construction of a theoretical explanation is too naive. As noted, the facts depend on the explanation, the explanation depends on the facts, and everything depends on common sense. Beyond that, we saw that the generalization from facts to theory is problematic. We also saw that some of the theoretical entities generated in scientific theory cannot be experimentally observed at all. More generally, a scientific theory is mainly open to refutation, and perhaps to confirmation, but not to full experimental verification.

It should be noted that the problems we raised in describing the process of scientific inquiry are only some of the most basic ones. There are many others that are not the subject here. For our purposes in what follows, it is important to state in advance that the root of the entire problematic character of scientific methodology is that proposing a general scientific theory is clearly a synthetic process, and therefore it cannot be justified analytically. Intuition plays a decisive role in scientific thought and methodology, and we already saw in the first book that the attitude toward intuition lies at the center of the analytic-synthetic controversy.

It is therefore not surprising that those who hold synthetic and analytic positions will relate differently to science and its methodology. We will examine the difference between these two attitudes in the next two chapters.

Chapter Three: An Analytic Explanation of Science: Analytic Solutions to the Methodological Problems

Introduction

In light of the range of problems presented in the previous chapter concerning the understanding of scientific methodology, it is only natural that different approaches should be proposed to solve them and to explain science in different ways. In this chapter we will examine analytic proposals for explaining science, but first we should remind ourselves what an analytic position is.

In the first book we saw that analyticity is a convenient kind of solution to a problem involving lack of justification, proof, or grounding for a natural mode of thought. In such a situation, the problem is attributed to incorrect use of language, and the problem is solved by repairing the language. Very often the analytic solution does not accord with ordinary human intuitions, but the analytic thinker tends to ignore this difficulty, because he assigns supreme—and perhaps exclusive—value to consistency, definition, and proof. In the eyes of the analytic thinker, intuition is a kind of subjective feeling, and therefore it should be suppressed, with recursive, logical-analytic reason made dominant over it.

In fact, over the past two centuries several analytic directions have arisen for explaining science and its methodology. These can be divided into two main directions:

1. The positivist-instrumentalist solution, formulated in several variants by several thinkers—Dewey, Comte, Mach, Schlick, and others—which we will encounter in the present chapter. This is an attractive analytic solution to the entire set of problems raised in the previous chapter. In the chapters that follow we will discuss the relation of this solution to basic human intuitions.

2. The Kantian solution, which explains that science does not deal with the world as it is in itself, but with the world as it is reflected in human cognition. In that world, according to Kant, one can formulate comprehensive theories without the same failures we pointed to above. We analyzed the Kantian approach in great detail in the first book, not in the scientific context but in the broader philosophical one, and there we saw that it does not provide an adequate response to the problems it faced. Beyond that, we saw that Kant’s solution is fundamentally analytic. For those reasons we will not deal here with Kant’s specific approach, different though it is from the previous solution.

The Context of Discovery and the Context of Justification

As a first step in the analytic treatment of the cluster of problems presented in the previous chapter, let us briefly introduce the distinction made by the philosopher of science Hans Reichenbach between the context of discovery and the context of justification.⁵⁷

When a scientist is confronted with a set of facts and must provide them with an explanation, many and varied explanatory possibilities stand before him, as we saw in the previous chapter. Some of the explanations indeed use different terminology and concepts, but do not differ from one another on the practical plane. That is, all of their predictions concerning what should happen in various experiments are identical. Therefore, among such explanations or theories there is no way to decide by experiment, and the choice will usually be made on grounds of simplicity of theory and convenience of use, or on other grounds.

Some explanations differ from one another on the practical plane as well, meaning that they make different predictions about what will happen as the result of at least one possible experiment. Among those theories there is experimental room for decision. One must perform the decisive experiment and see which theory’s predictions withstand the test of experience. At first glance, the decision in such a case seems simpler.

Yet on the practical level one cannot carry out experiments testing all possible theories, even those that differ empirically from one another, whether because of the complexity of the experiments required, because of the sheer number of such theories, or because of additional parameters that intervene in the experiment and affect its results. It is therefore clear that selection among the various theories must first be made on a primary intuitive level. That selection is part of what Reichenbach calls the context of discovery. The stage at which the theory is discovered and proposed comes before the stage of its testing.

But in fact this description does not exhaust the meaning of the context of discovery. The very formulation of the theory—any of the theories possible in advance—out of a set of facts is a creative process with no simple explanation. Beyond the intuitive selection among the various theories already on the table after their formulation or discovery, as described above, the very formulation of each of them out of the facts is not trivial, and it is difficult to see how one could describe it and indicate the methodology by which it is carried out. This is discovery, and as such it is difficult to offer a satisfactory explanation or grounding for it.

For example, Semmelweis’s formulation concerning the matter from the dead body that caused the mortality of women in childbirth does not arise directly from the facts before him. There is here an element of generalization that is not derived from the particular facts, but there is also a more fundamental problem.

The set of facts that stood before Newton when he formulated the theory of gravitation or his laws of mechanics likewise could not have led in any logical way to those theories. The mere observation of objects falling to the earth and of the motion of heavenly bodies cannot lead to a theory of gravitation. The proof is that thinkers in generations before Newton, all of whom were fully aware of the phenomenon of bodies falling to the earth and equally aware of the motions of heavenly bodies, never conceived of such a theory. Newton did not merely determine empirically that this was the correct theory; he discovered and formulated it, thereby making it available for experimental testing. He not only drew a general conclusion from the set of facts before him, but also invented, out of nothing, a new conceptual system: force, mass, acceleration, and he also determined various relations among them. As stated, the conceptual system does not arise from the facts in any way whatsoever. The facts concern objects and their visible behavior, whereas the conceptual system is a system of abstract concepts not directly observable. Hence there is no way to see how it grows out of the facts. It appears to be a purely creative invention.⁵⁸

This is even more striking in modern physics. In that field one tries to understand the factual world by using a conceptual world built out of abstract theoretical entities. One cannot simply invent the concept of a wave function, as it is used in quantum theory, or even the concept of momentum, which is used in classical mechanics as well, or even force, by observing some collection of facts. These are theoretical concepts with no empirical root in the simple sense, and they were discovered in a process belonging entirely to the context of discovery. Below we will sharpen the point that this process is synthetic in its very essence.

Everything said thus far shows that the invention of a theory relevant to a given set of facts—and, as we saw above, even the characterization of the relevant set itself—is a creative step with no clear logical rules, and as such it cannot be described logically.

By contrast, once some scientific theory has been proposed as a hypothesis, one can derive from it, logically, predictions about experimental outcomes. If one then carries out such an experiment and discovers that the prediction is wrong, one may say, logically, that the theory has been refuted. This is the stage of justification of the theory, which Reichenbach calls the context of justification.

Let us now try to classify these two stages on the analytic-synthetic plane.

The stage of justification of the theory is fundamentally analytic in character. If a theory states that a certain object should fall to the earth at a certain speed, and in the experiment it does not fall in that manner, then the theory has been refuted by simple logic.⁵⁹ This is an application of the principle of non-contradiction, one of the basic pillars of logic—in other words, a thoroughly analytic instrument.⁶⁰

The analytic conclusion in such a case is that the theory has been refuted. But if the experiment succeeds and the theory withstands the test, we have seen that the conclusion in such a case is less clear. The only analytic conclusion is that the theory has not been refuted. As we saw, Baconian experimental verification of the theory is not an option. The conclusion that the theory has been confirmed by such an experiment, as Carnap and Hempel maintain, is clearly not analytic, because, as we saw, it is not open to logical justification. We saw that the approach of confirmation does not answer the problems raised by Popper. The concept of confirmation does not belong to the realm of pure logical analyticity.

If we look at the context of discovery, we see that it is clearly synthetic in character. There is no analytic logical route from the facts to the theory, and there is not even a clear relation leading from the facts to the theory that explains them. We therefore cannot escape the conclusion that the context of discovery is a process of creation, and it is synthetic in essence. It is not open to logical-analytic description, nor to analytic justification. Of course, its result can be tested in the laboratory; that is the essence of the context of justification. Below we will see what proponents of the analytic position say about this.

Reichenbach’s model is sometimes called the hypothetico-deductive model. The context of discovery places before us a hypothesis. Then, in the context of justification, we test that hypothesis by experience. If experience refutes the theory, the conclusion that the theory is not correct is a simple deduction, that is, a conclusion derived analytically from the results of the experiment.

We thus discover that science too contains creative synthetic stages, and not only analytic thought, as many tend to think. One may say that these stages are the more important part of scientific progress; some would say: the place where scientific genius finds expression. In principle, the analytic part of the work can be done by anyone familiar with the facts and with scientific methods. Most scientists deal with that part. The creative part—discovery—takes place in the mind of the scientific genius, and there is usually no description or explanation of how he reached the theory he proposed. Even when such explanations exist, they may sound very strange. Some attribute it to dreams, revelations, or philosophical, religious, and other beliefs. This is the non-objective part of science, and therefore also irrelevant to criticism. Logically one can only justify, that is, confirm, or refute the theory, but one cannot criticize the way it was discovered.

The essence of Reichenbach’s argument is that there is no point in relating to the way the scientist arrived at the proposed theory. All discussion of the theory takes place on the plane of empirical testing, and therefore concerns only the context of justification. A theory discovered in a dream or prophecy and found to be correct is no less valid than a theory found by rational and accepted routes. Both will be tested in the furnace of empirical examination, that is, in the laboratory.

We must note, however, that as we saw, at least according to Popper, in the stage of justification one can only refute the theory, not support it. Add to that the fact that the context of discovery is irrelevant to the testing of theories. These two claims lead directly to the analytic picture of science.

Science According to the Analytic (Actualist) Picture

Proponents of the analytic position now take a further step and argue that scientific theory in fact says nothing about the world beyond the set of facts it describes, that is, beyond the set of its predictions. According to the analytic thinker, a scientific theory is nothing more than a formulation whose purpose is to include all the facts known to us within one comprehensive description. We do not claim that the theory is true in itself, meaning that it describes something in the world beyond the set of facts included in its formulation. According to this interpretation, modern physics does not claim that there is in reality a wave function or momentum, since these are the result of a subjective-creative process and therefore do not withstand stringent logical or empirical tests. According to the proponents of the analytic position, the meaning of a scientific theory is exhausted by the fact that, if we adopt the conceptual system discovered in the context of discovery, we can include all the facts in one simple system. According to this approach, the theory is nothing but a sophisticated description of the set of experimental facts it describes, and nothing more. The theoretical entities and laws are interpreted only functionally and operationally: what they do, not what they are in themselves. Their empirical content, what can be tested experimentally, is all that they are. One can easily discern here the traces of conventionalism, which generally characterizes analytic positions, as we saw in the first book, in the second gate.

It should be noted that this approach erases the common distinction between a substantive theory and a phenomenological theory, as defined above. The analytic thinker regards the substantive theory, which uses a set of abstract theoretical entities, as a sophisticated kind of description of facts—that is, as phenomenology. There is nothing in the substantive theory beyond a different description of the facts. We define theoretical concepts in order to describe the objective facts by means of them more conveniently and efficiently; those facts alone, in the eyes of the analytic thinker, point to actual reality. The theoretical entities do not exist. They are merely useful fictions for describing facts.

Accordingly, it is clear that experiments conducted in order to refute the theory are not meant to teach us anything comprehensive about the world. In such experiments we are merely trying to continue learning more particular facts. Once we have learned them, we again examine whether the theory still constitutes an adequate description of the totality of knowledge we possess. If it has been “refuted” in experiment, we discard it and seek a description, or a language, that will stand up to the test of the newly discovered facts. The new description is not more correct, nor does it describe the world more reliably, in any sense than the previous one. It simply succeeds in bringing under its linguistic wings the new facts revealed in the latest experiment.

This description sounds rather Popperian. But we must note that it contains an important extension of Popper’s claim. Popper argued that we have no way to prove, or even to confirm, a comprehensive scientific theory, only to refute it. But Popper’s argument does not say that a scientific theory says nothing about reality. Popper’s argument concerns only our inability to reach true reality. Even according to Popper, it is entirely possible that there is a true theory in reality, and it may be that the set of theoretical entities defined in scientific theory really does exist. He only pointed out that we have no experimental way of knowing this.

Thus the analytic picture—actualist, in Bechler’s terminology—presented here is indeed a natural continuation of Popper’s point, but it certainly takes an important step forward, or perhaps backward. It denies altogether the existence of abstract concepts or theoretical entities. In its eyes these are nothing but a convenient language for describing the facts known to us.

According to this approach, then, science does not discover general laws, nor does it uncover abstract entities that cannot be observed by the senses. Science merely proposes a theoretical language in which one can express the set of facts conveniently and elegantly, so that it will be easier for us to use them for various purposes. The experiment that tests a theory does not really teach us anything about the theory itself. The results of the experiment, which have taught us new facts, must of course be taken into account when we decide on the language in which we formulate our scientific knowledge. If the previous language cannot incorporate the results of the experiment just performed, it must be abandoned and a new, more suitable language must be found.⁶¹

From here the sociologizing revolution that overtook the philosophy of science in the wake of Thomas Kuhn follows naturally. In his book The Structure of Scientific Revolutions, Kuhn describes the process by which a ruling scientific theory, a paradigm, is replaced as a process that takes place on the sociological plane. The replacement is not the result of refutation or of a substantive preference for one theory over another, but only of considerations involving political-social mechanisms of power within the scientific community. If science says nothing about the world, then it cannot be refuted, and replacing a theory is nothing more than a change of language. We will discuss Kuhn’s subject further below.

The Solutions Offered by the Analytic Picture to the Problems Above

There is no doubt that such a picture, together with Reichenbach’s distinction, completely solves all the methodological problems raised earlier.

The scientist does not generalize at all, and therefore there is no room to ask what justifies the generalization of particular facts into a general law. The general law is nothing more than a unifying formulation, in a different theoretical language—see the discussion of the concept of paradigm in the next gate—of the set of particular facts, and no more than that. It says nothing in itself about the world, and therefore there is no point in seeking a justification for the claim embodied in it. This is an analytic picture of science, according to which it says nothing about the world beyond the facts that we have directly observed.

The stage at which a theory is formulated, the context of discovery, is indeed mystical in essence and cannot be justified or described consistently and logically. But there is no problem in that, because there is no discovery here about the world itself. It is hard to see any special problem in someone inventing an appropriate language by esoteric means, so long as that language does not pretend to say something about the world.

In Kantian terminology, one may say that the theory is not a synthetic a priori claim—that is, an a priori claim about the world—but an analytic claim, one that adds nothing to the world. In this process we discover a convenient language, not facts about reality. There is no importance to the question of where the language we use comes from. Language is arbitrary, and it is very likely that many other languages would be equally correct. We merely want to test whether the language we use is also correct—not whether it alone is correct.

One must note that a basic assumption here changes, an assumption we earlier took for granted. Above we noted that a theory about the color of ravens has no value if we have already observed all of them. We explained that the value of a scientific theory exists only when it has predictions about situations we have not yet observed. We assumed, implicitly, that a scientific theory does not come merely to summarize the knowledge already in our possession. On the contrary, the knowledge we already have serves as a means for formulating and testing the scientific theory, which expresses comprehensive knowledge about the world and which, not the set of facts derived from it, constitutes the primary goal of science. Here we see that proponents of the analytic position do not accept this approach. In their eyes the theory is nothing more than a comprehensive description of empirical facts, and no more.

Note 9: The Purpose of Torah Study

In the halakhic world it is very common to think that the purpose of Torah study is halakhic knowledge, and in the language of the sages: “to bring the discussion to a practical halakhic conclusion” (see, for example, Babylonian Talmud, Sotah 21a; Bava Kamma 92a; Sanhedrin 106b). Such knowledge enables us to act in accordance with the requirements of halakha (Jewish law): “An ignorant person cannot be sin-fearing, and an unlearned person cannot be pious” (Mishnah, Avot 2:6). In the yeshiva world, however, it is specifically accepted that study is an end in itself, and not merely a means to knowing how to act.

Some would say, even more radically, that the laws themselves are means rather than ends. According to this conception, the various laws are meant to serve as test cases for the underlying conceptual analyses. That is, the aim of study is the correct understanding of the Torah’s principles, and the laws are meant only to serve as cases by which one can test the different understandings.

There is here a dispute exactly parallel to the one we saw above in the scientific context: is the purpose of scientific inquiry empirical knowledge, that is, the set of items of knowledge in concrete, practical cases; or is the more basic goal the understanding of the principles themselves, with empirical testing serving only to purify and refine scientific principles? For this discussion see also the end of Note 7 above, and the passage shortly before it.

To sharpen this point, I would like to survey briefly earlier attempts to understand the philosophical principles underlying halakha, and to show what basic approach stands behind each of them.

In fact, every book dealing with the conceptual foundations of halakha tries, in one way or another, to understand its basic principles. By contrast, most responsa literature deals with principles as tools for clarifying the law in the particular case under discussion.

Yet from this distinction alone one cannot infer the existence of a conception that sees study as an end in itself, for one can still think that engagement with principles is intended to clarify the law in other cases that will come before the learner or legal decisor. One should remember that principles are an efficient tool for determining the law in cases not explicitly discussed in the literature of legal rulings. In the same way, in the scientific context as well, it is clear that even those who regard scientific principles as tools for clarifying the behavior of nature in concrete cases certainly also engage in clarifying the correct principles and generalizations, and not only in collecting particular items of factual information. The difference is that they regard those principles only as means.

There are a few halakhic books whose declared purpose is the clarification of concepts, such as Shev Shema’teta by Rabbi Aryeh Leib Heller, Sha’arei Yosher by Rabbi Shimon Shkop, Atvan de-Oraita by Rabbi Yosef Engel, and others. But, as stated, that still does not necessarily prove such an ideological approach.

There are other books, far fewer, that attempt to deal with the philosophical principles underlying halakha. Examples are the books of Rabbi Yosef Rozin, the Rogatchover, which try to ground halakha on a conceptual basis taken from Maimonides’ philosophical work Guide of the Perplexed.⁶² Another example is the books of Rabbi Amiel, the best known of which is Ha-Middot le-Heker ha-Halakha,⁶³ and which are devoted almost entirely to the philosophical principles of halakha.⁶⁴

All these books are still built around questions that are primarily halakhic. They ask questions such as: What is causality in halakha? What is halakhic time? They investigate the theory of halakhic substance, and the like. Therefore even in these works there is no clear proof of this ideological approach, although anyone who studies them can certainly gain the impression that this is indeed their tendency.

In one of my articles⁶⁵ I brought a clear example of such an approach from the writings of Rabbi David HaCohen, known as the Nazir. The Nazir argued that the thirteen hermeneutical principles have philosophical significance; that is, one can derive from them conclusions that are not confined to the Torah-halakhic context. If there is halakhic study whose declared aim is the clarification of a non-halakhic philosophical issue, that testifies to an approach that sees Torah study as having value on the intellectual plane, meaning that it is not merely a means of determining halakha in concrete cases.⁶⁶

There are quite a few explicit references to this approach, which sees study and the understanding of principles as valuable in themselves, in essays and introductions written by various sages. It seems that this tendency has become more central as the generations have advanced. As noted, in halakhic-analytical works it is hard to find clear evidence of this. I will therefore conclude this note with a passage from Rabbi Israel Salanter, the founder of the Musar movement in Lithuania in the late nineteenth century, in his essay “Law and Judgment,” which is devoted entirely to this issue.⁶⁷ This passage finds unequivocal expression in the words of the sages and reflects the approach just described, which is, as stated, very widespread in the yeshiva study hall. Rabbi Israel proves the principle from the words of the sages in the Babylonian Talmud, Sanhedrin 71a:

The stubborn and rebellious son never existed and never will exist. Why then was it written? Study it and receive reward—in the world to come.

Rabbi Israel raises the following question: what need is there for an entire passage whose sole purpose is to grant reward for study? The Torah is hardly lacking in passages, and no one can comprehensively finish all the passages that are already there. If so, it is not clear why we need an additional passage in order to increase the reward of those who study.

Rabbi Israel explains that the Talmud means to say that this passage was not written in order to increase the reward for study, but in order to teach the very principle of “study and receive reward.” This passage was written to teach us that study has value in itself, expressed through reward, beyond the fact that it teaches us what we must do on the practical plane.

In fact, one can see this from the very position brought in the Talmud, namely that the stubborn and rebellious son never existed and never will exist. If so, here is a passage with no halakhic value at all, and yet it is an inseparable part of Torah study. That itself teaches us that such study has value. The saying of the sages, as interpreted by Rabbi Israel, merely sharpens this point further.

The clearest expression, then, of a position that sees study as valuable in itself, beyond the halakhic instruction contained in it, is found in passages that have no halakhic ramifications whatsoever. That is precisely the situation in the law of the stubborn and rebellious son.⁶⁸

Let us now summarize how the analytic picture completely solves all the philosophical problems in the methodology of science that were presented in the previous chapter.

1. The problem of circularity that we saw above, according to which the facts precede the theory that explains them while simultaneously depending on it, does not arise at all according to the analytic explanation of science. The theory is nothing more than a description in a different language of a set of facts. Therefore there is no importance to classifying the facts into different groups. The classification is made according to whatever language currently seems most convenient to us. There is nothing miraculous in our ability to find a group of facts that fits a certain law even before that law is known to us. The reason is that this law is nothing more than a different formulation of the set of facts we chose. Had we chosen a different set, we would have found, or not found, a different “law of nature,” no more correct and no less correct, describing that other set of facts.

2. According to the analytic picture of science, the context of discovery is not discovery at all, but creation. The question of justifying theoretical discovery is similar to the question of justifying the conventions underlying language. It is an arbitrary occurrence that says nothing about the world and therefore requires no justification. It is true that it is not clear how it occurs; the formation of language involves many unexplained miracles. But we have no problem saying that we do not understand that “miracle.” If that miracle were saying something about the world, then we would be in trouble, because in that case we would not merely be admitting that we have no explanation for something; we would at the same time be claiming complete confidence that this is indeed how things are.

For example, if the statement about the wave function in quantum theory were a statement about the world—namely, that wave functions exist in the world—then the fact that we have no explanation of how we “hit upon” the fact of its existence would raise the obvious question: who told you that it really exists? But if it does not in fact exist in the world outside us, and we merely use it as a convenient language for describing facts, then we have no special difficulty acknowledging the miracles involved in the formation of quantum language, which creates out of nothing concepts such as wave function, whose origin in our consciousness we do not know.⁶⁹

3. As for the problem of the relation between the hypothesis and the facts, according to the analytic picture the hypothesis does not arise from the facts; it is a convenient and efficient language for describing them. As noted, there is a certain creative dimension here, but there is nothing problematic in its justification, since it says nothing about the world.

4. According to this picture, the problem of induction does not exist either, because there is no inductive step in the transition from the facts to the general law. The general law is nothing more than another formulation of the set of facts known to us, and it says nothing about other facts. So long as the comprehensive formulation has not been refuted, it is entirely correct. After its refutation, what has become clear is not that it was incorrect from the outset, but only that it can no longer be used, because the stock of facts available to us has grown and it no longer fits them.

