Gate Four: Parallel Planes of Explanation

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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Parallel Planes of Explanation

Introduction

In the second gate, we encountered some of the assumptions that form the methodological and philosophical basis of science. In the third gate, we encountered an expanded concept of myth, defined as the framework within which all human understanding takes place, in every field—in our terms, an a priori structure of meaning. We saw that science too is not exempt from a framework within which its activity takes place, whether we call it a paradigm, a myth, or any other name. This means that science too rests on a number of a priori assumptions, unexamined in themselves, and only within their framework can one understand the scientific method, and therefore scientific results as well.

We ended the previous gate by saying that if these two domains are indeed based on a similar logical structure (although there are, of course, significant differences between them), and both presuppose assumptions, implicit or explicit, that have not been examined, empirically or otherwise, then the contradictions between them appear even harder to resolve. According to the picture described thus far, one cannot easily dismiss one in favor of the other, and therefore both must be taken into account. We briefly noted that the contradiction between the domains arises in many contexts, and it is very common in thinking about problems connected to religion, especially problems dealing with the relation between religion and science.

In the present gate we will deal, in the most principled and general way, with situations in which there are different explanations for the same phenomena. There is a tendency to treat every such situation as if there must necessarily be a contradiction between the explanations, and therefore we must decide which one we choose.

Here we will point out that sometimes this is indeed the case, but not necessarily, and not always. There are many situations in which we adopt several explanations simultaneously, and there are different reasons for this and different meanings to it. In this gate we will encounter a number of different situations in which such a state of affairs can appear.

We noted that attitudes toward myth undergo upheavals throughout history, especially in the transition from modernism to postmodernity. Modernism, generally speaking, tended to negate myths and to treat them with contempt, and therefore also the religions based on them. But in the postmodern age, as we already mentioned, a certain rehabilitation of myth and religion takes place. This is based specifically on undermining certainty regarding truths that modernists regard as absolute, or at least as legitimate. Once it is seen that everything depends on basic assumptions, and therefore everything is context-dependent, the expected analytic conclusion is drawn: no system of assumptions is superior to any other. One may then return to myth as an equal member of the cultural system of human beliefs and knowledge. This is the “dance of differences,” an expression cited at the beginning of the previous gate from Rabbi Shagar’s book Broken Vessels.

According to this approach, science is not superior to myth, since at the basis of both domains lie assumptions that have not been examined and certainly have not been proved. The postmodernist concludes that these are both subjective systems, but that there is nothing wrong with that.

It is very important to emphasize that this rehabilitation addresses the legitimacy of the domain, not its degree of reliability. In fact, it remains in agreement with the critique that treats myth as arbitrary and subjective, but grants subjectivity legitimacy—out of lack of choice. According to postmodernism, human beings have no escape from life within uncertainty and total subjectivity. In terms of the maturity parable we presented in the first book (at the beginning of the third gate), this is nihilistic maturity, which preserves the basic assumptions of youth.

Above, we pointed out that a synthetic position aspires to a different kind of rehabilitation, a different kind of maturity. In the first book, reason and certainty in general were rehabilitated by proposing a possible mechanism for grounding our ability to attain certainty about some things there called auditory reason. In the previous gate it was called thinking-sense. In this book, we wish to continue that move and present a rehabilitation of myth and religion that is not based on undermining modernist assumptions (which trust science and morality), but precisely on taking them into account.

The claim that science rests on a priori, unexamined assumptions, something like a myth, is shared by the approach presented here and the postmodern approach. But unlike the postmodern way of thinking, here it does not serve to undermine science, but rather to strengthen myth and religion. The fact that there are basic assumptions at the foundation of every domain of thought does not necessarily indicate that all of them are arbitrary. The synthetic position holds that there are also ways to reach conclusions about the truth of basic assumptions. The essence of the synthetic argument was that if such ways do indeed exist, then they are what underlie our trust in scientific thinking, and therefore there is no reason to reject them when they concern myths or religions. This is a complete reversal of the analytic-postmodern conclusion, yet it rests on the same analysis of reality—namely, that we have no “proof,” in the analytic sense of the term, for any claim in any domain.

According to this approach, the various possibilities that postmodernism offers for understanding different domains are not seen as relativism, but on the contrary: all of them represent possible understandings, and in many cases there may be truth in all of them. Instead of rejecting them all, we propose adopting them all—or at least several of them—simultaneously.

In the closing chapter of the first book we called this constructive postmodernism. We must take the multiplicity of truths to which postmodernism points, but not draw skeptical conclusions from it; on the contrary, we should assume that all these truths are valid. Modernism was mistaken in assuming that there is only one truth. But postmodernism is equally mistaken in claiming that there is no truth at all, or that all truths are subjective. Maturity—or awakening from both of these conceptions—according to the synthetic position proposed in this book, holds that there are many truths, and all of them can be correct and objective simultaneously, in the classical, not postmodern, sense of the concept of truth.¹

This is the point at which the present gate joins the overall argument. In this gate we will examine the possibility of adhering to several planes of explanation at once, and we will reject the common assumption that we must necessarily choose only one of them.

In this sense, there is a continuation here of the first book. There we saw that much of the modernism prevalent in our circles is nothing but disguised postmodernism. It seems that the only possible genuine modernism—that is, a synthetic position—is not the abandonment of myths, but specifically their acceptance. This acceptance can be grounded in recognition of the existence of the auditory, or eidetic, faculty. The point is not merely recognition of its existence, or even of its legitimacy, but primarily recognition of its reliability. This faculty allows us to encounter things that the senses do not enable us to encounter. As we saw, this assumption is also required for the acceptance of science, and therefore there seems to be a broad consensus about it, even if not consciously. We saw in the first book, and again above in the present one, that the meaning of this claim is, in effect, giving up the sharp distinction between thinking and cognition.

As postmodernism rightly claims, explanations are always made within a cultural framework, that is, within a myth (or paradigm). Therefore there is no possibility of throwing the mythic layer away and focusing only on “the absolute facts,” as the analytic-positivist illusion suggests. On the other hand, if we are willing to give up the identification of certainty with truth, then here too we can continue to hold different truths in different domains, including religion or myth, while remaining aware that they are not certain—at least not in the pure logical sense of the concept of certainty. It should be stressed that this absoluteness is not the possession of science either, precisely because of the mythic (paradigmatic) framework within which it operates. This, of course, stands in contrast to the illusion of the absolute certainty of science, an illusion characteristic today only of those who are unaware of its philosophical, logical, and methodological limitations.

As stated, the main stage in presenting the alternative is to point out that different explanations are not necessarily contradictory, and therefore they are not necessarily mutually exclusive alternatives. Hence we are not necessarily obligated to choose only one of them. The discussion in the present gate will focus on this point.

The contradictions—and also the apparent contradictions—between science and various myths (or some religion), which are our point of departure, will be discussed in greater detail in the final gate. There we will apply the different modes of approach developed more generally in the present gate, and we will try to use them to present possible ways of understanding the relations between religion—at least Judaism—and science on different planes. That discussion, beyond its specific content, will also serve as a good demonstration of the tools discussed and developed in the present gate.

Chapter One: Presenting the Problem

We will begin the discussion with several examples that present different situations in which parallel planes of explanation appear. Let us say in advance that the point is not to exhaust the discussion of the examples themselves, but only to present and sharpen different situations in which parallel explanations appear.

On Apples and Wheels: Newton and Abraham

As a first example, directly concerning the relation between science and myth, let us take the well-known story of Newton and the apple. As is known, the legend—or myth?—tells that while Newton was sitting under a tree, an apple fell on his head. Newton asked himself why apples fall from the tree, and thus discovered the force of gravitation. Seemingly, as a believer—Newton was a devout Christian—he ought to have understood that this was punishment for some transgression he had committed. God was punishing him for that offense by dropping an apple on his head. Why, then, was he troubled by the question of the cause of the apple’s fall?

It turns out that even if the metaphysical explanation was indeed acceptable to him, and it was clear to him that the apple fell because he deserved punishment from God for some sin, Newton still asked himself how such a thing happened. That is, what was the scientific explanation of the phenomenon?

This is an example showing that the existence of a metaphysical explanation does not prevent Newton, or anyone else, from asking about the scientific explanation. And equally in the other direction: the existence of a scientific explanation should not prevent someone from asking what the metaphysical explanation of some event is.

A parallel example may be seen in Abraham, whom the midrash (rabbinic interpretive exposition) describes as wondering why the wheel in the heavens turns and never stops moving. Who turns it? Thus the midrash describes the line of reasoning through which Abraham arrived at faith in the Holy One, blessed be He. Here too one might say that his wonder stemmed from ignorance, for today we know that there are laws of nature that “cause” the wheel to move.² But this claim precisely conceals the same mistake to which we pointed above. Here too, the law of nature is not a substitute for the metaphysical-theological explanation, but a scientific description of it.³ The phenomenon would have astonished Abraham even if he had possessed a scientific explanation for it.

Becoming Religious and Becoming Secular

There is an interesting phenomenon connected to our attitude toward people—especially those close to us—who radically change their worldview, particularly along the religious-secular axis.

When someone becomes religious, his secular environment wonders what psychological crisis caused him to take such a step. His new religious environment, by contrast, understands that at last he has “discovered the light of pure truth.” That is, in such a case secular people tend toward psychological explanations, whereas religious people tend toward a philosophical mood.

Conversely, when someone leaves religion, his religious environment explains that his sole purpose is to permit sin to himself—in the language of the Sages, “to permit forbidden sexual relations to himself.” By contrast, his secular environment explains that at last he has “discovered the light of truth,” or alternatively, “escaped from primitive religious darkness.” In this case, then, the picture is reversed: the secular camp is seized by lofty philosophical inspiration, while the religious camp suddenly begins, for some reason, to take an interest in psychology and instinct.

At first glance, there is an interested interpretation here, with each side tending to operate on the plane that is more comfortable for it. When the step is taken against my worldview, I tend to be a “psychologist,” that is, to attribute it to nonessential explanations, so that my faith in my own worldview will not be harmed. My views are “perfect”; there is no flaw in them. Obviously, the person who behaves against them is merely giving vent to dark impulses. By contrast, when the step supports my views, I tend to see it as confirmation of them, and I do so by explaining it on the logical-philosophical plane.⁴

This analysis, however, could be rejected, and the interpretive paradox explained in a different, more sympathetic way. Since a person believes in his worldview and thinks it is correct and accords with the conclusions of rational deliberation, his natural tendency is to say that if another person disagrees with it, then that other person is probably acting out of motives that are not rational—in this case, psychological ones. After all, if he thought logically, he would arrive at the “correct” conclusions. By contrast, if the step taken accords with the observer’s worldview, he will explain it as a step stemming from logical and rational consideration. If this is the correct analysis, then there is no dishonesty or self-interest here, but simply adherence to the basic assumptions of each observer.

In any event, whatever the interpretation of the dispute between the two explanatory planes and the tendency to focus on one of them, there is no doubt that both explanations are correct, though they deal with different planes. A person acts simultaneously from psychological considerations and from philosophical-logical considerations. Clearly, both planes are relevant to understanding his behavior.

Thus, the plane of inquiry will dictate the character of the explanation that is found. If we are interested in his psychological motives, we will find a psychological explanation; if we are interested in his value-oriented and rational motives, we will find an explanation of that sort.

It seems to me that in general it is preferable to deal with the value-oriented, ideological, and philosophical plane, because this is the objective plane, one that concerns us as well, and therefore it is the one that can be useful to us. Psychological analyses, which discuss what is happening inside another person rather than raising questions and criticisms about our own worldview, may indeed be correct, but usually they serve as a means of escape from confrontation and self-criticism.⁵

The Jew and the Nobleman

There is a well-known Hasidic story dealing, naturally enough, with a Jew and a nobleman. When Mushke the Jew expressed his confidence that the Holy One, blessed be He, is the one who feeds and sustains him, the nobleman proposed an experiment: go out into the forest and wander there without any provisions, and we shall see whether your God will sustain you and whether you survive. When the Jew went out into the forest, the nobleman sent his servants with bundles of provisions and food to place beneath various trees in the forest, according to Mushke’s needs. When Mushke returned and related, with great excitement, how the Holy One, blessed be He, had fed and sustained him, and how every morning he found food packages under the trees according to his needs, the nobleman burst into loud laughter. He explained to Mushke that in fact it was he himself who had taken care of him, not God.

Not surprisingly, the story nevertheless ends with clever Mushke’s victory. After listening intently to the nobleman’s explanations, Mushke corrected him and explained that in this sophisticated way God had used him, the nobleman, in order to sustain Mushke, His faithful servant.

As in the two previous examples, here too both explanations are equally correct. The nobleman’s mistake was not that he believed he had fed Mushke. That was, of course, entirely true. His mistake was to think that this claim contradicted the explanation that God had sustained Mushke. He assumed, unjustifiably, that these two explanations contradicted one another.

It should be noted that a similar failure can sometimes be found in the opposite direction as well. Believing Jews sometimes protest against the conception of “my power and the might of my hand,” that is, the belief that steps based on human reasoning—such as relying on doctors, using an army, and the like—are what save us in various situations. Many in religious circles attack such an approach as a deficiency in faith. A Jew, they say, ought to believe that God alone saves us, not our own power and strength.

This is precisely the same mistake as the nobleman’s, only in the opposite direction. Indeed, on the physical-human plane of explanation it is true that the doctors, the shoemakers, or the soldiers are the ones who save us. But they receive their power from God and thereby serve as His agents. A Jew should be aware of the theological layer on which things occur, but it is wrong to demand, on that account, that he not believe in the plain human plane of explanation.⁶

Vitalism

Above we saw two ways of relating to the phenomenon of life. As we saw there, many researchers assume that science has no need for the assumption that there is a living substance, and that biological and physiological phenomena, almost in their entirety, can be explained scientifically even without such an assumption. On the other hand, there is a clear feeling that even if there is no scientific need for it, the phenomenon of life still contains an additional component beyond the physico-chemical components.

The relation between these two planes is more complex. According to extreme mechanism, there is no such additional component at all, and such a claim stands in complete logical opposition to the vitalist conception. But in a more moderate formulation one may say that science has no need for the assumption of vitalism, although adopting it is not necessarily contrary to scientific facts. The claim that there is such an additional component does not mean that the scientific dimensions of the phenomenon of life cannot be explained without it. Scientific explanation will not refer to vital substance, but an explanation on another plane—for example, a mythic one—may certainly make use of it.

True, Ockham’s razor (see above, in the second gate), the principle that leads us not to assume the existence of entities and principles we are not compelled to assume, can lead to the conclusion that there is no such component at all, and this would be a reasonable scientific conclusion. But of course this is not a necessary conclusion on the logical and ontological plane, but at most a rational method for guiding and managing scientific research.

The parallelism prevailing between these two planes of description is of a different character from the parallelisms we saw above. Here we are not really proposing two explanations, but two worldviews. The two do not coexist in parallel; rather, on the scientific plane we relate only to part of the phenomenon under discussion, while the spiritual-theological plane leads us to address other parts of it. This example leads us to a brief discussion of reductionism.

Reductionism, Again⁷

Reductionism is the view according to which the higher sciences can be reduced to the more basic ones. Biology, according to the reductionists, is nothing but a composition of very many chemical processes, and in fact these too are composed of physical processes. The whole is described as the laws of biology. The reductionist claims that if we had an enormous computer, capable of calculating the mode of operation of every particle in our bodies, we could extract the scientific description of our physiological activities from physics and chemistry, and thereby reduce biology to physics and chemistry. Some would extend this to psychology and beyond, but here we already reach other issues—such as materialism and determinism—which are not our concern here.

It is well known that Laplace said that if all the states of all the particles in the universe were known to him, then in principle he could tell us what the state of the entire universe would be at any time, future or past. This is an expression of extreme reductionism, which grounds the world, with all its phenomena, in physics. From this, of course, a deterministic conception follows—see the third book.

In any event, some would say that one can propose several parallel explanations for any phenomenon belonging to a higher science. Each explanation would use the terminology and principles belonging to one of the lower planes beneath it: the physical, chemical, biological, physiological, and so on. In principle, even such a conception does not contradict reductionism, since it may be possible to reduce each of these explanations to an explanation on a lower plane. But substantive parallelism—a truly parallelist conception—holds that one explanation cannot be reduced to the other.⁸

As an anecdote, let us note that Laplace himself claimed that he did not believe in God because “he had no need of that hypothesis”—for the reasons described here. He believed that everything was explained through the laws of nature, and that science leaves no gap requiring the introduction of any spiritual concept. But it should be noted that even if he is right, and everything can be explained in mechanical terms, it is still possible that there is a parallel explanation alongside the scientific one, even if the scientific explanation is complete. Thus, implicitly, Laplace is rejecting the possibility of two parallel planes of explanation.

Teleology

The observation given above regarding teleology and causality is also an example of two explanations that exist in parallel.

The mechanical description of a body’s motion through the action of a force on it, and the description of its motion through the presence of a potential in the space through which it moves, are equivalent descriptions. Likewise, the pair of descriptions in optics by way of Fermat’s principle—or minimum principles in physics generally—and by causal form—Snell’s laws in optics, and the like—are also equivalent descriptions. One can adopt both simultaneously without any contradiction. On the other hand, in an essential sense only one of them is correct. The fact that two scientific descriptions are equivalent in terms of their future predictions does not make them one explanation. The question of which of these explanations correctly describes reality remains open. Here we propose that it may be possible to say that both are correct descriptions of reality: one on the causal plane, and the other on the teleological plane. We will elaborate on this below.

This parallelism can be expanded if we examine our treatment of the concepts of cause and purpose more generally. I call my friend in order to arrange a meeting next Sunday, and therefore the desire to arrange a meeting is the reason, or explanation, for the telephone call I make. Seemingly this is a teleological explanation: an action for the sake of attaining a certain end. On the other hand, one can also describe the reason for it on a physiological plane, through the action of the brain on the muscles via the nerves, leading to lifting the telephone receiver and moving the mouth muscles, and so forth. This is, of course, a causal explanation. This discussion leads us to the problem of body and soul, or the psychophysical problem, which troubles a great deal of human reflection and thought, mainly because of the parallelism between two different explanations for human actions.

Another example in this context: certain mental problems can be treated by psychological processes, or by chemical means, such as various medications. This demonstrates that two entirely different mechanisms can lead to the same result. Thus our mental activity too takes place simultaneously on two different planes: the psychological and the physiological.

The Happy Prince

The final example of a situation involving two parallel planes of explanation—this time on the literary plane—is taken from Oscar Wilde’s story The Happy Prince. In this story, one swallow remains in cold Europe after its companions have left the region for warm Egypt. The swallow stays in order to help the Happy Prince in his acts of kindness. The prince is merely a golden statue, decked out with precious stones of various kinds, and through the swallow he distributes them to various needy people. Near the end of the story, the swallow is about to die from the fierce European cold, and it comes to the prince to bid him farewell. At this point the prince is already blind, because he has given his eyes—two precious stones—to the needy, and the following dialogue takes place between the swallow and the prince:⁹

“I am glad that you are going to Egypt at last, little Swallow,” said the Prince, “you have stayed too long here; but you must kiss me on the lips, for I love you.”

“It is not to Egypt that I am going,” said the Swallow. “I am going to the House of Death. Death is the brother of Sleep, is he not?” And he kissed the Happy Prince on the lips, and fell down dead at his feet.

At that moment a curious crack sounded inside the statue, as if something had broken. The fact is that the leaden heart had snapped right in two. It certainly was a dreadful frost.

There is an interesting conjunction of circumstances here. The prince’s leaden heart split in two at the moment of the swallow’s death—seemingly from sorrow and “heartbreak,” in both senses of the expression. But immediately afterward the author adds an apparently innocent clarifying sentence, attributing it to the material strength of the leaden heart: it was the frost that split the prince’s heart. The properties of lead were his undoing.⁹ So who really split the prince’s heart—the cold, or sorrow over the swallow’s death?

On the mechanical-scientific plane, it was the frost that split the prince’s leaden heart, but on the emotional-psychological plane, it was the heartbreak over the swallow’s death that split it. Perhaps one can say it differently: the prince’s heart split from grief. This is the “psychological” explanation. The scientific description explaining how such a thing actually happened physically—after all, it is a physical event—depends on lead’s inability to withstand the intense cold. We have here a literary description that makes use of the simultaneous existence of two causes, or two planes of explanation.

Pardes

We will conclude the chapter with an observation on parallel explanations in the interpretation of Scripture, where we encounter such a situation directly, together with its problematic character.¹⁰

Observation 27: Interpreting the Torah: Pardes¹²

As is well known, in interpreting the Torah it is customary to assume the existence of several parallel planes of explanation. We speak of “seventy faces of the Torah.” The Sages distinguish between four fundamental levels of interpretation: the plain sense (peshat), allusion (remez), homiletic exposition (derash), and secret or esoteric meaning (sod), known together as Pardes.

This is itself another example of a situation in which several planes of interpretation exist in parallel, all of them correct. But this fact is not at all simple, and in recent years it has aroused a number of fierce debates, among them debates concerning the relation between peshat and derash. The simple assumption is that peshat is the straightforward meaning of Scripture, whereas derash is not really interpretation, but rather an expansion of the scriptural meaning in certain ways.¹¹

At the root of the matter is the fact that the assumption that there are several interpretations of the same text, all of them legitimate, is not at all simple—especially if we operate from a hermeneutic conception according to which the desired interpretation is the author’s intention. Seemingly, an author can have only one intention. It is clear that if interpretation is simply whatever arises in the reader, then the present discussion has no significance.¹⁴

The question whether several different explanations of the same text can exist is closely parallel to the question whether several different explanations of the same events can exist, and that is what we will discuss later. Here we wish to point out that this difficulty is what led to the dispute between Maimonides, in the second of his “roots,”¹³ and Nahmanides, who criticizes him precisely on this point.¹⁶

There Maimonides introduces a far-reaching innovation that aroused waves of severe criticism, followed by a wide range of defenses among his commentators. We possess systems of hermeneutical rules by which the Torah is expounded—that is, ways or tools by which the Torah may be interpreted. The methods of halakhic (legal) exposition are generally divided into thirteen, following the baraita of Rabbi Ishmael at the beginning of the Sifra, and Maimonides at the beginning of the root under discussion deals with fourteen exegetical methods: Rabbi Ishmael’s thirteen principles plus the principle of inclusion.

Maimonides there determines that the laws deriving from these exegetical methods are not defined as laws of Torah status, but as rabbinic laws. There are different interpretations of what exactly Maimonides means by this, and this is not the place for it. But one thing clearly emerges from his words: these laws are not the result of interpretive procedures applied to verses, but an extension of them. In Maimonides’ own phrase: “like branches emerging from roots.”¹⁵

Nahmanides, in his criticisms there, finds this very difficult, since it is accepted that the hermeneutical rules by which the Torah is expounded were given to us at Sinai, and Maimonides himself writes so explicitly. In addition, the Torah itself was certainly given to us by the Holy One, blessed be He, at Sinai. If so, applying rules that were given to us at Sinai to a text that was given to us at Sinai should seemingly lead to laws of Torah status, as though they themselves had been given to us at Sinai and were written explicitly in the Torah.

Maimonides himself addresses this point, and writes as follows:

Their confusion reached an even more severe point than this. When they found an exposition on a verse from which one is obligated to perform some action or avoid some matter—and all these are undoubtedly rabbinic—they counted them among the commandments, even though the plain meaning of the verse indicates none of those matters, despite the principle taught to us by the Sages, namely: “A verse never departs from its plain sense”…¹⁸

Maimonides explains here that the plain sense of Scripture can only be one, and therefore the homiletic expositions cannot be included within the plain sense of Scripture.¹⁷

Nahmanides, in his criticisms there, also addresses this point, and writes as follows:

Behold, the Rabbi has hung this collapsing mountain by a hair. He said that the principle taught to us by the Sages is: “A verse never departs from its plain sense”… Heaven forbid! For all the midrashim concerning the commandments—the halakhic midrashim—do not remove the verse from its plain sense; rather, all are included in the language of the verse, though they expand it by inclusions. And the exposition that accords honor to Torah scholars from the phrase “You shall fear the Lord your God” does not remove the verse from its plain sense…

But Scripture includes everything, for the plain sense is not as the linguistically ignorant say, nor as the Sadducees say. For the book of the Lord’s Torah is perfect; there is not a single superfluous or missing letter in it; all were written in wisdom… And so too, in every place where they interpret it as parable and metaphor, they believe that both are true, the inner and the outer…

Rather, we possess its exposition together with its plain sense, and it departs from neither of them. Scripture can bear them all, and both are true.

