Gate One: Basic Concepts — The Analytic and the Synthetic

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This is an AI-generated English translation of a chapter from the book Two Wagons and a Hot Air Balloon (שתי עגלות וכדור פורח) by Rabbi Michael Avraham. Translated by OpenAI’s GPT-5.4 model with high reasoning effort. Read the original Hebrew (PDF).

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From the book Two Wagons and a Hot Air Balloon by Rabbi Michael Avraham. Translated from Hebrew using gpt-5.4 (reasoning_effort=high, batch API).

Basic Concepts: The Analytic and the Synthetic

This gate contains four chapters:

Chapter One: Pre-Kantian dogmatism and the challenge to it
Chapter Two: Kant’s solution: the synthetic a priori
Chapter Three: Another aspect of analyticity and syntheticity: forms of inference and axiomatic systems
Chapter Four: Forms of thought and positions: analytic and synthetic

Introduction

Over the past several decades, in various contexts, a new kind of critique has emerged. This type of critique is inseparably bound up with what is called “postmodernism,” “the new Enlightenment,” “political correctness” (PC), and many similar labels.

I would like to set out from one of the most salient features of this critique: the pointing out of hidden assumptions. For example, in this kind of criticism, feminists look for the male assumptions embedded in every argument or field of activity. Because of these assumptions, they claim, women find themselves at a disadvantage in various domains. Black people in the United States make a similar claim—or, more precisely, their white spokesmen mostly make it on their behalf—against the “white” mode of thought that dominates various fields, and because of which they find themselves at a disadvantage in achievement and status. The same applies to homosexuals, to Jews of Middle Eastern descent in Israel, to Eastern peoples as they are perceived in Western scholarship, and likewise to other disadvantaged minorities.¹

In arguments of this type, it is often enough simply to point to the assumptions that underlie prevalent positions or modes of thought in order to reject or endorse them. There is no need to address the validity or reasonableness of those assumptions themselves; for purposes of criticism, it suffices merely to indicate that they exist.

Such a mechanism of argument means that the mere existence of first assumptions makes the criticized position dependent on the critic’s grace, and therefore gives it only relative significance. If there are first assumptions, then obviously one may choose not to agree with them, since first assumptions are arbitrary and cannot be proven. Hence the possibility of accepting those positions is, at least ostensibly, completely undermined. As noted, in many arguments of this kind there is no reference to the assumptions themselves, apart from pointing to their existence and perhaps attaching pejorative labels to them—chauvinist, racist, and so forth—depending on the context. There are feminist positions, for example, that have reached the critical conclusion that Newtonian mechanics is based on masculine thinking. Those feminists who claim that Newton’s mechanics is masculine do not bother to justify the point, or even to argue that such assumptions are unreasonable in their view. It is enough for them merely to indicate the existence of such assumptions in the background of the field under criticism.²

It seems to me that here one can see most sharply the philosophical conception that underlies these new modes of thought. This conception holds that the assumptions of an argument are arbitrary things, which can be accepted or rejected without justification, since it is self-evident that axioms cannot be justified, and certainly cannot be proven. Consequently, any hegemonic position can be easily destabilized simply by pointing to the existence of axioms at its foundation. According to the proponents of political correctness, every principled position is created out of the interests of those who benefit from it, since there is no room for any other kind of justification for first assumptions. I want to stress again that this ideological-social outlook is based on a philosophical conception regarding the status of the first assumptions of arguments.

Let us briefly note that someone who supports these ways of thinking will apparently accept a position as valid and binding only if it can be justified without any first assumptions at all. But this is, on its face, an absurd position—unless that supporter is willing to admit that he is simply a complete skeptic. That is, he must concede that he recognizes no certainty whatsoever. At this point we move directly from this kind of critique to philosophical postmodernism, and to the mode of thought that will later be defined as “analytic.”

It is important to note that skepticism of this kind arises precisely from a demand for logical certainty and proof. The analytic thinker accepts only proven claims as true, and therefore—absurd as it sounds—demands a proof that is not based on any first assumptions whatsoever. When he reaches the unavoidable conclusion that no proofs meet this bizarre criterion, that extreme rationalist becomes an intellectual nihilist who denies the very existence of certainty. Schematically, we see here the strange reversal produced by such a worldview. One who seeks proofs and does not find them casts doubt on and destabilizes every position. Thus the maximalist demand with respect to truth and certainty leads directly to intellectual nihilism and skepticism.