5. The problem of identifying the relevant parameter in analogy and scientific induction does not exist at all in the analytic picture of science, because according to this approach science does not perform generalizations. The division of facts and occurrences into different classes is arbitrary and is not based on any real similarity among them.

6. The problem of the direction of correlations, discussed above, is likewise irrelevant according to this picture. The general laws say nothing about the world, and therefore the correlations are not really there either.

7. It seems that the most decisive argument in favor of the analytic picture is the criteria of simplicity and elegance in scientific theory.⁷⁰ As we saw, there seems to be no way to justify simplicity and elegance as criteria for choosing among scientific theories that fit the factual data. More than that, it is often difficult even to define what the simplicity and elegance of a scientific theory are.

Here the analytic approach comes into its own with greatest force. According to the analytic thinker, the whole question of the simplicity and elegance of science does not arise at all. We choose the simplest and most convenient theory for us, in the sense of ease of use and everyday convenience. Since the theory is not supposed to be true but efficient, there is no point in worrying about justifying the demand for simplicity and convenience, because this demand concerns the language we use, not the description of reality itself. Obviously we want to use the simplest language for us, and there is no need to justify or define that at all.

Let us sharpen this final point. According to the “naive” conception of science, the theory expresses truth about the world. On this view, clearly only one theory is true. Therefore, given a set of facts that we must explain, one must choose one of the theories fitting those facts and decide that it, and it alone, correctly describes reality. Thus arises the problem of choosing among theories, and with it the problematic criterion of simplicity and elegance. The core problem is this: on what basis do we believe that the world behaves specifically in the simplest and most elegant way? Why should simplicity and elegance be criteria for the truth of the description?

According to the analytic picture, there is no such thing as a true theory and an untrue theory. All theories that accord with the totality of facts in our possession are equally “correct.” All are possible languages for describing that set of facts. If so, simplicity and elegance are not criteria for describing the world, but for the arbitrary choice of one language among many possible ones for describing the body of facts in the relevant domain. Such a choice is arbitrary, and therefore there is no reason not to make it in the ways that intuitively seem most convenient to us. There is no need to justify an arbitrary choice among things equal in standing, value, and meaning. A simple and elegant language is undoubtedly the most convenient, and it is therefore obvious that we choose it to describe reality.⁷¹

Why This Description Is an Analytic Position

Let us now explain the connection between this explanation of science and an analytic position, as that was defined in the general context, as discussed above in the first gate and in the first book.

The analytic position was defined as an approach that accepts as true only what is directly observed by the senses or what is proven analytically. We saw that in fact there is no analytic proof of any proposition whatever, since every proof rests on axioms, and these are not proven in any way. The analytic position is therefore left empty of any claims whatsoever, since any statement that says something new about the world, that is, any synthetic statement, cannot be certain. Absolute certainty is achieved only at the price of emptiness, and in the eyes of the analytic thinker whatever is not certain is not true.

Thus the analytic position demands only consistency. All positions are equal in its eyes, provided they are internally consistent and accord with the directly empirical facts, and in the scientific context, with experience.

We also saw that the analytic position leads its proponents to attribute all philosophical problems to language. Every problem is analyzed as a linguistic failure, and the solutions proposed are corrections of linguistic usage. Some thinkers—Leibniz, Russell, and others—went even further and proposed the use of a precise formal language within whose framework all philosophical and other difficulties would simply disappear.

In the first book we saw that in the analytic world concepts have only use and not meaning. Their use derives from agreement within the community of users—baptism, in the terminology of the philosopher Saul Kripke. And certainly the concepts do not exist in the world in any sense.

In addition, we saw there that in a postmodern world the various positions are described in terms of power struggles among groups, and only power and interests operate in value-laden or ideological contexts. A person does not believe in a position because it is true—for there is no truth—but adopts it on the basis of various considerations and interests.

Let us now turn to the analytic picture of science as presented in this chapter and see that it contains exactly the same characteristics.

We have seen that according to this picture science says nothing about the world. It is merely a convenient language for describing the totality of facts known to us. Once again, as in general analyticity, the concern is with language. In addition, the only demands made of a scientific theory are internal consistency and fit with observed facts. As we saw, that is precisely the founding principle of the analytic position. This approach is called actualism in Bechler’s terminology, because it takes seriously only the facts actually present before our eyes.

We saw that in the analytic picture of science, abstract entities that cannot be observed do not exist. They are conceptual fictions meant to improve communication and to organize language and thought. This is an exact parallel to the fictional character of concepts—that is, to the conventionalism—we saw in the general analytic position.

Finally, we mentioned, and will discuss further below, that Thomas Kuhn describes the process of paradigm replacement, the replacement of the ruling theory, as a sociological-political process involving struggles for power and status within the scientific community. In such a process, it is not the quality or truth of the theory that decides matters, but various power relations. This reflects the postmodern outlook, according to which the adoption or rejection of an ideological, evaluative, or other position does not occur because of substantive acceptance of the principles it contains, but because of power considerations and interests. In the postmodern world every speaker or actor is described as driven by interests and schemes, motivated by power and by the desire to seize hegemony. This is the inevitable description in the absence of substantive dimensions to the competing positions. In a world where there is no truth, only power and interests speak.⁷²

Analytic Causality

Another very important point links this picture to the general analytic position. If, in the analytic picture of science, abstract theoretical entities do not exist, the conclusion is that the phenomena we observe occur without causes. For example, if there is no gravitational force in objective reality, but “gravitational force” is merely a linguistic way of describing the acceleration of a body in the presence of another mass nearby, then that acceleration has no cause. Normally it is accepted that the force is the cause of the acceleration. But if force is a fiction, or an alternative description of the fact that the body accelerates, then the emergence of the acceleration has no cause at all. Somehow, whenever an additional mass is present near some object, it simply decides to begin accelerating, but the other mass is not a cause of its motion in any substantive sense.

Likewise, if there are no force fields and no charges that are the sources of force, then force arises without a cause. The particle accelerates in the presence of field sources at some distance from it, and the field itself at the point of acceleration is only a fiction. If so, that motion too is caused by nothing. It follows that in the analytic picture science merely describes natural processes but does not offer explanations for them. We will return at length to this point below, in the fourth gate.

Here we return to the root of modern analyticity, which in the first book we saw to be grounded in the philosophy of David Hume. One of Hume’s central arguments was his challenge to causality. He argued that since we cannot observe the phenomenon of causing that exists between events, the concept of causation is a fiction. A cause, according to Hume, is nothing more than temporal succession. To say that event A is the cause of event B means: event B always occurs after event A. According to Hume, there are in truth no causes in the accepted sense, meaning cause as the active production of its effect.

Thus the analytic picture of science in effect returns to the roots of modern analyticity and relinquishes the principle of causality. The reason for relinquishing it is also similar to Hume’s reason: the cause itself cannot be observed empirically. One can observe events, but not the relation of causing between them. In the analytic world, what cannot be observed does not exist. This is one aspect of the fictional character of concepts and of the analytic demand for absolute empiricism, or actualism, in Bechler’s terminology.

Degrees of Confirmation, and Confirmation in General

Another point at which the analyticity of the picture presented here becomes evident is the meaning of different degrees of confirmation for a scientific theory. As we saw, even if the quantity of empirical evidence for the theory under discussion has some significance in the analytic picture—and that itself, as noted there, is not at all clear—the diversity of the evidence has no significance in this picture.

Already here one can hear echoes of Carnap and Hempel’s notion of confirmation. It should be noted that in the analytic picture the process of confirmation has no reasonable basis; it is nothing but semantics, as we already remarked above. In the analytic picture, confirmation means nothing more than absence of refutation.

As we saw in previous chapters, especially in Note 3, diversity in evidence certainly has no significance within the analytic picture of science. Diversity plays a role only in confirmation, not in Popper’s question of refutation or non-refutation.

Here too we see the analyticity of this description. In the analytic picture, a claim is true only if it is proven analytically, in something like the mathematical sense. As we saw, every proof rests on basic assumptions, which themselves have no proof, precisely because they are basic assumptions, axioms. Analyticity, then, does not really believe in proving claims, but only in refuting them. Aristotle already held that reasoning, that is, logic, is destructive. There is no logical way to prove arguments, only to refute them.

On the logical plane there is no way to verify any claim. All we can examine is its internal consistency. If the claim contains no internal contradiction, we will say that it is acceptable—not true, since it is based on basic assumptions, and in the analytic picture those assumptions are arbitrary. But if it contains an internal contradiction, then the claim in question has clearly been refuted. This is the general analytic analogue to Popper’s position. Here one can clearly see its analytic roots. Popper concentrates on the logical plane and is unwilling to accept synthetic additions within scientific methodology. Therefore he concludes that one cannot confirm, and certainly cannot verify, a scientific theory; at most one can attempt to refute it. According to the analytic conception, science receives a status similar to that of logic. As we will see below, this is probably the solution to the riddle posed at the beginning of this gate concerning the power and lofty status of science in the postmodern world.

Feyerabend

I would now like to present briefly the thought of Paul Feyerabend, an important thinker often regarded as expressing a sharp pole of the analytic method in explaining science. I will point out briefly that this characterization is not precise. For the sake of the discussion I will quote several passages from one of his essays, “How to Defend Society Against Science.”⁷³

Already in the title one can discern hints of the conception that emerges with great force throughout the essay. Feyerabend argues that science has become a substitute religion in the modern age. To that one may perhaps agree, as we saw above. But he tries to ground his attack in a nihilistic interpretation of the scientific process, one that strongly expresses what I have here called an analytic explanation of science.

Feyerabend brings several of the problems we pointed out above and argues that a theory is tested only against competing theories and not against any truth. There is no serious way to test a theory in itself. Every theory, he says, is full of defects, contradictions, and ambiguities, and therefore all the criteria of Popper and his colleagues for testing such creatures are pointless.

The core of his argument against the mythological-religious status of science is that science has no method—here he relies on Lakatos, the well-known Hungarian philosopher of mathematics and science, who in despair gave up the concept of scientific method and proposed using the term research programmes—and that it has no special results beyond what other ideologies have achieved. In Feyerabend’s eyes, science is an ideology and nothing more. In his words:

Science is only one among many ideologies that propel society, and it should be treated accordingly.

Feyerabend argues that there should be a separation between state and science, just as in many countries there is a separation between state and religion. School curricula should contain traditional-religious positions alongside scientific ones. In his words:

Three cheers for the California fundamentalists who succeeded in removing from textbooks a dogmatic formulation of the theory of evolution and left room for the account of creation based on the book of Genesis…

Because of statements like these, Feyerabend is regarded as one of the most extreme among analytic interpreters of science. But it is important to distinguish between rhetoric and content. The claim that the theory of creation should be allowed to be taught together with evolution in schools does not require an analytic explanation of science. Even according to synthetic interpretations one can understand the simultaneous existence of these two theories. We will discuss this below in the fourth and fifth gates.

Nor does his criticism of decision-making by scientific experts, or of staffing committees with such experts, together with his recommendation to staff them with ordinary people, necessarily testify to analyticity. As was described in the first book, in chapter 2 of the thirteenth gate, there is here a correct and profound conception of the meaning of decision-making. Here too the sharpness of the formulation is a form of protest against distorted conceptions of science that lead us to relate to it with mythological reverence and excessive trust, especially in areas where, by definition, science is not at all competent.

When we examine this essay of Feyerabend’s carefully, we do not find real nihilism there, as one might expect from the sharp and sarcastic language he employs. In fact he believes in the ability of science to achieve results, but he tries to limit the religious trust our society places in science, and especially in scientists. For that purpose, as he explicitly states at the beginning, he uses especially sharp formulations.

In all his remarks there, despite their sharpness, one does not find the claim that science is analytic, meaning that it makes no claims about the world. It seems to me that a careful reading of his words yields a fairly balanced picture, unlike those analytic thinkers cited by Bechler in his book.⁷⁴

An Ad Hominem Note⁷⁵

As an addition to everything said above, it should be noted that the thinkers who proposed the analytic explanation of science were mainly pragmatists, such as Dewey, whose solution is called instrumentalism, or positivists, such as Schlick, whose solution is called positivism. In the first book we saw that the sharpest expressions of a general analytic approach are American pragmatism, founded by Dewey, and positivism, represented by Moritz Schlick and his students.⁷⁶ It is therefore no surprise that these two joined forces in presenting the analytic explanation of science as well. Analyticity emerges quite naturally from the loins of positivism and pragmatism.

Back to the Paradox of Bokononism: The Root of Analytic Trust in Science

To conclude the chapter, let us return to the paradox presented in the introduction to this gate. There we pointed to the phenomenon that in the first book I called Bokononism: precisely in a world where certainty has lost all status and everything is in doubt, science enjoys supreme, almost idolatrous status. In our analytic world, scientific claims inspire absolute trust. Some have called this the Church of Science.

The great difficulty arises in light of everything we have seen, namely that science is fundamentally and clearly synthetic, since its purpose is to make claims about the world. If so, it is not clear why a world that is analytic in essence grants complete trust to a clearly synthetic discipline.

The solution to the paradox of Bokononism is the analytic picture of the explanation of science. It bridges this gap. We saw that according to this picture science does not make claims about the world, but merely constitutes a convenient language for describing facts already observed, or their logical derivatives. Such a science can indeed satisfy analytic criteria. It is no wonder that it inspires such great trust, for it says nothing at all. According to the best analytic tradition, the emptiness of science is what produces absolute certainty in it.⁷⁷

Chapter Three: A Synthetic Explanation of Science: Synthetic Solutions to the Methodological Problems

General Syntheticity and a Synthetic Picture of Science

In the previous chapter we dealt with the analytic explanation of scientific methodology and with the meaning of scientific theory. At the end of the chapter we discussed the relation between that picture and the analytic position in general.

This link to the analytic position in its broader senses, beyond the scientific context, allows us to use the entire “toolbox” that has served us until now, in our attack on analyticity in the third part of the first book, and thereby to reject the analytic position in the context of explaining science as well.

In the scientific context, however, the situation is more complex. As we saw, the analytic picture provides impressive solutions to the cluster of severe problems characterizing the methodology of science, as presented above in chapter 1. This is not surprising, of course, given the well-known fact that one who says nothing cannot be wrong. If science says nothing about the world, it obviously cannot err.

In addition, it is important to note that, in principle, one can hold a synthetic position in the general philosophical context and at the same time adopt an analytic outlook in the scientific context. In the first book we emphasized the obvious point that even those who hold a synthetic position are not committed to accepting every unsupported claim in every context. The innovation of the synthetic position is that it is possible to accept claims that do not reflect direct observation and are not open to logical proof. Obviously, someone with a synthetic position may still reject claims of that sort. The choice whether to accept or reject a claim that lacks analytic grounding depends on the degree of trust one has in it; the analytic thinker will always reject such claims, because they lack analytic grounding.

Thus, a person with a synthetic position may adopt an analytic position regarding the philosophy of science. And as we saw, there is considerable logic in adopting such a position, because it seems to solve almost all the difficulties we identified.

In this chapter we will discuss several main questions: Are there nevertheless good reasons to adopt a synthetic picture in the context of scientific methodology? If so, what is that synthetic picture? And finally, how does the synthetic explanation of science deal with the methodological difficulties presented in the previous chapters?

The Criterion: Intuition

In the first book we saw that the most basic characteristic of someone with a synthetic position is a strong trust in intuition. Intuition enables us to recognize the truth of our basic assumptions even though we have no proof for them. Someone holding the analytic position, by contrast, has no trust in his intuition and therefore demands logical proof or empirical observation as grounding for every claim. Basic assumptions, on his view, are subjective and arbitrary.

Therefore, in order to understand whether there are reasons to adopt a synthetic position with regard to science, we must examine ordinary human intuitions concerning science and see whether they are analytic or synthetic.

Intuitive Problems in the Analytic Picture of Science

The first intuition to which I will refer is the one mentioned earlier, according to which the theory that all ravens are black is insignificant if we have already observed them all. In other words, the intuition is that the main purpose of science is to say general things about the world, not merely to reorganize, linguistically or conceptually, bodies of facts already known to us.

A second intuition is the principle of causality. Here we are back on the track by which we rejected general analyticity as well. We saw that analyticity in general, and the analytic explanation of science in particular, in effect relinquish causality in its classical sense. Both in the first book and here we saw that the analytic position implies that there is no causation of events and occurrences, or at least that we do not know of any such causation; there is only a relation of temporal succession between them.

To give up the intuition of the principle of causality, we need very good reasons. This is a powerful intuition, and abandoning it should be our last resort. Obviously, the decision depends on whether we have another way out of the system of difficulties raised by scientific methodology, and that will be discussed below.

There is also a third strong intuition: that science progresses. Some theories are refuted and rejected, while others are confirmed and begin, or continue, to survive. In the analytic picture of science, this has no meaning whatsoever. Science does not progress except in the narrow sense of accumulating more particular facts. Such progress is trivial, because as time passes we manage to test more situations in the laboratory and therefore are equipped with more facts. This is a kind of progress comparable to the progress of job seniority—the same sense in which an employee’s seniority increases in the workplace. In language there is nothing more than a description of the growing set of facts. According to the analytic explanation, theoretical knowledge is not a deepening reservoir of understanding of nature itself, but an expanding reservoir of facts about it.⁷⁸

As we saw, the concept of confirmation, whether by quantity and certainly by diversity of empirical evidence for the theory under discussion, cannot have meaning in the analytic picture. The fact that this concept appears on the philosophical stage through analytic thinkers means that, for them, confirmation is nothing more than a semantic trick. As we explained in the previous chapter, in an analytic world there is no explanation why confirmation should be more than absence of refutation.

This too strongly contradicts our basic intuitions, which understand success in experiment as genuinely confirming the proposed theory. As it successfully withstands more experiments, our confidence in the truth of the theory grows.

Conversely, in the analytic picture of science there is no possibility for science to regress. It necessarily progresses, because more and more facts about the world accumulate as time passes. According to the synthetic explanation of science, however, science can certainly be in retreat, though that is rare, because of course the breadth of the factual basis is very important. Such a regression occurs when a theory is adopted that is less correct than its rival, even though it correctly describes more facts. In principle, it may turn out that the earlier theory is closer to the truth—that is, that the entities it uses are indeed the ones that exist in reality, not the entities of the new theory, and that the experimental failure was due to side factors requiring correction rather than to essential reasons. According to Popper’s own criterion, it is very plausible to say that a science that cannot regress cannot progress either.

Here we must point to another intuition, namely that science generally progresses, but in principle it can also regress, certainly temporarily. In light of the previous paragraph, this intuition too clearly points against the analytic explanation of science.

The central problem in the analytic explanation of science lies in its analytic claim that abstract theoretical entities do not exist. This raises the problem of causality mentioned earlier. Beyond that, and perhaps this is only another aspect of the same problem, the simple intuition is that there are forces in the world, especially gravitational force. The same is true of the electron and of other elementary particles, the wave function, and the like.

A strong indication of the existence of theoretical entities, and thereby of the realism of a scientific theory, is future consequences. This point can be divided into two aspects that are essentially one: correct predictions and technological applications.

First, the fact that a scientific theory withstands future empirical tests and does not always fail in them raises a major question regarding the analytic picture. If the theoretical description is only language, there is no reason for it to succeed in an experiment not yet known. Theories should have been refuted and replaced anew after every experiment. We saw that there are countless theories fitting all the facts, but clearly almost all of them will not survive the next experiment, which had not yet been performed when the theory was formulated. The fact that theories generally, though not always, of course, withstand future tests, and are only rarely refuted and replaced, indicates that the investigator has sound intuition in choosing the correct theory, not merely the useful one. In other words, his theory does say something, and even something true, about reality.

It should be emphasized that the overwhelming majority of ruling theories in the modern history of science were not refuted and not wholly replaced. The great break at the beginning of the twentieth century, with the discovery—or invention?—of quantum theory and relativity, is usually described as a crisis in the philosophy of science. But there is another side to the coin: in all these cases, the new theory shows that the previous one remains valid across broad domains. A considerable part of the conceptual world remains in place as well, with the same meaning. It is therefore hard to see how arbitrary languages succeed so impressively in predicting future results and in standing so impressively in empirical tests, if they themselves make no empirical claim.

Technological applications are another aspect of the same point. In the technological age in which we live, science provides the infrastructure on the basis of which various technological solutions are built; science has uses. If these uses were uses only of facts, we would not need the scientific formulation in order to reach them. If theory is nothing more than a sophisticated formulation of the system of facts, then there is nothing in it beyond the set of facts and nothing at all more. It has no predictive or generalizing power in itself. In principle we could derive all technological solutions from the bare set of facts itself, with no real need for the mediation of theory. Admittedly, theory may ease the task if it provides a concise and apt formulation of the facts. But the impression is that theory serves as something more than that, even on the technological plane.

Take as an example the construction of spacecraft. In that process people assume various things about “forces” and about the state of affairs that prevails among the different heavenly bodies and between the spacecraft and the earth. These phenomena have never been directly observed; they are only theoretical results. Can one rely on them and send spacecraft to the moon? Of course, because of this problem failure can occur, and sometimes does occur in practice. But does anyone really have trouble with the basic a priori trust we place in the scientific system in this context?

Take another example, this time from medicine. Physicians give patients medicines based on theoretical explanations of observed physiological phenomena. Decisions to use certain medicines are not always based on observed facts alone but often also on the theoretical explanations of those phenomena. If theoretical explanations are not statements about reality but only fictions that help describe the totality of facts, then we ought not take such medicines. According to the analytic explanation of science, a medicine can rely only on direct observations of the disease in question. One must not draw conclusions from a scientific generalization regarding cases not yet observed. Still less may one rely on fictive theoretical principles, which do not even pretend to say anything about reality, as a basis for practical decisions, certainly where human life is concerned. This is, of course, just the aspect of fulfilled predictions discussed above, from a slightly different angle.

Thus, the analytic explanation of science indeed offers a perfect solution to the methodological problems presented in the first chapter. As noted, one who says nothing about reality obviously cannot be wrong. But it does so at the heavy price of giving up several very basic intuitive principles. In addition, it generates new problems, such as the fulfillment of predictions and technological successes, which are very hard to explain within the analytic picture of science.

Again: Ad Hominem

The overwhelming majority of those engaged in science, and in chapter 5 I will note that the same is true of mathematics, believe that they are discovering and dealing with real entities. In other words, scientists themselves mostly have a synthetic approach to science, though not necessarily with respect to other areas of life. At this point the analytic philosopher of science will come and say, with considerable justice, that most scientists are not at all aware of the severe methodological problems that accompany their work. That is the area of expertise of the philosopher of science, and scientists are often entirely unaware of it.⁷⁹ The scientist, as scientist, is not a philosopher of science and certainly not an authority on that question.