Nahmanides, at the end of his words, stands precisely on the point at issue: contrary to Maimonides, Scripture can indeed sustain two different interpretations, and both are true.

Thus we see that even within the framework of Torah interpretation, it is not entirely clear that two correct interpretations of the same text are possible.²⁰

Summary

In all the cases we have seen, we explain a given phenomenon on several different planes of explanation. There is an instinctive tendency to reject one explanation and adopt the other. This tendency expresses the assumption that two explanations for the same phenomenon cannot both be correct at the same time. But at least in some of the cases, we would be unwilling to give up either explanation, and in such cases we will have to live in some way with both of them.

Thus far we have presented several examples of situations in which two parallel planes of explanation appear, and one could have thought that this was a natural and simple state of affairs. But in the observation we can already see that such a state is not simple at all. The fact that several parallel planes of explanation exist for the same phenomenon demands careful analysis, and in some cases, as we shall see below, it appears impossible.

A situation of parallel explanatory planes can arise between myth and science, but also between two scientific explanations, or between two myths. Therefore, methodologically, it is preferable to analyze these situations in the most general and abstract way possible. Examples of the possible implications of this analysis will accompany the discussion throughout the present gate, and will be discussed in greater detail in the final gate.

Chapter Two: What Is an Explanation? The Problematic Nature of Different Planes of Explanation

The Problem Itself

At first glance, there is a simple description of the state of parallel explanations. As we saw in the previous chapter, we often live on two such planes simultaneously, and two different explanations belonging to those two planes are both perceived by us as correct at the same time. Each is correct in its own terms and language, and the question is simply on which plane we are relating to the events under discussion: philosophical or psychological, theological (mythic) or scientific, biological or physical, teleological or causal, peshat or derash, and so forth. The plane in which we are interested will yield the kind of explanation we adopt.

But this description, at least in its simple and superficial form, is only an illusion. There is a serious problem in holding two different planes of explanation simultaneously, and therefore it is no wonder that our intuition—at least when we are aware of this point—tends to resist it.

Take, for example, the theological-scientific axis. The assumption is that a scientific explanation, or scientific cause, means this: from a given state, a certain result must necessarily emerge, because of certain laws of nature relevant to that state. Newton’s apple falls only if the mechanical conditions in its environment—the strength of its attachment to the tree, its total weight, the wind, air resistance, and so on—are ripe for its falling. If these obtain, it will necessarily fall, even if Newton did not sin at all. On the other hand, if these conditions are not fulfilled, then according to the scientific view—and it seems that ordinary intuitions agree with it here—the apple will not fall even if Newton is a “serial sinner.” This is the accepted understanding of explanations in general, and of scientific explanations in particular.

On the other hand, the same applies to the theological explanation. According to one who believes that everything is governed by divine providence, which arranges events so that each person receives what is due to him, if Newton is a sinner—even if not a “serial sinner”—then the apple ought to fall on him, even if the physical conditions described in the previous paragraph are not fully met.

Another example is the psychological-philosophical axis. If the psychological explanation is correct, then given the set of environmental conditions—the person’s private history and genetics—which cannot always be precisely defined because of their complexity, but are generally thought to exist in principle, the “value-oriented” result, namely the worldview or change of worldview, will necessarily be produced, even if these steps have no adequate philosophical justification. Conversely, if we discuss the matter on the philosophical plane of explanation, then whenever the philosophical justification is correct, regardless of the psychological data—crises and the like—the person will take the relevant step or change his outlook as seems right to him. So here too there is a similar problem in the relation between two different planes of explanation.

The assumption that makes the problematic nature of parallel planes of explanation so acute concerns the very concept of explanation. In its simplest form, an adequate explanation must be a necessary and sufficient condition—at least in the existing circumstances—for the occurrence of the result being explained.¹⁹ We will expand on this point below, when we discuss the relation between explanation and cause, and Karl Hempel’s deductive-nomological schema. In any event, if a given explanation in given circumstances is indeed a necessary and sufficient condition for the occurrence of the explained phenomenon, then once we adopt an explanation belonging to plane X, if it truly deserves the title explanation, then whenever the required conditions are present, the result is necessarily expected to occur. And if the conditions are absent, the result necessarily will not occur. But all this is equally true of explanatory plane Y.

Thus, seemingly, a frontal contradiction arises between any two parallel planes of explanation. It should be noted that the contradiction does not depend on the content of the different explanations or on the relation between them. We are not speaking of explanations belonging to content-wise opposing systems, such as religion and heresy, or scientific and anti-scientific views. The problematic nature arises inherently from the very definition of the concept explanation, and from the fact of the parallelism between the two explanations.

The Relation to the Problem of Determinism

It is important to notice that we are not dealing here with the problem of determinism. The question whether the world is deterministic—that every given state uniquely determines the next state—concerns the status of scientific explanations themselves: whether they are correct, or complete, or not, and it also bears on issues such as free will. Here we are dealing with a broader question, one that does not necessarily concern human beings: if the scientific explanation, or another explanation, is correct, what is the relation of this fact to other parallel explanations?

The problem of determinism, especially regarding human beings, will be discussed in greater detail in the third book. The present discussion leads us to ask what an explanation is at all, what a cause is, and what relation exists between these two concepts. We will now briefly discuss this issue, and afterward we will reexamine the possibility of parallel planes of explanation.

What Is an Explanation? Reduction to the Familiar

The accepted concept of explanation, in science no less than in everyday language, means reduction to the familiar. If there is a phenomenon we do not understand, we seek an explanation for it in terms of principles or phenomena familiar to us. For example, the phenomenon of tides can be explained in terms of the gravitational force that the moon exerts on the waters of the sea. The assumption is that we know gravitational force, and therefore we can use it to explain unfamiliar phenomena. The explanation consists in reducing the unfamiliar phenomenon—the tides—to the familiar phenomenon—gravitational force.

Of course, in science there are also stages in which a new force is discovered, one not previously known. One may see here a parallel to the stage Kuhn calls a scientific revolution. In such a state, we cannot make do with the accepted mechanisms of explanation—the paradigm—because we have no familiar basis on which to ground the new phenomena. This is the state in which a new scientific theory is created.

Let us continue with the example of gravitation. When the force of gravitation itself was discovered, this was done as a generalization from many situations in which it was seen that bodies were attracted to other bodies. This is not an explanation of natural phenomena, but the finding of an inclusive principle under which all the natural phenomena in question can be presented as derivatives or applications. It should be noted that in the situation described here, the phenomena themselves were familiar, while gravitational force was not. Thus the direction of explanation here is reversed: it is a reduction of the familiar to the unfamiliar.

Thus, after the discovery of gravitation, reducing tides to gravitation is a reduction to the familiar. But the generalization of tidal phenomena, and of the falling of bodies toward the earth, and likewise of the rotation of celestial bodies around one another, into an inclusive law speaking of gravitational force, is the reduction of the familiar to the unfamiliar.

In the discussion of Kuhn’s theory in the previous gate, we saw an example—which he himself gives at the beginning of chapter 6 of his book—of a similar situation, in which there are two different uses of the verb “to discover.” The statement that someone “discovered” spider webs above her bed does not use that verb in the same sense as the statement that Lavoisier “discovered” oxygen. We saw there that Lavoisier’s discovery required conceptualization, that is, the creation of the concept oxygen. So what did Lavoisier discover? Clearly, he discovered the possibility of a certain theory that would explain, by means of the theoretical entities it employs, a large set of phenomena within its framework—or in its terms.

Every fundamental scientific discovery is an explanation, or a discovery, in a sense different from the everyday one. Paradoxically, scientific explanation through the creation of a new theory is the reduction of the familiar to the unfamiliar. We know everyday natural phenomena, and reduce them to abstract principles that are new and not familiar at all.

One may say that the creation of a new scientific theory is the creation of a new mode of explanation. We find a new physical mechanism—a force, a field, a particle, and the like—by means of which many familiar phenomena can be explained. The finding of this principle itself is not an explanation in the ordinary sense. But if we adopt it, it enables us to propose explanations for the particular phenomena that led us to it.

This is the difference between puzzle-solving in a state of normal science—in which there is a paradigm accepted by all the researchers in the relevant scientific field—and a state of crisis, in which we must replace a theory, change a paradigm, or carry out a scientific revolution. In the normal state, explanations are reduction to the familiar—in Kuhn’s terms, puzzle-solving—whereas in the crisis state, in which a new paradigm is born, explanations are the reduction of the familiar to the unfamiliar.²²

One can ask more than this: why does a collection of familiar phenomena need explanations at all? If there is an unusual phenomenon—like a malfunction or an accident—or an unfamiliar one—like a new natural phenomenon—then it is reasonable to seek explanations for it. But it is not clear why familiar phenomena also demand explanations. Science, at least modern science, studies specifically the familiar and everyday, not the exceptional and strange. It seeks the fundamental principles according to which the world behaves in its ordinary routine. The exceptional serves only as a touchstone to test the theory in extreme cases and subject it to further tests. The goal of scientific research is specifically to understand the regular, not to explain the weird and exceptional.²¹ The greatness of a scientist is expressed in the ability to question and wonder about the most ordinary and everyday things.²⁴

It is no wonder, then, that scientific explanations are not explanations in the everyday sense. The phenomena before us do not require explanations in the everyday sense at all. We know them better than we know the principles that “explain” them.

Yet despite the sharp division between these two kinds of explanation—the scientific and the everyday—what has been said above makes clear that there is a close connection between them. When we find a general law that explains many phenomena, we may say that we have found a rule which, once it becomes sufficiently familiar and is adopted as true, can itself serve as an explanation for the particular phenomena. The law of gravitation is indeed the explanation for the mutual attraction between two masses, except that before the law was formulated we did not know it, and therefore were not convinced that it was correct. After some time, when it becomes clear that it is indeed true and that its predictions are verified, it becomes familiar and is accepted as a scientific law. It then reveals itself as an explanation of the phenomena included within it. But it should be carefully noted that, in fact, once the general law is confirmed, we discover that it has always been the explanation for those particular phenomena. The fact that we did not know this does not contradict the claim that it is their explanation.

Thus, a scientific law, in one sense, is an attempt to create new possibilities of explanation, and in another sense it is the finding of existing explanations that had not been known to us until now. This attempt turns out to be successful only retrospectively, when we become convinced that the law we found is indeed true. Then it becomes clear after the fact that this law really is—and always was—the explanation of the phenomena included within it.

But now we return to the starting point. Scientific explanation apparently is not reduction to the familiar, for as we have seen, the finding of a scientific law is the reduction of the unfamiliar to the familiar. If there is indeed something common to these two concepts of explanation, then it is clear that the explanatory element does not lie in the fact that we ground something in the familiar. So what is the basic element underlying the concept of explanation?

Put differently: the concept familiar, which appears in the phrase reduction to the familiar, is relative. An object or phenomenon that was not familiar to us in the past can become more familiar. It would seem, then, that the concept explanation is time-dependent. Yet there is something about the concept of explanation that seems objective, that is, not dependent on the information and conceptual world with which we are equipped at a given moment. One may know or not know a certain explanation, but the fact that it is an explanation seems always true.

We are seeking the objective and enduring element in explanations, and, as we have seen, reduction to the familiar does not give it to us. In the next section we will argue that this objective element is connected with the concept of cause.

Explanations and Causes: The Deductive-Nomological Schema

The accepted model of scientific explanation is the deductive-nomological model proposed by the philosopher of science Karl Hempel. In fact, this is also the principal structure of explanation in everyday senses as well.

The deductive-nomological explanation is the proposal of a system of assumptions from which one can logically derive statements describing the events we wish to explain. For example, when we want to explain tides in terms of gravitation, we say that the explanation is a valid logical argument whose conclusion is a claim about the existence of tides. The structure of the argument is as follows:

• General premise: Every two masses attract one another with a force whose magnitude is proportional to the product of the masses and inversely proportional to the square of the distance between them—the law of gravitation.
• Particular premise: Water has mass, and the moon has mass.
• Additional particular premise: At certain times, the moon comes closer to the earth.
• Conclusion: At those times, the gravitational pull exerted by the moon on the waters of the sea increases, because the distance between them decreases, and therefore the force, which depends inversely on the square of the distance, increases.

This is a deductive argument, containing at least one general premise—there may be several—and several particular premises—there must be at least one. The conclusion of the argument is a certain phenomenon. It may also be a particular fact, such as a particular bottle falling to the ground, or some natural regularity, such as tides. That conclusion is the fact being explained.

This is the deductive-nomological schema of explanation. Deductive—because of the character of the argument. Nomological—because at least one of the premises is a generalization that is a law of nature, a nomos. We seek a general law from which we can build a deductive argument, one of whose premises is that law, and whose conclusion is the fact being explained.

But one can propose similar argument structures that we would find very hard to call an explanation. I will point here to two such cases. The first, which Hempel himself brings, is the following argument. It is meant to explain why a certain stone contains iron:

• General premise: All the stones in this box contain iron.
• Particular premise: This particular stone is in this box.
• Conclusion: This stone contains iron.

The reason this argument does not seem to us an explanation is that the general premise is an accidental generalization, not a law of nature. An indication of this is that we could not infer from the premise the following conclusion: if some stone is placed in this box, it must contain iron. The premise points to a contingent general fact, not to a necessary one.

There is an additional element in this argument that prevents us from treating it as an explanation. To sharpen this, let us illustrate the point with an argument whose premise is a law of nature, and not a contingent generalization. This is the second kind of argument that we still would not treat as an explanation. The argument has the following form, and its aim is to explain why Simon is mortal:

• General premise: All human beings are mortal.
• Particular premise: Simon is a human being.
• Conclusion: Simon is mortal.

It is clear that here too we are not offering an explanation of Simon’s mortality. But unlike the previous case, here the general premise is indeed a law of nature, and certainly not a mere accidental generalization.

The reason it is difficult to see such an argument as an explanation of Simon’s mortality is that in an explanatory argument, the general premise is the cause of the event described, whereas in the argument about Simon, the general premise is not the cause of his being mortal, but a generalization of which he is a particular case. That generalization is a premise explaining why we should be convinced that he is indeed mortal, but it is in no way an explanation of his being mortal. Thus we see that a premise that is not a cause will not constitute an explanation.

That is, it belongs to the essence of explanations that they offer causes for the events or phenomena explained by them. Here we encounter in sharp form the intimate connection between explanation and cause.

In fact, here lies the objective, trans-temporal element we were seeking within the concept explanation. An explanation is not merely a generalization, accidental or essential, from which the explained fact is logically derived. An explanation is the finding of a general law that constitutes a cause—a physical cause, not merely a logical condition—that produces the particular events it explains. After the discovery of gravitation, we know that this is the cause—the explanation—of the cluster of events we had observed in the past as well. It is not reduction to the familiar, but it is the finding of a cause, and in that sense it is an explanation.

This is also the explanation for why the discovery of a new law of nature is itself called a scientific explanation, just like reduction to the familiar. The reason is that in both cases we find causes for the events being explained. Sometimes the causes are familiar to us, and sometimes they are not. The question how we verify that we have found the correct cause, if it is still unfamiliar to us, was discussed in the second gate—eidetic seeing, auditory reason, and so on. Here we are only pointing to the scientific process as it actually occurs.

It seems that the difference between explanation by generalization and explanation by cause lies in the fact that each belongs to a different stage of scientific activity. When we collect particular cases and generalize them into a general law, we find a law that is their generalization, not necessarily their cause. In the second gate—see there, chapter 1, in the section “Two Types of Theories”—we called this a phenomenological theory. Such a state is reflected in the statement that a certain body falls toward the earth because all massive bodies fall toward it. The claim that “all massive bodies fall toward the earth” is a generalization describing the collection of facts available to us by means of an inclusive law, but there is no producing cause here. The claim that at the basis of this phenomenon stands a law of nature—the law of gravitation—which causes it in a physical, not merely logical, sense, depends on whether we can find a theoretical explanation for that law.

When we explain an empirical law of nature, or a phenomenological theory, we do so by means of an essential theory. The theory states, in our case, that all massive bodies fall toward the earth because of the action of gravitational force. This is a theory that explains the empirical law, and at this stage we reduce the familiar to the unfamiliar, as we saw above.

A theory, as we saw in the second gate, generally uses theoretical entities that are not familiar to us, and therefore require conceptualization and definition. These entities are the unfamiliar upon which the theory rests. Here, of course, we already reach the concept of cause. The theory is supposed to provide causes for the phenomena described by the general empirical law.

Thus, scientific activity proceeds through the following stages:

1. We observe a collection of particular phenomena—for example, several massive bodies that we have seen fall toward the earth.
2. We generalize this into an inclusive empirical law—for example, the law of gravitation: all bodies with mass fall toward the earth. This is the phenomenological theory.
3. We explain the inclusive law by means of a theory: there is a gravitational force acting between any two massive bodies. This is the essential theory.
4. From this we can infer new conclusions, such as: a similar force will act between other massive bodies, not just the earth. In this way, for example, we can explain the motion of celestial bodies.
5. Finally, after we have become convinced that such a law of nature exists and that the theory is indeed correct, we can explain new particular phenomena, such as tides, by means of it. At this stage our relation to it is as reduction to the familiar.

At stage 2, it is not necessary that the generalization provide a cause for the particular cases. It may be a non-causal, descriptive generalization, like the claim that all human beings are mortal. At stage 3, we look for a theory, and here a causal element is already required. As we saw in the second gate, at least in the synthetic picture of science, the theoretical entities are the causes of the occurrences described by the general empirical law, and not merely an improved description of them.

This is a schematic description of the process of the formation of a scientific theory, as the summit of the process of scientific explanation.

Already in the second gate we encountered the problematic nature of scientific generalization and the doubt regarding the correctness of the theory chosen, since there are many other possible theories for any given set of facts. We also saw there the problematic character of assigning ontological status to theoretical entities. We will now try briefly to understand how these phenomena, which were described there in a general way, relate to the process of the formation of scientific theory as described above.

At stage 1, we must decide how to divide the facts into relevant groups. This is how scientific facts are determined—we saw this in the second gate also with regard to historical facts. We discussed this in the second gate, where we saw that it can be done in several ways. We pointed out that the division is theory-dependent, and therefore this stage depends on feedback from the stages that come after it.

At stage 2, the phenomenological stage, where we create an empirical generalization out of the set of particular facts belonging to a certain group—the result of the division carried out in the first stage—an inclusive empirical law is formed. As stated, this law is not an explanation of the relevant phenomena, but their generalization. Since this stage involves generalization, it is clear that its result is not certain. The reason may be an error in the generalization, or the possibility of other generalizations that could also be correct. At this stage in the process the problem of scientific induction appears.

At stage 3, the essential stage, we create a theory that will explain, in causal terms, the empirical law formed in the previous stage. Here there are countless different possibilities for constructing a theory that will fit the laws and facts. The choice among the different theories, all of which fit the facts currently in our possession, is another of the problems we pointed to in the second gate. Here too appears the problem of the existence of theoretical entities, since these are not necessarily observable entities—for example, gravitational force is not something that can be directly observed—and we encounter them only indirectly, through the results of their actions.²³

The last two stages in the process described above—4 and 5—do not belong to the process of creating the theory or the scientific explanation, but rather constitute the use of the theory. These are logically trivial stages—certainly not practically trivial—assuming the chosen theory is indeed correct. In the second gate we called them the context of justification, which is a purely logical stage, as opposed to the context of discovery, which cannot be reduced to logic. If we fail at these two stages, then the theory has been refuted, and we must abandon it.

Thus, these stages constitute an empirical test of the theory—the context of justification. Of course, in the discussion of Kuhn’s doctrine we saw that even this is a somewhat naive picture, since the stage at which we decide that no explanatory fruit can be expected within the framework of a certain theory cannot be sharply defined in logical terms.

Phenomenology and Essence

Now, in order to return both to the problematic nature and to the need for parallel planes of explanation, one must notice another interesting phenomenon concerning the relation between stages 2 and 3 in the description above. There is another difference between these two stages. The empirical law can be tested, and therefore refuted, empirically and experimentally. One can test empirically whether every body falls toward the earth. But the theory that is the result of stage 3 cannot be directly tested empirically. It is very hard to think of an experiment that would test whether there is a gravitational force that causes bodies to attract one another.

From this one can understand the fact that in the history of science empirical laws generally do not undergo change—see Open University, Unit 4, pp. 193–195. What turns out to be an empirical law almost always remains true, certainly among the fundamental laws. The reason is that such a law is nothing but a generalization of facts that we observed directly. The component that changes in scientific explanation is the essential theory that explains those laws. Within this framework one may sometimes become aware that the domain of application of empirical laws is only partial. For example, after the discovery of the theory of relativity, it became clear that the relations between mass, velocity, and acceleration, and the trajectories of bodies under the influence of forces, are somewhat different from what we thought under Newtonian theory. But in the domain of low velocities, Newtonian theory remains correct.

The same is true of quantum theory, which shows that Newtonian laws are not valid in sufficiently small systems—for example, a single elementary particle, or an atom, or even a molecule. But those laws remain correct in the domains of macroscopic bodies.

At first glance, this claim may seem obvious and simple, for if we verified something directly, clearly it will always be found correct. But in fact it is far from trivial. As we saw in the second gate, and also above in the present gate, empirical laws too are not mere collections of facts. Generalization involves sorting and classifying facts, and then generalizing them. These processes involve quite a bit of speculative thought, which cannot be justified on the purely logical plane. In the next observation we will see this through several examples.

Observation 28: Phenomenological and Essential Theories

The distinction between stage 2 and stage 3 is not always clear, not even to scientists working in the field. Here I will present a few examples of theories that are in fact empirical laws, yet involve no simple speculative element.

The first example is Zipf’s law.²⁶ George Kingsley Zipf, who taught German at Harvard, pointed out in the early 1930s a whole series of regularities that share a number of similarities. Zipf’s laws deal with ordered lists of quantities—for example, the population sizes of cities or countries, the distances between stars, the sales volumes of business firms, the most common words in a given text, and so forth—when these are arranged by size. It generally turns out that the quantity stands in inverse relation to the item’s rank in the list. For example, if we arrange the population sizes of cities in the United States in descending order, we discover that the number of inhabitants is inversely related to their place in the list. The ratio between the population sizes—or business volumes, or stellar distances—will be proportional to the following series: 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9…

These are called scale laws, and there is a whole family of such laws. Benoit Mandelbrot, the well-known mathematician and discoverer of fractals, pointed to a more comprehensive structure of such power laws. These more refined structures are obtained by adding a power (N), not necessarily an integer, and also a constant to the series, as follows:
1/(c+1)^N, 1/(c+2)^N, 1/(c+3)^N, 1/(c+4)^N...

These Zipf-Mandelbrot laws have no convincing general explanation. But in many fields, explanations have been developed as to why the behaviors are precisely of this type. The most prominent example is the field of chaos in physics, where a considerable part of the research is devoted to finding various power laws and explaining them.

Zipf’s law is an example of an empirical law, since it is a proposal to generalize phenomena into a general law, even though we have no explanation for that generalization. It is an attempt to organize the known facts within a more coherent and compact theoretical framework, even though there is no attempt here at explanation. Such a foundation is, of course, a more convenient framework for future explanation. Once there is an empirical law, it is easier to search for explanatory directions, because we need find one theoretical mechanism rather than a separate theory for each member of the series under study.

It is important to notice that this generalization could certainly have turned out false. Even if it is tested on the first ten cities in the United States, it could continue differently in the next ten cities, and so forth. The same is true of word frequencies, business volumes, and the like. That is, stage 2 discussed in the text above is not free of the possibility of error. Refutation and the need for change can appear at this stage as well, not only at the stage of theoretical explanation, stage 3.