Postmodernity appears in the world in many different forms. Most of them are not classified as postmodern philosophy at all, but as movements in contemporary art or literature. Many of those involved do not define themselves as “postmodern,” and often this characterization is applied to such phenomena by outside observers. In this book, however, we will treat postmodernity as a comprehensive worldview, even though it is not always such in practice, and we will do so only at the level of abstract philosophy. At this abstract level, one may say that all these manifestations rest upon the philosophical conception described here. This is not to say that every such manifestation is consciously aware of this philosophical dimension, and some of them may not truly embrace it to the end. But in order to examine this phenomenon, we must generalize to some extent, and address that generalization. On the status and meaning of generalizations, see the introduction to the second part.

At first glance, it would seem that this form of thought should not have many supporters, since very few people are truly willing to define themselves as complete skeptics. Later I will try to show that, despite vigorous denials, analyticity is one of the important roots of the being of Western culture as a whole, and that it is what leads to this kind of critique. In the sixth gate we will focus more sharply and see that even most of those who feel themselves to be “modernists” (Gadi Taub, for example) in fact do not go much farther than those whom they criticize, namely the postmodernists. This is an important point about which the reader may hesitate during the first gates. My claim is that this supposed “modernism” is the herald of postmodernism and stands at its foundation. It is not in complete opposition to it, as one might perhaps expect, and as Gadi Taub tries to argue in his book.

From this brief discussion it becomes clear that in order to deal with analytic and postmodern claims and arguments, one must begin at the philosophical level that discusses the relation between premises and conclusions, or more fundamentally, between what is known with certainty and what is inferred and claimed with lesser certainty. As far as I know, Kantian philosophy is the only philosophy that tries to confront directly, and without evasion, the question of how one arrives at first assumptions, or how one arrives at comprehensive truths about the world. Kant will accompany us until the seventh gate, where we will argue that he did not really solve the problems he presented.

For this reason, the first two gates of the book begin with a description and definition of the analytic and the synthetic in the Kantian sense, a characterization of transcendental arguments (which will be discussed at length in Chapter Two of the seventh gate), and several logical and epistemological implications of these positions.

In this gate we will describe two parallel tracks: the analytic and the synthetic. The description of each begins with concepts. The concepts “analytic” and “synthetic” (which originate in Kant’s thought) are later generalized in this book into two types of thought: analytic thought and synthetic thought. From this generalization we will continue on to two philosophical positions: the analytic position and the synthetic position. The relation between the two, on several levels, is the subject of this book.

This gate opens with a description of pre-Kantian dogmatism and its destabilization, primarily by David Hume. It then presents the answers Kant proposed to the problems Hume raised, through the definition of analytic and synthetic propositions, the distinction between that division and the division between a priori and a posteriori propositions, and the formulation of the problem of the synthetic a priori. We will then discuss the meaning of logic and present the concept of an “axiomatic system.” Finally, we will try to illustrate and define the two types of thought and the two positions that will serve us in what follows.

Chapter One: Pre-Kantian Dogmatism and the Challenge to It³

Historical Background

The achievements of mathematics and science at the beginning of the modern age created an atmosphere of confidence in the process of drawing conclusions and in scientific progress. This is the stage later called by Kant the “dogmatic slumber.” It seemed clear that the foundations on which scientific inquiry rested—those that Kant refers to as dogmas—were stable. And indeed, the fruits yielded by that inquiry seemed to justify this sense of confidence. Science advanced steadily, and our understanding of the world around us, and as a result also our control over it, continually grew.

Alongside scientific work, which was accompanied by rationalist confidence—rationalism being the view that grounds human cognition in reason—from the school of Descartes and Leibniz, philosophers reflected on the role of consciousness and intellect, and of sensory impressions, in the process of inquiry.⁴ In contrast to rationalism, which characterized continental philosophy at that time, three philosophers in Britain represented various forms of the empiricist approach, which grounds human cognition in empirical acquaintance with the world: Locke, Hume, and Berkeley.

At the end of the seventeenth century, the English philosopher Locke argued that sensation is beyond doubt and faithfully reflects reality. It is specifically the intellect that can lead to error. The intellect’s combination of sensory impressions is essentially the imposition of a subjective form upon objective reality. Another English philosopher, Bishop Berkeley, took this insight further and argued that reality is nothing but what is perceived by human beings through their senses. He then continued and claimed that the hypothesis of an external world is an illusion, and that reality exists only in human consciousness. “To be,” he said, “is to be perceived.” This is an idealist philosophy that denies the objective significance of sensory impressions,⁵ and an extreme expression of British empiricism.