Yet that argument would be correct only if we were adducing evidence from the scientist’s opinion about the correct philosophy of science. Many indeed argue in that way, but it is decidedly mistaken.⁸⁰ One must note that we are not taking that route here. Our claim is that there is a strong realist intuition according to which theoretical entities do in fact exist, and are not merely a fictive language for efficiently describing facts. The fact that a great majority of those engaged in the sciences believe in the existence of a widespread intuition concerning the existence of theoretical entities is highly relevant to confirming this ontological-epistemological claim.⁸¹

Of course, we must now show that the methodological problems can be solved in a way that does not contradict those intuitions. If we fail to present such a solution, we will find ourselves forced to abandon those intuitions, despite the fact that most scientists hold them. In that case, as stated, this fact will indeed no longer be relevant.

But if we do succeed in presenting a reasonable synthetic explanation of scientific methodology, then there is no reason to give up our body of intuitions and adopt an analytic position. Let me now spell out this important methodological point somewhat more fully.

A Preliminary Methodological Remark

Up to this point we have asked whether the analytic picture explaining science accords with simple intuition. As the discussion so far shows, the answer is categorical: absolutely not. On the other hand, it offers perfect solutions to an impressive set of methodological problems. Therefore, so long as we do not propose a real alternative explanation of science, we will not be able to rely on our intuition. True, even if we propose a different explanation that also solves all the problems, we still will not yet have a decision as to which of the two explanations to adopt. At that point simple intuition enters the picture, and we will adopt the picture that better accords with intuition—namely, the synthetic picture.

As we noted above, in order to give up the principle of causality and the simple insights presented above, we must become convinced that there really is no other solution to the methodological problems. The moment we propose another solution, even if the two solutions now stand on equal footing, we will choose—according to the principle of simplicity and elegance—the one that accords with intuition.

There is therefore a clear asymmetry between analyticity and syntheticity. It is enough to show that there exists a synthetic solution to the methodological problems, even without proving the superiority of that solution over the analytic proposal, in order to decide in its favor. The exact same situation prevailed in the first book in the general discussion of analyticity and syntheticity. There too the analytic solutions were sharply counterintuitive, but it seemed to the analytic thinker that there was no escape. The difficulties concerning the fit between the human being and the world, or what we called there the fundamental problem of epistemology, led him to the painful abandonment of intuition.

David Hume’s arguments were the first and most basic example of this process. Because of epistemological difficulties, Hume gave up two fundamental principles that were well grounded even in his own intuition: causality and induction. He described them as a human illusion or as fictive concepts.

Therefore, once an alternative way out of the epistemological difficulties is proposed, it should receive preference, because it accords with simple intuition. If there is no good reason to abandon our simple intuitions, it is very likely that we should not do so.

General Diagnosis

To understand the root of all the problems in scientific methodology that we mentioned, let us return for a moment to the argument presented in the first book. There we described the philosophical reason for adopting the analytic position as follows: intuitively, every person sees that there is a wondrous fit between the human being and the world. The concepts by which a person thinks fit the description of the world and active life within it in an astonishing and incomprehensible way.

The problem begins when a person asks what the source of this fit is. More than that: who told us that it exists at all? For example, the eye is an extraordinarily complex and intricate structure. One might perhaps believe that it came into being by chance, in a long evolutionary process. But how do we know that it correctly reflects objective reality? Who guarantees that this complex and astonishing mechanism does not distort what it reflects? Who says there is even something there being reflected, rather than that the mechanism creates the picture out of nothing? For further detail, the reader is referred to the first book.

There we saw, in the parable of the clocks in chapter 2 of the first gate, that the fit between the knowing human being and the world in itself can be explained in three ways:

1. The world acts on the human being: empiricism. This direction was rejected by Hume’s arguments.

2. The human being acts on the world: transcendentalism. This direction was adopted by Kant and rejected by many, including us in the first book.

3. The remaining possibility is that there is a coordinating factor between reason and the world. We called this factor God. This is the only reasonable basis for rationalism, and Descartes and Leibniz presented it in this way as well.

From this picture it followed, according to the analysis there, that the only possibility for a rationalism that is rational, and not merely rationalistic, is belief in a coordinating factor. That belief itself solves the problem of the fit between the human being and the world, since that factor is what ensures that such a fit exists.

As we saw there, the practical meaning of this claim is that the human being has an ability to make contact with the world not by means of the ordinary senses. One may call this a sixth sense, but we called it intuition. Unlike the analytic conception, the synthetic approach holds that intuition is a tool for arriving at truths about the world. It has the ability to create contact with the world beyond sensory contact, to observe it—in Rabbi Nazir’s terminology, to listen to it—and to understand what is happening in it.

According to Kant, the fundamental problem of epistemology can be formulated as follows: How are synthetic a priori judgments possible? That is, how can one know comprehensive truths about reality in a way that does not arise from our senses, from what is usually called empirical observation?

The practical meaning of the answer we proposed in the name of the synthetic position is that what is usually classified as thought does not necessarily take place only within the knowing and thinking subject. The common distinction between thought and cognition is not sharp. The mind not only thinks, but also knows—that is, it establishes its own contact with the world, not through the ordinary senses.

It is the mind that sees and understands concepts. Therefore, the fact that we do not know them through the senses does not mean that they do not exist. Nor does it necessarily mean that we have no way of knowing them. The holder of the synthetic position argues that there is no need to give up the simple intuition concerning the existence of concepts. We grasp them through the eyes of the intellect, in Maimonides’ terminology, or through auditory logic, in Rabbi Nazir’s terminology, and do not merely think them.

One should note that these two expressions, eyes of the intellect and auditory logic, combine terms from the world of thought—intellect, logic—with terms from the world of cognition—hearing, eyes. The terminology itself hints at the basic nature of the solution: abandoning the sharp distinction between thought and cognition. Thought too has an element of cognition. The mind observes concepts; it does not merely think them.

The solution we proposed to the problem of the synthetic a priori as formulated by Kant is that although there is no possibility of genuine synthetic a priori knowledge, unless one accepts Kant’s untenable positions, one can arrive at propositions usually classified as synthetic a priori by means of intellectual sensation.

It is indeed impossible to observe through the senses the causal relation between event A and event B, as David Hume pointed out. But we can observe that relation by means of intellectual sensation. In Kantian terms, which we accept in part, one would usually classify the claim that A is the cause of B as synthetic a priori. But it would be more accurate to say that this is a posteriori in one sense and a priori in another, and in any case synthetic. This insight is found within us, but it also correctly describes the world outside us because of the fit between the human being and the world, the fit to which we pointed earlier.

Induction and analogy, the most important forms of inference in science, can also be understood in this way. As we saw above, the problem with these inferences is that it is not clear what the relevant criterion of similarity is between the two things being compared. How can one know that green color is essential to frogs even before we have inferred this by analogy and induction, which are supposed to establish it? We saw that there is a circle here with its tail in its mouth.

In light of the synthetic picture, we may say here too that the initial seeing that gives us insight into the relevance of properties of similarity is indeed intuitive seeing, akin to what I earlier called intellectual sensation. It arises from the mind, yet matches the objective external world. Here too the solution is that we know the relevance of the property of similarity rather than think it. It is not the result of analogy, but of intellectual observation. Contrary to Hume’s assumption, one can observe relations, such as the relation of cause and effect, and not only objects. Once we have an intuition about what to observe, we can establish analogies through which these initial intuitions are formulated and elaborated.

What has been said so far is a concise description of the solution to the fundamental problem of epistemology as presented in the first book. Here we will see it in the context of the philosophy of science, and from that we will arrive at a synthetic explanation of science.

A Synthetic Explanation of Scientific Methodology

As noted, standing against the intuitions that support the syntheticity of science are the methodological problems we presented above, which in practice function as arguments for the analytic picture. As we saw, the point that seems to be the strongest argument in favor of the analytic picture of science is the demand for simplicity, convenience, and elegance. At first glance there is no reason why the world should operate according to what seems to us, human beings, simple or elegant. This is true even if we solve the very difficult problem of defining the concept of simplicity and finding criteria for it.

According to the analytic picture, there is indeed no such reason, and therefore the conclusion was that the world certainly does not operate that way. This is a property we demand of our language, not of the facts in objective reality. According to the analytic proposal, it is quite reasonable to choose in this way the language most convenient for our use from the infinite set of languages fitting the totality of facts presently in our possession.

But precisely this point, because of the difficulty it poses, leads us toward the synthetic solution to the methodological problems of science. It is with that solution that we will now deal.

As stated, the fundamental problem that arises here is why the simplicity and elegance of a theory, which are properties derived from the structure of our intellect—an intellect with a different structure would regard other structures as simple and elegant; taste cannot be argued about—should actually lead us to true descriptions of the world itself. We encounter here once again a problem of fit between the form and character of human reason and the world in itself.

Our present discussion is therefore only a particular image of the general discussion we encountered in the first book, which also dealt with this question of fit. Not surprisingly, the synthetic solution we will propose to the methodological problems raised here will be similar. In fact, the problems in scientific methodology, for the most part, and certainly all those leading to analytic explanations, derive from the same point: they all concern the epistemological problem of the fit between the thinking and knowing human being and the world. As we saw in the first book, the moment the analytic thinker encounters such a problem he raises the analytic solution; that is, he determines that the claims in question do not concern the world, since there is no way to ascertain the truth of such claims. He therefore says that they are analytic claims.⁸²

At the beginning of the present gate we pointed to the centrality of empiricism in the modern conception of science. That empiricism stands at the center of the methodological problems raised above. The main difficulty stems from the fact that one cannot point to a way in which we can observe theoretical scientific claims empirically. For example, one cannot see theoretical entities, and therefore the analytic conclusion is that they are theoretical fictions. One cannot see the relevance of a property of similarity in analogy and induction, and therefore these too are no more than an illusion. Above all, one cannot empirically ascertain the truth of a theory in any direct way, since we cannot know what will happen in situations we have not observed; this is the problem of induction. In particular, we cannot directly observe relations of causal production, and more generally we cannot observe relations between events, which are the main concern of science.

The solution to all these problems is the understanding that we possess additional means of observation: further empirical tools, means of cognition, besides the ordinary senses. Those “naive” conceptions of science according to which we can verify theories or ascertain the real existence of theoretical entities are based on observation. This observation is not done by the ordinary senses, but by the eyes of the intellect. This is the basis for the intuition that everyone feels, and once that basis is accepted there is no reason to surrender to rationalistic, though not rational, objections and give up those intuitions.

Here it is worth recalling the concept of eidetic vision developed by the philosopher Edmund Husserl, which was also mentioned in the first book. Husserl pointed to our ability to see general concepts through particular objects, as though directly observing Platonic ideas through the concrete objects in which they are embodied, or through which they are embodied. He described this as if the concrete events and objects present before us were “transparent.” Through them we see the general concepts and the ideas behind them. Through the “transparent” horse we see the idea of horseness that governs its form and essence. Through two adjacent occurrences we see the mechanism of causation that relates them to one another, beyond their temporal succession.

In the same way, through the particular cases we observe, we simply see the result of induction, the generalization. If we observe objects falling to the earth, we see through those occurrences the force of gravitation behind them. That is how inductive generalization is performed, and that is how analogy among similar phenomena or similar objects is made.⁸³

In the first book we already mentioned that Husserl attributes this ability to a transcendental process. That means it does not point to features of the world in itself, but is explained in a Kantian way as concerning only the phenomena that appear before our eyes and consciousness, not the objective world itself, and only to those does science relate. According to Husserl, the ideas do not exist anywhere in the world itself, as Plato thought. In that sense Husserl continues Kant rather than returning to Plato. He probably could not break out of the analytic framework that binds anyone unwilling to accept the hypothesis of a coordinating factor as the guarantee of a fit between the human being and the world.

Here we extend Husserl’s remarks and establish, as the basis for the synthetic explanation of science, that this fit exists between the human being and the world in itself—in Kantian language, the noumenon—and not only with respect to the world as it appears to our eyes, the phenomenon in Kantian terms.

Any reader who remains unconvinced and to whom this explanation appears “mystical” is invited to return to the first book and see that he too most probably implicitly assumes such a fit. That was the entire purpose of the argument in the first book: our simple intuitions indicate the existence of such a fit, and there is no need to give up what is self-evidently true in our eyes. We saw there, as we see here, that analyticity is much farther from common sense, even if it sometimes has no small amount of intellectual charm.

Let us now review once again the methodological problems we pointed to above and examine them in this perspective.

Let us begin specifically with the hardest problem: simplicity and elegance. As noted, in light of what has been said here, it seems that if we do indeed assume a fit between the human being and the world, then a theory that appears simple and elegant to us really ought to describe the world. Its simplicity indicates that it fits thoroughly the structure of our human intellect. And since, on our assumption, there is a fit between the human intellect and the structure and properties of the world, this is indeed an indication that we have found a theory that correctly describes the world. The intuition according to which a simple and elegant theory is true stems from the fact that simplicity means fit with the categories of the intellect, and therefore also fit with the world. Describing a theory as simple is precisely the indication of its truth.⁸⁴

Let us now turn to the problem raised by Popper concerning the inability to prove a scientific theory. We have already seen that for holders of a synthetic position, provability—that is, susceptibility to proof—is not the criterion of truth, certainly not an exclusive one. There is no need to abandon the demand that a scientific theory be true just because of the essential inability to prove it. In the first book we saw that not every truth is provable, and the logician Kurt Gödel proved this as well in his famous theorems.

It is indeed true that observing some ravens cannot prove that all ravens are black. But intellectual contemplation of ravens, and not merely looking at them, indicates that blackness is essential to them. This is similar to the ability to contemplate concepts in themselves described above. We contemplate the idea of the raven and see the black color.

This too sounds a bit “mystical,” but there is no doubt that it is the intuition accompanying most people. After seeing several ravens, it is clear to us that their color is characteristic of them. Doubts about that conclusion arise only despite the existence of this clear intuition, because it does not seem to people that there is any way to justify the assumption of a fit between our inner intuitions and the world. They do not see what justifies the assumption that this intuition, which seems to arise within their own minds, also correctly reflects external reality. In light of the picture drawn above, these are not merely our inner human intuitions but the results of processes of cognition and not merely thought. These intuitions are drawn from the world through a kind of synthetic move, which in certain respects is also a posteriori, and therefore there is no problem in expecting them to match what occurs in the objective world in itself.

Above we pointed out that in the view of some philosophers of science, a scientific theory can also be confirmed and not only refuted, as Popper thought. We also remarked there that those philosophers do not offer any explanation of why success in such experiments really confirms the theory despite Popper’s arguments. We saw that from a purely logical point of view, the fact that a theory has not been refuted is not enough to confirm it.

According to the synthetic conception, the process of confirming a scientific theory by empirical experiments occurs through direct recognition of its correctness. Success in an experiment does not mean only that the theory was not refuted, but also that it was justified. By means of the experiment we see with the eyes of the intellect, even if not with the eyes of flesh. The event itself is, as it were, “transparent,” and through it we see the general law that stands behind it. Our conviction is not merely a subjective illusion disconnected from external reality. It is the result of contemplation—of listening to the world—that leads to conviction in the truth of the theory under discussion. Again, once we assume a fit between human beings and the world, there is no need to give up our simple intuitions.⁸⁵

This is the sting in the concept of confirmation, as distinct from its older and more intelligible counterpart, proof. Confirmation is recognition of an unproven truth, or an increase in the degree of certainty of such a truth. For the analytic thinker, certainty is a binary parameter: every claim is certain, false, or entirely doubtful. By contrast, for holders of a synthetic position there are continuous levels of certainty. Movement to a higher level of certainty is called confirmation.⁸⁶

Let us proceed to the next point. We saw that the heaviest price exacted by the analytic picture is that it gives up causality, at least in its classical sense. If theoretical entities do not exist, then there are no causes for occurrences. If there is no gravitational force, then there is no reason why one object should be attracted to another. It simply moves toward it with no cause producing that motion.

The synthetic solution to the problems David Hume raised concerning causality was already described in the first book. Contrary to what Hume and Kant claimed, we do indeed directly observe the causal relation between events, and therefore we can certainly infer it from empirical observation of the world. It is true that one cannot see with the senses a relation of causality—in fact, one cannot see relations at all—but with the eyes of the intellect one certainly can. These “eyes” see the theoretical entities that stand behind events, and therefore they see the causes of their occurrence. It turns out that in the synthetic picture we retain our intuition that theoretical entities do indeed exist, and causality therefore remains in force. This is real rationalism. It is specifically the analytic approach, according to which events happen without any cause, that is pure mysticism.

A similar process occurs in the discovery of correlations. First we discover that there is a correlation between two types of events. Here, admittedly, we see this specifically with our ordinary senses. One can observe that the pair of events or phenomena under discussion always occurs together.⁸⁷ After that comes the stage at which we must identify the direction of the correlation. At this stage the eidetic capacity intervenes, and we see the relation between the events or phenomena, exactly as we see the relations of causation discussed by David Hume. In fact, the problem of the direction of correlation between pairs of events or phenomena is completely equivalent to the problem of determining causality, assuming the correlation is real and not illusory. Determining the direction of correlation means determining that event A affects B and not vice versa. That is precisely a determination of causal relation.⁸⁸

Let us now turn to the problem of the priority of theory over the facts that lead to it. Here too one must distinguish between two different stages. First, there is an eidetic contemplation of the world, and this gives us the initial direction. We “see” the theory directly, at least in a raw form. This intuitive sense points out to us the relevant facts from the totality of facts before us, and therefore we focus only on them. That is the stage of building the phenomenological theory. After that comes the analysis of the set of relevant facts, and that analysis leads to the explicit formulation of the substantive theory that explains the facts.⁸⁹

Again, the auditory capacity for “listening” to the world—in Rabbi Nazir’s terminology, or observing it with the “eyes of the intellect” in Maimonides’ terminology, or exercising eidetic vision in Husserl’s terminology—is what underlies our intuitive ability to classify facts according to laws that, apparently, are not yet known to us.

This model also helps us understand the problem of the relevance of properties of similarity, both with regard to analogical inference and with regard to induction. In both of these cases we observe facts with the eyes of the intellect, and only afterward, in light of the results of that eidetic observation, do we observe them with our bodily eyes and analyze them, and thus arrive at the generalization, the similarity, or the discovery of the relation, whether direction of influence or correlation.

In the comparison of frog A to frog B that we used above as an example of this problem, we assume that the frogs’ color is a relevant parameter, as opposed to their length. If so, even before we inferred that frog B is green like frog A, we already knew that color is a relevant parameter, and therefore we implicitly knew that all frogs are green. We pointed out that this is apparently begging the question. The same is true regarding an inductive generalization about all frogs. In that case too we must assume that we are observing a relevant fact and that we are allowed to generalize it to all frogs.

In light of what was said earlier, one may say that the discernment of the relevance of a property of similarity is a result of eidetic contemplation, and only after that contemplation and discernment do we formulate the similarity explicitly and reach the conclusion. We do indeed “know,” implicitly and in a very raw way, already from the beginning, that the color of frogs is a relevant parameter, but we formulate this for ourselves only at the end of the process. Only after this formulation does that knowledge become part of the scientific knowledge in our possession. Learning by analogy is the bringing of abstract insights, received through the eyes of the intellect, from potentiality into actuality.⁹⁰

Thus, the solution to all the problems is the adoption of the thesis of a coordinating factor and the restoration of trust in the built-in fit between the human being and the world, as described earlier. In other words, one must give up the sharp distinction between thought and cognition. This is our simple intuition, which in the analytic picture we abandon for reasons that are not necessary and not truly useful. The attempt to be “rational” and not to believe what we do not see with our ordinary senses leads us to a rationalistic picture that is very far from rational. In that picture there are no causes for events, theories are only language, and science teaches us nothing about the world. Gravitational force, and forces in general, do not exist, and many other similarly “mystical” determinations follow. All of this is done under the banner of pure rationalism.⁹¹

In summary, as we saw in the methodological note above, from every intellectual point of view the synthetic option is preferable to the analytic one.

A Looping Interlude: Back to God

We pointed out that the only possibility for rational syntheticity is the assumption of the existence of God—philosophical theism—as the coordinating factor between our reason and the world.⁹² We may now ask whether God, as a theoretical entity that explains the methodology of science synthetically, exists or not.

This is a theoretical entity created in order to explain the following observed facts: human beings see science and the world synthetically, that is, they think that science describes the world, not that it is a formal language for organizing information already accumulated. God, then, is a theoretical entity within the theory that explains how one can believe in the existence of theoretical entities.

The analytic thinker who looks at the course of our argument here will say that this meta-theoretical entity, like all theoretical entities, does not exist, because it cannot be directly observed. It is only a language designed to formulate elegantly our intuitions, which in the analytic thinker’s eyes are illusory, and not a statement about the world itself. Such constructions are sometimes called, as Karl Marx referred to religion, “opium for the masses.” In the first book we already saw the analytic foundations of communism, and it is therefore no surprise that we find them clearly in its theoretical founder as well.⁹³

Proponents of the synthetic position, by contrast, relate to God as an abstract but existing theoretical entity, like many other theoretical entities. True, we do not see Him with our ordinary senses, but we certainly discern Him with the eyes of the intellect. In the first book we pointed out that eidetic vision, or auditory logic, is the name given to the power usually called faith. Here we see that very identification from a somewhat different angle.

Of course, the entire discussion here is a second-order discussion. It takes place on the plane of meta-discussion relative to the epistemological plane on which the first discussion was conducted, the discussion that dealt with observing concepts and abstract entities. Here we are not dealing with epistemology whose purpose is to explain how we know the world. Rather, we are explaining epistemology itself as a fact that itself requires a “scientific” explanation. One may now continue to the meta-discussion of this discussion, and so on indefinitely.

We see, then, that belief in God requires the auditory faculty at a more abstract level. It is an auditory listening to our own auditory capacity.⁹⁴

Between Syntheticity and a Synthetic Picture of Science

Up to this point the discussion has proceeded under the assumption that anyone holding a synthetic position would necessarily adopt the synthetic explanation of science. We did point out, however, that this connection is not necessary. It is possible that someone with a synthetic philosophical position will adopt an analytic explanation of science.

We then rejected that possibility and saw that if there is indeed a synthetic explanation of the methodology of science, there is a clear preference for it, because it leaves us with several of our simple intuitions intact.

Still, there is room here to qualify things somewhat. One should say that it is indeed right to adopt the synthetic picture with respect to concepts and entities whose existence deeply convinces us. Syntheticity ratifies simple intuition. Yet it seems that often, especially with respect to modern science, there are situations in which we have no clear feeling or intuition as to the existence of the theoretical entities. In such situations it is entirely possible that the entities and concepts appearing within a given scientific theory are indeed useful fictions for theoretical explanation, and not necessarily statements about the world.⁹⁵

The process of understanding the methodology of science is dialectical. At its beginning a simplistic, dogmatic model is set up, and for our purposes that is the Baconian explanation of science. Within it there is an illusion that empirical observation necessarily yields the theoretical explanation itself.