Another example of an empirical law is the theory concerning the number of petals on flowers. There is a well-known thesis that their number generally belongs to the Fibonacci sequence.²⁵ Leonardo Fibonacci was an Italian mathematician in the twelfth century, and he played with a certain number sequence in the framework of his studies of rabbit populations. This sequence is defined as follows: it begins with 1, 1, and thereafter every number is the sum of the two preceding ones. The beginning of the sequence is: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…

It turns out that this sequence is an inexhaustible source of empirical laws. For example, the florets in a sunflower are arranged in two intersecting spirals, one clockwise and the other counterclockwise. In each species of sunflower, the numbers of florets in the two spirals are consecutive numbers in the Fibonacci sequence. There are sunflowers for which the numbers are 34 and 55, and there are others for which the numbers are different. A similar structure appears in the protrusions on the skin of the pineapple—its “scales”—and elsewhere.

There is a theory that the number of petals on flowers belongs to the Fibonacci sequence. This is not always true, but it is true for very many kinds of flowers, and therefore it is unlikely to be a coincidence.

All these are empirical laws, generalizations of the phenomena we know into general mathematical laws. It should be noted that this is a fairly speculative generalization, and therefore it could certainly turn out not to be correct.

A final example of an empirical law is Balmer’s law.²⁸ In the last quarter of the nineteenth century, physicists were looking for a theoretical explanation of the spectral lines in the absorption and emission of different gases. In 1885 a Swiss schoolteacher named Balmer proposed a formula for the wavelengths, lambda, of the emission spectrum lines of hydrogen. On the basis of measurements made by Angstrom, Balmer examined four spectral lines and proposed the following formula for them:

lambda = B * N^2 / (N^2 - 4)

Here B is a known constant, and N is an integer greater than 2. When one substitutes the numbers N = 3, 4, 5, 6, one obtains the four lines measured by Angstrom. Balmer claimed that if one substituted other numbers—integers greater than these—one would obtain additional spectral lines of hydrogen.

Balmer did not know that already in his own time several more such lines had been measured, and indeed all of them fit the formula he proposed. Today the first 35 hydrogen lines are known, and all are obtained by substituting larger integers into the variable N that appears in Balmer’s formula.

This is a completely nontrivial empirical generalization, and it turned out to be entirely correct. Only later was a convincing theoretical explanation of this formula proposed by the well-known Danish physicist Niels Bohr, who viewed the atom as built from a nucleus around which electrons move in circles of increasing radius. This theoretical explanation—stage 3—already contains many theoretical entities, and in fact laid one of the central foundations that led to quantum theory as developed in the first half of the twentieth century. The process continues from a more essential generalization—Bohr’s—to a truly essential theory—quantum theory. This is a multi-stage process, not necessarily the two-stage one that may have been suggested by the description so far.

This collection of examples can convince the reader that even the generalization in stage 2, the generalization that leads to the empirical law, is not trivial. Some even call it a theory as well, but it is a phenomenological theory—that is, a theory of describing phenomena, not of explaining them.

The generalizations that lead to the claim that all human beings are mortal seem more innocent, but even there too there is a speculative component, as in every generalization. Why decide that all human beings are mortal? Perhaps the claim applies only to human beings who have not eaten a certain food, or who lived in a certain period, or whose skin color is such-and-such. Here too the generalization is nontrivial, even if it appears more obvious at first glance.

The examples given here convince us that generalization is not a simple intellectual move, but one containing a great deal of speculation.

This character of generalizations leads to the conclusion that, unlike logical-deductive inferences, they are not certain, and therefore there is no necessity that they turn out true. And indeed, some empirical generalizations do not remain true over time and in all situations. But the fact pointed out above—and well known to anyone engaged in science or in its philosophy and history—is that many of the fundamental empirical, phenomenological generalizations in science have remained true over time, whereas only the essential theories, stage 3, are what change as science advances. This fact calls for interpretation.

We see here that theory is not merely a convenient description of reality, as the analytic position claims. There is no reason why a convenient description of partial reality should turn out true for additional cases that were not examined when the generalization was formed. The fact that the main phenomenological descriptions remain correct means that human beings have ways of discerning various truths, even if they are not proved logically. Generalizations are not necessary, but the fact that generalizations—and especially the central generalizations of science—often turn out true is itself a powerful confirmation of the synthetic position, which distinguishes between truth and provability. Somehow—through auditory reason—we discern truth even before it is formulated by us in the essential theory. In the second gate we pointed to this phenomenon, and discussed the support it gives to the synthetic position.

Our conclusion, then, is that empirical generalizations are not trivial. Precisely because of this, the fact that empirical laws remain correct even when the theory changes is a genuine confirmation of the synthetic position. If these generalizations were nothing but ways of viewing things, there would be no apparent reason for them to remain correct over time. The fact that these generalizations remain valid even with respect to cases not observed by us when we formulated them indicates that the generalization is based on an intuitive apprehension of reality, prior to the formulation of the law and the theoretical explanation of the phenomena. It is not merely a formal trick of presenting data. For further detail, see the second gate. Once again we encounter the fact that what appears to adherents of the analytic position as mere thinking, and therefore says nothing about reality, turns out to be an act of cognition, and therefore can indeed constitute a synthetic statement about reality itself.

Phenomenological generalizations are a type of synthetic-a-priori claim. Giving up the distinction between cognition and thought is what enables us to uphold a synthetic position and reject the claims of the analytic thinker. As we see once again, this is the only possible rational basis for our trust in science. In the next section we shall see that the coin has another side: the synthetic position is also the basis for the problematic character of adopting parallel planes of explanation.

Analyticity and Syntheticity: Returning to the Problem of Parallel Planes of Explanation

As we saw in the second gate, the analytic and synthetic positions interpret differently the meaning of theories and of theoretical entities. The analytic thinker treats theory as a simpler and more convenient form of describing the totality of facts. There is nothing in theory beyond the description of the facts. The theoretical entities do not really exist; they are only tools for the convenient—not necessarily correct—description of reality. The course of the formation of scientific explanation, as described above, leads to a more convenient and more general description of reality.

It should be emphasized that according to the holders of the analytic position, there is no difference at all between stage 2 and stage 3. Both are descriptions of phenomena, not explanations of them. On their view, even the essential theory is nothing more than a refined description of reality—that is, a refined phenomenological theory.

The synthetic position, by contrast, regards the phenomenological theory—the empirical law—as a generalization that hints at something about reality. In itself it is only a description of the facts, but adherents of the synthetic position regard this description as true, not merely as convenient and efficient. Only because of this conception is there any point in seeking a theoretical explanation for the phenomenological empirical law. The main evidence that these generalizations—stage 2—are true is that we find theoretical explanations for them—stage 3—which also constitute discoveries of a deeper layer of reality, not merely descriptions of the facts. The synthetic position treats theory, as well as the theoretical entities within its framework, as having ontological significance. The theoretical entities exist, and the principles of the theory reveal a depth-layer of reality, so long as the theory is valid. The course described above leads, according to the synthetic conception, to a deeper cognition of reality.

We can now understand the problem involved in the existence of parallel planes of explanation. If theory were only a convenient and refined, but fictitious, description of reality, and not a statement about reality itself, then the theoretical entities would not differ essentially from mythic entities, as the analytic position indeed views them. In such a state, one could carry out a postmodern rehabilitation of myth and religion. Myth and science would stand on the same platform of describing reality, and the choice between them would not be made in terms of true and false, but in terms of language and efficiency, in essence according to the answer to the question what interests us at the moment: a theological explanation or a physical one. According to the analytic position, explanation has no ontological status whatsoever.

Therefore, within such a framework, the problem of parallel explanations does not exist. We do not need explanations at all, but their existence creates no problem. They are different ways of describing the facts, different languages, as it were, and they do not touch reality. Obviously, according to such an approach, the two theories are not descriptions of two contradictory realities. We already noted that according to the analytic position, occurrences have no causes at all, only descriptions. Thus, according to the analytic conception, the objective component within scientific explanation—namely, the producing cause—does not really exist.²⁷

The problem of parallel explanations, as described above, arises from the synthetic conception because of two considerations:

1. According to the synthetic position, the theoretical entities in scientific explanation exist, and they are the producing cause of the phenomena. Therefore we must decide what the real producing cause of the phenomena is.
2. The conception of explanation as a necessary and sufficient condition of the explained fact is also derived from the causal dimension in explanation. If that does not exist, the problem disappears by itself.

But according to the synthetic position, theory describes reality itself, and the theoretical entities are the deep causes of factual occurrences and exist in reality itself. Thus, explanations contain an objective dimension of causes for occurrences. Therefore specifically according to the synthetic conception, the problem of parallel explanations arises in all its force. The question is who describes reality correctly, and what are the correct causes—the necessary and sufficient conditions—for the occurrences we observe: the mythic causes or the scientific ones. Which entities actually caused the occurrences we observe?

In logical formulation, this may be stated as follows: a necessary and sufficient condition must always be a unique condition. This is the essence of the problem of parallel explanations, as described above.

For example, on the physical-theological axis that we saw in connection with Newton’s apple, the analytic thinker holds that there is no gravitational force in reality. It is merely a convenient and efficient description of the phenomenon that massive bodies tend to attract one another. He thinks that we have no explanation of this phenomenon, and that the assumption that a force of attraction lies behind it is only a convenient option for describing the facts themselves—namely, that bodies attract. Thus it is entirely reasonable, on his view, to say that the true cause, or explanation, of the apple’s fall is punishment from God for Newton’s sins, for cause and explanation do not contradict description. This is the postmodern rehabilitation of religion and myth, which rests on undermining the status of scientific explanations and turning them into descriptions. Of course, the analytic thinker does not stop there. As we saw in the first book, he treats religion and myth in the same way: they too have no higher ontological status than the theoretical entities of science. Therefore talk about punishment from the Holy One, blessed be He, is likewise no more than a subjective description, not a statement about reality. As we saw in the first book, analyticity is generally—and not accidentally—associated with the social Left, and tends to oppose religions and myths. It is willing to accept them as legitimate, but only in their postmodern sense.

On the other hand, it is specifically for the adherent of the synthetic position—who, as we saw in the first book, is the one inclined to believe in religions and myths—that a problem arises here. The adherent of the synthetic position regards the explanation for the attraction of bodies, which attributes it to the existence of gravitational force, as a claim about reality, not merely as a description. Therefore the problem of parallel explanations presented above arises for him in full force. If gravitational force is indeed the explanation for the mutual attraction between massive bodies and for their falling toward the earth, then it is a necessary and sufficient condition for falling. The fall will occur even if Newton did not sin. And conversely, even if Newton did sin, if the physical conditions for falling are not met, it will not occur, even though he deserves punishment. The same is true of all the other examples.

Hempel on the Difference Between Mythic Explanation and Scientific Explanation

To sharpen the problem, let us now examine the differences between mythic explanations and scientific explanations as presented by Hempel himself in his book. The adherent of the synthetic position, who takes science seriously as a way of knowing the world, will seemingly have to reject mythic planes of explanation in favor of scientific ones. In his view, science is a systematic and rational way of understanding the world, and therefore it is preferable to mythic and religious alternatives.

In this section we will see that on the purely logical plane this way out is problematic, since there is no real difference between the two kinds of explanation. Therefore one cannot justify a priori the decision to reject one in favor of the other. This strengthens the need to accept parallel planes of explanation simultaneously.

At the beginning of chapter 5 of his book Philosophy of Natural Science, Hempel offers a brief description of mythic explanations and attempts to distinguish them from scientific explanation. Let us briefly follow his arguments there, and we shall see that they have no real logical justification.

Hempel begins by describing the bewilderment of ancient humanity as it tried to overcome its lack of understanding of the natural phenomena around it—a lack of understanding that also leads to fear and a sense of helplessness—by means of anthropomorphic explanations of the forces of nature. These explanations attribute natural phenomena to all sorts of hidden, mystical agents operating in the world—a creating God, or, alternatively, gods, demons, and so on—as if they were the ones operating natural phenomena. Hempel then writes:

Attempts at explanation of this sort may no doubt give the questioner a feeling that he has attained some understanding and that his bewilderment has been relieved, and in this sense they may “answer” his questions.

But however satisfying such answers may be psychologically, they do not satisfy the needs of science. For science seeks to develop a conception of the world that bears a clear and logical relation to our experience, and which can therefore be subjected to objective testing. For this reason, scientific explanations must satisfy two systematic requirements, which may be called the requirement of explanatory relevance and the requirement of testability.

Hempel distinguishes between answering distress and lack of understanding, on the one hand, and explanation, on the other. The purpose of explanation, he says, is to answer the scientific need. But he does not address a very important question at all: which of the explanations is correct, the scientific one or the mythic-religious one? Seemingly, if the mythic-religious explanation is correct, then the scientific explanation, despite satisfying all the parameters Hempel sets for it, could turn out false. Hempel assumes that the goal driving us toward explanation dictates the type of explanation we will accept, but he does not address the question of reality. Which explanation correctly describes reality?

Hempel next argues that the distinguishing feature of scientific explanation, as opposed to other kinds, is its satisfaction of two requirements: first, explanatory relevance, meaning that the explanation offers a framework from which the explained result naturally follows, generally by deduction; second, testability, meaning that the explanation is open to empirical testing.

As Hempel shows, the explanatory schema meeting these two requirements is the deductive-nomological schema, which we have already presented above and which Hempel proposes as the general schema of scientific explanation. An explanation is the finding of a general law from which the explained fact can be derived through deductive processes.

Again, however, we must ask ourselves: what about cases in which the explanation is not empirically testable? Can no explanation be supplied for them? Or are they simply excluded from science? Can such explanations coexist with scientific explanation, on the assumption that these are two different planes of explanation?

We will deal with the issue of parallel explanations later in this gate, but here we wish to point out that in fact Hempel’s two requirements also do not categorically distinguish scientific explanation from non-scientific explanation.

When Hempel wishes to illustrate an explanation that does not satisfy the requirement of explanatory relevance, he brings the objection of the astronomer Francesco Sizi, a contemporary of Galileo, to Galileo’s claim that Jupiter has moons orbiting it. Sizi writes as follows—see Hempel, p. 56:

There are seven windows in the head: two nostrils, two ears, two eyes, and a mouth. Likewise in the heavens: there are two favorable stars, two beneficent stars, two luminaries, and only Mercury is undefined and indifferent. Hence, and from many similar natural phenomena, such as the seven metals, etc., which there is no reason to enumerate, we infer that the number of planets must necessarily be seven…

Moreover, the satellites are invisible to the naked eye, and therefore cannot influence the earth. Therefore, if they existed they would be useless; therefore they do not exist.

Hempel chooses, not by accident, a very strange and esoteric objection in order to illustrate his claims,³⁰ and therefore we too shall focus specifically on Sizi’s claims, even though they certainly do not seem reasonable at all. Our goal is to show that the sharp criterion Hempel sets is not sharp at all, and if we reject Sizi’s claims, it is not because of some sharp scientific-philosophical determination, but because they simply do not seem plausible to us. In other words, this is not the rejection of myth in favor of science, but the rejection of an implausible myth in favor of a more plausible myth.

Sizi’s words are composed of two different arguments:

1. Every basic collection of objects or phenomena in nature consists of seven items, divided in their properties into three pairs plus one isolated item. Therefore the stars too must total seven. This is an empirical, phenomenological law, and therefore it need not possess explanatory relevance. Is the claim that the number of petals of some plant must belong to the Fibonacci sequence more scientific?
2. The second argument is apparently based on a theological conception: whatever is not useful to our globe—or to the human beings upon it—necessarily does not exist. Here too it should be noted that assumptions similar to this one exist in several scientific fields, including physiology and others.

Generally speaking, one may say that Sizi’s claim is fully testable. One can test and see whether there are more than seven planets. If Galileo did in fact observe them, then either Sizi is mistaken, or Galileo’s observational procedure requires correction. In any case, we are dealing here with a fully testable claim.

Beyond that, given the assumptions Sizi lays down, the conclusion that there are no more than seven planets follows deductively. True, not all the assumptions are made explicit in the argument, but Hempel himself points to many scientific arguments that are presented in incomplete form—he calls this “presentation of the bare skeleton of the argument”—and in order to arrive at a deductive structure, we must complete what is missing, that is, what remains implicit in the argument. This is a difference between everyday language and pure logical language. Thus Sizi’s explanation is of the deductive-nomological type, and therefore it also satisfies the requirement of explanatory relevance.

As stated, one may certainly agree that Sizi’s objection is not plausible. But from there to the claim that there is a sharp definition distinguishing his arguments from scientific explanations is a long way. Sizi’s arguments are implausible, and therefore it seems reasonable to reject them, but this does not necessarily apply to every non-scientific explanation.

Incidentally, Sizi could have been mistaken at several points and still left his principal metaphysical approach intact. In his second argument, for example, if we give up the assumption that objects must be useful specifically to the earth, one can continue to claim that only useful things exist. Very few believing people, it seems, would disagree with such a claim. One can also give up the assumption that what we do not see with the naked eye does not affect the earth, and again leave Sizi’s basic principles intact.

It should be noted that the latter assumption—that what we do not see with our eyes does not affect the earth—is not connected with any theology or myth. It is a completely scientific assumption, and one open to empirical testing—although today it is quite clear that it is false. Thus Sizi’s errors did not necessarily stem specifically from the myth he believed in, but from false scientific, empirical assumptions that he made. The mythic assumptions—such as the assumption that only useful things, or things that affect us, exist—could very well remain standing.

With regard to his first argument, one could say that indeed in every basic order of reality there are seven fundamental objects. The error in applying this to the number of planets could certainly arise from Sizi’s assumption that each such object cannot subdivide into several more objects. One could also claim that the number of planets does not belong to the fundamental structure of the universe, and therefore this general law does not apply to it.

Of course, the arguments we raise here in order to save Sizi’s worldview may be seen as ad hoc arguments. But as every philosopher of science knows, science itself is full of such claims, whose purpose is to save theories that science cherishes. There this is called articulation. See above, in the third gate. There we noted that myths too undergo articulation, and these are possible examples of such a process.

In addition to everything said here, it should be noted that Sizi’s claims are based on assumptions that are certainly testable. One can test whether every fundamental structure in reality contains seven items. One can also test whether only things useful to us exist in the world, and so on.

Thus, explanations that attribute various events and phenomena in the world to the activity of mystical agents can certainly possess explanatory relevance, and certainly can be empirically testable. Sizi’s first argument is admittedly not explanatorily relevant, but that is only because it is a generalization—an empirical law—not a theory. He generalizes the phenomena he sees into a general law saying that every basic structure in the universe is composed of seven items. This is a perfectly valid generalization, no less than Fibonacci sequences or Balmer’s formula—though probably false.

For example, the theological claim that God punishes transgressions can be tested empirically. The claim—apparently false—that there is no “righteous person who suffers,” or no “wicked person who prospers,” can also be tested empirically. The claim that if one prays, rain falls, is likewise completely testable. Hence empirical testability cannot serve as a sharp criterion distinguishing scientific explanation from mythic or religious explanation.²⁹ Such claims certainly also possess explanatory relevance: the One who created the world wants us to worship Him, and therefore He makes the abundance He bestows upon us conditional on our doing what is required of us. Is that not a good explanation of the phenomena under discussion? Of course, one must now examine whether it is correct or not, but the same is true of every theory. The discussion here concerns the scientific character of a theory, not necessarily its truth. For this distinction, see the discussion of Karl Popper’s doctrine at the beginning of the second gate.

Summary

The conclusion of the present chapter is that according to the synthetic position, a theoretical explanation refers to reality, and theoretical entities can indeed be existing entities within the synthetic picture. It is precisely because of this view that the question arises so sharply: how can one hold mythic-religious explanations alongside and in addition to scientific explanation? At the end of the chapter we saw that mythic explanations are not necessarily based on a logic different from scientific ones, and therefore one cannot determine a priori that one of them must be rejected in favor of the other.

The next chapter will deal with the question whether, and under what conditions, it is really possible to hold both kinds of explanation—or several parallel explanations in general—together.

Chapter Three: The Simultaneous Adoption of Parallel Planes of Explanation

Introduction

This chapter is the central part of the argument in the present book, and it will present various theoretical possibilities, or models, for adopting several parallel planes of explanation simultaneously. As we have explained, this is an important focal point of the synthetic alternative in its relation to science, religion, and myth. Some of the possibilities will turn out to be related to one another, and perhaps even identical with one another. Some may describe different situations—that is, for one kind of dilemma we may adopt one model of parallel planes of explanation, and for other kinds of dilemmas we may adopt other models.

We are speaking of a situation in which we have two different explanations for the same phenomenon, both formulated in different ways and attributing the explained phenomenon to different causes. We saw that according to the synthetic position such a state is problematic, because every causal-synthetic explanation contains a different necessary and sufficient condition for the occurrence of the result. More than that: in the synthetic picture, an explanation reflects an existing reality, not a form of description. We are therefore dealing with two different realities proposed as causing the phenomenon under discussion.

We will now present, one by one, the different ways that nevertheless make it possible to adopt two different explanations simultaneously, the meaning of each of them, the connections between them, and also their domains of application. The order in which the different ways are discussed runs from the lightest to the heaviest, or the more complex. One may view this order as expressing the degree of syntheticity in the different models—this will become clearer to the reader as the discussion proceeds. The discussion of some of them is broad and even divided into subsections. The different ways are therefore numbered in ascending order. For the reader’s convenience, we list them here:

1. Contradiction.
2. Description and explanation.
3. Model and explanation.
4. Cause and purpose.
5. Complementary explanations.
6. Local and global fit.

1. Contradiction

The first possibility is to adopt the initial intuition presented above and decide that there really is an intolerable contradiction between the different explanations, and that we must give up at least one of them.

This possibility divides into two: complete abandonment of one of the possibilities, or adoption of one of them as a description—a phenomenological theory—but not as a genuine causal explanation—an essential theory. The second possibility will be described below; it is connected to the next two modes, 2 and 3.

It should be noted that this possibility adopts toward the rejected theory a mode of action that expresses an analytic attitude to scientific theory. We give up the assumption that the theoretical entities included in that theory exist, and we also give up the ontological status of the principles of the theory. In other words, we treat the theory as refined phenomenology, not as a genuine explanation in the causal sense.

In the previous book we saw that analytic thinking is also used by adherents of the synthetic position. The difference between them and their analytic colleagues concerns only the exclusivity of this mode of thinking. The synthetic thinker is willing to give up proofs and analytic modes of thought, but he certainly uses them when he needs to do so. The application of this principle to our present issue is that adherents of the synthetic position do not become mystical believers in science. It is clear to them that science is indeed an important way of knowing the world, but like every other human activity, it can contain errors. Likewise, scientific theories do not deal only with ontology. Sometimes we must conclude that one of the theories in our possession does not describe reality itself, in the way the analytic thinker would treat it.³²

Sometimes there is no escape but to reject one explanation in favor of the other. Which one will we choose, and which one will we reject? That is already a question that cannot be answered in a general way. It depends on which of them we trust more, and which belongs to more basic planes within our overall outlook.³¹

It should be noted that although we open our survey of the different possibilities specifically with this option, as a rule this will be the last course of action we choose to take. Only if we conclude that all the other escape routes detailed below do not exist for us in the case at hand will we be forced to choose one explanation and reject the other. The reason is that a situation in which two planes of explanation both seem relevant to the phenomenon under discussion arises because we have good reasons to believe each of them. Several examples of this were presented at the beginning of the present gate. Therefore we will tend to “turn every stone” in order to examine the possibility of adopting both. Rejection of one of them will take place only when we have no other reasonable way out. This is an important point, and the reader should bear it in mind when examining the next possibilities that will be proposed below. Even if some of them seem forced in certain cases, he must decide whether it is not more problematic, in his eyes, to give up one of his fundamental beliefs.

2. Description and Explanation

Above we saw that the explanations of science itself divide into phenomenology and essence. There are scientific laws that constitute an inclusive description—admittedly somewhat speculative, by the very fact that we are dealing with generalization—of the experimental facts in the domain at hand. This is an empirical generalization, or phenomenology, which does not claim anything about reality, but offers it a convenient, inclusive, elegant—that is, simple—and efficient description. On the other hand, there is a theoretical plane that purports—at least according to the synthetic interpretation, as we saw above—to offer a causal explanation, and not merely a description, of the events described by the empirical law. This explanation is formulated by means of theoretical entities and theoretical principles, which generally are not directly observable.