David Hume: The Problem of Induction and the Problem of Causality

In the second half of the eighteenth century, the philosophical world was thrown into confusion by several problems posed to it, chiefly by a third English philosopher: David Hume. He too, like the other empiricists, tried to purify sense experience of the additions supplied by the intellect—this is the essence of the transition from rationalism to empiricism—and discovered that such a process raises quite a few difficulties. Two of the major problems Hume raised are the problem of induction and the problem of causality. Both will serve us later in our discussion, and the concepts they involve first began to take shape in Hume’s philosophy. We will therefore now describe in greater detail the problems he raised, as background for presenting those concepts. Readers should note that throughout what follows we will return again and again to Hume’s arguments and Kant’s answers to them.

Hume defined two kinds of knowledge, or two types of propositions:

Demonstrative propositions. These are items of knowledge, regarded by him as beyond all doubt, that clarify the relations among our ideas. Examples are the propositions of logic and mathematics.⁶ One well-known example, familiar to anyone who studied geometry in high school, is the principle that “two quantities equal to a third quantity are equal to one another.” Mathematical propositions too, in his view, belong to this category, since they indicate only relations among constructs of our mind and say nothing about the external world.
Propositions of fact. These are items of knowledge concerning the external world, whose validity Hume calls into question. As long as we are dealing with propositions that describe a known fact from the past, such as “the sun rose yesterday” or “this chair is green,” there is no principled problem in treating them as certain.⁷ The problem arises when we try to formulate a general factual proposition which, like every scientific proposition, purports to assert the truth of an entire set of facts, including facts that we have not directly observed. An example would be the factual proposition: “The sun rises every morning.”

With regard to the second type of proposition, Hume asked: from where can we draw confidence in the truth of such a general proposition? Our past experience teaches us only that up to now, on all the mornings on which we observed it, the sun did indeed rise. The conclusion that this process will continue in the future does not follow directly from those past observations, but rests on an additional principle, the “principle of scientific induction.”⁸ This principle states that under certain conditions one may infer general conclusions from specific examples one has encountered—in other words, that one may infer a general law from particular facts. The problem Hume raised is: what is the justification for this principle itself?

To better understand the problem, and the challenge it poses to scientific inquiry, let us consider another factual proposition, one closer to the world of scientific determinations: “Every piece of wood placed in a fire burns.” To test the truth of this proposition, three stages are required:

Formulating the hypothesis: every piece of wood placed in a fire burns.
Experimental testing: every time we placed a piece of wood in a fire, it burned.
Conclusion: whenever we place a piece of wood in a fire, it will burn.

Hume argues that in this process the conclusion is far broader than the test, and therefore cannot be justified. For this reason, the entire process of accumulating scientific knowledge cannot be justified either. In his view, we assume the principle of induction to be true without any real justification.

At a superficial level, one is immediately tempted to justify the principle of induction by relying on experience, since it indeed works. For example, in the case above, if we continue to examine reality we will discover that we were right: pieces of wood really do always burn when placed in fire. If so, one may infer that we are entitled to continue using the principle of induction in other cases as well, and with respect to other phenomena. But this justification begs the question. Its real meaning is that because the principle of induction has worked in the past with respect to certain phenomena, it will continue to work in the future with respect to other phenomena. But this claim itself rests on a second-order inductive principle, according to which a principle that has worked until now will continue to work in the future. Yet if we still have no confidence in the principle of induction itself—since that is precisely what we are now trying to justify—then obviously we cannot use it to justify itself, and so on without end. At best, this leads to an infinite regress, which cannot count as a reasonable justification for the principle of induction.

Hume then went on to consider whether the use of induction might be justified by relying on the concept of cause and on the principle of causality, the principle that every event has a cause that brought it about. This argument works as follows: every reasonable person assumes that every event has a cause. That is the principle of causality. A further assumption is that analysis of the concept of cause shows that the same cause will always produce the same effect (see the appendix to this book for a fuller discussion). From this it follows that if we discover that the cause of wood’s burning is its being in fire, then this cause will always produce the same result. And from this one can apparently infer the general factual proposition, “whenever we place a piece of wood in fire it will burn,” on the basis of the experimental test. That is, inductive generalization rests on the principle of causality and an understanding of the concept of cause. This is an argument that seemingly does not require the principle of induction itself, and therefore appears not to be circular—that is, not to beg the question.⁹

In order to reject this possibility as well, Hume turned to analyze the concept of cause and raised the following problem. Every concept derives from sense impressions, and usually we can point to its source. But what is the source of the concept “cause”? At first glance, we discover it through experience, when we see the relation between certain forces and their effects, or between causes and results. Hume thinks this is mistaken. When we observe two events, what can be sensed in experience—beyond the mere occurrence of the two events—is only the temporal succession of the two phenomena, not any causal connection between them. For example, when we see that whenever someone kicks a ball, the ball flies, our senses see only that the kick always precedes the ball’s flight. We cannot see that the kick is the cause of the ball’s flight.¹⁰ The causal relation—if it exists at all—between the kick and the motion of the ball is not something observable by means of our senses. It is an addition that consciousness adds to the immediate data supplied by the senses. Hume, as an empiricist, concludes that the concept of cause, which expresses a necessary causal connection between cause and effect, is a fictitious concept. Since it does not emerge from observation but is derived from human reason, his conclusion is that it is a fiction of our consciousness. It should be noted that Humean skepticism is a clear expression of the transition from rationalism to empiricism. For the empiricist, products of thought that are not based on experience are mere fabrication.¹¹