After that comes skepticism, represented in the present discussion by the methodological problems in the explanation of science. This is a healthy skepticism not found in Bacon. Following these challenges, holders of the synthetic position reach the rational and healthy conclusion that science is not all-powerful and all-knowing, whereas holders of the analytic position continue to cling to it while emptying it of factual content.

As we saw in the first book, at the beginning of the third gate, Western civilization, like many of the individuals who live in it, undergoes a three-stage maturation process. In the first stage, childhood, there is dogmatic and naive faith in the principles handed down by the adults. In the second stage, adolescence, a rationalistic revolt appears, challenging those childish dogmas and demanding proof. In the third stage, after one comes to see that nothing can be proven, maturity appears. This takes one of two forms: first, the analytic-postmodern form, which is a total abandonment of truth and certainty, essentially a remaining in the adolescent stage; second, synthetic maturity, according to which one disconnects provability from truth. At this stage one understands that not every truth can be proven.

In the scientific context as well, we see that the analytic thinker matures in the first way, or in fact remains in adolescence. He remains with the skepticism and finds analytic or postmodern solutions and formulations for it. The synthetic alternative, also in the field of explaining science, is more mature. We accept certain things as true even though we have no proofs for them.

Yet even if we have become synthetically mature, we have learned something from the adolescent stage through which we passed. It is clear that in the third stage we certainly do not accept every unproven thing as true merely because there are truths without proof. We have definitely left the first, childish, dogmatic stage. From now on we will consider seriously whether we are convinced of something or not. If a high degree of conviction arises in us about it, we will adopt it despite having no proof for it. But if we genuinely are not highly convinced, then even if we are told that it is a “scientific determination,” we will not hesitate to doubt it.

Contrary to the claim of the adolescent, the holder of the analytic approach, in the mature stage we do not return to the childish and dogmatic Baconian stage. Observations are open to interpretation, and it is indeed not simple to claim that a theoretical statement has been experimentally proven. In that the analytic critics are entirely right. They called our attention to the fact that scientific theory has no analytic proofs. Nevertheless, as we saw, we need not surrender, as they themselves do, and conclude that science says nothing about the world—that is, that its claims are analytic in character. Rather, we can preserve healthy skepticism and sometimes cast doubt on and examine what science says.

Perfect syntheticity is the proper balance between skepticism, represented by the analytic thinkers, and dogmatism, represented by Bacon. The synthetic position is that science is true to the extent that we trust it. The methodological problems presented here certainly arouse a measure of doubt, and therefore the synthetic attitude toward science is sane and balanced. In a similar and more general way, we saw in the first book that a synthetic position uses analytic tools, and its dispute with the analytic position concerns only the exclusivity of analytic thinking, not its legitimacy.

Back to the Paradox of Analytic Trust in Science

As we have seen, the analytic thinker has perfect trust in science. Above we presented this as a paradox, but as the discussion progressed the paradox was explained by pointing out that in the analytic picture science says nothing about reality. Many laypeople adopt analytic certainty with regard to science but without the necessary background—namely, the view that science says nothing about reality itself, that is, without the assumption of its analyticity. Such laypeople sometimes criticize claims from other domains in the name of science, battering them with the assertion that they contradict scientific truths. But note thoroughly: if science says nothing about reality, there is no possibility of contradicting it, meaning of its contradicting something else empirical. One can contradict known facts, which are the true and only content of scientific statements, but not theoretical scientific determinations. These are only a language for expressing the totality of facts. If, on the other hand, science does say something about the world, then it is not proven, and to say of something that it is “unscientific” is not enough to reject it.

Any conflict between a scientific result and some other claim must be examined on its own merits: from what perspective are we viewing it, analytic or synthetic, and is this a contradiction to scientific theory or to some known facts? From a synthetic perspective, even a contradiction to the results of the theory itself is significant and requires examination. But from that same perspective, the claims of science are not certain, and therefore they are sometimes worthy of renewed examination. One may reject them, and one may also treat them as an analytic language for describing facts, a language that says nothing about the world—in other words, locally adopt the analytic approach. See, for example, Note 6 regarding potential.

These remarks concern situations in which the scientific explanation contradicts a different conceptual system or alternative explanations of various phenomena—for example, religious or mythic determinations. As stated, in such a case there opens up the possibility of accepting the scientific explanation as a theoretical language and not necessarily as a statement about the world. Perhaps, according to this approach, the scientific theory itself acquires the status of myth. We will discuss this further in the next gates. It will become clear there that there are options for the simultaneous acceptance of several different planes of explanation, including planes that appear contradictory.

Chapter Four: Two Planes of Analyticity and Syntheticity: On the Meaning of Scientific Equations

Introduction

To summarize what has been said so far, and in order to sharpen the meaning of the two pictures presented for explaining the methodology of science, this chapter will briefly discuss the meaning of the scientific equation. The analytic and synthetic aspects of the language of equations itself, namely mathematics, will be discussed in the next chapter.

As is well known, science, at least natural science, speaks the language of mathematics. One of the important elements of that language is the mathematical equation. In this chapter we will not discuss the mathematical meaning of equations, but their scientific meaning. This discussion will lead us to distinguish between two different planes of analyticity and syntheticity, distinctions that many thinkers in various contexts tend to overlook.

Newton’s Laws

Take, for example, the equation of Newton’s second law, which we saw above: the force acting on a body equals the product of its mass and the acceleration it develops. On one side of the equation stands force, and on the other side stands the product of acceleration and mass. The force is proportional to the acceleration, with the coefficient of proportionality, the constant number expressing the relation between them, being the mass of the body. One can relate to this equation as a synthetic proposition, a law of nature that determines a relation among these three magnitudes. But one can also relate to this equation as an analytic identity: the existence of force is a different kind of description of the situation in which a certain body of mass M is in a state of acceleration. For this description it is convenient to use the fictive magnitude “force,” whose size, and in effect whose definition, is the product of the mass and the acceleration.

From the description here we immediately see that these two ways of looking correspond to the synthetic and analytic positions, at least with respect to science. The holder of the synthetic position regards the mathematical equation as a law of nature that says something about reality, a synthetic proposition. The holder of the analytic position, by contrast, regards force as a language that more conveniently describes the reality of a mass in a state of acceleration. In other words, in his view this equation is nothing but a definition of the concept “force,”⁹⁶ and like every definition, it is a proposition that says nothing about reality: an analytic proposition.

Another example, which will sharpen this point further, is the relation between the field of force and its sources, and it will be presented in the following note.

Note 10: The Field of Force and Its Sources

As we saw in the previous note, the field is in a certain sense an abstraction of the concept of force, but it is a more abstract entity than force because it exists in space all the time. Its presence is not dependent on the existence of an object upon which it acts.

We also saw there that the sources of force are charges. Electric force acts on electric charges, but it is also a product of the existence of such charges in space; they are its sources. The same is true of all other forces, each with its relevant charges.

In other words: in every situation in which there is some distribution of charges in space, there is also in space a force field of a definite character. That character is derived from the spatial distribution of the charges by means of a mathematical equation linking them. In the electromagnetic context, the equations linking the distribution of charges with the force field are Maxwell’s equations, which constitute the full and general description of classical electromagnetic theory.

One of those equations links the electric field with the distribution of electric charge:

div D = rho

The div is a mathematical operation, divergence, such that if one applies it to the electric field, denoted here by D, one obtains the charge distribution, rho. This too can be seen either as a law of nature linking charge distribution and field, or as an analytic proposition deriving from the very understanding of what an electric charge or electric field is. Let us spell this out a bit more.

According to the analytic interpretation of this equation, what we have here is simply an identity. Instead of saying, “there is in space an electric charge whose distribution is rho,” we say, “there is in space an electric field D.” Everyone agrees that these are two equivalent forms of description of one and the same physical state. Both have the same empirical content, and when one is refuted, so is the other, and vice versa. But it should be noted that two equivalent forms can still refer to a state of affairs in which there are in space two real systems of entities. That is, one could say that in the relevant space there are both a field and a charge. The analytic thinker’s claim is that there is nothing there apart from a charge distribution, an array of charged particles, and the field produced from it is only an alternative fiction for describing that very reality, one that may be convenient for certain purposes. Put differently: the mathematical-physical concept charge has a correlate in the real world, which the mathematical concept represents in the equations. But the concept field has no correlate in the real world. Accordingly, according to the analytic position, this equation defines the fictive concept—that is, the theoretical entity—force field by means of the concept charge distribution, which, if anything at all, is real.

I once saw a work by a physicist from Boston named Solomon Schwebel, which seems indirectly to prove the analytic picture described here. When physicists calculate energies in field theory, they use a factor consisting of the charge at point X multiplied by the intensity of the field at point Y, divided by the distance between them. One must remember that the force acting between two points, whether the electric force or the gravitational force, is inversely proportional to the square of the distance between them. They then sum this over all space. Clearly, when the sum reaches the term in which the charge and the field are calculated at the same point, that is, X equals Y, the denominator becomes zero, because the distance between the points is X minus Y equals zero. The quantity then “blows up”; that is, it becomes infinite, and in effect it is mathematically undefined.

Physicists encounter “blow-ups” of this kind in almost every calculation of the energy of a force field. Such a phenomenon cannot be accepted as a reliable description of “sane” reality, and therefore they usually seek some response to it. Very often the solution makes use of various complicated mathematical techniques to which it is very difficult to give a systematic mathematical justification. The assumption is that if one arrives at some finite result, it is probably correct, because in nature no quantity truly “blows up.” Nature is “sane,” and therefore the values of physical variables are not infinite.

In a series of articles, Solomon Schwebel proposed an explanation according to which these blow-ups, arising as stated from the product of a charge at a certain point and the intensity of the field at that same point, represent an error of interpretation. The product of the charge at point X by the field at point Y represents an action of the force field on that charge point. But if the field is nothing more than a description in a different language of that same distribution of charge, then there is no justification for speaking of interaction between them. A charge, or a field, does not interact with itself. Interaction exists between entities, not between two descriptions of the same entity.

According to Schwebel, we take the mathematical identity between charge distribution and field and mistakenly treat its two sides as though they represented real entities. But that is incorrect, because only one side, and usually the tendency is to say this is the charge, though for purposes of the calculation it is not important which one, has reality. The second is only an alternative description of that same reality.

Accordingly, Schwebel argues, multiplying a charge by the intensity of the field at the same point in space is parallel to calculating the force that a given electric charge exerts on itself. Apparently, according to the equation of electric force, every charge exerts an infinite force on itself, since the distance between it and itself is of course zero. Thus the denominator in the force equation is zero, immediately leading to an infinite result. Clearly this is absurd, because a particle does not exert force on itself, only on other particles.

In the eyes of the analytic thinker, this very absurdity shows that the particle and the force field created by the particle are two descriptions of the same reality, namely the presence of a charged particle at a certain point. Therefore there is no place to speak of force exerted by the particle on itself. It should further be noted that according to the synthetic picture it is not really clear why a field, which actually exists throughout space, does not act on a charge that actually exists at the same point, even if the charge is what causes the field, according to the causal-synthetic understanding of the equation above.

One should note that this is an exceptional case, because here we see that the analytic assumption becomes an empirical determination. Here it is not merely a different formulation or interpretation, but has real consequences that can be checked empirically, or by calculation.

Incidentally, Schwebel himself apparently later retracted his proposal, but the reasons for this cannot be explained here. They are probably connected to what was said in the previous paragraph. For our purposes it was important to sharpen the meaning of the difference between these two approaches to scientific equations.⁹⁷

Let us note once again that even if Schwabl were right, with respect to a force field one can adopt the analytic picture without giving up the basic synthetic outlook. As we saw at the end of the previous chapter, the synthetic outlook does not require the existence of theoretical entities; it merely allows for the possibility of such entities.

It seems to me that the intuitive view regarding the relation between a charge (or a particle) and the force field it induces in space is as follows: both are indeed phenomena that exist in reality, yet they are still two phenomena derived from one underlying reality, and therefore it makes no sense to count both the charge and the field it creates at the same point as though they were two separate things.⁹⁸ Even so, the matter still requires clarification.⁹⁹

Let us now return to gravitation. As a child, I once saw a performance by a hypnotist who raised a child lying above a bench by means of hypnosis, and left him suspended in the air above the bench. Some time later I wondered whether he had done something contrary to the laws of nature. A gravitational force was acting on the child, drawing him toward the earth, and yet he remained where he was instead of accelerating downward and falling.

As a physicist, I am inclined to answer that the hypnotist—if he was not a fraud in some way that I failed to notice—applied an opposing force to the child, thereby neutralizing gravity. In such a case, the net force acting on him was zero, and therefore he remained suspended in the air. The fact that we do not know how such a force could have been produced by hypnosis should not deter us. Many forces were identified long before science knew how to explain or even coherently describe them. Before the emergence of electromagnetic theory, electric or magnetic force was no less mysterious than such a hypnotic force.

The interesting fact is that the scientist is usually convinced that there is such a hypnotic force, and dismisses out of hand the possibility that Newton’s law has actually been violated here. All this despite the fact that he has no positive indication at all that such a force is present. Such a view is based on an analytic conception of Newton’s first law: “When the net force acting on a body at rest is zero, the body will remain at rest.” According to the analytic conception, the two halves of this statement are analytically equivalent. To say that the body is at rest is already to say that no net force is acting on it. Thus, when we see a body remaining motionless under hypnosis, we can immediately say that there is a force balancing the gravitational force pulling it downward. It is the very same statement, in different words.

Exactly the same holds for the law of conservation of energy. It can never be violated, because whenever some quantity of energy appears to be missing in a process, we define a new kind of energy that exactly balances the deficit and add it to the equation. In fact, some would say that the concept of “energy” is defined precisely as a conserved quantity. When a body accelerates down a slope and its speed increases, its kinetic energy increases, and the law of conservation of energy would seem to be violated. In such a situation we define a new concept, potential energy, and say that as the body descends its potential energy decreases and is converted into kinetic energy. Total energy is then defined as the sum of kinetic and potential energy. And behold: suddenly the law of conservation of energy—of total energy—is not violated after all, despite the change in kinetic energy. We can continue in the same way and say that if we discover that the increase in kinetic energy is not equal to the decrease in potential energy, then the missing energy results from the transformation of part of it into heat energy, which must also be included in the conservation equation. Energy will now be defined as the sum of three quantities: kinetic, potential, and thermal energy. Obviously, energy is once again conserved, and the conservation law has been rescued by the narrowest margin. These ways of treating the conservation law, as well as similar basic scientific principles, view them as logical tautologies—analytic identities, or definitions.

Thus, simple intuitions regard scientific equations and laws as analytic identities.

Two Types of Analyticity and Syntheticity

We have seen two ways of understanding scientific equations: the analytic camp sees them as tautological identities, that is, as definitions, whereas those who hold the synthetic position understand them as synthetic laws of nature. We must now pay attention to a very important distinction, one that is very easy to overlook.

It is entirely clear—and the analyst too must agree—that we arrive at laws of nature in a synthetic way. We do so by means of generalizations and analogies, as described above. As we have seen, no one disputes that scientific equations, that is, laws of nature and theories, are not derived analytically from observations. The analyst merely claims that this generalization does not suffer from the shortcomings of generalization—such as the problem of induction and analogy—because it says nothing about reality. It is only a convenient and efficient organization, something like a language, for the collection of particular facts available to us. In analytic language, the way one arrives at such a generalization is given the honorable title the context of discovery, and this context is then classified as something that may perfectly well be mystical, since it does not lie within the empirical-rational domain of science. In this way, from the standpoint of the analytic position, the methodological problem of scientific generalization is solved. That is, according to the analytic picture, generalization is indeed induction, but it belongs to a mystical context that says nothing about reality, and therefore we are permitted to make use of our imagination, despite having no guarantee whatsoever that it corresponds to reality as such.

Once this generalization has been made, we arrive at the scientific theory, a general proposition which, according to the analyst, does not deal with the world but only organizes the totality of facts. To organize the facts means that all the facts currently known to us that are relevant to the field in question are analytically derivable from this proposition. The moment there is some fact that is not analytically derivable from the theory, we replace the language with one that is more efficient and convenient, and that fits all the facts in our possession perfectly.¹⁰⁰

Here it is important to note that even in the synthetic picture of science, it is entirely possible that the nature of the laws of nature, including those not seen as mere definitions, is akin to that of analytic propositions. We may clarify this by presenting Saul Kripke’s critique of Kant’s classification of propositions.¹⁰¹

As we have already seen, Kant divided propositions into four principal kinds along two axes: the analytic-synthetic axis, which is a logical axis depending on the structure of the proposition and on the logical relation between its concepts, and the a priori-a posteriori axis, which is an epistemic axis, determined by the ways in which we know the proposition in question.

Although in such a situation we would expect to encounter four kinds of propositions, since the two axes are ostensibly independent, Kant claimed that we can encounter only three. The type analytic-a posteriori does not exist. The reason is that analytic propositions cannot be known a posteriori, since they are derived not from reality but from the structure of the concepts involved in them.

Kripke, in his book Naming and Necessity,¹⁰² argues that since these are two independent divisions, they clearly generate four types of propositions, not only three as Kant claimed. That is, he argues that the fourth category—analytic-a posteriori propositions—must exist.

Kripke offers as an example the fact that most children learn the rules of elementary arithmetic, which are fully analytic propositions—though Kant treated them as synthetic a priori, a classification that has generated broad debate in the philosophical literature—in an a posteriori fashion. For example, the teacher demonstrates addition by means of objects: “Let us imagine a basket containing two oranges, and we add another three oranges. If we count the number of oranges in the basket, we will discover that there are five oranges altogether.” This is an a posteriori learning of an analytic rule. Therefore, according to Kripke, the proposition 2 + 3 = 5 is analytic-a posteriori.¹⁰³

Kripke’s argument appears formally valid, but it is entirely clear that he is speaking about concepts of apriority and posteriority different from Kant’s. Kant’s classification remains valid, provided one preserves an appropriate meaning for these concepts.

Kant apparently means to say that the a posteriori character of our coming to know analytic propositions is not essential. That is, if we were well acquainted with the concepts involved in analytic propositions—in our example, the numbers 2, 3, and 5, the operation of addition, and the relation of equality—we would not need experience in order to arrive at the analytic claim. The need for experience derives from the fact that we do not yet know the concepts; but in principle, no observation is needed in order to arrive at the analytic proposition itself.¹⁰⁴

This is similar to the question whether, in order to know the proposition “Every bachelor is unmarried,” which is also a very common example of an analytic proposition,¹⁰⁵ we require an a posteriori process. On the face of it, it is obvious that someone who does not know the meaning of the term “bachelor” must first become acquainted with it. In a certain sense, this too is an a posteriori process. Yet it would be very difficult to say that this proposition is analytic-a posteriori. The reason is that once one knows the concepts “bachelor” and “married,” it is clear to all of us that the relation between them follows from their very meaning—that is, it is analytic.

Kant’s classification is meant to classify propositions in themselves, and not to offer a classification dependent on the relation of this or that individual to the different kinds of propositions. Therefore, a proposition that does not essentially require any a posteriori dimension in order to be known will be called a priori, even if someone nonetheless needed an a posteriori demonstration in order to be convinced of its truth.¹⁰⁶

Note 11: The Dispute between Maimonides and Nahmanides on the Thirteen Hermeneutical Principles

We have seen that the epistemological status of a proposition does not necessarily depend on the way in which we know it. There can be an analytic proposition—and therefore, automatically, also an a priori one—that we come to know in an a posteriori way. If we know the proposition 2 + 3 = 5 a posteriori, this does not necessarily mean that it is itself an a posteriori proposition. In this note I wish to illustrate this principle through a dispute between Maimonides and Nahmanides.¹⁰⁷

At the beginning of Maimonides’ Sefer HaMitzvot appear fourteen “roots” that define the rules for counting the various mitzvot (commandments) in the enumeration of the commandments. In the second root, Maimonides discusses the status in halakha (Jewish law) of laws learned from scriptural texts by means of the thirteen principles through which the Torah is interpreted.¹⁰⁸ There he determines that their status is not that of laws of Torah origin. According to Maimonides’ definition, these midrashic (interpretively derived) laws stand to their biblical sources as “branches that emerge from roots.”

Nahmanides, in his glosses there, disputes Maimonides and argues that if we learn these laws by applying derashot (interpretive derivations) that were given at Sinai to a text that was also given at Sinai—that is, the Torah—then clearly the results of those derivations were also given at Sinai. Put differently: these are laws implicit in the scriptural text itself, and therefore they are laws of Torah origin.

Underlying Maimonides’ determination is the fact that the hermeneutical principles by which the Torah is expounded are not deductive-mathematical forms of inference, but various forms of analogy. As we have seen, the conclusion of a deductive inference is already contained within its premises. This is the very essence of the emptiness of the analytic that we discussed above. By contrast, the conclusion of an analogical or inductive inference is not contained within the premises themselves, but rather constitutes some expansion of them.

If so, it would seem that Maimonides is right. As we have seen, drawing analogical or inductive conclusions does not uncover what lies latent in the premises, but creates something beyond them; in fact, it expands them. This is what Maimonides there calls “drawing roots out of branches.” These conclusions are not contained in the premises, and the act of inference is not an act of uncovering but an act of creation.

There is no doubt that Nahmanides also understood that the principles by which the Torah is expounded are analogical rather than deductive in character. Yet, contrary to Maimonides’ position, he claims that despite this, the conclusions of halakhic exegesis are contained in the scriptural text itself. According to Nahmanides, the methods of derash, despite their analogical character, uncover contents that are actually present in the texts.

Here we have an illustration of the process we saw above. Even though we use tools of an expansive character—that is, synthetic—and not merely revealing tools—that is, analytic or decompositional—this does not necessarily mean that the conclusions are not already present within the text to which those tools are applied. There can be situations in which we use analogical-inductive tools in order to uncover contents hidden within the text being deciphered.¹⁰⁹ We are performing analysis by means that are synthetic in their essence.

We saw above, and will sharpen the point further below, that scientific laws too can be understood in this way. There is no principled obstacle to viewing the laws of physics as analytic, despite the fact that we learned them in synthetic ways. For example, it may be that the law of gravitation follows analytically from the nature of the concept mass. Whoever fully understands what it means to possess mass could infer from the fact that two objects have mass that they attract one another through a gravitational force. The fact that we need analogies and inductions—that is, generalizations from empirical experiments, which are of course synthetic tools—in order to arrive at the law of gravitation derives from the fact that we apparently do not fully understand the concept mass, and therefore do not know what follows from it. Through this very route—empirical-synthetic scientific inquiry—we come to know it better.

According to this suggestion, Nahmanides is claiming that by means of the methods of derash we come to know halakhic concepts better. If we fully understood the concept nevelah—the carcass of an animal forbidden for consumption—it would be obvious to us that one may not eat a nevelah. But since we do not know the concept fully, we are unable to derive this conclusion on our own, and we therefore require the Torah’s command. That is the ordinary case. Yet there are times when we also need scriptural exegesis in order to arrive at the characteristics of a halakhic concept. For example, we need exegesis to derive the thirty-nine primary categories of labor forbidden on Shabbat. It may be that full acquaintance with the concepts Shabbat and melakhah would render this inference analytic, but apparently we do not possess such acquaintance.