Sometimes, when we encounter two different planes of explanation, we can distinguish that one is a theoretical explanation and the other is an empirical generalization. Therefore there is no contradiction between them, for the theoretical explanation is nothing but the causes of the validity of the empirical law. Niels Bohr’s explanation does not contradict Balmer’s formula; it is the explanation of why it is correct. Balmer’s formula is not an explanation but a description, and therefore its very existence does not stand in opposition to any other explanation, theoretical or mythic.

A full adoption of such a doctrine with respect to contradictions between science and myth or religion leads to an analytic conception of science. According to this conception, science and scientific theories are always descriptions of reality, not explanations of it, and therefore they can never contradict religion or myth.

A blunt and extreme example of such a position is psychological behaviorism, discussed above at the end of the second gate. The more extreme adherents of that approach claim that all theoretical entities in psychology describe no reality whatsoever. The purpose of psychology is to describe the facts, and nothing more. The psychologist ought to analyze a case before him only through the phenomena he sees directly, without recourse to planes that are not open to observation.

This places psychology under a stricter standard than that of the natural sciences. Those sciences, as we have seen, do not hesitate to use entities and principles that are not directly observable—abstract theoretical entities—in order to explain and describe various natural phenomena. Without the use of theoretical means, science could not have advanced at all. The illusion that psychology could progress in a more formal way than the mathematical natural sciences is rather naive.

As stated, adherents of the synthetic position described in this book do not proceed along this path exclusively, although sometimes this direction is also possible for them. Clearly, full adoption of this mode of approach takes us back to complete analyticity, or to apologetics. In what follows we will present more substantial synthetic alternatives.

This distinction, between phenomenology and essence, leads us to a brief discussion of the relation between semantics and syntax—a relation that will continue to accompany us later as well, and therefore deserves some brief treatment here.³⁴

Preliminary Discussion: Semantics and Syntax

It is well known in various fields that many systems can be described on two different planes, one called syntax—structure or form—and the other called semantics—meaning. We will illustrate this distinction by means of an entertaining game presented in the following observation. The reader is advised not to skip it, in order to better understand the discussion that follows.

Observation 29: The MIU System³³

The MIU system is an example created by Douglas Hofstadter in order to show the importance of isomorphic correspondences. For our purposes one can see here two parallel planes of reference, whose relation to one another is that of semantics to syntax.

In the next chapter we will see that when we are unable to make progress in understanding certain phenomena, it is specifically ignoring meaning—the semantics—and focusing on external, formal shape—the syntax—that can advance us. The MIU system is a good example of this claim as well.

The MIU system is a kind of language,³⁶ whose alphabet consists of three letters: M, I, and U. The language itself is a set of “words,” which are simply strings built from these three letters. As in every language, not every string built from these letters is a legal word in the MIU language. The way to determine whether a given word is legal—that is, belongs to the language—is by means of word-construction rules. These rules work as follows: one word is known to us as a legal word in the language—an axiom. From it one can derive additional words by means of four derivation rules. These rules may be applied to every legal word in the language, not only to the axiom. Every word produced in this way—by applying the derivation rules to some other legal word—is also a legal word in the MIU language.

The given word—the axiom—is MI. The four derivation rules by which one may create new words from existing words are:

1. XI -> XIU
2. MX -> MXX
3. XIIIY -> XUY
4. XUUY -> XY

Here X and Y symbolize any string whatsoever. The first rule means: if a string ends in I, one may add U to its right. The second rule is: if a string begins with M, one may duplicate everything after the opening letter. The third rule is: whenever three consecutive Is appear, they may be replaced by U. And the fourth rule is the possibility of deleting two consecutive Us.

As an example, let us show a proof that the word MUIIU is a legal word in the language; that is, we derive it from the axiom by means of the derivation rules:

• MI — axiom
• MII — by rule 2
• MIIII — by rule 2
• MUI — by rule 3
• MUIU — by rule 1
• MUIUUIU — by rule 2
• MUIIU — by rule 4

It is worth noting that every string we created during the proof—or derivation—is also a legal word in the language. Now that we know the system, Hofstadter poses a riddle: is MU a legal word in the language? Before continuing, the reader is advised to try solving the riddle.

This system can be symbolized mathematically. If we adopt, for example, the following notation—write 3 in place of M, 0 in place of U, and 1 in place of I—then the words in this formulation are strings of digits, that is, multi-digit numbers. For example, the word MUU is represented by the number 300. We are now playing the same game with other symbols. As long as the symbols have no meaning, it makes no difference which symbols we use. Therefore, if we solve the riddle as formulated in these symbols, that also solves the original riddle. In other words, if we succeed in showing that the number 30 is legal in the algebraic system, then obviously the word MU is a legal word in the MIU language, and vice versa.

The advantage of the new numerical symbols is that we know another way of relating to them: arithmetic. In other words, we can now relate to the derivation rules of the words—now represented by numbers—as arithmetic operations. For example, the first rule above means in the new language: if a number ends in the digit 1, we may add 0 at the end. In algebraic terms, this means that we may multiply it by 10. The same applies to all the other rules.

The form in which we represented the system of derivation rules above is called a typographical form. Such a form deals with the external appearance of the words, the strings. Clearly, the rules can also be represented typographically in the language of the digits 0, 1, and 3, and this would still be a typographical presentation. The advantage is that in the formulation that represents the words by numbers, one can also represent the derivation rules in algebraic form, not only typographically. For illustration, here is the algebraic representation of the four rules of the MIU language:

1. 10m + 1 -> 10(10m + 1)
2. 3*10^m + n -> 10^m(3*10^m + n) + n
3. k*10^(m+3) + 111*10^m + n -> k*10^(m+1) + n
4. k*10^(m+2) + n -> k*10^m + n

Here k, m, and n are arbitrary integers.

If one relates to the algebraic formulation of the rules, one can use algebraic methods to solve Hofstadter’s riddle. For example, one can use a computer to check whether there is a mathematical way to begin with the number 31 and end with the number 30 by means of the four operations defined here.

Now let the reader imagine the riddle as if it had been given to him in this arithmetic form: the four algebraic rules listed above are given, as well as the number 31. The question is whether one can reach the number 30 by means of these rules. It seems that if the riddle had been given to us in this form, there would be no real chance of solving it—unless we used a computer, and even that is not at all certain.³⁵ It is not even easy to notice the typographical fact that all the numbers legally derived from these four rules are composed only of the three digits 0, 1, and 3. It is not clear that every one of us would have noticed that.

On the other hand, sometimes specifically the arithmetic form makes it possible to solve such riddles, if only because in that form one can use a computer that will systematically scan all the possibilities.³⁸

Now let us turn to the solution of the riddle. The solution specifically in its typographical formulation is very simple. The word MU is not legal in this language because it contains no I at all. One can see immediately from the derivation rules that such a state is impossible. Since the initial word contains one I, one cannot arrive at a word that contains no such “letter.”

The proof is quite simple. Use of rules 1 and 4 does not change the number of Is in the word. Therefore we need check only the two remaining rules. Now let us note that the number of Is in a word cannot be divisible by 3. The reason is that using rule 2 doubles the number of Is, and in this way it cannot arrive at a number of Is divisible by 3—the proof of this is very easy, and this is not the place for it. And using rule 3, which replaces a consecutive triple of Is with U, certainly cannot change the fact that the number of Is is not divisible by 3. This property is not affected by that rule.

Thus we have an algebraic riddle that seems extremely complicated, perhaps even unsolvable, even by a computer—but only if we relate to the algebraic meaning of the derivation rules. Paying attention to a formal-structural feature of the legal numbers in the system allows us to solve it in a very simple way, using purely formal considerations.

This is an amusing illustration of the relation between semantics and syntax. The semantics of the system is the meaning of the derivation rules and of the words in the language. That meaning is algebraic—that is, it is a collection of numbers and mathematical rules defining how to generate them from one another. But one can relate to the very same system on a structural-formal plane detached from that algebraic meaning. This is the plane of syntax, the formal structure.

In fact, all of mathematics can be described in semantic terms and in syntactic terms. Number theory can appear as a collection of derivation rules for words in a language from other words. For example, one can relate to the rule 1+2=3 as a rule expressing the arithmetical relation among the three numbers 1, 2, and 3. Proof that this is a correct statement will rely on the rules of arithmetic. But one can also relate to it as a legal collection of signs—a “word”—in a language, and then proof that it is legal will rely on a collection of formal formation rules, without any connection to arithmetic meaning. For further detail, see the first book, chapter 3 of the ninth gate.

To conclude, and for our purposes later on, it is important to note that the hierarchy between semantics and syntax can itself be relative. There are situations in which one plane of reference, considered semantics in relation to a second plane, can itself be considered syntax in relation to a third plane.

Let us give a brief example. Above we saw that the algebraic formulation of the derivation rules in the MIU system—or in number theory—is regarded as semantics relative to the typography, which served as syntax. We could now continue and say that the algebraic terminology itself constitutes formal syntax if we relate to it in a wholly technical way. But if we relate to the algebraic meanings of those formulae—seeing in them multiplication, addition, and division—and think of them as such, then we are relating to them on a deeper semantic plane. For example, we could arrive at an algebraic proof that the number 30 is not legal in the language defined above, not by using those rules directly, but by thinking about those algebraic rules and relating to them on a plane beyond them themselves—in their meta-language. Such a proof would show that with such algebraic rules we could never arrive at the number 30, on the basis of more abstract mathematical considerations—as opposed to a computer, which would test this by checking all possible results through the use of those very rules.

A fine example of the relation between semantics and syntax is the Chinese room example—see the first book, Observation 12—proposed by the British philosopher John Searle.³⁷ Imagine a Hebrew speaker who does not understand a single word of Chinese, sitting in a closed room with two windows. In the room there are containers filled with letters from the Chinese alphabet. Through the first window—the input window—questions written in Chinese are passed to him, and he must assemble answers from the letters in the containers and pass them out through the second window—the output window. Every time he gives an irrelevant answer, he receives an electric shock. Assuming the poor fellow has unlimited time to try, fail, and correct himself, it is very plausible that in the end he will learn to assemble relevant chains of symbols in response to every question, in such a way that he no longer receives electric shocks.

If enough time passes—billions of years—it will be possible to conduct an intelligent conversation with this person through the two windows in Chinese. Every question put to him by passing strings of letters through the first window will receive a relevant answer by means of output strings he passes through the second. Searle’s question is: does this person really know how to speak Chinese?

Seemingly, anyone who did not know how that person had acquired his conversational ability would be convinced that he spoke fluent Chinese. He would greatly enjoy conversing with him and be deeply impressed.⁴⁰ But we, who know everything about the “language training” that person underwent, are convinced that he does not know Chinese at all. At most, he knows how to avoid electric shocks optimally.

In fact, what we see here is that there are two ways to learn a language: through its semantics and through its syntax. The person in the Chinese room learned Chinese through its syntax. He knows all the legal and illegal structures in the language, but he has no idea what they mean. By contrast, the person who learns Chinese in a proper language course learns it through its semantics. He absorbs the language through the meanings of the words and sentences in Chinese.

Searle’s purpose in this example was to show that a computer can never think like a human being. A computer operates only in syntactic ways, and therefore “knows” how to multiply in exactly the same way that Searle’s man “knows” how to speak Chinese. Thinking, by contrast, is engagement with semantics, with the meaning of words and sentences in language, and therefore it is a specifically human property.

For example, a computer multiplies numbers through typographical rules—see the previous observation. Anyone familiar with computer logic knows that every gate—a basic logical unit—in a computer is built as a set of typographical rules. For example, the multiplication gate in a computer operates by the following rules—the reader should remember that the computer’s alphabet has only two letters, 0 and 1:

1. If you receive the string 00, output the string 0.
2. If you receive the string 01, output the string 0.
3. If you receive the string 10, output the string 0.
4. If you receive the string 11, output the string 1.

The computer, of course, has no idea what the symbols it receives as input mean, nor what the symbols it outputs mean. It operates only on the syntactic plane, exactly like the person sitting in the Chinese room.

Another example I once saw, I think, in Yeshayahu Leibowitz, concerns the description of a computer as such. Suppose an alien creature arrives on earth, a being totally unfamiliar with human culture, and suddenly sees a computer. Suppose also that this being has perfect scientific knowledge and infinite calculating ability. It looks at the computer, analyzes the way it functions, and knows the location and speed of every particle of matter composing it. It will be able to know exactly what will happen at every future moment through precise calculations of all the mutual effects among the particles.

The question, of course, is whether it knows what a computer is. The obvious answer is that it does not. Even if it knows all the details about the trajectories of the electrons composing the computer, it has no idea what a computer is. Until someone explains to it the meaning of the computer’s operations at the macroscopic level, it will have no such understanding. That is, describing the computer at the scientific level does not illuminate, even slightly, the meaning and role of the computer as a calculating machine. There are here two levels of explanation and description, both correct, but with no connection between them.³⁹ There is a description of the computer in physical terms—the trajectories of electrons, and so forth—and there is a description in logical terms—the arithmetic or logical operations of the computer.⁴²

In the next observation we will describe a common confusion between semantics and syntax in discussions of logics different from the conventional one.

Observation 30: Logic and Reality⁴¹

In contemporary mathematical theory, several mathematical systems have been developed that receive the title “logics.” One can indeed offer a precise mathematical definition of this concept, but this does not necessarily reflect anything connected to the everyday concept of logic. Nevertheless, in some cases modern thought tends to describe certain domains as if they were characterized by a logic different from conventional logic, namely one of those “logics.” There is deontic logic, which deals with desires and values. There is fuzzy logic and multi-valued logic, continuous⁴⁴ or discrete, quantum logic, and more besides. Usually one points to some principle of ordinary logic that is broken in that domain. It is very important to understand the context in which such statements are made, and therefore to avoid drawing hasty conclusions from them.

I will refer here to two such contexts, though there are several others as well: quantum logic and three-valued logic.

The proponents of quantum logic try to explain the strange phenomena that arise from observations and from quantum theory in physics by saying that in quantum theory we discover that its logic is not the ordinary logic familiar to us from everyday thinking.⁴³ Without entering into the details of the arguments of the different schools, it is important to understand that this statement cannot be accepted literally. There is no doubt at all that thinking about quantum theory takes place within the framework of ordinary logic. A given quantum result is either true or false, no more. The mathematics leading to the development of quantum theory, and underlying the measuring devices on which that theory is based, is ordinary binary logic. The mathematics of quantum theory, and the physics of those devices, are based on proofs by contradiction, as well as on other procedures that themselves rest on conventional logic. In other words: one cannot measure logic in the laboratory. Logic—the conventional sort—is a condition underlying scientific activity, and therefore it cannot change as a consequence of scientific results.⁴⁶

We must therefore understand such statements, at most, as a description of quantum phenomena, not as they are generally presented: a dramatic conclusion that bears on the rules of logic meant to guide us. There are physicists and philosophers of science who try to make such claims, but this is a complete misunderstanding, for the reasons described above.

The meaning of this misunderstanding is a confusion between semantics and syntax, or between description and explanation, and therefore it is related to our topic here. Those scientists or philosophers present their claim as an explanation of the meaning of the results of quantum theory. They argue that if we understand—so they say—that our logic is not correct, then we will no longer be surprised by the well-known oddities presented by this esoteric theory. But their claim can at most constitute a syntactic description of those oddities, not an explanation of them. At most one can say that this is a phenomenological theory describing quantum theory. The very oddness of the results is precisely that they behave as if there were a different logic here. The statement that there is a different logic here is only a certain kind—and not a necessary one—of describing the fact that there is a strangeness here requiring explanation. This is the syntax of quantum theory, meaning the external form of the quantum formalism, but not its semantics—that is, at the level of meanings and empirical and verbal content, a different physics is impossible. To construe quantum logic as a possible “explanation” of the results of quantum theory reflects a confusion between semantics and syntax.

Footnotes

One might invoke friction, since in its presence a force is needed even to maintain uniform motion, but in space there is no friction. Beyond that, one might ask: who initiated this uniform motion? Yet even that should not really trouble us, for even if we observe motion at a constant speed, every speed, including zero, is equally probable. Every wheel must have some speed, and so it is no wonder that it has a particular speed that is not zero. A speed of zero, that is, rest, is no more plausible than motion at any other speed.

If we set aside all these casuistic considerations, which are of course anachronistic, the problem that really ought to have troubled Abraham is: what causes the law of inertia itself to operate? Inertia too is an occurrence, or a phenomenon, and therefore it too should have some cause. Science says nothing about that. We will elaborate on this problem below.

Thus the main difference between the “mythic” explanation he cites and the scientific explanation does not lie in the explanation’s scientific or unscientific character, but in the subject being explained. The existence of objects, such as Jupiter’s satellites, cannot be explained in a deductive-nomological scheme. By contrast, the “mythic” explanation for the existence of the rainbow would be God’s promise to Noah that after the rain there would be a rainbow in the cloud. This is an explanation that can be tested empirically, like the scientific one, since one can examine whether a rainbow always appears after rain or not. Again, the difference lies in the subject explained, not necessarily in the method. Below we shall see this more sharply.

There is an assumption here, quite a reasonable one, that the syntax of a spoken language cannot fully overlap with its semantics, and therefore a computer will never be able to converse in a spoken language like a real human being. There are indeed more sophisticated ways of teaching computers non-formal language, such as neural networks. The subject will be discussed in the third book. Such a thing can happen only in a formal language, that is, a language with a strict mathematical structure. On the aspirations to create such a language, Leibniz, Russell, and others, see the first book, especially the section beginning on p. 118, and chapter 4 of the eighth gate.

For our purposes this distinction is not important. The Chinese Room exercise can be performed with a language of perfect mathematical structure, and still the person in the room will not know that language, but will only know how to “converse” formally. For example, if we try to teach some mathematical field in this way, we will discover that the student does not really know mathematics. To a considerable degree, schools try to do precisely this, and indeed students usually do not know mathematics. Searle’s argument remains valid even if the Chinese Room exercise is carried out in purely formal languages.

The next example we will present here, that of three-valued logic, does not require prior knowledge (the previous example required familiarity with quantum theory), and therefore it can be presented more fully. Its presentation may also help clarify somewhat further the distinction made above with respect to quantum logic.[^47]

Three-valued logic was proposed by a Polish logician named Lukasiewicz, who wanted thereby to explain the Aristotelian paradox of the sea battle. Aristotle asked: what can we answer to the question, “Will there be a sea battle tomorrow?”

Clearly we cannot answer: yes, and neither can we answer: no. Both of these are unfounded predictions. But one may continue and ask: what about an answer such as, either yes or no? Apparently that is certainly a correct answer. Yet if so, does that not itself contain a statement about the future? Can one today say anything with certainty about the future? Alternatively, one may ask: assuming there will be a sea battle tomorrow, is the truth value of this statement already positive today? The fact that we do not know the answer does not change the truth value of the statement: there will be a sea battle tomorrow.[^48] Apparently that value is already positive today, if a sea battle will indeed occur tomorrow, even if we do not know this.[^49]

From such considerations, Lukasiewicz argued, we are forced to conclude that with respect to questions concerning the future, the logical law of the excluded middle is not valid. That is, it is not true that only one of two answers is possible: yes or no, with no third option.

The law of the excluded middle is an essential component of ordinary, classical logic, which is binary in essence (the truth values of propositions within it are yes and no. There is no third value). Because of the difficulty presented here, Lukasiewicz proposed a logical system in which every statement has three possible truth values rather than two, as in classical logic.

By way of illustration, let us present the truth tables for logical conjunction, the operation of and, in the two logics.[^50] The following table presents the truth values according to classical logic (T = true, F = false):

R	Q	R and Q
T	T	T
T	F	F
F	T	F
F	F	F

The meaning of this table is that when both statements R and Q are true, their conjunction (R and Q) is also true. In all other cases the conjunction is false. This table has four rows because there are four combinations of truth values for the basic statements that make up the conjunction.

The following table presents Lukasiewicz’s proposal for three-valued logic. This table will, of course, have nine rows, since each of the two statements (Q and R) can receive three truth values (T, F, P), and therefore there are nine possible combinations of truth values for the two basic statements. The symbol P was chosen for POSSIBLE:

R	Q	R and Q
T	T	T
T	F	F
T	P	P
F	T	F
F	F	F
F	P	F
P	T	P
P	F	F
P	P	P

This table reflects one possible proposal, though not a necessary one. In the rows that contain values belonging to conventional logic, the result is identical to the conventional one. In the other rows, however, one must take into account considerations that depend on the precise meaning of the truth values of the conjunction itself, and we will not enter into them here. In general, according to Lukasiewicz’s proposal, the lower of the two truth values determines the result.

Now that we understand the basic meaning of three-valued logic, let us note several references in contemporary literature that make use of Lukasiewicz’s logical theory. What they all share is recourse to Lukasiewicz’s doctrine as a solution to paradoxes.

Generally speaking, one may say that all of these propose a solution to the paradox they face by suggesting that we adopt a many-valued logic, in which the value P this time in the sense of PARADOX is a legitimate value. According to these approaches, the problem with paradoxes is that we remain bound to binary logic; but if we adopt a different logic, the paradox will dissolve on its own. One formulation of such a solution appears in Belnap,[^51] who relies on four-valued logic. So too Kripke, who proposes an attempt similar to that of Lukasiewicz.[^52]

Let us turn to several more specific examples. Benjamin Ish-Shalom, in his book on Rabbi Kook,[^53] resorts in two places to three-valued logic. First, in note 71 to chapter 1, where he compares Rabbi Kook’s thought, which treats philosophical oppositions as legitimate, to Lukasiewicz’s logic. He does so in greater detail at the end of chapter 3, in note 133. This is a grounding of a claim about the legitimacy of paradox, a grounding that relies on Lukasiewicz’s logic.

Feuer, in his book on Einstein,[^54] discusses Lukasiewicz’s logic in greater detail. He describes there how the disillusionment of the First World War led to a search for ways to overcome the deterministic outlook.[^55] At the end of his remarks there he says this:

Thus the anti-determinist approach found in three-valued logic an instrument that banishes from reality events in the past. The logician was supposed to preside over the last public ritual of erasing the past. Militants among the younger generation of anti-determinist physicists were drawn to the idea of three-valued logic, and in particular to the erasure of the ineradicable past.

After quoting the well-known physicist Werner Heisenberg, Feuer continues and brings us back to quantum logic, saying:

It was therefore natural that Heisenberg supported replacing classical logic with a new logic in which the law of the excluded middle is fundamentally altered. “In quantum theory this law, the law of the excluded middle, has to be changed.”

In his view, the relation between classical logic and quantum logic is like that between classical physics and quantum physics. Classical logic is valid at the level of everyday objects, which natural language addresses, but not at the level of atoms and electrons.

Feuer then describes the opposition of the older sages of the scientific generation to these ideas.

As stated, there is a fundamental error in all of these ideas. Physics cannot examine, and certainly cannot change, logic, for the reasons raised above.

The same error appears in the quotations we brought in the more philosophical contexts, those that try to solve paradoxes by adopting a different logic. Just as in the physical context, here too we may ask: is not the discussion of Lukasiewicz’s tables themselves conducted in the terms of classical, binary logic?

Can we say about some value in his table that even if we proved it is T, that does not necessarily mean it is not P or F? Clearly, the discussion of the table itself, in the metalanguage, is conducted in the ordinary logical-mathematical way. If so, how can one infer from this table that we must change our logic? If we change it, then the construction of this very table is itself incorrect, for it is itself based on classical logic.

Put differently, with regard to Lukasiewicz’s own basic argument: when he proves that there is no sea battle tomorrow, does that mean that the statement that there is a sea battle tomorrow is false? Perhaps both are true together? Clearly, the discussion itself is conducted in the terms of ordinary classical logic. If so, it is obvious that one cannot infer from it any necessity to change the logic of our thinking, something that is in any case impossible for us.

More generally, with respect to the overall scheme proposed by Belnap, Kripke, and others for solving paradoxes in this way, we would raise a similar argument. Is not the discussion about adopting three-valued logic itself formulated in the terms of ordinary logic? Can one adopt three-valued logic and at the same time not deny its falsehood? Why not, if we give up ordinary logic?

The argument raised here against these proposals exactly parallels the argument against the advocates of quantum logic presented above.

Therefore, the meaning of all these “logics” is nothing more than a description of results in a particular domain, results that behave as if there were a different logic here. But we are not dealing with the logical plane itself; rather, only with the structure of those specific results. There is here a borrowed expression, as though there were a different logic, whereas in fact there is a dependence among parameters connected to the domain under discussion, and this dependence is metaphorically called the logic of that domain. This has nothing whatsoever to do with our fundamental logical forms of thought, which cannot change in any way. These “logics” are higher-level language rules, relevant to a very specific domain and context, and all of them are built on the basis of the good old binary logic, always correct.

As we noted above, all of these are confusions between semantics and syntax, since the logical description is the syntax of the results in the domain under discussion, but it cannot serve as an explanation for them. Explanation can be given only in terms of the semantics of the system, and that is always understood in the terms of ordinary logic. Put differently, this is a confusion between concepts of description and concepts of explanation. The logical structure describes results, but it cannot serve as an explanation for them.

In fact, there was no need to dwell at such length on this subject, were it not for a number of scientific and philosophical discussions, some of which were mentioned here, that rely incautiously on arguments of this kind.

One may hope that this collection of examples clarifies the relation between semantics and syntax reasonably well. One can speak in a language, prove mathematical theorems, or engage in any other formal system, on these two planes. If it is a logical-mathematical system, then work on the two planes will be completely equivalent and will lead to exactly the same results. The difference lies only in the meanings of the two planes, see, for example, the Chinese room example. In other cases this usually will not happen.[^56]

Back to the Planes of Description and Explanation: A Possible Solution to the Problem of Parallel Explanations

By analogy to these examples, all of nature is also a system of phenomena, with systems of rules governing the relations among them. If one stops trying to explain them, symbolizes them by mathematical means, and deals only with their syntactic description, one can advance to a more comprehensive description of the system. Sometimes attempts at understanding, or engagement with semantics, actually hinder progress. Mathematical description makes possible a more sophisticated and detailed treatment, even though the mathematical form has no essential or meaningful connection to the real law of nature, since it merely describes the structure and adds no understanding. As we shall see in the next chapter, the abandonment of semantics and the concentration on syntax are what made possible the great scientific progress of the past several centuries.

There are, then, cases in which we adopt two parallel planes of explanation, and this can be done because one explanation is formulated on the semantic plane while the other is formulated on the syntactic plane. One is description, a phenomenological theory, and the other is explanation, an essential theory. There is full correspondence between the two “explanations,” but in fact both are only two sides of one and the same explanation.

For example, the theory of evolution is a theory that offers a scientific explanation for the emergence of our universe, including life and human beings. Creation by God is a different theory, and probably not a scientific one. But if we regard evolution as God’s instrument in creating the world, the contradiction disappears. The description of creation is given in terms of evolution, and the explanation is given in terms of divine will and decision. For further discussion of this point, see the final gate. We saw the same thing in the discussion of the relation between vitalism and the life sciences. There we suggested the possibility that the laws of the life sciences are a scientific description of the action and properties of vital matter.

Let us now bring another example of this possibility of reconciliation between two different planes of explanation. In the terms of Aristotelian physics, one can say that stones fall to the earth because of their striving to return to their place of origin. At the same time, one can say that they do so because of the law of gravitation. The law of gravitation is a description of the phenomenon, whereas the Aristotelian explanation is an essential explanation, one that gives the reason and meaning of the stones’ fall toward the earth.

Here we must qualify the direction proposed here, and distinguish between phenomenological theories and essential theories. The law of gravitation, as a phenomenological explanation, does not contradict the Aristotelian explanation. But the force of gravitation, which is an essential theory giving the cause of the fall, does seem at first glance to contradict that explanation. The contradiction arises because the law of gravitation is a phenomenological description of the phenomena, whereas gravitational force is their cause. And as we have seen, one cause does not tolerate another.

Yet here too one may argue in the same way that there is no contradiction between these two claims. According to this proposal, gravitational force is nothing more than a mechanical and quantitative, that is, mathematical, description of the degree of the stone’s striving to return to its place of origin.

It turns out that the farther stones are from the earth, the less their “striving” to return to it, in inverse proportion to the square of the distance. On the other hand, large stones “strive” all the more strongly to return to their place of origin. Fat ones, as is well known, are not only good-hearted but also home-loving creatures.

If so, according to this proposal, the relation between the two planes of explanation is like the relation between explanation and description, or between semantics and syntax.

The difficulty presented at the beginning of the second chapter, according to which two different explanations cannot both be accepted at once, was based on the fact that each of them is a necessary and sufficient condition for the occurrence of the explained phenomenon. But when we are dealing with a pair of explanation and description, this problem clearly does not arise. There is no situation in which the physical conditions for the stone’s fall to the earth are present, but the stone lacks the “striving” to return to its place of origin. The physical conditions arise by virtue of that “striving.” These are two faces of the same phenomenon.

Relative Levels of Semantics and Syntax

Our conclusion is that phenomenological theories, which are in essence descriptions, cannot contradict metaphysical, mythical, or religious explanations. But as we have now seen in the discussion of gravitational force, even essential theories do not necessarily contradict such explanations, if with respect to them too we adopt the relation between description and explanation.

At first glance, however, this seems to contradict the synthetic position, which sees scientific theory as making claims about reality, and theoretical entities as actually existing entities. According to the analytic position, which regards scientific theory as an elaborate description, it is clear that no contradiction can arise, and we already addressed this above. But now we are proposing to treat even essential theories as phenomenology, that is, as description. At first glance, this seems to be a return to the analytic position.

This point will be clarified if we notice an important point discussed above, at the end of note 29, and one to which many do not pay attention. The hierarchy between semantics and syntax is relative. We saw there that a plane considered semantics relative to a second plane may be considered syntax relative to a third plane. So too here. The synthetic position indeed understands that an essential scientific theory, not a phenomenological one, makes claims about reality, and is therefore an explanation and not a description. But that is true only relative to a phenomenological scientific theory. Relative to a metaphysical, mythical, or religious explanation, by contrast, one may treat it as syntax.

Let us take as an example the discussion above about gravitation. The law of gravitation is pure syntax, description without any explanation. The theory about the existence of a gravitational force is the semantics of that syntax; that is, it is the causal explanation of the phenomenological law, or as it was called above, an empirical generalization. At first glance, this is a synthetic claim about the world. Yet despite what was said here, one may understand the Aristotelian explanation, according to which massive objects strive to return to their place of origin, as the explanation for the existence of the force of gravitation. Relative to it, the claim about the existence of gravitational force is no more than a description, syntax and not semantics. Likewise, the explanation that God makes objects fall downward by using, or more precisely by creating, the force of gravitation is an even deeper semantics, relative to which gravitational force is syntax. These are two demonstrations of the principled hierarchy among different planes of syntax and semantics, and of the double aspect of the intermediate plane: relative to a higher plane it is semantic, and relative to a deeper plane it is syntactic.

In the context of the distinction between explanation and description, one may raise an additional argument.[^57] When we explain the fall of the stone to the ground by means of gravitational force, one may continue and ask what explains the existence of gravitational force itself. More generally, with respect to any scientific law, one may ask what explains the existence of the law and its power to impose itself on inanimate nature. This question can be answered on mythical or theological planes, because on the scientific plane one can continue searching for explanations all the way down.[^58] God is the entity at which the questions and the search for explanations stop. The illusion that everything can be explained on the basis of science, without recourse to a force outside it, cannot stand, because we will always stop at some point for which we have no further, more basic scientific explanation.

Let us sharpen this important remark a bit further. We saw that science, unlike the everyday concept of explanation, seeks explanations that ground the familiar in the unfamiliar, that is, in the theory. This distinction reflects a deeper layer of difference. Whereas everyday explanation tries to find a familiar explanation for an unfamiliar phenomenon, science tries to find the causal explanation, even if the theory is not yet familiar. Thus, while the chain of everyday explanation can stop at the point where everything sounds familiar to us and our wonder subsides, scientific explanation cannot stop there. Scientific explanation seeks causes even for familiar things, and therefore arriving at a familiar plane is not a sufficient reason to stop the process of scientific inquiry and explanation.

Precisely in the scientific context, then, the question arises with greatest force: where does scientific explanation stop? Every plane of explanation that is found, even if it is familiar, and as we saw, usually it is not, will itself require explanation and cause. Thus this is an infinite process. In philosophy it is accepted that an infinite regress cannot constitute a sufficient explanation, and therefore it is reasonable that there is a plane at which this entire process stops. That plane is a plane of a priori meaning, myth or theology. At the end of every such chain stands God.

According to this view, mythical or theological explanation not only does not contradict scientific description or explanation, it is their necessary conclusion, and for two reasons:

1. At the basis of scientific explanations, which are always descriptions relative to explanations on a deeper level, there must stand one fundamental level beyond which one cannot continue. This is exactly where myth, theology, and all our a priori systems enter, as the basis for the chain of scientific explanations.
2. Because of the character of scientific explanations, which seek causes also, and perhaps chiefly, for familiar phenomena, one cannot define a plane at which scientific inquiry stops. That plane must be located outside science, and must include a factor with respect to which we no longer ask for causes of its actions.

Thus, from these considerations as well, it clearly emerges that the world of “explanations” is built as a chain of links, each one explaining the next. This is the same claim we encountered about relative levels of semantics and syntax, seen from a somewhat different angle.

Problems with the Semantic-Syntactic Schema

Up to this point we have seen that the schema of description and explanation is one of the possible schemas for reconciling two parallel planes of explanation. We shall now see that there are cases in which this schema does not work.

The discussion so far dealt with general theories, phenomenological and essential. The models we proposed may assist us in the context of generating, or more correctly discovering, new scientific laws and theories. This schema provided a reasonable way to reconcile them with other essential explanations. But above we saw that there is another concept of explanation, more basic, one resembling everyday explanation: grounding in the familiar. In the scientific context this is the explanation of an unfamiliar particular case by subsuming it under a familiar theory. The example we gave was the explanation of the not-yet-understood phenomena of tides by means of the familiar law of gravitation.

Let us take a sharper example and illustrate through it the difficulty in the relation between description and explanation in this context. We assume that an object falling on someone acts by virtue of gravity. On the other hand, the theological assumption is that this event constitutes a divine response of punishment for sins committed by that person. But the law of gravitation is already embedded in creation by God Himself, as we saw above. Therefore, it would seem that if that person had not sinned, and the physical conditions, the strength of the apple’s attachment to the tree, its weight, and the like, were the same as those that prevailed when the apple fell on Newton, then the apple would fall on him even without the sin. Conversely, if the theological explanation is correct, then if Newton did indeed sin, the apple ought to fall on him even without the scientific conditions required for that from a scientific point of view.

With explanations of the type of grounding in the familiar, then, it is very hard to see how a relation of semantics and syntax could arise between the theological explanation and the physical explanation. It is difficult to say that God embedded the law of gravitation in creation in order to punish sinners. We assume that this law operates even when there are no people around, and certainly also when there are people who are not sinners. It therefore seems that in such a situation the solution of semantics and syntax does not work.

One might indeed broaden this approach and say that for Newton to receive his punishment, it is not enough that the apple fall from the tree; Newton himself must also happen to be sitting under the tree. In addition, the force of the wind must not deflect the apple away from Newton, and so on. According to this proposal, the fit between the theological explanation and the scientific explanation, as presented above, was too narrow. We must see a fit not between the law of gravitation and the divine principle of reward and punishment, but between the whole complex that led to Newton’s sitting under the tree and the falling of the apple, on the one hand, and the principles of reward and punishment, on the other.[^59]

This possibility is problematic, because even the whole complex is nothing but a physical system, though a more complicated one and one harder to analyze scientifically in detailed and precise fashion. Newton’s arrival under the tree, the force of the wind, and so forth, are all physical events that should have a scientific cause, and therefore the relation between the physical system and the theological one remains unanswered. An answer of this type is at most a case of “one who wishes to deceive should make his testimony remote.” A much more complex system is simply harder to examine.

In the next note, which will conclude the discussion in the present subsection, we shall discuss Torah and halakhic aspects of the relation between semantics and syntax.

Note 31: Definition and Rationale

Bishop Berkeley, a seventeenth-century Scottish philosopher, argued that science asks only “what happens,” not “why it happens.” That is, as we argued above, it deals with the description of reality, syntax, and not with its explanation, semantics. An exactly parallel phenomenon appears in the Torah world, where it is said in the name of Rabbi Hayyim of Brisk that the student must not ask why the Torah or the Talmud say a certain thing, but only what they say. Put differently: we are interested only in the formal definition of the law, and not in its rationale.

In fact, already in the Talmud the tannaim, Rabbi Judah and Rabbi Shimon, disputed whether one derives law from the reason of the verse, or not.[^60] As halakha (Jewish law), we rule, contrary to the opinion of Rabbi Shimon bar Yohai, that one does not derive the law from the rationales of the verses. In the accepted terminology, one says that within halakhic discussion and analysis we try to clarify the formal definition of the laws, not their rationale. In the literature of Jewish thought there were those who tried to clarify the rationales of the laws as well, but usually such inquiries have no real halakhic standing.

The relation between the rationale and the definition is not clear. In fact, as we showed in note 13 in the first book, it is difficult to draw a sharp line separating understanding the law from defining it. On the one hand, it is not plausible that we could define a law if we do not understand it. Definition depends on understanding. We saw there that when the Torah speaks of tort law, it speaks of an ox that gores another person’s ox. Can we infer from this that one would also obligate the owner of a dog that bit someone else’s chicken to pay? If we do not deal at all with the rationale of the law and its understanding, but only with its formal definition, how will we know whether the law applies only to domesticated animals, or perhaps only to goring and not to biting, and so on? As we saw there, the definition of the primary categories of damages is based on subtle considerations involving an understanding of the rationales of the laws that appear in the Torah.

Let us now bring another example, also concerning the laws of damages. Rabbi Isaac Alfasi, at the beginning of Tractate Bava Kamma, explains the exemption of “tooth,” damage caused when an animal eats another’s food, and “foot,” damage caused when an animal causes damage in the course of walking, in the public domain, on the grounds that “this is their normal way.” The commentators explained that his words are based on the reasoning that the owner of the animal has the right to let it walk there, and one cannot obligate him to follow it around all the time. Therefore one should not obligate him to pay; rather, one should obligate others to guard their own property, see Rabbi Asher ben Yehiel there, section 1, who also cites a halakhic consequence of this rationale. Several commentators object to him, see for example Pilpula Harifta on the Rosh there, note 9: how can he determine a rationale for the laws of the Torah and even draw a halakhic conclusion from it, given that as a matter of halakha we do not derive law from the rationale of the verse? There are other such examples, but this is not the place to detail them.

Usually such exceptions are explained by saying that Alfasi does not mean to explain the rationale of the verse, but only to give its halakhic definition. This definition is needed because it has halakhic consequences, or practical differences, in the yeshiva terminology. But as we noted above, the problem is how Alfasi knows that this is indeed the correct halakhic definition of this law, if he does not presume to understand and explain the rationale of the verse.

Here too we see the blurry line separating definition and description, syntax, on the one hand, from explanation and giving a rationale, semantics, on the other. What counts as syntax in one context will count as semantics in another. As we saw above, semantics and syntax are relative concepts. Therefore, even in the context of the exemption of tooth and foot in the public domain, Alfasi is explaining a principle that certainly rests on understanding the law that appears in the verse, but this is semantics of a lower order, which can also be defined as syntax. It seems that in his view there is a deeper explanation, deeper semantics, with respect to which the rule was said that we do not derive the rationale of the verse. The explanation proposed by Alfasi is syntactic relative to that deeper explanatory plane, and therefore does not violate the rule that we do not derive the rationale of the verse.[^61]

In the body of the book we presented the distinction between empirical generalization, phenomenological theory, and theoretical-causal explanation, essential theory. There too we could ask ourselves how we can propose an empirical generalization before we know the principle underlying the set of phenomena in question. Even the division and classification of the phenomena, stage 1 above, which precede the empirical generalization, stage 2, cannot be done without essential understanding, stage 3. This too can be seen clearly from the example brought in the second gate concerning Semmelweis and childbed fever.

Our conclusion there was that every empirical generalization rests on an implicit essential understanding. Only after formulating the phenomenological generalization do we succeed in formulating explicitly the understanding that already stood beforehand at the basis of the generalization we made, and in bringing it to the level of a properly built theory, see also the discussion of Semmelweis and of Carr’s arguments in the second gate.

A similar process occurs in understanding the Torah. Here too we make generalizations based on implicit understandings. The explicit formulation of those understandings is a further step, but it is only a refinement and explicit definition of intuitions we already had in earlier stages. What we are not supposed to do in halakhic discussion is only the final stage, the creation and explicit formulation of the essential theory.

The meaning of the rule that one must not derive the rationale of the verse is that we may rely on implicit understandings, but not on formulated insights. The implicit understandings are a kind of prophecy, Rabbi Ha-Nazir relates auditory reason to a prophetic power, and perhaps one may say that this is what the Holy One placed in our mouths, or in our heads, and therefore we may rely on it.

To summarize: phenomenology is the formal definition, in the halakhic sense, and it is syntax, in the logical sense. These three concepts are almost overlapping. Semantics is the rationale, or the essential theory, and it is usually already implicit in some form within the phenomenological theory.

Let us conclude this note with a brief discussion of the dispute between Maimonides and Nahmanides on this issue. Maimonides states, in the Fifth Principle, that when the Torah explicitly states the reason for a commandment, we do not count that reason as a separate commandment. In the course of his remarks there, a dispute arises between him and Nahmanides, in his glosses there, and that dispute seems very relevant to our discussion here.

The Mishnah cited in Babylonian Talmud, Sanhedrin 21a, discusses the prohibition on a king to multiply wives for himself, “lest they turn his heart away.” This is a case in which the Torah itself explicitly writes the rationale for the law. The Mishnah there says:

“He shall not multiply wives for himself” — only eighteen. Rabbi Judah says: he may multiply them, provided they do not turn his heart away. Rabbi Shimon says: even one who turns his heart away he may not marry. If so, why is it said, “He shall not multiply wives for himself”? Even such as Abigail, who was righteous.

Nahmanides, in his glosses on the Fifth Principle, understands this Mishnah literally: there is here a dispute between two tannaim, Rabbi Judah and Rabbi Shimon, as to whether the prohibition is determined by its rationale or not. According to Rabbi Judah, the problem is that multiplying wives will cause the king’s heart to turn away from God. If that is not the case, there is no prohibition on the king to multiply wives. According to Rabbi Shimon, multiplying wives is prohibited even if they do not turn his heart away.[^62]

Maimonides, in his commentary on the Mishnah there, ends with a surprising statement: “The halakha follows neither Rabbi Judah nor Rabbi Shimon.” It follows from his words that he understands the Mishnah to contain a third opinion as well, namely that the statement at the beginning of the Mishnah is the view of another tanna, the first anonymous tanna, and the halakha follows him. From Nahmanides’ words it clearly appears that in his understanding the Mishnah contains only two tannaitic opinions. According to Maimonides, then, the halakha is that the king is forbidden to multiply wives even if they do not turn his heart away; that is, we do not derive the rationale of the verse at all.

This is at first glance very puzzling. Simply put, the reason one does not derive the rationale of the verse is that perhaps we will not understand the depth of the matter and will therefore err in our interpretation, see Encyclopaedia Talmudit there. But in the case under discussion here, the Torah itself writes the rationale, and therefore there is no interpretive problem in determining the rationale of the law. Why, then, should we not derive the rationale of the verse here?

It follows that Maimonides understands that one does not go after the rationale of the commandment even when it is written explicitly in the Torah. This is a more extreme opinion than that of Rabbi Judah, and Maimonides attributes it to the view of the anonymous first tanna, and even rules accordingly.

It appears from this that Maimonides holds that the prohibition against deriving the rationale of the verse does not stem from fear of error. If so, how does he nevertheless understand the principle that we must not derive the rationales of the verses but content ourselves with the formal definition?

It is likely that he understood this principle in the opposite way: one must not derive the rationale of the verses not because there is fear of error, but because the rationale adds nothing beyond the formal definition. The semantics completely fits the syntax, and therefore there is no point in seeking it, since no new law will ever emerge from it. In cases where following the rationale changes the law that emerges from the syntactic interpretation, the formal definition, it is clear that there is an interpretive mistake somewhere. Maimonides apparently assumes that the formal definition should fit the rationale completely. This is the scriptural principle of adequacy and congruence.[^63]

Maimonides thus radicalizes even further the conception we saw above, according to which in determining the formal definition we implicitly assume the rationale. His claim is that if that is indeed so, then the rationale cannot possibly add anything to the formal definition. It has already served us in determining the definition itself.

By contrast, the conception of Nahmanides and the other commentators, who understood that the reason one does not derive the rationale of the verse is fear of interpretive error, requires explanation. If we indeed presuppose the rationale at the basis of our use of the formal definition, how can it be that we use the formal definition without fear, whereas the rationale cannot be used because of fear of interpretive error? If we erred about the rationale, then the formal definition based on it is not correct either.

Nahmanides and those who follow him apparently understand that we have an intuitive grasp of the rationale, and as long as it is not formulated explicitly one may rely on it. Our intuition, so long as we do not interfere with it by analytic considerations, works correctly.[^64] It is precisely explicit formulation and precise definition of the rationale that can lead to errors.

This is very much Husserl’s conception of eidetic intuition, and perhaps even more radical than his. Eidetic intuition is more precise than formulated thought, which tries to grasp the abstract “by the horns.”

Nahmanides’ conception fits the formulation we saw above regarding the superiority of intuitive apprehension, which is a kind of prophecy, and therefore, according to his view, it is permissible to use it in interpreting the verse.

This subject is very deep, and it bears essentially on our entire discussion here, but we have already dwelt at sufficient length on this note.

3. Model and Explanation

Until now we have examined one way in which two parallel planes of explanation can be adopted simultaneously: by treating one as description and the other as explanation, in the causal-essential sense. In situations where this way can be applied, one explanatory plane will be phenomenological, syntactic, and the other essential, semantic. Put differently: one will be an empirical generalization, and the other an essential theory, a theoretical explanation.

In the present section we shall broaden this possibility to a situation in which we have before us two explanations that are both built like theories and not like empirical generalizations. It therefore appears that both lay claim to the crown of the essential explanation of the phenomena. Both explanations contain theoretical entities and principles, whose operation, at least according to the synthetic interpretation, is the cause that actually produces the explained phenomena. In such a state, according to the synthetic position, it seems at first glance that we have no way to classify one as description and the other as explanation.

The way we shall now present for the simultaneous adoption of different explanatory planes is based on the fact that in the scientific context there are certain theories which, even according to the synthetic position, are not claims about reality but rather models of reality. One should note that these are explanations of a third type: they are neither phenomenological descriptions nor essential-causal theories. These are theories that look essential, since they not only describe but also explain, yet they constitute only a model of the real situation. The structure of such a theory resembles that of essential theories, except that from the outset and explicitly it is not meant to be an explanation, but rather a sophisticated description.[^65]

Let us now clarify the concept of a model by means of several concrete examples.

We mentioned above the model proposed by Niels Bohr for Balmer’s formula of the hydrogen emission spectrum lines. Bohr described the hydrogen atom as a positively charged nucleus around which smaller particles, carrying a negative electric charge, move in circles. Each such circle describes a state of a certain energy of the electron, and the larger the radius of the circle, the higher the energy of the state described. Bohr’s main innovation was that electrons occupy orbits with very specific radii, and they cannot move around the nucleus at just any radius. According to Bohr, the emission lines of hydrogen describe transitions of electrons from one permitted orbit to another. The energy difference between the orbits produces an emission of energy, which is the source of the emission lines we measure in hydrogen.

It should be noted that this explanation does not have the character of an empirical generalization. It does not provide an overall description of a set of phenomena familiar to us. Bohr, and for that matter no one to this day, has ever seen an electron revolving around the nucleus of an atom. Bohr proposes a theoretical model that is supposed to explain the phenomena we do see, namely the hydrogen emission spectrum lines.