According to Nahmanides, the need to employ methods of derash—methods that are analogical in character—stems from insufficient acquaintance with the halakhic concepts. We improve our acquaintance with them by using those very methods of derash. But once we reach a full understanding of the concepts involved in the halakhic principles, the laws that pertain to them can be derived analytically from their very meaning.¹¹⁰

Whether we accept Kant’s definition or Kripke’s, there is no obstacle to understanding scientific equations in precisely this same way. One may call this analytic-a posteriori—following Kripke—or synthetic-a priori—following Kant—but it is clear that there is a situation in which we pass through a route that is empirical, that relies on experience, while the result we discover—the law of nature, the theory—is an analytic proposition. According to this, it is possible that we discover the law of gravitation by means of inductive, hence synthetic, generalization, and only afterward realize that what we have actually discovered is the definition of the concept mass. Had we understood that concept fully, we would not have needed experience at all.

One must pay close attention here: we are distinguishing between the analyticity of the proposition itself and the route by which we come to know it. Kripke would say that this is precisely the distinction between analyticity and apriority. The route can be synthetic—Kripke would say: a posteriori—yet the result may be an analytic proposition. In such a case it becomes clear that the route was merely a catalyst that helped us discover the essential properties of some scientific concept. Thus even one who holds the synthetic position, and believes in our ability to discover truths about the world by means that are synthetic in their essence, can still hold that the truths discovered are analytic. For example, one may understand the discovery of Maxwell’s equations or Newton’s laws as a generalization that makes claims about the world, and whose concepts have correlates in reality—they describe real entities, not merely linguistic-logical fictions—and yet still claim that the two sides of a scientific equation are an analytic identity. More than that: this truth is disclosed to us by synthetic means. The essential dispute is not about the character of the equation, but about the possibility of discovering truths about the world in ways other than direct empirical observation.

This distinction is extremely important, and many fail by ignoring it.¹¹¹ One possible consequence of it concerns the question whether it is meaningful to speak of other worlds in which bodies would have mass like that which they have in our world, while at the same time the law of gravitation would not hold there in the same way. More generally, this bears on the question whether there is any difference between a law of nature and a law of logic. In the next note we shall sharpen this point somewhat further.

Note 12: Logical and Physical Necessity

Philosophers are accustomed to distinguishing between two kinds of necessity: logical and physical. Logical necessity is absolute necessity, without escape and without exceptions. In no world, not even an imaginary one, can it be violated. Physical necessity, by contrast, is a necessity that follows from the laws of nature prevailing in our world, but there is nothing to prevent us from thinking about a world in which different laws of nature prevail and in which this necessity does not exist.

The necessity that a log placed in fire be burned is a physical necessity. This necessity follows from the laws of the world in which we live. It is possible, at least theoretically, to imagine a world in which fire would not burn logs—as with the bush in the Book of Exodus, of which it is said that it “was not consumed.” By contrast, the necessity that a bachelor not be married is a logical necessity. This necessity will exist in every world, however imaginary it may be—of course, so long as we preserve the meanings of the concepts bachelor and married.¹¹²

If we adopt an analytic interpretation of scientific equations, or of scientific laws, then the principles expressed in them become analytic, as we saw in the previous note and will see below. In other words, we have thereby removed the very distinction between logical necessity and physical necessity. According to this interpretation, the statement that a log placed in fire must burn speaks of logical necessity, and not merely physical necessity. It is impossible to imagine a world in which a log is placed in fire and yet does not burn.¹¹³ The empirical result is implicitly contained in the very definition of the objects involved in the event. We discover this analytic fact by synthetic means, but it is itself an analytic conclusion from the very essence and definition of the concepts.

The Meaning of All This for the Principle of Causality

All that now remains is to reexamine, in light of this last distinction, the relation of the advocate of the complex synthetic position to the principle of causality. At the end of Chapter Two we pointed out that, according to the analytic picture, the principle of causality disappears in its usual sense. If theoretical entities have no reality, then there is nothing that causes events. If in objective reality there is no force, then there is no cause of acceleration in motion. We have now seen that even advocates of the synthetic position can adopt an analytic interpretation of scientific equations, and indeed of scientific theories in general. After theoretical generalization, we arrive at the general principle, the law of nature, which identifies two physical entities. If we interpret this identification as an analytic tautology, then we return once again to a situation in which there is no force, only acceleration. Thus the advocate of the synthetic position too finds himself giving up the principle of causality. Yet, as we have already noted many times, intuition—which is what guides the advocate of the synthetic position—points to causality as one of the most fundamental principles in our view of reality.

How, then, can one preserve the intuition regarding the validity of the principle of causality within an interpretation of science that expresses a complex syntheticity? It seems that several avenues of solution are possible.

First, the advocate of the synthetic position is not obligated to adopt an analytic interpretation of theoretical equations; he merely can do so. By contrast, the advocate of the analytic position must interpret them in that way. Therefore, the first way out of the dilemma of causality is simply not to adopt such an interpretation.

A further way out is to hold that science does not explain phenomena, but only describes them. That is, science truly does not find the causes of phenomena; it only describes their occurrence. The real causes exist, but they lie in other domains, or perhaps in other equations within the same domain.¹¹⁴ In the fourth gate we shall see in more detail this form of approach, which treats science as syntax—that is, as formal description—and other planes of reference as semantics, that is, as essential and contentful explanation. Such a direction would, in principle, characterize specifically the advocate of the analytic position, but for other reasons he will not adopt it. A consistently analytic thinker does not accept non-scientific planes of explanation and classifies them, pejoratively, as mythical or religious. In his eyes only observational entities are legitimate topics of discussion, and only such entities can exist in the world. The advocate of the synthetic position, even if he adopts an analytic interpretation of scientific equations, does so only with respect to the equations of science; in his general outlook he remains synthetic, and therefore this route remains open to him.

A further route is to maintain an analytic interpretation while at the same time understanding that both sides of the equation exist together. We saw such a direction at the end of the note on the relation between charge and the electric field it creates. Let us now elaborate a bit more.

When there is an equation that describes a relation between force and acceleration, or between charge and the force field it creates, we understand that these are indeed two equivalent forms of description of the same phenomenon. Wherever there is charge, if we understand the concept of charge well enough, there must also be a force field. But this does not necessarily mean that the two descriptions do not exist simultaneously. The relation between the two sides of the equation is indeed necessary and a priori, but both of its sides describe existing entities. In the example of the force field and its sources, for instance, there are particles, and there are also force fields that they create. In other words, it is a relation of identity, but both sides of the equation exist simultaneously.

Here the advocate of the synthetic position does adopt an analytic interpretation of the equation, but unlike the advocate of the analytic position, both sides of the equation—the real one, the charge, and the abstract-theoretical one, the field—can exist together.¹¹⁵ In such a case there is cause and effect, but the causal relations between them are “upgraded” to the logical level. As we saw above, if one adopts an analytic interpretation of equations, then there is no difference between the logical level and the physical-natural level; there is no difference between the laws of logic and reason and the laws of nature. Therefore, the causal relation expresses logical necessity rather than physical necessity. If the cause exists, then it is a logical necessity—and not merely a physical one—that the effect occur as well. There is no possible world in which this cause exists and yet the effect does not follow from it.¹¹⁶

For a more detailed continuation of the discussion, and for an expansion of its different possibilities, see below in the fourth gate.

Chapter Five: Between Science and Mathematics¹¹⁷

Is Mathematics a Science?

The basic question with which we must deal in order to understand the meaning of mathematics in the context of our discussion is this: is mathematics a science?

There are various fields within mathematics, and in certain respects they differ in character from one another. Logic, which many regard as the most basic of all—since it deals with the most fundamental modes of our thinking—appears to be the farthest removed from experience; it is a priori, and therefore far removed from empirical science. Geometry, by contrast, appears to occupy a field in which it is not entirely clear whether it belongs to mathematics or to physics, since it describes space, and physics does the same. In this chapter we shall move back and forth between these two poles, logic and geometry, and we shall see what they share and why, in the final analysis, it is justified to include both within mathematics. The main discussion will illustrate these ideas through geometry, for two reasons: its uniqueness—namely, its apparent similarity to empirical-synthetic science—and the fact that the average reader has a reasonable acquaintance with geometry from high school.

According to the accepted definition of science as presented thus far in the present gate, we must examine two different questions, which are very easy to confuse:

1. What is the status of mathematics with respect to experience?
2. What is the status of mathematics with respect to reality?

Another way to formulate the first question is: is mathematics empirical—that is, is it a priori or a posteriori? A parallel formulation of the second question is: does it say anything about the world—that is, is it analytic or synthetic?

As we shall see, according to all views mathematics is not empirical; that is, it is a priori.¹¹⁸ The remaining question is whether it says something about the world, that is, whether it is synthetic. Here there are two basic approaches, which will be discussed below: the approach that regards mathematics as analytic—that is, as saying nothing about the world—and the approach that regards it as synthetic, or more precisely as synthetic-a priori, that is, as saying something about the world despite its apriority.

Let us begin with the first question: is mathematics empirical, or a priori?

Mathematics Is A Priori

According to the Popperian definition we encountered above, the criterion for the scientific status of a theory is that it be open to empirical refutation. We saw that Carnap and Hempel add an intermediate level, according to which a theory’s standing up to empirical tests constitutes confirmation. We proposed a synthetic conception, parallel to Bacon, according to which a theory can even be confirmed inductively, that is, grasped as true through a kind of eidetic seeing. What all three approaches share is the basic principle: for a theory to count as scientific, there must be some possibility of putting it to the test of experience, where it can fail or succeed. The dispute among these approaches concerns only what can be inferred from a theory’s success or failure in such tests.

Let us take a simple, almost banal example. Suppose we try to think of an experiment that could test the following algebraic theory: 2 + 3 = 5. The first idea that comes to mind is the experiment Kripke proposed as an objection to Kant, mentioned in the previous chapter: let us take two oranges from a basket, add three more oranges to the basket, and then count the total number of oranges in it. If the total number is five, the theory has passed the empirical test. If the number is different, the theory has failed the empirical test—that is, it has been refuted.¹¹⁹

Let us assume that we performed the experiment, and to our surprise we found that no fewer than seven perfectly ordinary oranges were resting in the basket. At first glance, we have refuted the algebraic proposition 2 + 3 = 5. Experience has shown us that the sum in question is 7, not 5.

In principle one might think that this is indeed the meaning of the result we obtained. Yet it seems to me that a more correct interpretation is the following: the failure of a theory in some experiment forces us to give up one of the assumptions on which it is based, but not necessarily any particular one. In order to explain a failed experiment, or alternatively to preserve a revised theory, we may abandon any one of the assumptions that led to the theoretical prediction that has been disproved. If we discover that some additional hidden assumption underlay the mistaken theoretical prediction, we can say that it was not the algebraic assumption 2 + 3 = 5 that was refuted, but rather the additional assumption. It turns out that we would prefer to give up any other assumption before abandoning such a basic and intuitive algebraic truth. As we have already seen, those who hold a synthetic position do not readily give up their basic intuitions.

As noted, in this experiment we assume the algebraic proposition 2 + 3 = 5, but we also assume something else, tacitly: namely, that adding oranges to a basket is correctly described by the operation of algebraic addition. It is therefore possible that, in order to explain the result of finding seven oranges in the basket, we should abandon this assumption rather than the immortal algebraic proposition 2 + 3 = 5. Thus, the result of the experiment has not necessarily refuted an algebraic proposition; it may instead have refuted the assumption that what occurs in the real world is described by that mathematical proposition.

Let us try to classify the nature of this additional assumption we have uncovered. It seems that such an assumption belongs to physics, or to applied mathematics, rather than to algebra, because it involves an assumption about the nature of reality. Physics, or the natural sciences in general, deals with claims about the character of the natural world, and therefore a claim of this sort belongs there rather than to mathematics. Physics is what deals with determining the correspondence between mathematical theories and empirical occurrences—in other words, with determining which mathematical theory is appropriate for describing some empirical domain.

Let us clarify the argument with another example, one that will make clear why the assumption that algebraic addition describes a kind of occurrence in the real world is not trivial. For that reason there is room to include it in the system of considerations as a separate assumption, and at times perhaps even to give it up.

Suppose we consider a force of magnitude 10 units acting on some body and pushing it southward. We now add another force, also of magnitude 10 units, acting westward. We now ask: what is the sum of the forces—or, in the usual language of physics, the resultant force—acting on the body? At first glance, the simple assumption is that the summation of forces, just like adding oranges to a basket, is described by algebraic addition, and therefore the resultant force is 20 units. But it is obvious that this is true only when the two forces act along the same axis and in the same direction—for example, when both point southward. In such a case the resultant force is the algebraic sum of the two forces. But when they act along the same axis in opposite directions—one northward and the other southward—we must subtract one from the other. The total force acting on the body in that case is zero; no net force acts on it. In other words, the magnitude of the resultant force in that case is 0 units. Even in such a case, one could still preserve our tacit assumption—that the addition of forces is correctly described by algebraic addition—if we define a force acting northward as a force of negative magnitude: -10 units. Under that definition, we can continue to add the forces algebraically and obtain the correct result: 0 units of force.

However, in the case described above, where one force acts southward and the other at right angles to it, that is, westward, the two do not act along the same axis at all. In such a case the two forces cannot be added algebraically under any definition. Here we must abandon entirely the assumption that forces combine by algebraic addition of their magnitudes. Mechanics tells us—and this is, of course, an empirical result—that in such a case the resultant force has a magnitude of a little over 14 units, directed southwestward. It is obvious that such a conclusion cannot be described as the result of any relevant algebraic addition.

This kind of addition of quantities that have both magnitude and direction is called in mathematics vector addition. Although this operation too is called addition, this is only because of a superficial resemblance to ordinary addition, that is, algebraic addition. In fact it is a different operation. Thus, forces are not really combined according to the same rules of addition—the algebraic rules—that are valid for adding oranges to a basket, but according to the rules of vector addition. The conclusion, then, is that the algebraic sum of the forces is indeed 20 units, but that is merely their algebraic sum, and it has no physical value or meaning. A physically meaningful addition of forces is the calculation of the resultant force, and that calculation is performed differently from algebraic addition.

We therefore learn that one must not take for granted the connection between an algebraic operation and an occurrence in the real world. This assumption requires examination in every case, and it clearly belongs to physics rather than to mathematics. The claim that forces are combined according to the rules of vector addition—which is itself, of course, a purely mathematical operation—belongs to physics rather than to mathematics. The rules of vector addition themselves belong to mathematics, just as the rules of ordinary algebraic addition do. Physics, as noted, deals with determining the relation between the facts in the world and the relevant mathematical theories.

Let us now return to adding oranges to a basket. If an experiment were to yield a result that refuted the hypothesis that adding the oranges should leave five oranges in the basket, it would be more reasonable to say that what has been refuted is the physical claim that adding oranges to a basket is well described by algebraic addition. It would be less reasonable to claim that we have refuted the algebraic proposition 2 + 3 = 5 itself.

Mathematicians sometimes refer to the relation between these two levels as that between theory and model.¹²⁰ Arithmetic and vector addition are two mathematical theories. Mathematics—in this case, algebra—is what deals with the properties of the addition operations included in these theories. By contrast, the addition of forces in physics is a model for the mathematical theory of vector addition, and adding oranges to a basket is a model for the mathematical theory of simple algebraic addition. A model is a particular concrete situation or structure correctly described by a certain mathematical theory—that is, the propositions of that theory are true of it. In such a case we say that the situation in question is a model of the mathematical theory under discussion.

In these terms, the claim that some real state of affairs is, or is not, a model for a given mathematical theory is a claim belonging to physics—or perhaps to applied mathematics—and not to pure mathematics. The principles that characterize the mathematical theory itself, and only they, belong to pure mathematics.¹²¹

The meaning of this claim is that the determination that some situation is a model of a given mathematical theory is a proposition that can be refuted—and also confirmed—empirically, like any proposition in empirical science, such as physics. That is what we saw in the examples above. By contrast, the claims of the mathematical theory itself are not open to empirical refutation. If we conduct an experiment to test the claim that some situation is a model for a certain theory, what we are testing is the claim that the situation is a model of the theory, not the claims of the theory itself. If the theory “fails” in the experiment, what has been refuted is not the mathematical theory but the physical assumption that this reality is a model of the mathematical theory in question. That is what we saw above in the examples of oranges and forces.

In common terminology, perhaps under the influence of the Jewish religious context,¹²² the mathematical structure itself is sometimes called a doctrine, whereas the empirical claim that some reality is a model of this doctrine is called a theory. Quantum mechanics as such is primarily a mathematical system. Only the claim that the world, or some part of it, is a model of quantum mechanics is itself a theory in physics; in the terminology used above, it would be more accurate to call this quantum theory. If that theory is refuted by experiment, what has been refuted is the theory, not the doctrine.

In fact, one can see this distinction even at the methodological-practical level. Pure mathematicians, unlike physicists or other scientists, do not engage in empirical experiments. Mathematical results are a priori, and in the opinion of many also analytic. This is another aspect of the fact that mathematics is not science by the accepted definitions, and certainly not empirical science.

This can also be described from another angle, one that will allow us to rank our attitude toward the apriority of mathematics along several levels of dogmatism. If we take geometry as an example of a mathematical doctrine, we see that its basis consists of axioms, from which we derive theorems. If we discover that geometry does not correctly describe the space of the world in which we live, we will not conclude that its theorems are false in the mathematical sense, but rather that it does not correctly describe the world. Put differently: geometry does not claim that its theorems are true simpliciter. It claims that if someone adopts the assumptions—the axioms—on which it is based, he will be compelled to accept the conclusions derived from them. If the geometric description of the world is refuted, we will conclude that the axioms underlying the geometry do not correctly describe our world, and therefore the conclusions derived from them do not describe it either.

Yet one might still think that an empirical test could be proposed that would refute mathematics. For example, suppose we were to find that the axioms do indeed describe the world correctly, while the conclusions do not describe it correctly. If, for instance, we discovered that the axioms of Euclidean plane geometry describe the world correctly, and yet the sum of the angles of a real triangle is not 180 degrees, then the connection between the premises and the conclusions would have been refuted. Such a refutation already concerns pure mathematics.

But it is obvious that such a thing should not happen at all in mathematics done properly. The situation is entirely different in empirical science, which remains open to refutation even when it is constructed properly. Even if we were nonetheless surprised, and the connection between the axioms and the conclusions drawn from them were in fact refuted, we would still tend to look for another explanation of this surprising experimental result—such as the intervention of some additional factor in the experiment that had not been taken into account. In any case, we would not give up the mathematical conclusions, that is, the necessity of the connection between the premises and the conclusions derived from them. This is already the phenomenon of saving a scientific theory—in this case, a mathematical one—by means of ad hoc additions, and we shall not deal with it here.

Let us go one step farther. Even if we were to discover no relevant factor at all, it still seems to me that one could say that the mathematical theory would continue to be regarded as correct, and that the experimental results describe the world rather than the mathematics. Put differently, we would distinguish between the theory and the claim that actual space is its model. For example, we might conclude that the logic underlying the derivation of the conclusions from the premises describes only us and not the world. It should be noted that in such a situation there would be those who would dispute this dogmatic attitude toward mathematics.

In this sense geometry seems, at first glance, exceptional, since it is harder to imagine a parallel experiment that would test set theory or algebra. Yet above we saw an experiment that places an algebraic claim under empirical test—adding oranges to a basket. In the same way, one can take elements of different sets and test empirically their intersection, or the results of other logical operations on them, and see whether the predictions of set theory stand up to reality or not. In all these situations one can repeat the analysis we made above with respect to geometry. It can always be argued—with varying degrees of persuasiveness—that the failed experiment refutes the fit of the proposed model to the theory, and not the mathematical doctrine itself.

Thus we see that the apriority of mathematics can be understood at several levels of dogmatism. This description already touches more directly on the second question we raised at the beginning of the chapter: is mathematics analytic or synthetic? There is indeed a connection between the questions, for if mathematics says nothing at all about the world, then clearly it cannot be subjected to empirical test. But if it does say something about the world, that still does not necessarily mean it can be tested. This claim leads to the approach that regards mathematics as synthetic-a priori. A synthetic-a priori claim, by definition, says something about the world but does not stand empirical test.

Beyond that, there may be a situation, like the one Kripke describes, in which we learn a mathematical rule from empirical experience, but afterward discover that it is a rule of our own thinking—in other words, that it is in fact an analytic proposition. Here the empirical experiment is a methodological tool that helps us examine ourselves, and not an epistemological tool that helps us learn something about the world. This adds another layer to the distinction between the two questions: the one concerning the apriority of mathematics and the one concerning its syntheticity. According to this proposal, we arrive at the conclusion that mathematics is analytic—and perhaps also a posteriori—and not synthetic-a priori, as follows from the previous approach.

We shall now discuss this question—the logical character of mathematics, whether it is analytic or synthetic-a priori—in greater detail.

The Problem of Platonism: Is Mathematics Synthetic?

As noted, up to this point we have concluded that the propositions and concepts of mathematics are not empirical but a priori. As we saw above, there is still room to ask whether the concepts and propositions of mathematics are analytic or synthetic. That is: do they describe something in the world, or do they concern only the structure of the intellect and the forms of human thought? This is the problem of Platonism in mathematics, and that is what we shall now discuss.

It should be noted that if we indeed say that the propositions and concepts of mathematics have correlates in the world as it is in itself, and at the same time adopt the conclusion reached above that mathematics is not an empirical domain but an a priori one, then the propositions of mathematics will be classified as synthetic-a priori.¹²³ In other words, we are here reformulating the Kantian problem of the synthetic-a priori—namely, whether it is possible—within the field of mathematics.

This is precisely the point of departure for the connection between the present discussion and the previous ones. According to the analytic position, because the propositions of mathematics are a priori, they must also be analytic. The category of the synthetic-a priori is not accepted here—except perhaps in the Kantian, essentially subjective, sense.

Therefore, according to the analytic position, mathematics is an analytic domain. Put differently: because mathematics is necessary, it is clear that it says nothing new—and in fact nothing at all—about the world. It is merely the extraction of conclusions that were already hidden within its premises, and this is exactly the principle of the emptiness of the analytic. The validity of mathematical propositions lies in their being necessary conclusions for anyone who adopts their premises. They make no claims about the world itself.

From another angle, one may say that the analytic interpretation of mathematics is based on the fact that it is very difficult to see where in the world there exist correlates of the concepts and propositions of mathematics. Quite simply, they appear to concern only our forms of thought and not the world as such.

According to Plato, general concepts, or mathematical terms, are Ideas that exist, in some sense, in the world—or more accurately, in the world of Ideas. Therefore, the approach that regards mathematics as describing properties of the world and regards mathematical entities as real entities that exist in the world is called Platonism.

Logic

The most basic mathematical domain is the one that deals with our most fundamental forms of thought: logic, that is, the theory of reasoning. At first glance, this is the domain most remote from a synthetic interpretation. Put simply, it appears to say nothing about the world, but only about our modes of thought.

Bergmann, in his book Introduction to Logic,¹²⁴ discusses the question of the syntheticity of logic at great length. Here I wish only to make a brief remark on this matter.

In fact, the relation between logic and the world lies at the foundation of the discussion of the ontological argument—see the first book, p. 66, and Avraham Zvi Braun’s The Question of Being.¹²⁵ Steinitz, in his book The Tree of Knowledge,¹²⁶ also addresses this issue, though from a different angle. His claim there is that if we have proved that some phenomenon or object contains an internal contradiction, then we infer from this that it cannot exist. If so, this is apparently an inference from our logic to the world. There is an assumption here that logic makes claims about the world.

But here too one may reply that logic as such says nothing about the world. What we do is add an additional assumption: that the world is a model of logic. Therefore, even when we discover a contradiction in some claim or concept, our conclusion that such a contradictory thing cannot exist in the world is based on the empirical assumption that the world is a good model of logic, and not on any assumption pertaining to logic itself.

Two examples from the opposite pole from logic, closer to empirical science, are game theory and geometry. Both are usually classified under mathematics, yet both touch the real world in a significant way. This feature leads to many confusions in the treatment of these two theories, and so we shall now discuss them briefly.

Game Theory¹²⁷

In this section I shall present two points that illustrate the confusion surrounding game theory. Game theory deals with the question of how games—and conflicts in general—are conducted among human beings, or among groups of human beings. At first glance this is an emphatically empirical field. Yet it turns out that game theory is rather weak in predicting the actual behavior of human beings, and it is repeatedly criticized for that.

Moreover, many people argue against game theory that it educates those who study it toward a cold, rational, and self-interested attitude toward the game and toward the opponent, whereas reality does not necessarily dictate such an attitude, and such an attitude does not always lead to optimal results either. Here the claim concerns the allegedly negative influence of game theory on reality, and not, as before, its rather poor ability to describe reality.

The truth is that both claims stem from a misunderstanding of the nature of game theory. This theory is classified under mathematics because its subject is the abstract and purely rational concept of a game. Real games, and international conflicts as well—the latter being another subject with which it deals—are usually not like this. Game theory assumes wholly rational behavior and decisions, that is, a desire to derive maximal utility from the game or the conflict. But the data by means of which it determines what utility is are not necessarily money or positions of power. If the players value honesty, loyalty, and trust, then those too will enter the utility function, and game theory will take them into account.

Thus, what determines the outcome that game theory will recommend is the players themselves. It can only help them make a wiser decision in light of their own assumptions.¹²⁸

A considerable portion of these criticisms, and perhaps all of them, stems from a failure to understand the difference between game theory as a branch of mathematics and the question of how it describes, or affects, the real world. Questions of the latter kind belong to the social sciences, which try to make use of game theory, and not to mathematical game theory itself.

Geometry A: An Example of a Mathematical Field That Appears Synthetic

Another example of a mathematical field that appears, at first glance, close to syntheticity is geometry. In this section we shall try to clarify why geometry is indeed close to the analytic pole. Yet in two later sections of this chapter we shall continue to discuss geometry and see why it is nevertheless an analytic domain, like the rest of mathematics. In the first of those sections—“Geometry B”—we shall “move” it toward the analytic pole, and in the second—“Geometry C”—we shall place it between the two poles: analytic, yet “quasi-synthetic.” At the end of the process, geometry—precisely because of its uniqueness—will serve as a good illustration of what it means to regard mathematics as a priori, and of the general relation between apriority and analyticity.

The various branches of mathematics deal with abstract constructions, and one may debate whether they have correlates in reality. Geometry, by contrast, appears almost to be a description of reality itself. It deals with the description of space, and for that reason it is, in many ways, closer to physics—the most fundamental science describing the world—than to mathematics, which, at least according to the proponents of analytic positions, deals only with products of the human spirit and not with the actual world.¹²⁹

According to the analytic position, the proposition that in flat Euclidean space the sum of the angles of a triangle is 180 degrees is not a statement about the world, but a statement about what follows from adopting the assumptions of Euclidean plane geometry. One who does not adopt the assumptions of plane geometry is not obligated to adopt its conclusions. According to this conception, geometry has nothing to say about the world itself—in other words, about the question what the sum of the angles of some concrete, real triangle will be.

With respect to geometry in particular, this approach seems utterly absurd. It is entirely clear that in a Euclidean plane the sum of the angles of a real triangle is indeed 180 degrees, and it is very hard to deny this. In principle, one can measure the sum of the angles in any triangle and verify it. In fact, one can say even more than this: it is clear that the entire collection of axioms—that is, the basic assumptions—of Euclidean plane geometry are all true, meaning that they describe the space around us very well. Anyone who does not accept them, at least up to errors due to the limited precision of the senses or of measuring instruments, would by many be considered mentally unsound—or at least deficient in perception.

Other Areas in Mathematics

Up to this point we have dealt with geometry and game theory, which, as noted, are the fields most prone to a Platonist—that is, synthetic—interpretation. With regard to more abstract domains, the question is much more complicated. In what sense is a statement about the properties of sets, of the kind described by the propositions of set theory, also a statement about the world? Do sets—or in another terminology, species and genera—exist in the world? Bertrand Russell attempted to answer this question by means of an elegant analytic argument, which will be presented in the following note.

Note 13: The Ontological Status of Sets

In the first book—in the fourth gate, Chapter Two, and Note 15 there—we dealt with the ontological status of the universal as opposed to the particulars that compose it. We discussed there the question whether the universal is an entity that exists in itself, or whether it is an ontological fiction and the truly existing entities are the particulars. We saw there that those who hold synthetic positions tend to see the universal as a genuine entity, whereas analysts generally understand it as an ontological fiction, like every abstract and ideal entity, and like concepts in general. See also the next chapter, especially the section on species and genera and Note 14 there.¹³⁰

All these are implications of the philosophical problem concerning the ontological status of universals. In this note I wish to present a sophisticated argument of Bertrand Russell against attributing real existence to sets in the mathematical sense. To the extent that it is correct, this argument may have far-reaching philosophical consequences for the question of Platonism. A reader who finds it difficult to follow the mathematical argument may, of course, skip the present note.

Set theory deals with the properties of sets. Every set is a collection of objects. Within any set one can also define the notion of a subset—or partial set. A subset is a collection of objects all of which belong to the parent set. Such a collection is called a subset of the parent set. For example, if we consider the set of natural numbers, {1, 2, 3, …}, then the collection {4, 8, 19} can be defined as a subset of the set of natural numbers. So can the collection {11, 55, 10, 11110, 339678}. The set of natural numbers has, of course, infinitely many possible subsets.

There is a theorem in set theory that determines the numerical relation between the number of elements in a set and the number of subsets of that set. Let us begin, for example, with the set {1, 2, 3}. Let us now try to list all the subsets of this set. For any given set, one of its subsets is the empty set—that is, a set containing no elements at all—and another is the set itself, namely the subset that contains all the elements of the parent set. Besides the empty set, there are three one-element subsets: {1}, {2}, {3}. After that come three two-element subsets: {1, 2}, {1, 3}, {2, 3}. Then there is one additional subset with three elements: {1, 2, 3}. Altogether, there are 8 subsets.

It is quite easy to see that in this counting process we have assumed nothing at all about the nature of the elements. The only assumption was the number of elements. In every set of three elements one can label one element 1, another 2, and the third 3, and repeat the whole procedure. Thus, every set with three elements has 8 subsets.

More generally, the counting process of subsets can be described as follows: if the number of elements in the parent set is N, then we construct one possible subset as follows. We decide, with regard to the first element of the parent set, whether it will be included in the subset or not. We then decide the same with regard to the second element, the third, and so on. The total number of possibilities—and thus the total number of subsets—obtained in this way is 2^N. Our general conclusion is therefore that if the number of elements in the parent set is N, then the number of its subsets is 2^N.

At this point we must emphasize two things:

1. Every subset is itself also a set.
2. The number of subsets of the parent set will always be greater than the number of elements of the parent set. This is a very simple result, if we note that for every element there is at least a corresponding singleton subset, and beyond those there are also subsets containing more than one element.

Now Russell continues and argues as follows. Let us take all the entities in the world and form from them one set, containing infinitely many elements. Let us denote the number of elements in this set by N. The number of subsets of this set is 2^N.

As we noted, each of these subsets is itself a set. But here a difficulty arises. We noted that the number of subsets is greater than the number of elements in the set—that is, 2^N > N. Thus we encounter sets in a number greater than N itself. But if N is the number of all entities in the world, then these sets cannot themselves be entities. For if they were entities, then the number of entities in the world would be greater than N, contrary to our original notation.

At this point the reader may object: how are we applying these theorems to sets whose number of elements and number of subsets are both infinite? In our case N is infinite, and therefore the number of subsets is also infinite. In what sense can one speak of the number 2^N and claim that it is greater than N, if both are infinite?

Here we must use another mathematical theorem, this time Georg Cantor’s theorem on infinite cardinals, which states that even for infinite sets one can define the magnitude called the number of elements of the set—its cardinal number, or cardinality—as well as the number of its subsets. More than that: it can be proved—and the proof is not even especially difficult—that with respect to infinite sets too, the following relation holds between the cardinalities:

2^N > N

Thus, we have before us a mathematical proof that sets are not entities—or, put differently, that sets do not really exist in the world.

This problem can be discussed from several aspects. For example, one can repeat the whole process even if sets are not entities in the world. They would still be abstract objects that have no ontological expression, yet they can still be counted in the same ways. At first glance, the paradox still remains.

One can also examine the extent to which Cantor’s own definitions of infinite cardinals describe entities in the world, or relations between such entities. Perhaps they themselves are no more than mathematical fictions. If that is so, then it may be forbidden to infer from them anything at all regarding ontological problems, or to draw any philosophical conclusions from them.

Beyond that, this paradox appears to be very closely connected with another and better-known paradox raised by Russell with respect to set theory. See the first book, ninth gate, Chapter Five, footnote 68.

In his famous paradox, Russell divides sets into two types: a set that contains itself as an element—for example, the set of all sets—and a set that does not contain itself as an element—for example, the set of natural numbers, which is not itself a natural number and therefore is not an element of itself. He then defines the set of all sets that do not contain themselves as an element, and denotes it by G. Russell then asks: does the set G itself contain itself as an element or not?

If it does contain itself as an element, then it is not one of the sets that make up its elements, and therefore it does not contain itself as an element. If it does not contain itself as an element, then it is one of the sets that make up its elements, and therefore it does contain itself as an element. This is an inescapable loop, which forced mathematicians to reconstruct set theory in a more careful and precise way. In the end, the more precise axiomatic set theory replaced the naive theory.

Here too we see that when we apply the concept set to an overly inclusive domain of elements—always in the infinite realm—we run into problems. The set of all sets was one of the concepts at the root of the difficulty expressed by Russell’s famous paradox, and here we saw that the inclusion of all the entities in the world within one set—the set of all things—creates a difficulty of a different kind.

Moreover, in axiomatic set theory, which was developed after Russell’s famous paradox, neither of these two kinds of paradox can be formulated. Therefore it is not clear whether formulating them within naive set theory has ontological significance—that is, whether one can draw philosophical conclusions from them, and certainly not ontological conclusions. At most, we encounter here yet another paradox of infinity, which usually arises from incautious use of that concept.

In any event, as has emerged from the discussion thus far, Russell’s argument does not necessarily lead to the denial of the existence of universals, or of sets in general. Of course, there is no proof here in the opposite direction either, and the ontological-philosophical question will have to find its solution by other means—apparently not mathematical ones.

In Note 15 below we shall discuss another aspect of the present issue. At its beginning we shall suggest, as a matter of principle, a direction according to which Russell’s proof presented here is not valid.

If we continue to examine the more abstract branches of mathematics—those that deal with groups, rings, mappings, models, and the like—we find ourselves increasingly hard pressed to defend the position that the claims of these branches concern the world.

Thus, despite what was said in the note above, those who hold the analytic position in mathematics can certainly argue that, at least with respect to the more abstract branches of mathematics, it is even harder to maintain that their propositions are claims about the world.

Of course, one is naturally led from here to draw conclusions also about the less abstract branches of mathematics. There is a strong intuition that mathematics is one unified whole in this respect. If its claims are claims about the world, then this is true of the claims in all branches of mathematics. And if we oppose mathematical Platonism, then we must oppose it in every branch of mathematics.

Geometry B: The Analyticity of Geometry

In order to place geometry on the same analytic plane as the rest of mathematics, one can argue, as we saw above, that geometry does not deal with actual space. It is a completely formal theory, and actual space is at most a model of it. But that no longer belongs to the concern of pure mathematics.

There are entirely formal formulations of geometry that do not even include a single diagram. The theory begins with basic abstract concepts—A, B, C, and so forth—which are defined as concepts with no intrinsic meaning, and geometric doctrine determines only the relations among them. From these basic relations, which are laid down as axioms, there follow the formal conclusions about the more complex relations among them.

Once the theory is formulated, we are equipped with a perfect mathematical theory devoid of all concrete content. We then discover empirically that if, in place of the concept A in the theory, we substitute the concept point in Euclidean space, and in place of B we substitute line in Euclidean space, and in place of C we substitute angle, then all the axioms of geometry are satisfied. We may therefore infer that actual space also satisfies all the propositions derived from these assumptions.

We conclude that the geometry of our actual world possesses the formal properties of the abstract mathematical doctrine. If so, actual space is a model of the abstract geometric theory. One must pay careful attention here: unlike the abstract mathematical theory itself, this is an empirical result, one that arises from observations and their analysis. For example, one can measure the sums of the angles in various triangles one draws, and if one indeed gets 180 in every measurement, one can make an inductive generalization about all triangles and formulate an empirical law: the sum of the angles in every triangle in our real world is 180 degrees.

The conclusion is that geometry as such indeed says nothing about the world, and one proof of this is the existence of many geometries. On the other hand, it is clear that there is only one correct geometry that describes the actual world itself, and therefore there is no relativism with respect to the description of the world.

One may say that geometric theory belongs to mathematics, and in this sense it is analytic, that is, non-Platonist, and therefore its truth or existence is not dependent on the world itself. Its results are not derived from observations of the world, which is to say they are a priori. If so, geometry is a priori—that is, not subject to examination by experiment—and also analytic—that is, it says nothing about the world. By contrast, the claim that space in the actual world is a model of some geometry belongs to physics, like the examples we saw above. It is this claim, and not the various geometries as such, that must be tested empirically, and it is obviously synthetic.¹³¹

Today it is known that various measurements of actual space indicate that it is not precisely Euclidean; that is, the sum of the angles in a triangle in actual space differs slightly from 180. One can offer various interpretations of this, but on its face it would seem to be evidence that the proposition “the sum of the angles in a triangle is 180” is an empirical proposition that can be refuted, and as such does not belong to mathematics.

In light of what we have said here, we would answer that this is not a refutation of Platonism, but neither is it evidence for it. What is at issue in such measurements is the physical proposition, not the mathematical one: namely, the proposition that our world is a model of Euclidean geometry. No one can refute the claim that in a world satisfying Euclidean assumptions the sum of the angles in a triangle would be exactly 180. What has been refuted, if anything, is the claim that our world does in fact satisfy the Euclidean axioms—that is, that it is flat Euclidean space.

The same arguments can be repeated with respect to other branches of mathematics. We saw this with algebra in the experiments dealing with the addition of velocities or the addition of oranges. We saw it with set theory in the experiment involving operations on sets of real objects, and so forth.

From what has been presented thus far, a clear conclusion would seem to follow: all branches of mathematics are a priori. More than that, geometry is not essentially different from any other mathematical field. The claims that concern the world itself—whether it is or is not a model for various mathematical domains—belong to empirical science, and only they deal with the world, and therefore only they are open to confirmation or refutation. Mathematics itself is immortal. It cannot be refuted, and therefore, according to the Popperian criterion, it is not science in the usual sense of the term.

It therefore appears that the dispute between Platonists and analysts in mathematics is, to a large extent, illusory. If we accept the distinction between theory and model, then we may say that the Platonists include within mathematics the claim that the world is a model of some geometry—that is, applied mathematics—whereas the analysts exclude this from mathematics and treat it as part of physics. In their eyes, mathematics is only pure mathematics.

If so, this is seemingly a semantic rather than a substantial dispute. All mathematicians would agree that if we measure the angles of a real triangle and find that they do not sum to 180, we have not refuted any part of mathematics. Euclidean geometry has not changed in the slightest as a result of such a measurement. What has changed is only the mathematical description we will use for the geometry of our actual world.

The obvious conclusion is that with respect to geometry, no less than with respect to logic, even those who hold the synthetic position will agree to the analytic interpretation. We saw in the first book that those who hold the synthetic position accept the analytic mode of thought; they too use basic logic. The difference is that they claim there is also another mode of thought, and that it too can yield certainty, at least to some degree. Here we are extending the domain of analytic thought beyond logic to all pure mathematics.

In the next section we shall take one further step in the pendulum swing that has characterized this chapter, and we shall nonetheless qualify this conclusion. We shall see that, in a certain sense, there is still room for a Platonist interpretation within pure mathematics.

Geometry C: Still Syntheticity, or “Quasi-Syntheticity”

To conclude, let us point to a certain aspect that partly explains the strong intuition that geometry is nonetheless essentially different from the rest of mathematics. This perspective will help us understand the relation between apriority and analyticity in general.

It would seem clear, independently of the interpretations mentioned above, that there is an imaginary or theoretical space, very similar to actual space, in which the sum of the angles of a triangle is undoubtedly 180. Even if our own actual space is curved to some degree, and the propositions of Euclidean geometry are therefore not valid of it, with respect to that imaginary space the propositions of geometry are not subject to refutation. It seems that even if this theoretical-imaginary space does not exist in actuality, it is still a real entity. One can perceive it in imagination, or with the mind’s eye.¹³² In this sense geometry is the result of “observation,” and not only of abstract thought. It can be presented in the form of an abstract mathematical theory, but it is clear that it emerges from a relation to abstract space, which is some kind of being, and in fact geometry describes that being.

Therefore, with respect to geometry, it is hard to free oneself from the understanding that its propositions concern, at least in some sense, reality—that is, that they are synthetic. Perhaps not reality in our actual world, but certainly in a theoretical world that resembles, in some way, the Platonic world of Ideas. Beyond that, these propositions even seem to be a posteriori. We do not create this imaginary world ex nihilo; rather, it is present in our thought, in the thought of every person, from birth. Thus we are, as it were, observing this space as a real entity, and its description is the result of a quasi-epistemological activity.

The same may be said of logic, which we discussed above. We saw that we tend to infer from the existence of a logical contradiction in a concept, or in an event, that it is impossible or nonexistent. We saw that there is here an apparently unjustified leap from logic, which occurs in our minds, to conclusions about the objective world. Even if we deny the possibility of doing this—just as the positive ontological argument is usually denied, one may deny a negative ontological argument for the same reason—it would seem that the claim concerning the nonexistence of contradictory concepts or events is true in some theoretical or imaginary world. Here too one senses the presence of a claim about some sort of theoretical reality. Thus the propositions of logic too are not merely abstract propositions. The claim that actual reality is a model of the theoretical-imaginary world of logic is certainly a synthetic claim, and perhaps one can even imagine empirical ways of testing it.

Thus, these entities—or worlds—are not merely conceived by us in an abstract fashion, but this act of thought also has certain visual dimensions; they are apprehended by means of something like observation. Therefore, they seem to exist in some kind of visual sense.

On the other hand, plane geometry, as well as logic—and unlike the various branches of physics—will certainly never be refuted, at least with respect to the imaginary space mentioned above. Thus there is in them something of the analyticity of mathematics. We saw the same with respect to adding oranges to a basket. At first glance this too can be tested experimentally, but it is perfectly clear that it will never be refuted. This is true at least with respect to the theoretical act of addition that we imagine, and very likely with respect to the actual world as well. We therefore have here a proposition that is certain, and yet can be tested experimentally, which is to say that it says something about reality.

If so, geometry is a synthetic domain—it makes claims about a world, or more precisely about a world, without the definite article—and yet at the same time it is a priori, in the sense that it is not empirically falsifiable.¹³³ It is a kind of synthetic science, but one open only to proof, and in no way to refutation. There is here a kind of syntheticity, and perhaps even a kind of empiricity. But because the objects of thought and knowledge are located in the Platonic world of Ideas, there is no possibility of refutation.

One can say even more. For every given space there is only one correct geometry. The various geometries relate to different realities, that is, to abstract spaces with different properties. Actual space is a model of one among those theoretical spaces. To the question which of them actual space is a model of, there is only one correct answer. For further elaboration, see the first book, Chapter Three of the eighth gate.

The Relation to Analyticity and Syntheticity in General

What follows from our discussion, then, is that the dispute between Platonism and analyticity in mathematics is not merely semantic. It is true that, according to all views, the propositions of mathematics are not claims about our world, and therefore are not empirically falsifiable in the ordinary sense. Yet there remains a real dispute as to whether mathematical entities, and the relations among them, have existence in some Platonic world of Ideas. That is, the question is whether what we have here is merely a description of what takes place within the human spirit, or whether mathematical claims too are claims derived from “quasi-observations.”

We saw in the first book that the logical dispute between the analytic position and the synthetic position finds expression also in the context of epistemology. The analyst treats concepts as fictions grounded in agreement among a community of speakers. According to his view, concepts have only use and not meaning, and certainly not reference—that is, a correlate in the real world. The concept as such exists nowhere, whether in the actual world or in a Platonic one.

By contrast, we saw that within the synthetic approach one must assume the existence of concepts as such. There we proposed the distinction between phenomena and noumena also with respect to concepts. According to the synthetic position, the concept is not merely the collection of all its properties; it is the entity that bears those properties.

As we saw there, the root of the dispute lies in the difficulty of becoming aware of the existence of concepts. A concept cannot, it would seem, be observed by the senses, and therefore we have no indication that it exists. The synthetic position claims that we do in fact have such a mode of “observation.” It is not carried out by the senses, but by the mind’s eye. This is the essence of auditory reasoning, or eidetic vision. In previous chapters we saw this in the scientific context, and now we are repeating the distinction in the mathematical context: according to the synthetic position there is a way to apprehend, in a non-sensory manner, “existing” phenomena. Mathematical concepts, like all other concepts, are entities. They do not exist in the actual world, but they do genuinely exist in a world of Ideas. This is the essence of Platonism, and here we see how it follows from the synthetic position.

The “observation” we described above of Euclidean mathematical space—which is not a space in the actual world—is a process that occurs both in life, with respect to ideological concepts, values, and other concepts, as discussed in the first book, and in science, and now we encounter it in mathematics as well. Mathematics provides us with a context in which the meaning of claims about the existence of concepts, or Ideas, in some world of Ideas—that imaginary space we defined above—can be perceived with particular sharpness. Here too one can grasp the meaning of eidetic vision, or intellectual listening, which we encountered above in broader contexts.