The main purpose of this explanation, then, is not to describe known reality by way of generalization, but to propose a theoretical physical mechanism from which the measured phenomena will be derived, that is, which will be their cause. In Bohr’s theory theoretical entities appear, such as electrons, the nucleus, orbits, permitted energy states, and the like. It is therefore clear that this is a theory and not merely a phenomenological empirical generalization. Balmer’s formula is the phenomenological generalization, stage 2 in this context, and Bohr’s theory comes to explain it as stage 3. Since it is a theory and not phenomenology, according to the synthetic approach we cannot say that it is description and not explanation.

On the other hand, today we know that this theory is not correct. The little material spheres called electrons in that theory do not exist. The electrons revolving around the nucleus in the eigenstates, that is, the permitted states, as we describe them today, are waves spread out over all space, not pointlike little spheres. Thus the theoretical entities that make up that theory do not really exist, at least not in the sense Bohr had in mind. Yet Bohr’s description remains a convenient and good description in a wide variety of contexts and applications even today. Therefore, many times, especially in popular presentations of atomic physics, this description is presented as though it were a true description of quantum physical reality.

Bohr’s theory, then, is a successful model of real reality, but it does not truly describe it. Treating it as a model stems from two levels:

1. Even if it were a correct description of the phenomena known to us in reality, it is very hard to claim that all this elaborate description is a faithful description of atomic reality itself. As stated, no one has seen electrons revolving around a nucleus, and what Bohr argued was that if we nevertheless assume that this is reality, then we will thereby have an explanation for the hydrogen emission lines. Therefore it is very reasonable to treat this description as a model and not as a description of reality itself.
2. In practice, we know that this model is incorrect. The fact that it correctly describes certain phenomena stems from the fact that it provides an averaged description of real reality.

A model is judged, then, by whether it provides a good description of the experimental results in the domain that interests us, not by the question whether it correctly describes reality.

It should be noted that the first claim, that reality itself is not known to us and therefore it is hard to assume that such a detailed description of it is a correct description of it itself, is precisely the claim of the analytic position. They argue that every scientific theory is a model, since it arises from an attempt to explain the facts and not from direct observation of reality itself.

In such contexts, even some proponents of the synthetic position may adopt such an attitude, because the description here is highly speculative, and the hydrogen emission lines are not a sufficient basis for deciding that this is a description of reality itself. Gravitational force, by contrast, is a simple description of reality, since the phenomenological generalization of the set of experimental facts known to us, that every body with mass falls toward the earth, is not far removed from that description. Therefore, in this case the proponents of the synthetic position adopt the theory as a faithful description of reality itself and not as a mere model. Extreme analytic thinkers would argue that here too there is an unfounded generalization, and therefore it cannot be accepted as a claim about reality itself; it is only a model explaining the empirical phenomena.

As far as I know, Bohr proposed his theory as a genuine explanation of the physics of hydrogen atoms. Treating it as a model is mainly correct for a later time, when quantum theory was already known in fuller form. The first claim we brought above, that as a claim about reality this is far too speculative a theory, could have been directed at Bohr himself. The second claim, that it is not a correct description, is of course anachronistic, that is, it is made toward him only retroactively, since today we know things Bohr himself did not know.[^66]

This example clarifies the way the theoretical scientist operates. In light of the facts, the phenomenological generalizations, he proposes a theory that will explain them through theoretical principles and entities. He must then ask himself whether this description is a model or a theory. The answer depends on whether there will be additional experimental results that are explained by this theory. The more facts it fits, the more we will tend to assume that it is a theory and not a model. In any case, even if it is clear that the model is not correct, it has great scientific value. Sometimes it is very difficult to think about bare reality, and we need a model to help us relate to the set of experimental facts within some coherent causal framework. Many scientific explanations are today known to be models, and this does not prevent scientists from acting as if they were theories. With respect to some theories there are disputes about this very point: are they theories or models?

In many cases the model is connected to the true theoretical explanation. In the case of Bohr it constitutes a kind of average of it. In other contexts it constitutes one or another abstraction of the full explanation, and this is part of the secret of the success of models in describing reality, even though they are not correct, or at least not exact.

Another example of an explanation that looks like a theory, but is probably only a model, is the way we think about our systems of thought, or about the brain. We are accustomed to thinking of the brain as composed of different units. One is responsible for memory, another for information processing, a third for learning, and so on. In fact, today it is already known that the brain probably does not function this way. There is no clear geographic division in the brain between parts responsible for its function as a remembering system and those responsible for its function as a learning, analyzing, and similar system. Today it is accepted that the entire brain is involved in all these functions. The currently accepted description of the brain’s mode of operation is called a neural network. This subject will be discussed in the third book. Even the division into functions is only a convenient way to describe the brain’s operation, but it is not clear that it has a root in reality. In this connection it is worth thinking about the synthetic thesis that there is no sharp distinction between thought and perception, and trying to examine how that would find expression in the physical model. This division, then, is a model and not a description, and certainly not an explanation.

This model of thought has undeniable usefulness. For example, the computer is a result of this mistaken model of human thought. In the computer, at least in classical architecture,[^67] we build each such unit separately, and thus succeed in reconstructing part of the functions of human thought. This is reconstruction by means of a model, even though it is known not to correspond to reality, that is, to our brain.

In the first book, and in several contexts above, we saw that we must give up the traditional division between thought and perception. This division is anchored in part in our physiological structure. Our senses are distinct, and therefore it is plausible to view them as a separate unit.

Yet we know clearly that seeing is not done only by the eyes, and hearing not only by the ears. Part of the work is done in the brain. Therefore it is clear that there is no necessity to view the distinction between thought and perception as a theory, and it is possible to relate to it as a model only. As we saw, in certain contexts it is not even such a successful model.

On the other hand, it is important to notice that even the challenge to this distinction uses a conceptual system that refers to functions of thought and perception. Even after giving up this distinction, it is still very important to distinguish between those functions, if only in order to see them distinctly and to understand what we mean when we say that these are not separate functions. This is another reason why a model is very important. It creates a conceptual system, and thereby helps us distinguish between different functions, even though in actual reality the distinction between them is not so simple, and perhaps does not exist at all.[^68]

Models, then, are very important for the activity of science, and we must use them, but do so carefully and correctly. On the one hand, we must not neglect the advantages we derive from using them, and on the other hand, we must not take them too far. One must be careful not to see a model as a real theory and draw from it conclusions that are not valid. Every conclusion drawn from a model requires careful criticism as to whether the conclusion was reached by overstepping the bounds of the model’s relevance.

In the next note we shall deal with another aspect of models of human thought.

Note 32: Do We Think in the Form of an Axiomatic System?

In recent years there has been a considerable theoretical effort whose goal is to arrive at a reasonable description of that part of the brain engaged in thought. Among those working on this are physicists, psychologists, biologists, neurologists, philosophers, mathematicians, and computer scientists.

One of the questions dealt with mainly by mathematicians and philosophers, the two fields that contain logic, is whether human thought proceeds in the form of an axiomatic system, or whether we think in some other way. Even if we were to know that human thought indeed takes place in the form of an axiomatic system, one could still ask what those axioms are, and even how many there are. Is the number of axioms of thought finite? Infinite? If infinite, of what kind? Countable or larger? And so on.

One of the main implications that interests researchers is whether Gödel’s theorem, which determines limits on proving claims within axiomatic systems, see the ninth gate of the first book, is relevant to human thought. In other words, the question is whether human thought is more powerful than a computer.

The question discussed here can be formulated as follows: is an axiomatic system a theory of thought, or only a partial model of it?

We are so captive within the framework of axiomatic systems that it is difficult for us even to see another form of orderly thought. Even our very engagement with this question is conducted using mathematical or philosophical tools, that is, by using axiomatic systems. There is a certain difficulty here, but it is a difficulty characteristic of all reflective thought about thought as such. Beyond that, even if an axiomatic system is only a model and not a theory, it can certainly be used in scientific research. That does not disqualify it outright as a research tool in this field. True, this distinction does require great caution regarding the conclusions of such research.

There are two main approaches to this question:[^69]

1. One approach argues that if we were to collect all the axioms of the human brain, we would be able to find true propositions that could not be proved in the axiomatic system thereby created. Therefore, some researchers conclude, it is proven from this that the human brain is limited.

There is, of course, a blatant begging of the question here. Those who claim this assume that the human brain is a large axiomatic system, and from this they prove that it is limited. If it were such a system, then it would be limited even without applying Gödel’s theorem to it. Beyond that, the very application of the theorem to it is problematic, because Gödel’s theorem has several conditions required for its proof, and it is not clear that the brain, even if it functions as a large axiomatic system, satisfies them, for example, that the number of axioms be at most countably infinite.

2. According to the proponents of the second approach, the claim is the opposite, and it is called Lucas’s argument. The argument proves that the human mind will always be stronger than any machine, since a machine can always be presented as an axiomatic system, and therefore for the machine there will always be true propositions that are not provable. One can construct such a proposition using the technique on which Gödel’s theorem is based, see the first book, ninth gate. By contrast, we as human beings will know those propositions, and can even prove them, just as Gödel himself did in his proof. Therefore, some researchers conclude, the human mind is stronger than any axiomatic machine.

The question here is whether, for every machine, we can identify all the assumptions and principles on which it operates, hardware and software. This claim is not simple at all, and certainly cannot be proved, see Avron there.

Yet this challenge does not seem essential. The point at which our thought is stronger than that of any machine is precisely the ability to go outside the system defined for us by some set of axioms. This principled mode of operation is what distinguishes our thought from machine calculation. Even if we do not always succeed in doing so, the principled possibility of performing such an act is unique to human thought. This is exactly the claim presented in detail in the ninth gate of the first book.

One can apply this to the human brain itself and say, as a challenge to Lucas’s argument, that even if it functions as an axiomatic system, like a machine, it is impossible to gather all its axioms, and therefore impossible also to construct the Gödelian proposition that lies outside it. In other words, we will never be able to understand our own thinking fully. The representation of the brain as a machine is not simple, and it is entirely plausible that it is not possible at all. One may therefore argue that the brain is a sophisticated and complicated machine, but still a machine. Our claim here is different: this impossibility is not technical but essential. As stated, the claim is based on the principled possibility of creating Gödel-type propositions, not on the ability actually to create them.[^70]

Up to this point we have become acquainted with the concept of a model. Let us now return to the possibility of using this concept in order to reconcile two parallel planes of explanation. The possibility presented here is to determine that one of the two explanations is a model and not a theory, and therefore the theoretical entities within it do not really exist. In fact, it is not a causal explanation but a sophisticated description, like a phenomenological theory, except that it is built with the structure of an essential theory. As stated above, descriptions do not contradict explanations. In effect, we reject this explanation, at least as an explanation, in favor of its counterpart. Although on its face it looks like a theory, we determine that it is nothing more than a sophisticated phenomenological description. Of course, we continue to use this explanation as a model, and do not give it up at all. Therefore we classify this possibility as a way of remaining with both planes of explanation together, even though that is not truly the case. In fact, we adopt an analytic stance with respect to the matter under discussion, and treat it as a description, that is, as a model.

As noted, the analytic thinker treats all theories as sophisticated descriptions, including theories that in the synthetic picture are considered essential theories. These will receive from the analytic thinker the status of models. We already noted that in the analytic picture there is no recognition at all of the distinction being discussed here between theory and model. Such a distinction exists only in a synthetic world of thought. In the present section we are proposing a possibility in which proponents of a synthetic position adopt an analytic attitude toward certain theories in order to reconcile them with competing or contradictory explanations.

As noted at the beginning of the section, this possibility is logically equivalent to the previous possibility, but it broadens the distinction between explanation and description so that it can be applied even in a situation where we possess two explanations with the structure of essential theories. In the end, the logical solution to the contradiction between the planes of explanation is like that in the previous possibility: one is explanation, semantics, and the other is a sophisticated description, refined syntax.

A solution of this type seems applicable, for example, to the case of teleology and causality in science, a case discussed above. There one can say that teleology is the real reality, for example potentiality, and the causal description, which certainly also includes theoretical entities such as force, is nothing but a model.

A similar approach can be adopted with respect to reductionism. Whoever thinks that physiology can be grounded in physics and chemistry is in effect claiming that the essential explanation of the phenomena is the physical explanation, and the physiological, biological, description is nothing but a model. Here too the model is a generalization and abstraction of the physical description, because dealing on the physical plane requires us to deal with the physical parameters of every particle in the system, something impossible in the context of macroscopic systems that contain billions upon billions upon billions of particles. Beyond that, even if we were to succeed in addressing all the degrees of freedom, it is still not clear how we would define from them the macroscopic parameters relevant to the domain under discussion.

We therefore make an averaged simplification of the physical theory, and arrive at a macroscopic, biological, or physiological model.[^71] This situation resembles the proposal to view Bohr’s atomic model, which is the result of averaging the “real” theory, quantum theory, as a model.

To conclude the discussion in this section, let us examine the relation between model and theory in mathematics, and see that although it appears opposite to what was described here, there is nevertheless something shared between mathematics and the natural sciences on this point as well.

Note 33: Theories and Models in Mathematics

In mathematics too we distinguish between theories and models, and this distinction underlies the difference between pure mathematics and applied mathematics.[^72]

For example, mathematical group theory, as part of pure mathematics, deals with the properties of an abstract structure characterized by a set of abstract properties. This theory tries to derive theorems from those properties of the system under discussion. The same is true of sets, rings, vector spaces, and the like. All of these are systems with abstract properties from which various theorems can be derived.

Applied mathematics finds concrete systems characterized by exactly the abstract properties required by the pure theory, and uses the theorems found by pure mathematics. Those theorems will of course be true for all such concrete systems as well. These systems are called by mathematicians a model of the pure theory.

For example, it is known that symmetry operations in various areas of physics constitute a mathematical group. This means that these groups satisfy the abstract properties of the concept group as it is defined in pure mathematics, in group theory. We can therefore adopt automatically, without any detailed rechecking, all the theorems proved in group theory, since they are derived solely from the group-properties, and be sure that they are also true for the domains in physics under discussion, namely symmetry groups.

The same is true of the various states in quantum space. These satisfy the properties of the abstract mathematical space called a Hilbert space, and therefore conclusions can be drawn about them from various theorems about Hilbert spaces, conclusions that characterize quantum states. This is the way one passes from mathematics to reality.

The group is one of the basic mathematical structures from which other structures are derived, and therefore we will take it as a simple illustration of our topic. Let us take a set of elements, a, b, and so on, among which a binary operation is defined, whose result also belongs to the set. This set will be called a group if it satisfies the following properties:

1. The operation among any three elements is associative: a(bc) = (ab)c.
2. The set contains an identity element e, such that for every other element in the set: ae = ea = a.
3. For every element a in the set there exists an inverse element a^-1, such that: a a^-1 = a^-1 a = e.

Additional restrictions that we impose on the group will create more specific structures, constituting other domains in mathematics. For example, there is an Abelian group, named after the mathematician Abel, defined as a commutative group, that is, a group in which switching the order of the elements in the operation does not change the result of the operation. One can continue in this way to restrict the group by special requirements regarding the character of the operation and the character of the elements making it up, and arrive at rings, fields, vector spaces, and the like. Each of these is a domain unto itself in pure mathematics.

As stated, rotational operations in the plane are an example of an application of group structure. The elements of the group are rotations by some number of degrees. A rotation by 13 degrees is one element, and by 16.74 degrees another, and so on. Clearly this is a group with an infinite number of elements. The binary operation of the group corresponds to the physical operation of composing two rotations. It is customary to define it as applying one after the other, and it is clear that the result is a rotation which is itself an element in the group. The inverse element is the rotation in the opposite direction, and the identity element is a rotation by 0 degrees, that is, remaining in place.

The last paragraph is, in fact, a proof that rotations in the plane constitute a group. One can say more than that: they constitute an Abelian group, since changing the order of the rotations does not change the result. This is a physical model of the mathematical theory of groups.

As a counterexample, the integers under multiplication are not a group, since they do not contain an inverse element. There is an identity element here, the number 1, but from the number 5, for example, one cannot reach the result 1 by multiplication unless one multiplies by the number 1/5. But that does not belong to the group, since it is not an integer. Therefore 5 has no inverse element belonging to the group.

By contrast, it is easy to see that under addition one can certainly regard the integers as a group. Alternatively, if we expand the set of numbers from the integers to the rational numbers, we obtain a group under multiplication as well. These two examples are nonphysical, arithmetic, models of the mathematical theory of groups.

All the theorems that can be proved for the abstract concept group are immediately applicable, without any further checking, to each of these examples. We can infer at once that all those properties characterize two-dimensional rotations, the integers under addition, and the rational numbers under multiplication. The reason is that the truth of these theorems depends only on the abstract characteristics of the group, and those exist in all the examples, the models, listed here.

It is quite clear that the relation between theory and model in mathematics is not like the relation between theory and model in the natural sciences. In mathematics the theory and the model are not descriptions of the same thing, but rather relations to a concrete structure on more abstract planes.

In a certain sense the relation is reversed. In science the model is a sophisticated and abstract description of the facts, and the entities composing it have no meaning in themselves. It constitutes the pure form of reality, whereas the essential theory is supposed to describe reality itself in its concrete form. By contrast, in mathematics the theory is the abstract form of the model, which is the real reality.

This does not seem to be merely an accidental semantic difference. It follows from the relation between science and mathematics as we explained it above at the end of the second gate. Mathematics does not deal with entities, and certainly not with entities from our concrete world. It deals with abstract, Platonic entities, and the concrete belongs not to it but to the natural sciences. Therefore, from mathematics’ point of view, what exists is not the concrete object but the Platonic being. Reality is only an “unreal” model of that being. If we look at it this way, the relation between theory and model in science is preserved in mathematics as well. The theory describes reality, and the model proposes an alternative system that corresponds to reality in some way, in terms of factual results in science and in terms of theoretical properties in mathematics.

Cause and Purpose

The next possibilities we shall present for the simultaneous adoption of parallel explanatory planes are based on treating one as describing a cause and the other as describing a purpose. These possibilities involve discussion of the relation between cause and purpose, but we shall address that only briefly, insofar as our discussion requires. This is the topic of the present section.

We saw above the relation between the two kinds of physical explanation, teleological and causal. There we argued that in fact only one of them is correct, while the other is a fictive mathematical model that fully corresponds to the parallel theory. But as we already noted, in the physical example there is no real teleology involved. Even the teleological explanation does not claim that the particle or the stone “thinks,” “wants,” or “decides” how to move. It is only a form of description that uses purpose as a parameter rather than cause. In both of these descriptions it is clear that the occurrence is completely deterministic. The stone cannot suddenly decide that it no longer “strives” to return to its place of origin, that is, to the ground.

Yet in the course of the discussion we also encountered contexts in which genuine teleology is involved. For example, Newton’s apple falls on his head because of gravity, but the purpose of the fall is to punish him for sins he had previously committed. The scientific explanation therefore describes the cause of the apple’s fall, whereas the theological explanation concerns the purpose.

This distinction does indeed seem correct, but it is doubtful whether it solves the difficulties raised by the existence of two parallel planes of explanation. As we already noted, the question is what would happen if Newton were deserving of punishment but the physical conditions for the operation of gravity were not present, and vice versa.

In the literary example we saw above, the lead heart of the Happy Prince broke in two because of the cold, but the breaking also expressed the sorrow he felt over the swallow’s death. Here too one can speak of a relation between cause and purpose. One may extend this literary personification to actual human beings. We saw that different actions and decisions of human beings can be described in terms of psychological influences, as well as in terms of value-based, volitional, philosophical decisions, see the example of repentance at the beginning of the gate. Here too, these two planes refer to cause, psychology, and purpose, philosophy.

In this context it seems that our approach accepts cause and purpose as two planes existing side by side. A person gives charity to the poor so that the poor person’s condition will improve, and this is a purposive description. On the other hand, the functioning of that person’s limbs is driven by natural processes, and in that sense even human action has physical, physiological, causes.

Another example we saw is a person who extends his hand in order to make a telephone call and set up a meeting with his friend, and this is a purposive description. Yet the movement of the hand is also a physical event, and as such it has its own physical causes, and this is a causal description.

Let us now ask: if a person wants to speak to his friend on the phone, but the physical conditions that produce the lifting of the hand do not obtain, will he succeed in doing so? And conversely: if those conditions do obtain, but he is not interested in speaking to his friend on the phone, will he nevertheless be compelled to do so?

Here the answer seems more difficult. In this case many of us would tend to adopt both descriptions, the causal and the purposive, each on its own. Each of them constitutes a necessary and sufficient condition, in the sense we defined above, for the explained occurrence.

Before we explain this fit in somewhat greater detail, let us present a Torah aspect of the relation between cause and purpose.

Note 34: Cause and Purpose in the Creation of the World

At the beginning of Etz Chaim, the Ari opens with two inquiries that clarify the intended purpose in the creation of the worlds. At the beginning of the first inquiry he writes:

The first inquiry is that which the earlier and later sages investigated, to know the cause of the creation of the worlds, for what reason it was.

And they concluded and decided that the reason for the matter was that He, blessed be He, must be perfect in all His acts and powers and in all His names of greatness, exaltation, and glory. And if He had not brought His acts and powers into actual deed, then, as it were, He would not be called perfect, neither in His acts nor in His names and appellations…

And if the worlds and all that is in them had not been created, the true meaning of His eternal being in past, present, and future could not be seen, and He would not be called by the Tetragrammaton as above. And likewise the name of lordship…

But when the worlds are created, then His acts and powers, blessed be He, come into actuality, and He is called perfect in all kinds of His acts and powers, and also perfect in all names and appellations, without any deficiency whatsoever, Heaven forbid.

The Ari begins by clarifying the cause of the creation of the worlds. Yet his explanation deals with the purpose for which the world was created: in order, as it were, to complete the names, appellations, and actions of the Holy One. He does not address the cause at all.

In fact, since we are dealing here with an event so primordial, it is quite clear that it is difficult to speak here of a cause in the usual sense. Here the world was created, and therefore there cannot be a cause for it in our present sense. What could have constituted a cause for God to create all reality? Beyond the problem of a cause governing God’s actions, there is an additional problem here: nothing besides Him existed beforehand, and therefore nothing could have caused Him to act as He did.

The required conclusion is that even the relation between cause and effect, a relation that underlies the world as we know it, was itself created at the beginning of creation. The very principle of causality was created together with the creation of the world. Therefore one cannot ask, with respect to the beginning of creation, what cause brought it about. In an initial state one can ask only what the purpose of creation is, not what its cause is.[^73]

As Rabbi Elyashiv notes in his book Leshem Shevo ve-Ahlamah, in Sefer Ha-Beurim, which is part of the Leshem and deals with the interpretation of Etz Chaim, despite all this the Ari still uses the term cause. He explains it as follows:

…and He illumined them, that is, the worlds, all with His light alone, for He is the cause of all and the one root and one cause of them all, and He is also the purpose of them all, for they all return to their cause and their source, and as is known, such is the order of all creation, that every thing returns to its root and source, and that is its purpose…

This teaches us that His names, which are His revelations, are the cause of all, and therefore they themselves are also the ultimate purpose of all. For the cause itself is the purpose, since the end of the act lies first in thought…

The distinction between cause and purpose is a narrowed form of our way of looking. But from a comprehensive perspective, the purpose is the true cause, and everything is purposive rather than causal. For this reason we tended above to adopt teleology as a more correct description of physics, although teleology is deterministic from the inanimate object, yet volitional from the operator of nature, that is, from God, even though the causal description is also possible with respect to some domains.

Logical Introduction to What Follows: Three Possible Mechanisms of Relation between Cause and Purpose

In general, one may say that the fact that two planes of explanation, such as cause and purpose, exist together can stem from three mechanisms:

1. One causes the other.
2. Both combine into a single cause.
3. They both occur in parallel and correspond to one another, without any direct relation between them.

In logical terms, one can view this from a different angle. There is in fact room for hesitation concerning the logical dimension in the definition of the concept cause. There is an additional dimension, that of physical causation; see the appendix to the first book. Is the cause that produces the effect a necessary and sufficient condition for it, or only a sufficient but not necessary condition, on this too see the appendix to the first book.

Both of these are approaches found in the philosophical literature. But there is also a third possibility: that the cause is only a necessary condition, but not a sufficient one, for the occurrence of the effect. I am not aware of such a conception in the philosophy of causality, see Steinitz, Tree of Knowledge, part 2, chapter 4, but in the case of parallel planes of explanation it will appear below that this is a plausible possibility. To a certain extent, there is here a surrender of the idea that each plane is itself a full explanation in the complete sense of that concept.

If the cause is a sufficient condition for the occurrence of the effect, then it may be that the effect also has another cause. This is possibility C. If the cause is a necessary and sufficient condition for the occurrence of the effect, then it is the only possible cause of that occurrence, and then it is what also causes the formation of the second cause. This is possibility A. And if the cause is a necessary but not sufficient condition for the occurrence of the effect, then an additional condition is required, one that is also necessary but not sufficient, or another cause in that sense of the concept cause, for the effect to occur. This is possibility B, in which the two causes join together in order collectively to produce the event caused by them.

Option A, according to which purpose causes causality, will be described in the continuation of this section. The second sub-option within A, that the cause produces the purpose, contradicts the concept of purpose itself,[^74] and is therefore irrelevant. The next two possibilities, B and C, are included in the mechanisms described in the two sections that follow.

In the course of our discussion we shall see that these relations describe possible mechanisms for the simultaneous adoption of parallel explanatory planes, not necessarily only in the context of cause and purpose. We present them in the context of cause and purpose because very often the dilemma of parallel explanatory planes arises in that context.

4. Option A: Purpose Causes Cause

As stated, one possible explanation of these phenomena, which are also connected to the mind-body problem, or the psychophysical problem, is that the human will creates a force that acts on the person’s bodily system, and that force causes the movement of the hand toward the telephone, see the third book. At first glance this is a straightforward causal description, but in fact it is nothing of the kind. At the basis of the whole process lies a desire for something future, that is, a purpose. The purpose of the meeting with the friend causes the beginning of a physiological process, causal in character. There is here a chain in which the purpose creates a cause that moves the effects, which bring one back to the purpose. The first electron in the brain that set the whole physiological process in motion moved without any prior cause, solely because of the purposive desire that something should happen in the future as a result of that movement. The future moved the past, which leads back to it.

True, the cause of the beginning of this process is the desire to create the meeting, not the meeting itself. That desire as such is a cause of the whole process, not its purpose. The meeting is the purpose, but the desire for the meeting is seemingly a cause rather than a purpose, and it exists before the action.

But this is a description that we would usually classify as purposive. For example, Aristotle’s stone falls toward the earth because of its desire to return to its place of origin, and this is a purposive, teleological description that the modern person tends to reject. The reason, as stated above, is that the first electron that set the whole process in motion seems to move without any prior cause. The purpose is what created the force that moved it. Therefore we classify such an explanation as purposive.

We see, then, that in certain contexts cause and purpose are planes that can “live together,” and they do not necessarily contradict one another. Therefore in other contexts too one may examine whether one plane of explanation refers to cause and the other to purpose, and infer from this that they do not contradict one another.[^75]

It should be emphasized that the essential problem has still not been solved, since it is not clear what will happen if the purpose is present and the cause is not, or vice versa. In light of what we said above, it appears that the purpose creates the causal chain, and therefore these two planes will always both obtain. In the absence of one of them, the whole process will not obtain.

5. Option B: Explanations Combine

At the end of the previous section we saw that human action begins with desire, that is, with some purpose, and that desire creates a physiological chain with a causal character. We asked ourselves how these two planes live in parallel, and what happens when one of them does not occur. It turns out that when one of them does not occur, the result does not occur. It is therefore clear that both occur together. The first possibility we proposed for understanding the parallel existence of these two planes, cause and purpose, was that the purpose causes the causal process. The second possibility, which we will now discuss, is that the two together create the explained occurrence, and thereby constitute a kind of “overall cause.”

To clarify this, let us return for a moment to the example of the person who becomes religious or becomes nonreligious, presented above. We saw that one can explain the step such a person takes on two planes: the psychological plane, some crisis or circumstances that caused the person to decide as he did; and the evaluative plane, that is, the decision that this way of life is better or truer than the previous one.

It turns out that these two planes work together to create the final decision. The cause of this step is neither psychological alone nor philosophical alone, but composed of the two planes together. If the psychological conditions do not obtain, the process will not occur. If the required philosophical conditions do not obtain, it will likewise not come about. Both conditions are required for the result to arise.[^76]

One should note that this is not mere wordplay. There is here a surrender of the concept of cause as we have understood it until now. The concept of cause, according to its simple meaning, is an exclusive condition for the occurrence of the result. If the cause occurs, the effect will occur, and generally, if it does not occur, the effect will not occur, depending on whether the cause is only sufficient or also necessary. When I say that a stone moved because it was kicked, the meaning is that the kick is a necessary and sufficient condition for the stone’s flight. It flew only because of the kick, and without the kick it would not have flown, under the conditions that prevailed there.[^77] This is the essence of the conception of the notion of causal explanation.

According to that description, the cause is a necessary and sufficient condition for the occurrence. In the accepted conception, the cause is at least a sufficient condition, even if not a necessary one, for the occurrence of the effect. Sometimes it is also a necessary condition, and then there can be only one cause of the occurrence described.

Now we are giving up part of this meaning of the concept of explanation, or cause. We allow an explanation to describe only part of the conditions for the occurrence of the explained event. Explanation, in the sense proposed here, expresses a condition that is not sufficient for the occurrence of the effect. Thus both the psychological explanation and the ideological explanation, each on its own, fail to meet even the minimal demand of the concept of explanation, or cause: they do not even constitute sufficient conditions for the appearance of the behavior in question.

According to this approach, if a person undergoes some crisis, that will not be a sufficient condition for a dramatic change in his way of life and thinking. It may be that even if he decides that this change is correct, that too will not suffice. The explanation of the decision is based on both planes together. Becoming religious or becoming nonreligious occurs only if both the psychological conditions and the ideological conditions are present. According to this, the psychological explanation is partial, and the philosophical explanation is also partial. Only together do they provide the full explanation, or cause. The full cause, that is, the combination of the two partial explanations, is what constitutes a necessary and sufficient condition for the occurrence of the effect, and it does indeed obey the accepted definitions of the concepts of causality.

There are several possible scenarios for describing the actual formation of such a state of affairs. For example, psychology may be what caused attention to be directed toward the new direction. Although the new path seems to that person more correct, until now he had not noticed this. The redirecting of attention toward the new direction occurred because of psychological crises, and the final decision was made on philosophical and evaluative grounds.

Another possibility is a case in which that person had not dared to make the change in his way of life, even though he knew, at least implicitly, that it was the right change and that he believed in it. The psychological crisis brought him to a mental state in which he is prepared to make changes of this kind. There are other possibilities, and this is not the place to dwell on them.

This detail can point us to additional situations in which two explanatory planes are partial, and only their combination is the full cause of the described result. For example, in the case of Newton’s apple. The question is not only why the apple fell downward, but why the apple fell on Newton. The gravitational explanation explains why the apple fell, but it does not explain the fact that Newton was sitting there below, and that fact is no less important for answering the question. Newton’s timing may certainly depend on theological explanations: God directed him there in order to punish him for his sins. This is especially true when human actions are involved, where the laws of physics are not exclusive rulers, because a person has free will,[^78] and we have already seen that his spirit influences his behavior and his steps. Here too, then, there is a mechanism similar to the one we saw above: only the joining of the two explanatory planes provides a full and exhaustive explanation of the event.

It should be emphasized that the very definition of the concept of cause does not require that the cause be based on only one principle. One may certainly propose a cause based on the joining of several principles together, such that only their combination will create the described result. In this sense there is no principled problem in what is said here. The difficulty in this possibility lies in the fact that we usually tend to regard either psychology or philosophy, each on its own, as a full explanation of the actions and decisions of human beings. That assumption must be abandoned if one chooses the mode of treatment proposed here.

6. Option C: Local and Global Correspondence

The third possibility for the simultaneous adoption of parallel planes of explanation, which we also saw in the introductory section on cause and purpose, is the existence of a mutual correspondence between purpose and cause, without any hierarchical relation between them. We shall now examine this possibility, and afterward elaborate and generalize it.

First, let us recall what we saw at the end of section 2 above: the solution that distinguishes between description and explanation is of no help when we are dealing with the explanation of a particular case by means of a known law of nature, or with explanation by subsuming under the familiar. For example, with respect to the falling of the apple on Newton, it is not enough to say that the force of gravity caused it to fall, because we must also reconcile this with divine justice, that is, with the fact that Newton sinned. On the one hand, without Newton’s sin the apple would not have fallen on him, either the apple would not have fallen, or he would not have happened to be there. On the other hand, without the physical conditions, the apple would not have fallen.

The question is how the existence of such physical conditions depends on Newton’s sins. We saw that one can explain that general laws of nature are merely the result of God’s action, or one can give them some other mythical-religious explanation. But a specific case, which is an expression of the operation of a general law of nature, is hard to make dependent on an accidental state of affairs, that is, whether a person sinned or did not sin, and certainly on a person’s free choice, which precisely as free might also not have occurred. Would a different choice by Newton, had he chosen to do good rather than evil, have changed the laws of gravitation? Would it have changed the strength of the bond between the apple and the branch, which is apparently also a result of laws of nature no less rigid than the law of gravitation? A similar problem will always arise when the phenomenon being explained is not the sole implication of the law of nature that explains it.

One can adopt here the statement that God loosens the connection between the apple and the tree following Newton’s actions, just as we saw in human actions, where every causal chain begins with an act of will and decision. But such a statement is equivalent to saying that the laws of nature were broken here, and therefore they do not constitute an explanation of what occurred. This, then, is not an acceptable position for someone who concurrently accepts natural-scientific explanations in addition to theological ones.

One can say that it is the timing at which Newton happened to come under the tree that depends on Newton’s deeds. According to this direction, we will have to say that Newton’s arrival under the tree is not an ordinary physical event, and is not derived from laws of nature, for otherwise we have gained nothing. We can always return and ask what would have happened if the natural conditions had obtained, but Newton had not sinned. Therefore we must say that a person’s actions derive from choice, and hence are not subject to laws of nature. God arouses in him a desire to sit under that tree because of his sins. Thus, in an essential sense, the laws of nature are deterministic, but human actions depend on choice, and the punishments that befall a person are derived from the operation of laws of nature upon him in accordance with the intention of his own deeds.

According to this line of solution, we reconcile the two planes of explanation by saying that Newton wondered why apples fall downward, not why this particular apple fell on him. The fact that the apple fell on him derives from his deeds, but it is the general law that arouses his wonder.

Yet the direction proposed here is still problematic. It is not clear what we are to say when the event concerns inanimate objects, when the dimension of choice does not enter the event at all. Sometimes an event occurs that has consequences for a person, but directly concerns only inanimate objects, for example when a person’s property is damaged naturally and not by another person. We will therefore propose here a further direction, that of a general correspondence between different planes of explanation.

The meaning of the correspondence option is that the two relevant planes of explanation are simply two valid descriptions of the same phenomena or occurrences. These are like two different languages for describing reality. For example, scientific language will say that the apple fell because of physical conditions. Theological language will say that it fell because of considerations of reward and punishment. These are two corresponding languages for describing reality.

Yeshayahu Leibowitz used to say that as a believing Jew he held that the world was created about 6,000 years ago, whereas as a man of science he maintained that the world is about 15 billion years old.

Of course, the obvious question now arises: what will happen if the physical conditions obtain but the theological conditions do not, or vice versa? It is not enough to say that these are two languages, since they describe different causal mechanisms.

The answer proposed according to the model of correspondence is: such a situation is impossible. The reason is that there is a full correspondence between the two planes, and whenever the physical conditions obtain, appropriate theological conditions obtain as well, and vice versa.

There is indeed no principled obstacle to the existence of such a correspondence, but under random conditions, without external influence, such a correspondence is patently improbable. Why should we assume such a correspondence that arises by chance?

The situation resembles the problem of the correspondence between human thought and the world, which also seemed improbable without external intervention. There we argued that we must assume the existence of a coordinating factor, God, who ensures the continual existence of the correspondence. The same is true here. God ensures the existence of these correspondences as well.

We should note that in that context too three possibilities arose for explaining a general correspondence between two things, A and B: A influences B, B influences A, or a third factor synchronizes the two. Up to this point we have seen the possibility that theology dictates physics, and now we turn to the description of the last proposal, namely that there is a factor coordinating the two planes, without any direct mutual influence between them.

Let us note that we adopt such a picture only because both planes of explanation seem necessary to us, and therefore we have no other way to reconcile them. Clearly, had we not been constrained in this way, and had we been able to give up one of them, we would not have chosen such a view. The same is true with respect to the issue of the correspondence between thought and the world.

For example, one who believes in human freedom will explain each of our decisions on a value-based and philosophical plane. Yet on the other hand, it is clear even to him that we are also influenced by psychological considerations. When we find ourselves in a state in which we have full confidence in both planes of explanation, we are forced to choose solutions of this type, or one of the previous ones.

Let us now try to sharpen further the difficulty inherent in such a view, and bring a few more examples. First, let us distinguish between two kinds of correspondences.

A. Local Correspondence

Local correspondence is the matching of a particular principle within explanatory language A to a particular principle in explanatory language B, and vice versa. Here there is a one-to-one correspondence between the two explanatory languages, and such a state does not raise any special difficulty.

An example is the possibility we raised above with respect to the general laws of nature, that God is the one who operates the laws of nature. Here there is a divine will underlying every law of nature, and therefore the correspondence is created in a trivial way. These are two different languages for describing the same thing. On the theological plane, God operates nature, and this is the theological explanation of natural occurrences. On the natural plane, laws of nature provide the explanation. Or, in the context of Aristotelian physics: the stone’s desire to return to its place of origin is nothing other than a statement in a different language to the effect that the stone tends to be drawn toward the earth because of the force of gravitation.

Since there is a correspondence between the basic principles of the two systems, we are guaranteed that all the results will be identical. The explanatory systems will yield exactly the same results. We saw examples of such correspondences above when we discussed the MIU language, and the correspondence between the fundamental principles of the system guarantees that the same system of legal words will be produced in the language.

B. Global Correspondence

Global correspondence does not match a specific principle within explanatory language A to a specific principle within explanatory language B for each of the principles existing in the two parallel languages. The correspondence here is created globally. The collection of events or occurrences is explained by both explanatory frameworks. The correspondence exists only at the level of the facts, that is, in the collection of results. In the parable of the MIU language, one could say that we create another parallel system, that is, another language, in which at a direct glance no correspondence can be seen at all between its principles and those of the MIU language, and yet the two sets of words produced by the two systems, the two languages, are identical. Local correspondence matches a principle from system A to another principle from system B one-to-one. Global correspondence is a state in which there is a correspondence at the level of the entire system, but we cannot see how it is created. There is no correspondence between the individual principles of the two systems. The correspondence is created only from the combination of all the principles, a macro-level correspondence.

Let us spell this out a bit more. For the purposes of the present discussion, let us define a set of explained phenomena, X, Y, and Z, for which we propose an explanation, that is, a theory, by means of a system of several theoretical principles, A, B, and C, whose combined action creates the explained events. There is also another theoretical explanation for those same events themselves, but it is based on a different system of principles, a, b, and c.

A local correspondence between the explanations exists when principle A is simply principle a in another language, and vice versa. One can reduce the one to the other, like the transition in the MIU language from letters to the digits 013. The same is true of principles B and C. In such a state, it is no wonder that a full correspondence arises between the explanations, and if event X follows according to the first explanation, it necessarily also occurs according to the principles of the second explanation.

A global correspondence, however, is different. There is no possibility of matching any one of the principles that make up the first explanation to some principle in the language of the second explanation, or vice versa. Principle A does not correspond, that is, it cannot be treated as a description in another language, to principle a, nor to b, nor to c. Yet despite all this, the first system of principles explains the occurrence of the set of events X, Y, and Z under certain circumstances, and the second system of principles also explains the occurrence of exactly those same events under exactly the same circumstances.

This kind of correspondence we shall call global correspondence, since the overall operation of these two systems of principles leads globally to the same results, despite the absence of any detailed correspondence between the principles of the one system and those of the other.

Let us continue to elaborate. Even a global correspondence can arise on several levels. There are situations in which one can explain the system of laws of system A by means of the system of laws of system B, or vice versa. True, one cannot match one specific principle to another specific principle, but one can explain how the system of principles as a whole creates the second system of principles in general.

According to the prevalent reductionist conception, this is exactly the situation among the various domains of science. We saw that some hold that biology or physiology are nothing but a macroscopic average of the microscopic laws of physics and chemistry. The laws describing the behavior of large and complex systems are a summary of the behaviors of the small systems that compose them, for example the particles composing the body or the macroscopic object.

Even according to such a reductionist view, it is clear that a given biological law cannot be derived from one specific physical law; rather, the totality of biological laws is a result of all the laws of physics and chemistry relevant to the components of biological systems. In this sense, there is here apparently a global correspondence. But unlike the definition of global correspondence above, here the correspondence pertains also to the laws of the domains in question and not only to the phenomena explained by them. Through the correspondence at the level of laws, a correspondence to the explained facts is created as a matter of course.

According to this description, the laws of biology are a result of the laws of physics, that is, in principle one can derive them from the laws of physics. It therefore follows that the phenomena in biology can in principle be explained by both of these systems. Even in such a state, it is no wonder that this happens. This is in fact a correspondence that is local in essence, between systems whose numbers of laws, or principles, are not identical. Clearly one cannot derive physics from biology, and therefore this correspondence is essentially one-way. One may say that physics is the more basic and fuller explanation of biological phenomena, whereas biological explanation is only a shorter and more efficient language for describing the physical explanation. This state, with all its complexity, is the simpler one. It is reasonable to assume that there may also be a state in which the relation is symmetric, when sometimes there are the same number of principles, though that is not necessary, and from each system one can create the laws of the other system, so that identity in the facts arises as well.[^79]

But there is a more extreme state of global correspondence, in which no relation at all can be shown, even a global one, between one system of laws and the other. In such a state, the correspondence is created only with respect to the explained phenomena, and does not pertain at all to the theoretical laws that explain them. This is the case in the context of the theological and physical explanations of the apple’s fall on Newton. The principles of reward and punishment do not constitute an explanation of the laws of physics, gravity, the strength of the bond between the fruit and the tree, and so on, but only of the explained occurrences, the apple’s falling on Newton’s head, and the like. Thus there is no relation at all between the systems of theoretical laws of the different explanations, and yet a correspondence arises between the results of their operation. The same phenomena are explained by both systems.

Note 35: Mathematical and Physical Correspondences

To sharpen the distinction between global and local correspondences, one can look at mathematical coordinate systems. Let us take as an example the Cartesian and polar systems in two dimensions. Every high-school student knows the Cartesian way of representing points in the plane by means of a system of two axes perpendicular to one another, usually denoted X and Y. Every point in the plane is marked by an ordered pair of numbers, for example (2 ; 5.3). In the Cartesian representation these two numbers describe the size of the projection of the point onto the two axes. In this example, the projection on the X axis is 5.3, and the projection on the Y axis is 2. This is a common way of representing points in the plane.

One can vary Cartesian forms of representation in countless ways. For example, one may describe the set of points in the plane by means of coordinate systems shifted relative to the original. One can also choose coordinate systems rotated relative to the original.

And in general, the Cartesian form is not the only possible representation. There are many other ways to represent points in the plane. One of the common ones is the polar form, in which every point is represented by another pair of numbers. We will always need two numbers, since the dimension of a plane is 2. In the polar representation, the first number represents the distance of the point from the origin, that is, the meeting point of the X and Y axes, and the second number represents the angle formed between the line connecting the origin to the point and the horizontal axis.

Insert here a graphic depiction of the three representations.

Add conversion formulas: global and local.

There is a one-to-one relation between any two such forms of representation. This means that from any Cartesian pair of numbers one can reach the corresponding shifted pair and the corresponding polar pair, and vice versa. As can be seen from the imagined diagram above, the line connecting the origin with the point described, together with the two Cartesian projections on the X and Y axes, forms a right triangle. In such a triangle one can easily describe the lengths of the two legs in terms of the hypotenuse and the angle, and vice versa.

If we treat the different representations, Cartesian and polar, as explanations, and the points in the plane described by them as the facts explained by those “explanations,” then we have here two explanatory languages, in fact two descriptive languages.

The correspondence between the Cartesian representation and the shifted representation is simple, and it is clearly local. By contrast, the correspondence between the Cartesian representation and the rotated Cartesian or the polar representation is apparently global, since there is a correspondence with respect to all the facts, both languages describe all the points in the plane well, but there is no specific correspondence between one axis in the original Cartesian representation and another axis in the rotated or polar one, and vice versa. From the two axes of one representation together, one can reach each of the two axes in the parallel representation.

Yet, as we saw above, this too is not a genuinely global correspondence. The fact that there is a correspondence between the explanatory systems themselves, that is, the axes, even if it is not local, indicates that this is local in essence. The explanations reflect one another, and the correspondence regarding the facts arises by itself in a trivial way.

In fact, in mathematics correspondences between such systems are proved precisely through correspondences between the laws constituting them. There is no way to check the correspondence with respect to infinitely many points in the plane, and therefore it is proved through its existence with respect to the basic laws that generate that set of points. The correspondence with respect to the points themselves, the facts, is automatically produced as a result of the correspondence between the systems of laws that generate them. A genuinely global correspondence would arise only if we were to find an additional representation for all the points in the plane, one that admits no correspondence at all with the parallel representation, and yet nevertheless describes, or “explains,” all those points.

To see that the description here is not a merely mathematical phenomenon but a prototype for correspondences in various fields, let us discuss briefly two different physical-chemical descriptions of the substances that exist in physical reality, and see their relation in light of what was described here.

In ancient thought, four elements were used to describe the material objects in the world: earth, water, air, and fire. To modern eyes, after Mendeleev, this seems like a rather primitive conception of the elements of material reality. This ancient division seems wholly implausible, because these do not appear to be the basic characteristics or the basic parameters for describing material objects.

Yet one should note that concepts such as heat and cold, temperature, dryness and moisture, lightness and heaviness, mass, can indeed be understood as a basic system characterizing material objects. If so, one can see that the four elements also span the same space of characteristics. Each of the elements has characteristics that already appear in the ancient literature. For example: earth is heavy and dry, water is cold, heavy, and moist, fire is hot, light, and dry, and air is light and dry.

Any object in which we see heat and heaviness and moisture in some dosage can be described by different combinations of fire, air, water, and earth. For example, a great deal of moisture means a great deal of water. But heaviness is also present here, and therefore if the object is not as heavy as water, one must add a good deal of air to the description to offset that, and so forth.

In a certain and very limited sense, one can see this ancient description as simply a “rotation” of the modern coordinate system. Instead of describing matter as a combination of chemical elements such as oxygen, hydrogen, and the like, we describe it in coordinates composed of combinations of these. Dryness is a certain dosage of oxygen and hydrogen and nitrogen and carbon, and so forth; moisture is another combination, and likewise cold and heat and the other Aristotelian parameters. In effect, there is here a kind of “rotation” of the coordinate system, creating new axes, each of which is a combination of all the axes in the parallel system, as in the formulas relating the rotated representation to the original representation in the imagined diagram above. Seen this way, the matter seems much more reasonable. Instead of saying that a given object is heavy, we say that it contains much earth relative to the other components. One should note that according to our interpretation, the element earth is probably not identical with literal soil. Literal soil is only one representation of it.

We see, then, that a correspondence parallel to the rotation of a coordinate system can translate for us an ancient and otherwise unintelligible explanatory system into another system that sounds much more reasonable. If so, this demonstrates that if we notice such correspondences, we will not necessarily need to reject one explanation in favor of the other.

Yet, as we have seen, the correspondence between rotated coordinate systems is local in essence, and not a genuinely global correspondence. Here, however, one can continue and see a truly global correspondence if one notices that the ancient description does not descend to the same level of detail as the modern description. It is satisfied with parameters such as heat and cold, moisture and dryness, and the like, whereas the modern description analyzes every substance into Mendeleev’s elements according to several parameters. Thus the ancient description describes reality correctly, but it addresses only certain of its characteristics, whereas the modern description addresses other characteristics. Here we have a genuine global correspondence, but the two explanatory languages refer to different planes of reality. Both describe the same reality correctly, but in different parameters.[^80]

Let us now return to the discussion of genuine global correspondence, that is, a full correspondence regarding the facts that is not derived from a correspondence between the theoretical laws. The emergence of such a correspondence is very puzzling and unusual; in fact it is patently improbable. For if the systems of laws are really not merely fictions for description, but the physical causes of the emergence of the facts, then how did a correspondence arise between them if there is no relation between the causes of system A and the causes of system B? Here there is no visible mechanism of direct influence between the two systems, and therefore it is reasonable that we require the claim that there is a coordinating factor that does this. This factor must be powerful, for otherwise it is not clear how it can create such a global correspondence between intricate systems of laws pertaining to complex systems of events and occurrences, and also possess the will and ability to create such a correspondence.