We see, then, that those who hold the synthetic position do not necessarily agree to the analytic interpretation of mathematics, as might appear at first glance. Even in the context of mathematics there is a synthetic position, and it can find expression in Platonism. According to this interpretation, the principles of mathematics concern some world, but it is not our actual world; it is an ideal world.

Mathematics and Empirical Science

In this section we conclude the pendulum movement that has characterized this gate between the analytic and the synthetic, and we return to empirical science.

If we return to the description of empirical science that we saw above, we arrive at a somewhat surprising conclusion: mathematics is not so different from empirical science. The theoretical entities of empirical science are likewise not present in actuality in the world, in a sense very similar to the entities of mathematics. Both fields deal with observation of a non-sensory kind, although only science also makes use of sensory observation and is therefore usually classified as empirical, whereas mathematics is not.

Science does indeed deal with the behavior of real and concrete objects, but once it passes beyond the phenomenological stage, it describes those objects and their behavior in terms of theoretical entities. These do not necessarily exist in the actual world, and at the very least it seems clear that they cannot be observed by the senses. A concept like force field, which we encountered above, may perhaps exist in the actual world—or perhaps not—but it is clear that it cannot be observed. By definition, the moment one observes it—by placing a test charge on which the field exerts a physical force—it manifests itself as an actual force and ceases to be a theoretical force field. The quantum wave function is a concept even more remote from observation; according to quantum theory it cannot be observed at all, and it is also remote from simple existence in the world.

Our conclusion, then, is that according to the synthetic position presented here, the claims of empirical science are synthetic-a priori, whereas the propositions of mathematics are “quasi-synthetic-a priori.” The former make claims about the world, while the latter make claims about a “quasi-world,” namely a Platonic world of Ideas. The former are inferred by means of sensory observation of concrete objects, together with generalization and observation of ideal theoretical entities; the latter are inferred by means of “quasi-observation” alone.

The difference that follows from this is that scientific claims are open to refutation, whereas mathematical propositions are not.

It is important to note that “quasi-observation” of the imaginary mathematical space can certainly yield erroneous results. In that sense, the claims of mathematics are open to correction, just as ordinary observation is. The role of mathematical proof is to verify the results of “quasi-observation.” “Quasi-observation,” just like observation in the actual world, is not considered a sufficient mathematical source.

Thus, although mathematical discoveries are grounded in “quasi-observations,” the field itself is clearly based on analytic proofs. The fact that “quasi-observations” stand in the background is highly important. Its importance lies in the fact that the results of mathematical doctrine, as well as its assumptions—the axioms—are not arbitrary, as many suppose, but rest on a reliable, though not absolute, external foundation, one that is “quasi-objective.”

We saw that the claims of mathematical doctrine concern the abstract “quasi-world.” According to the account proposed here, geometry asserts the axiom itself—“Through two points there passes one straight line”—and not only the relation of implication of the theorems derived from it, that is, not only the claim that if we accept this axiom it follows that the sum of the angles in a triangle is 180. “Quasi-observation” belongs to the mathematical context of discovery. Mathematical proof parallels the context of justification, which is of course deductive in nature, and not empirical confirmation or refutation as in the natural sciences.

It may be for this very reason that the axiom does not possess the same full level of certainty that we have regarding the implication of the theorems derived from it. Yet it is clear that it is not arbitrary either. It is very hard to say that the proposition “In Euclidean space only one straight line passes between two points” is arbitrary, even though it has no proof—it is the result of “quasi-observation.” Some attribute to it full certainty, just like the certainty of the implications of the theorems derived from it.

Chapter Six: The Social Sciences and the Humanities, or, Is Empiricism an Exclusive Criterion?

Introduction

In the previous chapters we saw general characteristics of fields associated with empirical science, and the discussion focused on the natural sciences. We then examined the scientific and empirical status of mathematics. In this chapter we shall try to discuss briefly the status of the social sciences and the humanities.

The social sciences include fields dealing with the human individual and with human society, such as psychology in its various branches, sociology in its various branches, anthropology, economics, administration, international relations, and the like. The humanities include fields concerned with human creativity, such as literature, art, philosophy, Judaism, and the like.

The classification of philosophy within the humanities reflects certain assumptions that have changed over the course of history. At one time philosophy was classified as part of the natural sciences—indeed, it was itself the natural sciences—because it was then believed to be a tool for knowing the world. Today there is a tendency to regard it as yet another kind of human creation, because confidence in philosophy’s ability to describe the world as it is in itself has greatly diminished. See the first book for an extensive discussion of this.

Most of the disciplines included within the social sciences are clearly empirical. Theories are formulated in them and then tested by experience. In the humanities the problem is more difficult, and therefore their classification as science is far more problematic. But even the social sciences do not enjoy an unequivocal scientific status. In this chapter we shall try to examine this issue briefly.

Initial Intuitions

There is a sense that the social sciences and the humanities are not “scientific” in the same sense as the natural sciences. Confidence in the results of research in these areas, and in the precision of their methods, is significantly lower than confidence in the results of research in physics and chemistry, for example.

On the other hand, it is very difficult to define precisely why these fields do not possess a scientific status similar to that of the natural sciences. On the face of it, they too—at least most of them—are built on the formulation of theories, followed by the empirical testing of those theories by experiments, whether in the laboratory or in the field.

Yet if we continue along this line of argument, we discover a not very surprising fact: many of the fields in which we operate meet these criteria. A cobbler, who specializes in repairing shoes, also formulates rules for himself—sometimes tacitly—tests them by experiment, for instance whether repairs made according to these rules withstand actual wear, and draws conclusions when necessary. The same is true of building cleaners, carpenters, firefighters, and so forth.

It would seem that what causes us to define the natural sciences as scientific, while not treating the fields just mentioned in the same way, is the degree of sharpness, complexity, and sophistication of the principles and methods employed in the field in question. That is, what distinguishes clearly scientific disciplines from ordinary human thinking is not necessarily everything described in the previous chapters. The distinction depends mainly on complexity, rigor, and sophistication.

The conclusion, which may surprise some readers, is that science is nothing more than a careful and rigorous use of the ordinary and widespread principles of human thought, coupled with an attempt to avoid unsupported speculation. Fields that satisfy these criteria, and whose complexity is high, are what we call “scientific.”

In light of the above, another matter becomes clear: scientific determinations do not possess any greater validity than determinations in other human domains. A cobbler may make precise determinations in his field, and the degree of reliability of his determinations may be similar to that of scientific determinations, if not higher. The complexity of a field is certainly not a criterion for the reliability of the principles governing it. It is true that rigor of thought characterizes scientific fields more strongly—probably because of their complexity—and not every cobbler is endowed with such rigor. Cobblers also do not generally insist on feedback, criticism, and exact examination of the principles they discover by parallel professionals. Yet the distinction between the domains does not seem sharp.

Beyond that, as we have seen throughout the present gate, science does not define its determinations as true, but only as open to refutation—and at most confirmation. In light of this distinction, it is difficult to understand the common attitude that identifies “scientific statement” with “true statement.” See also our remarks above at the end of Chapter Three.¹³⁴

Quantification and Falsifiability

We have already noted that the social sciences and the humanities do not enjoy the same prestige as the natural sciences. This fact gives rise to apologetics. Quite a few courses in fields associated with the social sciences open by trying to define what science is. The lecturers then point out that the basic definition is openness to empirical testing, that is, to refutation, and conclude from this—unsurprisingly—that their own disciplines also belong to the scientific enterprise, since theories in the social sciences too are, at least in principle, subject to empirical examination. But as we have already seen, by that criterion cobbling too belongs, to a considerable degree, to the same discipline. It is interesting to note that courses in the natural sciences hardly ever deal at all with the question of the scientificity of the subject being studied. There is enough self-confidence there that no such apologetics are required.

Another common phenomenon, deriving from the same apologetic root, is the attempt of the social sciences and the humanities to become quantitative, or falsifiable, like the natural sciences and mathematics. This takes several forms, sometimes banalizing the results and sometimes even distorting them.

The banalization of the results means that “deep” and “scientific” research, using various statistical tools, arrives at the astonishing conclusion that among seventy percent of orphaned women who have undergone a third childbirth and suffered from a domineering mother, feelings of anxiety develop during a terrorist attack.¹³⁵ A considerable part of the results of “scientific” research in these fields can be predicted in advance in a trivial way, without any research at all.

Let us add that many times a result that does not belong to this category—that is, one that is not banal and predictable in advance—is simply false. The scientific-quantitative tools often do not help at all in these fields.

Here we come to the second part of the argument. As noted above, quantitative methods often lead to distortions and deceptions. People are easily misled by a quantitative argument—if there is mathematics, then it is “science” beyond dispute—and fail to notice the additional assumptions embedded at the base of that “scientific” result. One may find abundant examples of this in treatments of game theory, which confuse the pure mathematical field with the empirical claims derived from it—whether justifiably or not—with respect to the social sciences. See on this the relevant section in the previous chapter, and Poundstone’s book cited there. It is easy for social scientists to hide behind mathematics, a field free of such biases. But as we already noted there, this cannot really be done.

Another illuminating example, one of thousands, of such distortion of results appears in Samuel Huntington’s book The Clash of Civilizations.¹³⁶ Huntington is a world-renowned scholar from Harvard University in the field of political science. The book deals with the global historical processes of the late twentieth century and the beginning of the twenty-first.

Huntington argues against the widespread belief, in the academic world and beyond, that at the beginning of the twenty-first century the world is a “global village,” in Marshall McLuhan’s well-known phrase. He claims that after the Cold War the world was divided into eight different civilizations, and adds that only a consideration of all of them can yield a full understanding of what is taking place in the world during this period.

In the course of his discussion he addresses the counterargument that Western civilization has taken over the world, and that it influences all the others because of its political, economic, and cultural advantages.¹³⁷ Against this claim, Huntington argues that the West has been in continual decline since the beginning of the twentieth century. He supports his argument with quantitative data regarding most of the relevant variables: the territorial variable—where in 1920 the West controlled 48.5 percent of the world’s territory and in 1993 only 24.2 percent; the demographic variable—where in 1920 the West ruled over 48.1 percent of the world’s population and in 1995 only 13.1 percent; the economic variable—where in 1950 the West’s share of gross world product was 64.1 percent, whereas in 1992 it stood at 48.9 percent; and even the military variable—where in 1920 the West’s share of global military manpower was 48.5 percent, and in 1991 it stood at 21.1 percent; and so on.

At first glance, we have here a rather persuasive “scientific,” quantitative argument. Yet, as noted, we must examine the argument at the principled level as well, and not be carried away by the mathematics.

The question is whether the concept West, in the context of civilization rather than geography, really remained constant throughout the entire period under discussion. Some would argue against Huntington—and I count myself among them—that what was once called non-Western civilization has become increasingly Western in character. This, precisely, is the heart of the phenomenon of the global village. Who today does not enjoy American-European cultural products—beginning with films, continuing with music and technology, and extending to the values powerfully transmitted through all these media? Does a non-Western society today look as it did at the beginning of the twentieth century?

If so, Huntington assumes that the conceptual division between West and non-West has remained intact, and therefore he uses quantitative indices to confirm his claims. But he apparently fails to notice that the distance between the poles Western and non-Western has greatly shrunk today, and of course not toward the middle but toward the Western pole. One can continue to draw a line between very Western and less Western, but that line is arbitrary. Hence Huntington’s results seem to have almost no objective quantitative meaning. The quantitative illusion dissolves very quickly when one examines the conceptual assumptions embedded beneath the formal argument.¹³⁸

Another example of the same thing—not unrelated to the previous one—is the political analysis of the State of Israel. Here too many would say that the division between right and left—on diplomatic matters, not socioeconomic ones—is stable, and perhaps some would say that in certain periods there has been domination by the right. Both sides will support their claims with numerical data, such as election results, polls, and the like. But here too one must notice that the concepts right and left themselves have shifted over the years.

Here this is very easy to see if one examines political positions in terms of objective variables: what does a given person think about the transfer of territory to the Palestinians, or about civil and human rights for minorities, and so forth? And from another angle, within the Jewish sphere: what is the attitude toward religion and religiosity? If we examine these parameters, we will discover that what was once called the left—a willingness to speak with the Palestinians and neighboring Arab states, opposition to closing businesses and movie theaters on Shabbat, not to mention equal rights and opportunities for women—is today to the right of center. The positions of the group that was once called the right no longer exist at all, apart from negligible fringe positions in the religious public. What is called today “the center,” or “the sane center,” is fairly radical left by the standards of the past.

Here too the entire map has drifted leftward, and one can ignore this and present objective quantitative data to the effect that the eternal tie between diplomatic right and left in Israel continues forever.¹³⁹ Here too we draw an arbitrary, relative line between the groups on the current map, and decide that whoever lies to the right of the line is right, and whoever lies to the left of the line is left. In such a case, the tie is a trivial mathematical result that expresses nothing more than our way of measuring.

Another kind of distortion related to the previous one is the defining of concepts and the substitution of the concept by its definition. This phenomenon exists in all the sciences, but it characterizes the social sciences and the humanities more than anything else. We shall discuss it in Chapter Four of the fourth gate. There we shall see that this very process, which as noted mainly characterizes the social sciences and the humanities, lies circularly at the basis of their own definition of themselves as “sciences.”

In the following note we shall encounter a striking example of the results of the attempt to quantify the social sciences and the humanities, and to make them falsifiable.

Note 14: The Criterion of Assessability

A prominent expression of the effort of the social sciences and the humanities to meet Popperian scientific criteria of falsifiability is the criterion of assessability for academic articles. In this note I shall discuss this criterion a little, and the reader will forgive the personal tone that may occasionally emerge in the discussion, for reasons that will soon become clear.

Editorial boards of scientific journals insist that articles be assessable. In the natural sciences this issue usually does not arise, because the articles are such by their very nature. But in the social sciences and the humanities the editorial boards insist on this openly and explicitly. For example, once the editor of an academic journal in Jewish studies approached me and suggested that I write an article for the journal. Since that editor had read my first book, he added that what he had in mind was not something like that book, but rather an assessable article, in his terms.

What does the term assessable mean? An academic article cannot make claims about values—ethical or aesthetic—or express the author’s own positions on various issues, because such claims are “not assessable.” In an analytic world, everyone may have his own position on a given issue, but such a position is not a matter for objective scientific determination. It is not assessable, because everyone has his own stance, and in an analytic world there is no objective truth with respect to such stances. Therefore, an academic article almost always deals with the claims—whether value-laden or otherwise—of others, and with their interpretation. Such an article claims that the views of this or that thinker mean thus and so, or that their intention is such and such, or that they contain contradictions, or that there is some relation of dependence between one claim and another. On the one hand, such an article is required to make claims; on the other hand, it may not express the author’s own positions.¹⁴⁰

In this way, the academic system empties itself of content and occupies itself only with structures and empty formalism. These are the only kinds of statements that can be assessable. If someone claims that Kant held a certain position on some issue, he must bring evidence, and that evidence can be examined “objectively.” Such an article is assessable, because in principle one can examine the evidence in Kant’s writings and decide whether his position is indeed as the article claims. But Kant’s own claims are, of course, not assessable, because they are positions about which one can certainly argue, and as the saying goes, there is no disputing taste. In fact, Kant himself, as well as almost all the other classical philosophers, could not publish anything today in philosophical journals. His writings are saturated with his own positions, and therefore they are not assessable.

The matter becomes absurd in contexts where the discussions are “charged” at the value-ideological level, and the authors present positions that are highly relevant and meaningful to society as a whole, but carefully do so by way of interpreting the positions of others. For example, a scholar will not argue in a scientific article that Judaism ought to be pluralistic, but will instead try to prove that Maimonides’ view—or that of some other thinker—was that Judaism is such.

In connection with pluralism, a more sophisticated technique has developed in recent years for expressing positions under the guise of academic research. One can take some Torah or halakhic issue—no matter which—and survey the range of opinions stated about it over the generations, and stop there. If the scholar is careful to ensure that every option has some source—and with a bit of interpretive creativity there is no real difficulty in doing this, by softening the boundaries that determine which texts express canonical Torah positions—then the pluralistic message has been conveyed without ever having been explicitly mentioned, and certainly without any stance toward it having been stated. Yet in fact a very definite stance has been expressed, only without violating the rules of the ritual of “assessability.”¹⁴¹

In such contexts in the social sciences and the humanities, it is clear to everyone that these are the author’s own value judgments and ideological positions, but the rules require that he not state them, and instead prove that they were the positions of others.

This description pertains mainly to the humanities. In the social sciences a similar process takes place. There too there is an attempt to focus on the objective and quantitative, on what can be measured. Only this counts as “scientific.” This process too dries up those disciplines, because, as we have already noted, unlike the natural sciences it is not correct in these contexts to focus only on the quantitative.

For that reason, various forms of “circumventing” these draconian rules have begun to appear, in the form of “qualitative” or “individualized” studies. This terminology refers to research whose goal is to examine the individual case and, in effect, to cast off the quantitative-objective criteria.¹⁴² Such research can therefore be conducted only by someone with a sufficiently strong standing in academic politics—usually someone who has already proved himself in the quantitative field—or, of course, in “gender studies,” where “anything goes.”

Such restrictions greatly thin out academic writing and flatten it. They do not permit the presentation of any innovative, significant, and far-reaching idea that pertains to content. Academia in the social sciences and the humanities deals mainly with interpretation and with describing processes from the outside, and therefore it cannot really understand them as they are.

As noted, there is much room to criticize this state of affairs, but it can certainly also be defended. It is clear that in an academic world that wants to be scientific, one must preserve some degree of grounding, argumentation, filtering, and judgment. Academic forums are not intended for journalism, and they also wish to publish material that passes an objective filter, with some standards of quality—of course, in the field of gender there is no such aim. On the other hand, an absurdity is created here, because these forums are sustained by research into the writings of people who themselves could not have published their own work in them, and by interpretations of those writings.

If such restrictions were to apply to all writing, and not only to articles intended for publication in academic forums, then no philosophical systems would ever come into being, and academia would have nothing to discuss.

It may be said, however, that these forums are intended for scholars of philosophy and not for philosophers, or for scholars of literature and not for writers. And indeed these are distinct domains. As in other areas, so too in philosophy and in Jewish studies, science is parasitic on genuine creation. Incidentally, even in the natural sciences, science is a description and analysis of nature itself. There is no creation of nature here, only its description and analysis. The determinations of the natural sciences are assessable by their very nature, because they must satisfy the criterion of standing up to empirical experience. Their fit to experimental facts is the basis of their assessability. In the humanities and social sciences, the “facts” are somewhat more fluid—what counts as a better economy, or a better story or poem, or a truer value stance, and the like.¹⁴³ Hence the somewhat artificial situation described above.

The description up to this point indicates that the demand for assessability has some justification. It is very difficult to find standards of quality for the expression of positions. Yet one of the foundations of a synthetic position is that positions too, like axioms, are open to dialogue, examination, and decision. Not always, but it is possible. This can be done if we relinquish the logical criterion; that is, if we do not insist on deciding matters by way of an analytic knockout, by means of a proof that refutes the position under discussion at the logical level.

But if we want to make possible the creation of high-quality academic research that deals with positions and not only with claims about positions—an alternative humanities—then it is necessary to formulate standards of synthetic quality: standards for evaluating value-laden and ideological positions, or at least for evaluating, not merely in a literary sense, the ways in which they are presented and argued for.

There are synthetic criteria for judgment—by such criteria we judge literary works, poetry, Torah essays, and more—and the creation of an alternative humanities requires applying them to additional domains.

To sharpen the point, let us focus on literature. Such a proposal in the literary context seems very problematic, because it is clear that there is no place for publishing a story or a poem in an academic journal of literature and poetry. There is an obvious difference between analysis and interpretation on the one hand, and creation on the other. By contrast, in philosophy, for example, the situation appears different, because professors of philosophy are generally assumed to be “philosophers” as well. As everyone in philosophy knows, this is usually not the case, but such an illusion does exist—apparently by design.

The same is true of the field called “Jewish studies.” Here too there are “scientific”—more accurately, academic—forums, and here too they deal with interpretation rather than creation. On the other hand, here too, as in philosophy, large parts of society, and many of the scholars themselves, do not understand themselves in this way. Many feel that the articles published in Jewish studies are the contemporary form assumed by engagement with Judaism itself. And indeed the boundary is blurred, since Torah study has always had a dominant interpretive component. But if that is indeed the status they wish to grant themselves, they cannot continue to adhere to “scientific” criteria that do not allow substantive claims to be made. See below on this matter in more detail in Chapter Three of the fifth gate.

Returning to our subject: the criterion of assessability is one of the most prominent expressions of the attempt of the social sciences and the humanities to adapt themselves to the scientific criteria of falsifiability. As we shall see below, they do not really succeed in doing so, because scientificity is not exhausted by the question of falsifiability. In any case, if these fields were indeed understood as concentrating on external description and limiting themselves by strict methodological rules, there would be no real problem here, and there is certainly room for that as well.

The conclusion from our discussion is that it is important to pay attention to which claims are objective and quantitative and which are qualitative. But it is equally important to understand that these criteria are not identical with the distinction between correct claims and arbitrary claims. The problem arises when these fields substitute for the objects of their own study. When Jewish studies becomes Judaism itself—the title of a conference at Bar-Ilan University in 2002 was “There Is No Judaism without Jewish Studies”—and when the “science of philosophy” becomes philosophy itself. As we saw in the first book, this is a marked characteristic of an analytic world, in which discussion about contents replaces the contents themselves.¹⁴⁴

The synthetic alternative has significance in the domain of the social sciences and the humanities as well. One can argue and express positions, and one can also judge claims and positions. The “criteria” may not be so sharp, but that does not mean there is no possibility of such synthetic judgment. This is a necessary step for the sake of the social sciences and the humanities themselves. They must be rescued from the analytic emptying they impose upon themselves.

The question is why, in fact, it is so difficult to quantify research in the social sciences and the humanities, and why the results of research in these fields do not enjoy full scientific status. In what do these fields differ from the natural sciences?

The Root of the Problem

Some would say that these fields are simply too complex, and therefore it is impossible to isolate individual influences and understand them thoroughly, or to put theories to a sufficiently “clean” empirical test, as is done in the natural sciences. According to this approach, the psychology of a human being is very complex and influenced by many parameters, and therefore there is no way to test and describe it precisely.

The feeling is that this is not the whole picture. We do not succeed in describing even a single psychological phenomenon by means of precise mathematical tools. If the problem were merely the complexity of these fields, then we ought to be progressing more slowly but still in the same direction—toward mathematization and complete precision in these fields. If so, the problem is not an overload of parameters but the essentially different character of those parameters. At times it seems that they simply do not lend themselves to mathematization.¹⁴⁵

It seems that what is common to all the social sciences and the humanities is that they deal, in one way or another, with the human being and with human creation. Psychology and sociology deal with human phenomena. They examine characteristics of individuals and of human societies. Literature and art, by contrast, deal with the characterization of human creations, which are likewise the result of human behavior and of the human spirit.

It seems that the fundamental “problem” of the social sciences and the humanities is that the direct or indirect object of their study is the human being. Of course, nothing follows from this with respect to medicine, since medicine deals with the human body, which is a material entity, and not with the human being as such.

This matter is intimately bound up with the essence of the human being, and with the relation between spirit and matter in general, and this will be the subject of the third volume of the trilogy. In the present chapter I wish to deal only with a few points directly relevant to the current discussion.

Reductionism¹⁴⁶

At first glance, one can classify the different branches of science along a single axis, from the most basic to the most complex. Physics, which is the most basic, deals with the study of inanimate nature. Chemistry is a phenomenological description of complex physical phenomena—systems with many particles or components—and therefore it too is a branch of physics.

Biology is the study of living things, which are more complex. The basic unit of biological science is the living cell, and therefore biology operates at a much higher level of integration than physics and chemistry. A very complex physicochemical system is the basic building block of this science. Biology deals with the relations among different physicochemical systems, and with the whole that they compose.¹⁴⁷ Some would say that in biology an additional component enters the picture: vital substance, that is, a life-material. In other words, biology is not merely a complex physicochemical system; in that complex system there is also another component, one fundamentally different in kind. Some call it “soul,” or “life-force.”¹⁴⁸ The next level in the hierarchy is physiology, which deals with the interaction among biological systems and with the whole formed by their combination.

After physiology there is some sort of leap to psychology. Some believe that psychology too is nothing more than a property of complex physiological systems, and not the result of a distinct component, namely a spiritual one. Others would say that at this point another fundamentally distinct component is added, sometimes called “spirit.” See on this the third book.

If we ignore the next level, which includes intellect, emotion, and will, and together constitutes what is called the “soul”—a topic that will be discussed in the third part of the trilogy—we must move on to the highest level of integration, namely society. This is the object of study of most of the social sciences, apart from most branches of psychology.

Some would add here the study of divinity, or the study of the spiritual. These are domains of philosophy and religion, which today tend to be regarded as non-scientific. I shall therefore not address their status on the axis under discussion here.

In recent years various fields of research have been added that fill in, in greater detail, the intermediate domains between the sciences just described—for example molecular biology, which lies between chemistry and biology, and so forth. It should be noted that despite this increasing “density,” each such science uses its own terminology and discovers its own rules, and therefore the question whether reduction between them is possible remains open.

Reductionism is the approach that believes that all these levels of integration—or at least some of them—can be reduced to one another. That is, a higher field can be stated as a complicated description of the lower, more basic field. According to this approach, physiology is nothing but an upper-level language, or a set of shorthand rules, for complex physicochemical phenomena. If we wished to describe physiology, and even biology, in terms of the laws of physics, this would be possible. To do so would require an enormous computer, tremendous computational ability, and very powerful techniques of calculation. But in principle, if we could describe the state of every particle in a living body, we would know with complete precision what the body as a whole will do.

According to extreme reductionists, the entire human world, including the social sciences at the highest levels of integration—such as international relations—is nothing but highly complex and highly intricate physics. If we knew how to describe what every electron and every atom in every human body in the world would do, and likewise in the inanimate environment, which also affects human behavior, we would know with complete precision everything that would happen at the level of international relations: who would initiate each war, who would win it, which prime minister would be elected in each country, what its economic condition would be, and so on.¹⁴⁹

It is clear that total reductionism is deterministic. If everything is indeed physics and chemistry, then everything is governed by the laws of physics, and there is no room for uncertainty—apart from quantum uncertainty, which is not relevant here.

The opposing approach holds that, at least in some cases, there is something in the higher domain beyond the components described in the lower domain. In the description above I already noted that some people—today they are very few—maintain that between physics and chemistry on the one hand and biology on the other, a vital substance is added to the picture, a “life-material,” or soul. If so, the biological description is not merely complex physics; one must also add the properties of a component different in kind from inanimate matter.

So too in the transition from physiology to psychology, where others say that the human spirit—perhaps also the animal spirit—is added. Above that, the human soul is added, bound up with cognitive activity, intellect, and will. As we shall see in the third book, this is an addition that carries us from psychology to volition. Precisely because of it, we are able to engage in philosophy and values, to create them, and to live by them.

According to Kabbalah, the nefesh, ruach, and neshamah together compose the nonmaterial part of the human being. If so, in order to arrive at a full formulation of a science of the human being, we must take them into account in addition to the material parts.

To the best of my understanding, the reason the social sciences do not succeed in attaining a high degree of reliability is that they have failed to develop tools capable of dealing with the description of these nonmaterial components.

Later in the present book we shall discuss parallel planes of description and explanation, and from a different angle we shall confront reductionism in its broader philosophical form.

Classification and Categorization

A prominent feature of the social sciences and the humanities is that they—though not they alone—deal extensively with classification and categorization, and less with mechanistic explanations.

This feature raises the problem of the syntheticity of these sciences. Do the results of research in fact say something about the world, or are they merely our own forms of observation? This discussion is closely tied to the question of the ontological status of species and genera.¹⁵⁰

In Note 13 above we discussed the existence of sets. Also in the first book—see Note 15 there and the surrounding discussion—we discussed the ontological status of universals composed of particulars. We asked there whether a society or a nation is something that actually exists, or whether these are only our subjective and fictive definitions that allow us to refer to a collection of individual particulars. We saw that those who hold a synthetic position tend to understand a society as an entity that exists in itself, whereas those who hold the analytic position explain that it is merely a fictive definition that allows us to refer efficiently to collections of individuals.

If sets really are not entities that exist in the world, as Bertrand Russell argued—see Note 13 above—then it would seem clear that species and genera too are only forms of ordering of our own, which we impose on objective reality. If so, classification and ordering are principles of the human intellect, and not statements about the world as such.

But if that is indeed the case, then in the same way one could say that the behavior of complex inanimate bodies is also merely a statement about us. Why should we not say that in the objective world only elementary particles exist, and that the coherent groupings of them, which we call “macroscopic objects,” do not really exist? It would then follow that the behavior of macroscopic objects is likewise not a statement about the world. This is complete mystical absurdity. Incidentally, human consciousness itself—or at least the bodily-material tools it uses, such as the brain, which organize its picture of the world in that manner—is itself one of those macroscopic creatures.

But there is a mistake in the argument just presented, even if we adopt an analytic position. Even if species and genera are our categories, and those categories do not represent entities existing in reality, statements that employ them can still be statements about reality.

For example, if we say that a charged sphere of a certain size, with a certain distribution of electric charge, upon which a given magnetic field acts, will move in one way or another, then we have said something about reality. This is a statement about reality even if there is no sphere, but only a collection of atoms composing it, and no charge distribution, but only a collection of units of charge, and so forth. The general terms are a language for describing reality as it is. They are a way of expressing statements that describe the motion of the particles composing the macroscopic aggregate. The fact that the language is human, and has no correlates in the world, does not mean that the reality it describes is not real.¹⁵¹

Note 15: Halakhic and Other Expressions of the Ontology of Species and Genera

In Note 13 we discussed the ontological status of sets, or of universals as against particulars. In this section we are discussing species and genera, and here an interesting point arises that is related also to the previous question: must a decision regarding the ontological status of sets be sweeping? Put differently, when someone decides that sets may indeed be entities that exist in the world, must he adopt this approach across the board with respect to all sets, or is a view also possible according to which some sets are existing entities and others are not?

It is interesting to note the significance of this question with respect to Russell’s argument, presented in that note. If we do indeed accept that there are sets that are existing entities and other sets that are not entities, then Russell’s argument collapses on its own. For example, if we assume that sets are entities, but the subsets of the set of all sets are not entities, then the paradox does not arise at all. The same would hold for other possible divisions of all sets.¹⁵² Russell assumes that the decision regarding the ontological status of sets must be sweeping, but this does not seem necessary.¹⁵³

This note is somewhat speculative in character, and the reader is invited to judge the matter for himself. We shall discuss a hierarchy of universals and sets, and try to show hints from various sources concerning the ontological status of the different links in this hierarchy and the relations among them.

Let us begin the discussion from a linguistic angle. We saw in note 13 that any set can be divided into subsets, in the accepted mathematical terminology. It is clear that each subset can in turn be divided into its own subsets, and so on ad infinitum. In certain contexts in logic, we say that a particular individual, usually together with several other individuals, is included in a species, and the species, usually together with several other species, forms a genus, which itself is part of a division of some larger set into its genera. From a top-down perspective, we would say that a given set is divided into genera, each genus is divided into species, and each species is divided into individuals.¹⁵⁴

For example, one common division of all beings in the world classifies them into the following genera: inanimate, vegetative, animate, and speaking beings. Each such genus can be divided into species. Thus, for example, the genus of animate beings includes reptiles, mammals, birds, fish, and so forth. Each species, after being divided into further subspecies, is divided into specific individuals. Of course, the division can be extremely detailed, and here we are merely illustrating certain portions of it schematically.

We must now note an interesting linguistic phenomenon. In Hebrew, and likewise in other languages, two special terms were coined for such divisions: species and genera. There is no special terminology for higher or lower divisions in the logical hierarchy. It is self-evident that any division into species and genera captures only a certain segment of the chain, and therefore has no real uniqueness to it. There is no reason to assume that every division must consist of three stages: a general reference set > genera > species > individuals.¹⁵⁵ When one focuses on a certain segment of the chain, one can define a particular link as the general reference set and construct from it a more detailed division, in which the two higher levels will be genera and species. As stated, the language has no special names for still more detailed divisions.

One might have thought that language could not possibly coin a different term for every such stage, since in principle there are infinitely many possible levels of division, and it is impossible to give each a special name. But if that is so, then one could have sufficed with the single term genus and its repeated application. The term species is, on the face of it, superfluous, since it is nothing more than genus, if the general reference set is itself a genus within a more inclusive set. Dividing the genus into subgenera then yields the species.

One might perhaps say that language wishes to express the more complex relation of subdivision, and therefore could not make do with a single term for division. But here too one may ask: why not express a more detailed division as well, of three stages and beyond? What is so special specifically about two stages of division?

The hypothesis I wish to point to is that this terminology reflects an intuitive perception, one that points to something special and more essential in the division into genera and species, as opposed to more detailed stages of division, which do not reflect anything essential. Since division can, in principle, be infinite, that is, consist of an infinite number of stages, I do not mean to say that genera and species really exist while other levels of division do not. What I mean here concerns only the distinctions in the relations between the various levels. The claim is that although there is indeed a logical relation between any two levels of division, regardless of the distance between them, there is a real relation, or interaction, between the upper level and its sublevels only down to the third level of division: to genera, species, and the individuals included within the species. It should be noted that this rule, if it is correct, does not reflect a logical relation, since such a relation exists equally at every level of specification. What is meant here is a relation that is more like a law of nature, a synthetic proposition, or a fact that describes reality.

Processes of specification appear in many contexts. It is interesting to note that many of them display a similar structure. For example, in Hebrew there is no name for a relation higher than grandfather. The only names in the family hierarchy are father and grandfather. In another context, in mechanics, the only names in the hierarchy of motion, that is, position and its mathematical derivatives, are velocity, the rate of change of position, and acceleration, the rate of change of velocity. There are no names for higher derivatives, such as the rate of change of acceleration, and so on.

In Kabbalah, all reality is divided into sefirot, that is, emanative spheres, such as Hokhmah, Binah, Da’at, and so forth. Each of the sefirot is also subdivided into sub-sefirot, called in kabbalistic terminology particular sefirot: Binah within Hokhmah, Hesed within Hod, and the like. Usually one does not find in kabbalistic literature any reference to great-grandchild sefirot, that is, to a subdivision or specification more than two sub-levels removed from the parent level.

So far we have seen the linguistic aspects of the phenomenon. But in the Torah and in halakha (Jewish law) there are various hints that there are also essential aspects, of which the linguistic ones are merely a reflection. I will now present several examples.

A first example appears in Rashi on the verse in Genesis 21:23:

“If you deal falsely with me, or with my son, or with my grandson.”

Rashi writes there that a father’s compassion for his son, that is, his special relation to him, extends as far as a great-grandson. This is derived by the Sages from the talmudic passage about “one who comes in by the tunnel,” that is, the law of the burglar.¹⁵⁶ Incidentally, the biblical term itself appears only in the sense of son, and not in the sense in which we use it today. The Bible has no term for the son of a grandchild.

In the kabbalistic context, some explain the absence of any reference to a great-grandchild sefirah by saying that at such a hierarchical distance there really is no direct influence. The influences between such levels are only indirect, through the intermediate levels.¹⁵⁷

In mechanics, too, one may say in general that the character of motion depends only on velocity and acceleration, and not on higher derivatives, except in special cases. The reason is that Newton’s second law includes acceleration, and not a higher derivative of position.¹⁵⁸

Let us now bring an example that bears directly on the relation between levels of classification and categorization. In the Yom Kippur prayers there appears the confession of the High Priest, and there one sees a similar phenomenon. There are three confessions: one for himself and his household, the second for the entire priestly tribe, and the third for all Israel. In the first two confessions, the High Priest confesses and says that he and his household sinned, or that he and the entire priesthood sinned. But in the third confession he says that Israel sinned, and there he does not include himself. It would seem that he does not include himself in the third circle outward from him in the hierarchy of inclusions, beyond species and genus.

There is an entire series of halakhic contexts in which one sees the absence of a relation between remote levels in a hierarchy. Let us bring a few examples:

1. In the laws of Shabbat there are thirty-nine primary categories of labor prohibited on Shabbat. Each primary category has derivative forms. There is an opinion among the medieval authorities, see Rabbi Hananel’s novellae to Babylonian Talmud, Shabbat 73b, that second-generation derivatives were not prohibited.
2. In the laws of vows, a person must utter the vow in the language of a vow. If he does so in a different formulation, called a substitute expression or an allusive formulation, it is still effective. Yet halakha rules, see Maimonides, Mishneh Torah, Laws of Vows 1:17, that a substitute for a substitute is ineffective.
3. In the laws of vows there is a special way of creating a prohibition on an object, similar to uttering a vow regarding it. If a person transfers the force of a prohibition from one object to a second object by saying, “This is like that,” this serves as a substitute for the ordinary language of a vow, and some say it is actually the more basic form. This device is called attachment. Ritva, in his novellae to Babylonian Talmud, Nedarim 14a, Mossad Harav Kook edition, p. 110, and see the editor’s note 134 there, writes that in the case of an oath, an attachment based on an attachment is ineffective. That is, one cannot transfer the force of a prohibition to a new object from an object whose own prohibition came into being through attachment. From his wording there, it seems that this principle is connected to principle 2, namely, that there is no substitute for a substitute, and no allusive formulation for an allusive formulation.
4. Perhaps one may connect to this also the halakhic principle of tarti le-rei’uta, that is, the combination of two adverse factors; see Babylonian Talmud, Kiddushin 79a, and Babylonian Talmud, Niddah 2b-3a.
5. The same applies to the halakhic distinction between a person’s direct force and the force generated by that force. Something that comes about through a person’s own force is regarded as though he himself did it. But if that which comes about through his force activates something else, so that the result is already only a second-order effect of his force, then in certain halakhic contexts it will no longer be considered the person’s own action.¹⁵⁹

This collection of examples raises the hypothesis that at the logical level the more detailed divisions relate to one another in the same way all along the chain, but in practice, both linguistically and essentially, the range of direct relation between them extends only across a distance of two levels of division, and sometimes three.

This is a quasi-ontological aspect. To be sure, there is no claim here that species and genera have a different ontological status from the other levels of division. Yet on the plane of relations and influences between the various levels of division, we do indeed see that there are direct channels of influence only between levels that are not too remote from one another. If so, the special terminology in language points to a special reality, and is not merely an arbitrary linguistic phenomenon created for convenience alone.¹⁶⁰

Phenomenology and Correlations

Another characteristic of the social sciences and the humanities, closely connected to the previous one, is that they deal extensively with correlations. This fact is intimately related to what we saw above, namely, that the principal occupation of these fields is the classification and categorization of phenomena and people.¹⁶¹

The explanations for the formation of genera and species, and for the relations among them, are far more complex than in the natural sciences of inanimate matter. Therefore, theories in the social sciences are often phenomenological rather than essential. Phenomenology is, in its essence, the establishment of correlations. The role of an essential explanation is to propose an account that explains the formation of those correlations. In the first chapter we discussed the relation between these two planes and the problems it raises, and now we see that these problems may arise with even greater force in the social sciences and the humanities.

As stated, because the social sciences and the humanities deal with complex domains, the tendency is to engage more with phenomenology and less with essence. In any case, it is commonly accepted that phenomenology is the more scientific component of these fields, while essence, being more speculative in character, is naturally less open to empirical testing.

An extreme expression of this claim is found in the behaviorist approach, which instructs its adherents to engage only in phenomenology and not in essence. The more radical among them argue that psychology should describe behaviors and the relations between them, that is, correlations, and not explain them in terms of abstract entities and forces that are not directly observable, such as the superego, libido, repression, and the like. According to behaviorism, any engagement with essence, which by its very nature involves abstract concepts and forces that cannot be directly observed, is speculative and therefore does not belong to science. We shall discuss behaviorism and phenomenology later on, in the context of semantics and syntax.

An emphasis on phenomenology intensifies the many problems we saw above in chapter 1 regarding correlations. This is another reason why the reliability of research results in the social sciences is low. We saw there that so long as we do not possess a well-grounded essential explanation and deal only with phenomenology, two principal kinds of problems arise: we have no way of determining the direction of the correlation, and no clear way of deciding whether it is apparent or real.

This argument indicates that the essential theories of the social sciences, and certainly of the humanities, not only employ abstract concepts that are not directly observable, something that also characterizes the natural sciences, but are also based on shaky phenomenology. That phenomenology is built on correlations and classifications that cannot be established reliably.

This is not the place for a broader discussion of the scientific status of the social sciences and the humanities. Here I have merely tried to point to several additional aspects, some of which express points that arose throughout the present gate. It seems that precisely these aspects distinguish the natural sciences from what is called the social sciences and the humanities.

Summary of the Discussion in the Second Gate

This gate discussed the characteristics of modern empirical science, as well as scientific method and the scientific mode of thought. We saw that from the analytic point of view, scientific methodology suffers from several fundamental defects, almost all of them deriving from the synthetic character of science. In order to save the analytic worldview while at the same time preserving confidence in science, analytic thinkers developed an analytic interpretation of science, according to which science says nothing whatsoever about reality, that is, analytic science, and is nothing more than a sophisticated linguistic and conceptual form for organizing the specific information accumulated in our possession.

This interpretation is seemingly supported by the methodological rule according to which science adopts the most elegant theory. But we saw that according to this interpretation we are forced to arrive at conclusions that are patently unreasonable. Among other things, the following conclusions follow from this view:

1. Scientific theories say nothing about the world.
2. The aim of scientific study and research is the collection of facts. The theories are nothing but an organization of the known facts. Therefore an experiment is unrelated to the testing of theories, even though its results may influence them.
3. Science cannot, even theoretically, retreat; it can only advance.
4. Scientific entities do not really exist in the world.
5. Scientific equations are tautologies, or logical identities.
6. Finally, the principle of causality does not exist at all, except perhaps as a merely methodological principle. According to the analytic interpretation, events occur in the world without a cause, or at least we cannot in any way discern a cause.

As stated above, critics of this approach, Bechler and those like him, lament the absurdity inherent in the analytic, or actualist, conception of science, but they do not offer any reasonable grounding that would resolve these embarrassing points.

Because of these difficulties, it seems far more reasonable to adopt a synthetic position also with respect to the methodology of science, even though, as noted, this is not strictly entailed by adopting a synthetic position in other contexts. According to the synthetic interpretation of science, scientific theory is indeed learned through observation, and it is not merely a generalization carried out by the intellect. There is here a combination of intellectual generalization and observation of ideas. We arrive at the conclusion that theoretical entities exist not only because they explain reality, but because we also see them in reality itself.

After that, we dealt with the solutions that the synthetic picture can offer to the fundamental methodological problems surveyed in chapter 1. If science really does make claims about the world, then we must explain why the elegant claim, or theory, is also the true one. How do we classify facts as historical facts or scientific facts before formulating the theory that serves as the criterion for this distinction? It is not clear how we ground the principle of causality and the ability to generalize by means of induction, and so forth.

We emphasized that the basis of the solution to the perplexities in the philosophy of science, as was already noted in the previous book, lies in giving up the sharp distinction between thinking and cognition. The meaning of this assumption is that there is a thinking sense, which some call auditory logic and others call eidetic vision, and that this faculty succeeds in creating direct contact with the reality outside us, even before we formulate the insights that emerge from that contact.

This faculty discerns historical facts even before the formulation of historical theory, and it is what enables us to formulate such theory. It also discerns correct and incorrect generalizations even before the theory is tested in the context of justification. That is why many of our theories turn out to be correct to a considerable degree, despite the a priori improbability of arriving at that result by chance.

This sense, or faculty, was called faith in the previous book. By means of it we also discern spiritual entities and events. The existence of God, as well as various myths, comes to our knowledge through it. In the next gate we will discuss the manifestations of this thinking sense in relation to science and myth. There too, its existence will have implications that alter our superficial conceptions and restore our basic trust in our intuitions.

The gate concluded with two specific chapters whose significance for the overall course of the argument is marginal. The first discussed the status of mathematics as a non-empirical domain. The discussion was conducted from the perspective of the debate over mathematical Platonism. We saw that neither side of the debate is quite so simple, and that even the distinction between an analytic interpretation and a Platonist interpretation of mathematics is not so simple.

At the end of that chapter, a strong intuition was presented that mathematics nevertheless does say something about reality, that is, it has a synthetic aspect. We tried to explain this by means of an as-if epistemological mechanism, in which we contemplate an ideal world, in an abstract theoretical space, and we distinguished between this process, which is cognitive or epistemic, and what is usually called thinking. This was a sharp demonstration of fundamental concepts of synthetic epistemology, such as Husserl’s eidetic vision and Rabbi HaNazir’s auditory logic, which we encountered in the previous book and in the chapter on the synthetic interpretation of science.

We concluded with a chapter discussing the scientific status of the “social sciences” and the humanities. These fields seemingly satisfy scientific criteria of empiricism, the formulation of theories and their refutation, and so forth. Yet we saw that these are only necessary, not sufficient, conditions for characterizing a discipline as scientific. We noted briefly the basic difficulty these fields face in reaching a level of precision and sophistication comparable to that of the natural sciences and mathematics, and we attributed this chiefly to the fact that they all touch, in one way or another, on the human being and on human society. Because of this, there is a concentration on classification, categorization, and correlations, in fact on phenomenology rather than essence. Descriptions and predictions in fields that concern the human being cannot, by their very nature, attain a level of precision as high as that found in the natural sciences of inanimate matter. This point will stand at the center of the discussion in the next book, the third part of the trilogy.

Footnotes