On the other hand, it must be emphasized that such a correspondence is not impossible. We saw in the second gate that every system of facts necessarily has infinitely many empirical generalizations, which may differ from one another with respect to some other facts, and consequently also infinitely many possible theoretical explanations. Thus any two such theoretical systems constitute a pair of the kind we are discussing here. These are two systems of causes that create exactly the same macroscopic collection of facts. In the next note we shall bring two examples of this mode of thought.

Note 36: Theories of Synchronicity

The first example we will bring is taken from the book of the American logician Raymond Smullyan, The Silence of the Tao, and it deals with astrology.[^81] Astrologers are often asked how a star located at an enormous distance, many light-years away from us, can influence our lives. Usually they give vague, nonempirical explanations, and at times explanations that are downright foolish.

When a rational person examines most of the explanations proposed for the validity of astrology, he tends to reject them out of hand, and thereby to reject astrology itself. But as we shall see here, that conclusion is not necessary. The fact that the explanations are unconvincing is not in itself enough to reject the facts.

Smullyan proposes a theory from the school of the psychologist Carl Jung, called synchronicity. The stars do not influence me, but for some reason there is a correlation between the arrangement of the stars in the sky at the time of my birth and the history I will later have. The correlation may stem, for example, from the fact that there is someone who ensures such a correspondence simultaneously, and such a correlation can indeed occur without any mutual influence. Incidentally, theoretically it may be that there is no one who ensures such a correspondence at every moment, but that there are parallel processes in heaven and on earth whose occurrence is governed by equivalent laws, and therefore the correspondence is preserved all the time. These are two mechanisms parallel to the two kinds of global correspondence we defined above: a completely global correspondence, and a global correspondence in which the systems of laws are already corresponding and not only the facts.

Smullyan tells there a story about a Zen sage who was sunk in thought, and suddenly felt danger and turned around. He saw behind him only his young servant, and nothing else. It finally turned out that at that very moment the boy had indeed been thinking about the possibility of stabbing his master with a sword, and about the fact that despite the master’s great skill he would not be able to defend himself in such a case.

The explanation Smullyan proposes, since he does not accept the possibility of reading thoughts, is that the very event that caused the boy to think his thought at that moment also caused the master suddenly to feel danger. The event could have happened earlier or at that moment, but it is a third factor that created the correspondence between them. Again, the boy’s thought is not the cause of the master’s feeling; rather, there is a correspondence arising from some external cause.

The conclusion is that even if we cannot point to a mechanism of influence between two corresponding systems, that does not necessarily mean we must reject the claim that they correspond. They may do so without any direct mechanism of influence. To ascribe such a description to random miracle is not plausible, though in principle it is possible, but perhaps the correspondence is created intentionally and not by chance, either by a factor that continually coordinates the operations of the two systems, or because their laws are equivalent. The first possibility is what we called global correspondence, and what Smullyan calls synchronicity. The second possibility is what we called local correspondence, one of its two types.[^82]

The second example we shall bring is described in a book by Arthur Koestler,[^83] who discusses at length the synchronic theory of the psychologist Carl Jung together with the well-known physicist, Nobel laureate Wolfgang Pauli. Before them came an Austrian biologist named Paul Kammerer, one of the last adherents of Lamarckism, who had great influence on Jung’s thought. This is a generalization of the principle described above as a proposal for explaining astrology.

Kammerer attributed significance to coincidences. He kept a meticulous record, rigorously classified into different types, of coincidences that happened to him between the ages of twenty and forty.

Kammerer’s general claim was that there is an attraction between events similar in certain respects, which he classifies there, along the axis of time. The coincidences we directly encounter are only the tip of the iceberg of a broad natural phenomenon, namely the tendency of similar events to cluster near one another in time. He likens this to the force of gravitation that draws masses toward one another, and claims that there is a similar attraction between similar events.

Jung and Pauli argued that causal explanations constitute only a small part of our understanding of the ways nature operates, and synchronic explanations are mechanisms no less important for such understanding. Some events occur because of causes, and others because of synchronic attraction.

It is important to emphasize that the expression attraction sounds like a physical force, but this is only a phrase. We have here a description that is not causal at all. In fact, it is an alternative mechanism to the principle of causality. The fact that we are captive to causal thinking is what drew Pauli, Jung, and Kammerer to such quasi-causal descriptions.[^84]

Koestler himself notes there a resemblance between this attraction and another Pauli principle, this time in physics. Pauli is known mainly for his exclusion principle, Pauli’s principle, which states that in certain known kinds of particles, called fermions, after the physicist Enrico Fermi, see the first book, note 5, two particles cannot be in exactly the same quantum state. Pauli’s exclusion principle has no mechanistic explanation, that is, there is no force that ensures the repulsion of the particles from the same state. It is an ad hoc principle, and therefore very similar to the attraction between similar events in Pauli’s synchronic theory.

In our discussion above we proposed a mechanism of correspondence by means of a third factor in order to explain a correlation between two explanatory languages. We saw, in the first book and also above here, that this is also the explanation of the correspondence between human thought and the behavior of the world itself. One should note that in this case we are not dealing with synchronicity, but with a hidden causal mechanism that we do not always discern. Leibniz’s monadology, which we described in the first book, see note 107 in the fourth gate and elsewhere, is closer to a genuine synchronic theory in the Jungian sense.

Let us also note that Kant and Hume did not even conceive of the third possibility, that of a noncausal correspondence between the human being and the world, because they were captive to the principle of causality. They simply did not take into account the possibility of a noncausal explanation. As we saw, they also rejected hidden causality, which we proposed as a third alternative.[^85]

Defects in Global Correspondences

Because of the difficulty inherent in the very nature of a global correspondence, there are situations in which global correspondences will not be perfect, that is, they will contain “bugs.” In such cases we may still relate to them as essential correspondences, but say that they are imperfect. Let us bring two examples:

1. In the correspondence between the theological description of physical events and their physical description, cracks may appear. What the laws of nature dictate does not lead to the state required by the theological laws. In such a situation God must decide whether to let nature proceed according to its fixed laws, even if the theological considerations do not lead to the same result, or whether temporarily to neutralize the natural system in order to create a correspondence with the theological principles. The second mode of action is called a “miracle.” In a miraculous situation the world reaches a state where theological considerations indicate that occurrence A is required, but the laws of nature do not lead to that.[^86] In such a state, God sometimes decides to suspend the laws of nature, which ordinarily correspond to His theological tendencies, in order to prevent theological injustice. Of this we say that He performs a miracle.
2. Another example of a bug in a global correspondence is the concept of a “scriptural decree.” We saw above that in interpreting the Torah there are four basic modes of approach, peshat, the plain sense; remez, allusion; derash, interpretive exposition; and sod, esoteric meaning. Usually all of these correspond to one another, and every plain-sense consideration is correct independently of the esoteric consideration or the allusive consideration. Each such plane is examined independently. This is a sharp example of the concept of a global correspondence between planes of explanation.[^87]

It is important to note that this is a genuine global correspondence, because in the context of Torah interpretation there is no correspondence at all between the different systems of theoretical laws, but only between the explained facts, that is, the laws themselves. For example, the plain-sense explanation of why we eat karpas on Passover is to create unusual acts that will prompt the children to ask questions on the night of the seder. On the plane of sod, by contrast, the eating of karpas contains very lofty secrets, which are not connected at all to the children or their questions. There is no correspondence between the explanations, and it is created only at the level of the laws, that is, the explained facts.[^88]

In several places in the Talmud and its commentators we find references to some law as a scriptural decree. The meaning is that this is a law without a known reason. According to the approach that God determines nothing without a reason, we are compelled to say that there is here a secret reason, but on the plane of peshat we do not succeed in explaining it. If so, we have here a law that constitutes a defect, a bug, in the global correspondence between sod and peshat.[^89]

Let us emphasize here that such defects cannot arise in local correspondences. Since the laws correspond one-to-one, one can prove that the correspondence with respect to the facts they bring about, explain, is perfect. As we mentioned above, this is how mathematicians usually prove the existence of a full correspondence. Only when the correspondence is not based on a correspondence between the laws, that is, when it is global, can such bugs arise. In such cases, the correspondences between the facts are generally correct, apart from exceptional cases.

Let us illustrate this through the mathematical representations we saw above. Clearly there is no defect at all in the correspondence between Cartesian representations, and even those that are shifted and rotated, with respect to one another. But in the correspondence between the polar representation and the Cartesian one there is a bug, only one, admittedly. In mathematics it is called a topological defect, and it is located at the origin. At that point, whose Cartesian representation is (0 ; 0), the polar representation is problematic. The distance from the origin is of course 0, but the angle can be anything at all. That is, infinitely many polar representations correspond to this point. Such a mismatch can appear only because the correspondence here is not simple; in mathematical language, the dependence between the representations is nonlinear.[^90] As we saw, this correspondence is not entirely local, even though it is not entirely global either.

In correspondences of this kind, defects may appear, and they may also not appear. Truly global correspondences, because of their inherent difficulty, are more prone than any other correspondence to the appearance of such defects. As we saw above, miracle and scriptural decree are two clear examples of this. Even God Himself, as it were, cannot create a perfect global correspondence with laws as rigid as those in the created world, especially when the laws are affected by human free choice, as we noted above. In the next note we shall see a halakhic implication of the various concepts of correspondence.

Note 37: The Logic of Two Distinct Laws

The sage who influenced more than anyone else the style of learning in the contemporary yeshiva world was Rabbi Hayyim Soloveitchik of Brisk. He introduced a new logic and methodology, and perhaps above all a new mode of presentation, in the study of Torah. One of the basic tools in Rabbi Hayyim’s halakhic logic is the concept of “two distinct laws,” which expresses the existence of two different principles hidden under one heading, so that it is often difficult to discern their existence. By means of this mode of thought, Rabbi Hayyim explains many difficulties that are very hard to understand without this distinction.

This topic requires detailed analysis and research, but even without such research it is quite clear that there are several situations in which two distinct laws appear, and in each of them the concept appears with a somewhat different meaning. As we shall see in this note, these different situations may be connected to the different ways in which parallel planes of explanation appear, as we presented them in this chapter.[^91]

The Torah contains a prohibition of unfair commercial overreaching. This prohibition concerns Reuven, who sold Shimon an object at a price different from the market price. If the price is higher than the market price, if the difference is one-sixth of the price, the seller violated the prohibition of overreaching, and if the price is lower than the market price, the buyer violated the prohibition of overreaching.

Rabbi Chaim explains that from the prohibition of ona’ah (commercial exploitation through overcharging or underpaying) we learn two distinct legal principles:

1. A factual-legal principle: the money involved in ona’ah is someone else’s money.
2. A prohibitory principle: someone else’s money is forbidden property. The Talmud derives the prohibition of theft from ona’ah; see Babylonian Talmud, Bava Metzia 61a.

There are objects to which the law of ona’ah does not apply: slaves, promissory notes, and land. In light of what was said above, we must now ask: which of these two principles does not apply to them—the first, or the second? Or perhaps both? In other words: does the fact that there is no prohibition of ona’ah in the purchase of a slave mean that this is not someone else’s property, or that there is no prohibition of theft here even though it is someone else’s property?

Rabbi Chaim resolves the question through a simple logical consideration. As noted, the prohibition of ona’ah applies both to the seller, when he sells at too high a price, and to the buyer, when he buys at too low a price. Thus, when the slave is bought below market value, the buyer is in possession of a stolen slave; and when the price is above market value, the seller is in possession of stolen money.

Now, Rabbi Chaim adds, the Torah did not prohibit under the rubric of ona’ah even the second case, in which the “stolen” property is money rather than a slave. But with respect to holding another person’s money, it is obvious that the prohibition of “You shall not steal” applies. From this it is proven that the novelty regarding slaves was not merely with respect to the first principle—that in their sale the prohibition of theft does not apply. Rather, the novel point is that in the case of slaves sold at a price different from the norm, the money is not considered someone else’s property, and therefore no prohibition of theft applies to it either.

Let us note that with this argument Rabbi Chaim removes Tosafot’s difficulty in the passage there in Babylonian Talmud, Bava Metzia 61a; see Tosafot, comment beginning “Rather, is it not…?”

The structure of Rabbi Chaim’s reasoning is very common in his method. First, he analyzes the topic and finds that under the single heading of ona’ah two different principles are concealed. He then examines, in each context, which of them is the object of the halakhic (Jewish-legal) discussion, and in this way removes many confusions.

There is here something like a correspondence between two explanatory planes. The money involved in ona’ah is forbidden to us because of the prohibition of theft, but it is also forbidden to us because it is not ours. The fact that Tosafot, in the approach mentioned above, clearly understood the matter differently shows that each of these planes can be treated independently. They understood the prohibition of ona’ah as speaking only on the plane of halakhic prohibition—that it is forbidden to steal money involved in ona’ah—and not on the plane of legal facts, namely ownership of the money.

In this context, however, it is somewhat difficult to see that there are really two parallel explanatory planes here, because in this example the two principles contain one another, at least in one direction: if the money is not his, then obviously the prohibition of theft also applies. Rabbi Chaim introduces here a fundamental novelty: the converse is not necessary.[^92] Such a situation parallels a correspondence in which one system causes the other, as in the correspondence between a theological explanation and a physical explanation of a general law. In such a case, we saw that one can say that God activates the physical law.

An example in which one can see two principles that are foreign to one another, yet both exist simultaneously, is Rabbi Chaim’s discussion of the essence of prayer, in his comments on Maimonides, Mishneh Torah, Laws of Prayer 4:1.[^93] Such an example parallels global correspondence.

With regard to all commandments, we are required to intend to fulfill our obligation when performing them. Some halakhic authorities hold that without such intention one does not fulfill the obligation at all. Rabbi Chaim explains there that prayer involves two different kinds of intention: intention with respect to the words, that is, understanding the meaning of the words and directing one’s mind to that meaning while reciting them; and the general intention required in all commandments, namely to fulfill the obligation of the commandment. See there for details regarding the implications he derives from this and the difficulties that fall away in light of this distinction.

Even here, however, we are dealing with two different acts, both of which we are commanded to perform together, and not with two theoretical principles that explain a single law. This is not really global correspondence. Global correspondence, as we defined it above, parallels a situation in which one law is explained in two different ways, and both explanations are correct, each independently.

An example of this appears in the reasoning of Rabbi Chaim’s son, Rabbi Velvel, the Griz, concerning drinking before havdalah (the ritual marking the end of the holy day) on Yom Kippur.[^94] It is well known that Rabbi Velvel was uncertain whether it is permissible to drink water before havdalah on Yom Kippur, even though on other festivals it is clearly permissible to drink water before havdalah. His explanation of the doubt is that the prohibition against drinking before havdalah on an ordinary festival or on the Sabbath stems from the prohibition against engaging in acts that distract one’s attention before fulfilling the obligation incumbent upon us. This is the accepted explanation, though some disagree with it. From this standpoint, it is permissible to drink water before havdalah exactly as on any other festival, since this is not a significant act and it will not distract us from our obligation to recite havdalah at the close of the day. On Yom Kippur, however, there may be an additional reason that drinking before havdalah is forbidden: so long as we have not recited havdalah, the prohibitions of Yom Kippur, that is, the obligations of the fast, still apply to us, for the day has not yet fully departed. In such a situation, it is obviously forbidden even to drink water, since according to the laws of Yom Kippur any drinking whatsoever is prohibited, including water.[^95]

Here again, two principles are hidden behind a single law, and in this case they are two explanatory principles, that is, two halakhic theories. In this case both explanations exist, and it is possible that in certain circumstances, and usually this is so, each of them will have different implications.

In general, when there are two such principles, it is important to distinguish which of them is determinative. There are situations in which only one of them is really present, and after analysis it becomes clear that the second principle in fact does not exist at all, as in the case of ona’ah according to Tosafot, who apparently did not understand the matter as Rabbi Chaim did. By contrast, there are situations in which both principles are present, and both must be taken into account, as in the case of drinking before havdalah, and also in the case of ona’ah according to Rabbi Chaim.[^96] There may even be extreme situations in which the two principles contradict one another, and yet both exist simultaneously.[^97] There are also situations in which the two principles together create a third reality. Such situations parallel mode 5 above, that is, the case of cumulative explanations.[^98]

It would seem that there are additional situations of two legal principles as well. Indeed, almost Rabbi Chaim’s entire aforementioned book is composed of different examples of this principle. But we need not prolong the discussion here.[^99]

Summary

In the course of the present chapter, we saw a variety of ways of dealing with situations in which several explanatory planes appear convincing to us, and we try to adopt them in parallel, that is, simultaneously. The first way was the banal solution, namely giving up one of the planes, which means that the contradiction between them appears to us insoluble. We then moved on to models that make possible the simultaneous acceptance of several explanatory planes together.

As background to the distinctions presented in this chapter, we sharpened and illustrated the relation between semantics and syntax. We saw that one can relate to phenomena in at least two different ways: through syntax, or form—that is, description—and through meaning, that is, semantics. This distinction is essentially similar to the distinction between the thing in itself and its form, which was discussed at length in the second gate of the first book, and also at the beginning of the present book. We saw that our grasp of phenomena, and our inability to grasp noumena, is not a cognitive limitation. It follows from the very distinction between them. Every act of cognition is conditioned by instruments of cognition, and therefore it necessarily takes place on the phenomenal plane.

In light of this distinction, we presented the first way of holding on to two parallel explanations: by classifying one as description and the other as explanation. This mode worked in the context of the relation between scientific theory and theological explanations and the like. However, in an explanation of the type of reduction to the familiar—that is, grounding a certain phenomenon in a familiar theory—this model fails.

The next mode was the distinction between model and explanation. Logically, it is identical to the previous one, except that a model is a kind of theory rather than a kind of empirical generalization. That is, this distinction allows us to treat a situation involving two theories, and not only a theory and a phenomenological generalization, in terms of description and explanation.

The next mode was the distinction between an explanation of cause and an explanation of purpose. Here too, the matter sometimes appears similar to the distinction between description, identified with cause, and essential explanation, identified with purpose. We shall see some examples of this in the final gate. In any event, a teleological explanation and a causal explanation do not contradict one another. Sometimes causality is a description of the mechanism through which the desired purpose is achieved.

The next mode was a situation in which no explanation stands on its own, but all the explanations together combine to provide an overall explanation of the phenomena under discussion. This solution shows that sometimes what appears to us to be an explanation is only part of the picture. The whole picture—the full explanation—is the totality of the relevant explanatory contexts.

The final mode presented was correspondence between different explanatory planes. Local correspondence between the laws of parallel theories is nothing more than the same description, or the same explanation, of the phenomena in question in two different languages. This is, once again, a version of the distinction between explanation and description. But global correspondence is a new model, problematic by its very nature, and therefore prone to certain deviations in the fit between the explanations. As noted, we need very strong reasons in order to adopt both explanations together despite the problems involved, and not give up either of them.

What Does All This Have to Do with the Analytic-Synthetic Controversy?

In conclusion, let us note that distrust of the possibility of parallel explanatory planes, in almost all the modes discussed, stems from an analytic stance. The conception of scientific explanation as description follows from a distrust of our ability to provide, or to find, explanations at all. It is therefore clear that such a stance will also tend to avoid classifying different theories, in any field, as explanations, and will treat them all as descriptions. The theories will be models, and the empirical laws will be at most efficient and convenient generalized descriptions of a complex reality.

An analytic stance also does not recognize purposes, only causes, and therefore it generally refuses to recognize teleological explanations as well. As we saw, this is the reason for the rejection of teleology in the modern philosophy of science.

The next mode was the combination of two explanations into a single explanation. Here, in principle, there is no obstacle preventing proponents of an analytic stance from joining such a view, but in practice this usually does not happen. For example, those who hold analytic positions generally tend to explain a step of value judgment as though it takes place entirely on the psychological plane. The reason may be that if explanation refers to reality, one can acknowledge that reality is complex and that each field deals only with part of it. But if scientific theory and explanation are viewed only as refined description, then there is little likelihood of adopting a theory that requires additions from outside in order to describe reality correctly as the most efficient description. Convenience and efficiency push this sort of combination aside. According to the synthetic stance, since the study of reality shows that reality is indeed complex, then despite the preference for a single comprehensive theory—because of elegance, simplicity, and convenience—reality sometimes compels us to adopt a combination of two theories as explanation.

In other words: when we recognize the existence of constraints, we sometimes choose a more complicated description because we have no alternative. But according to the analytic stance, which holds that there are no constraints at all, and that scientific theory, or scientific description, is arbitrary, it is obvious that one may choose other descriptions of reality as one wishes, according to criteria of simplicity, elegance, and efficiency. If so, to those who hold the analytic stance there appears to be no reason at all to choose a complex description.

With respect to the final mode, the one that describes correspondence between different explanatory planes, it is clear that there is no possibility at all for proponents of an analytic stance to adopt it. The reason is that such correspondence appears impossible, at least if it occurs by chance. Local correspondence is nothing more than description in two languages, and there is no substantive novelty in it. Nor does it occur by chance, since at bottom there is really no correspondence here between two things at all. But global correspondence is impossible, at least probabilistically. The only way to adopt a conception of global correspondence is to recognize the existence of a coordinating factor, namely God, who possesses sufficient will and power to create such a complex correspondence.

As we saw in the first book, the connection between belief in God and a synthetic stance is twofold. On the one hand, the only way to recognize His existence is through synthetic considerations. Analytic proofs of His existence do not exist; see also the next gate and the final gate on this point. On the other hand, God Himself is also the only guarantee of the validity of synthetic assumptions. That is, He constitutes a condition for knowledge of the world. There we saw this with respect to the correspondence between human thought and the world in itself.

Here we see it with respect to the correspondence between different explanatory planes. Above, we compared in greater detail the considerations leading to these two claims. On the one hand, the way to arrive at the existence of God and the existence of such correspondence—in fact, the way to attain genuine trust in scientific conclusions—is only by synthetic means, whether through testimonial reasoning or eidetic seeing. On the other hand, such correspondence, in and of itself, necessarily rests on belief in the existence of God as a coordinating factor.

Summary of the Discussion in This Gate

In this gate, we discussed the possibility of accepting two explanatory planes, or interpretations, simultaneously, even in cases where they contradict one another.

From a discussion of the very concept of explanation, we saw that such a situation involves an inherent difficulty. Every explanation is supposed to be a necessary and sufficient condition for the consequent, that is, for what is explained. But it cannot be that there are two different conditions, even contradictory ones, that are both necessary and sufficient for one and the same occurrence. A necessary and sufficient condition is, by its very nature, unique.

The various possibilities for nevertheless adopting parallel explanatory planes require distinctions between cause and purpose, between explanation and description, between exclusive explanations and cumulative explanations, and finally between local and global correspondence among the different planes. In such a correspondence, two factors can be considered necessary and sufficient conditions for the same occurrence, even though they are different and perhaps even contradictory. What makes this possible is the correspondence between them, which ensures that whenever one condition is present, the other will be present as well.

Such correspondence requires a guiding hand. We have already seen that a synthetic stance cannot stand without belief in God, at least in the form of philosophical theism.

Another point that emerged in the course of the discussion is that the relation between semantics and syntax is not unambiguous. There is a whole hierarchy of semantic-syntactic planes, such that one plane can function as semantics relative to a higher plane, and as syntax relative to a lower plane.

For example, science is syntax relative to myth, since it deals with the outer side of things. But, as we saw, science itself relies on semantic processes and not only on syntactic ones. For example, it is clear that behaviorism is syntax in relation to ordinary psychology, even though both of them are syntax in relation to an essential-religious conception of the things themselves. More generally, phenomenological theories, parallel to behaviorism, are always syntax in relation to essential theories that bring out the meaning, that is, the semantics.