The new situation created by these two problems—the problem of induction and the problem of cause—cast a heavy shadow of groundlessness over the entire scientific enterprise. If there is no way to generalize rationally, as follows from the problem of induction, then the foundation of scientific progress as such collapses. The challenge to the concept of cause likewise threatens a basic element of scientific activity. The chief concern of science is to offer explanations, or causes, for the occurrence of phenomena. If the concept of cause is fictitious, what point is there in scientific activity at all?

Chapter Two: Kant’s Solution: The Synthetic A Priori

Two Classifications of Propositions

Kant reformulated the problems Hume had raised. In Hume, the problem of causality and the problem of induction were two separate problems—though, as we saw, they are related—and each of them separately, as well as both together, threatened the rational foundation of science. In Kant’s formulation, their common source comes to light. For that reason, the solution Kant proposes is likewise a common one. In order to present the common basis of these two problems and their derivatives, Kant defined and distinguished between two different classifications of propositions in language, which before him had been regarded as identical:

A priori propositions versus a posteriori propositions.
Analytic propositions versus synthetic propositions.¹²

The first classification divides propositions according to their source. An a priori proposition is one that does not require the confirmation of experience, whereas the truth of an a posteriori proposition does require such confirmation. In other words, adopting an a priori proposition is a process that precedes experience, or, in Kant’s language, is “purely rational.” By contrast, the cognitive process that leads to adopting an a posteriori proposition necessarily involves experiential elements. For example, the proposition “the sun is rising now” is a posteriori, since confirming it requires observing the present state of affairs in the world. This is, of course, an experiential process that uses sensory observation. By contrast, the proposition “5+7=12” is a priori. We have no need whatsoever for observation in order to confirm it. A teacher may sometimes illustrate this proposition by adding together actual objects, but that is only a pedagogical illustration, not a genuine proof. The algebraic proposition requires no empirical proof.¹³

The second classification, along the analytic-synthetic axis, divides propositions according to their structure. An analytic proposition is a proposition composed of properties of the subject that are implicit in it by definition. No additional information—empirical or otherwise—is needed beyond the definition of the subject in order to recognize the truth of such a proposition. For example, “this ball is round” is an analytic proposition, since the property of roundness follows from an analysis of the definition and properties of the concept “ball,” which is the subject of the proposition. Another well-known example is: “Every bachelor is unmarried.” Here too, the conclusion arises from an analysis of the subject of the proposition.

By contrast, a synthetic proposition is one that attributes to the subject a characteristic or property that is not contained in the subject by definition, and therefore cannot be discovered merely by analyzing the definition of the subject. An example is: “This ball is heavy.” A proposition of this kind adds information about the subject that we did not previously possess, since the concept of a ball does not contain the property of heaviness by definition. The synthetic proposition asserts something about the subject, and therefore also about the world. The analytic proposition asserts nothing about the world; it merely clarifies the meaning of the concepts we use.

The Importance of Synthetic Propositions

All scientific progress depends on synthetic propositions. These are the only propositions that, at least in principle, add knowledge about some subject beyond what we already had. An analytic proposition can only clarify information we already possess—for example, that a ball is round. To be sure, there are cases in which the information is very deeply embedded in the definition of the subject, and extracting it can be quite complicated. Mathematics provides many examples of such a phenomenon, where considerable skill and intelligence are required in order to analyze the concept under discussion in a proof, understand its features, and reach the conclusion, namely the mathematical theorem. Even so, the problem is technical in essence. In an analytic proposition, all the information is already in our possession by virtue of the definition of the subject. Sometimes we simply have difficulty “cracking the safe” that is already ours.

Here lies the difference between mathematics and the sciences. Unlike mathematics, the important stages in the progress of scientific inquiry are, as noted, the synthetic stages. The scientific generalizations that move from the particular to the general, as in the example above concerning wood burning in fire, are what make possible the discovery and formulation of general laws of nature.¹⁴ These generalizations are factual claims about their subjects, and they cannot arise merely from analyzing the subject. They also require the addition of information beyond what is included in the subject by definition.¹⁵

The Relation Between the Two Classifications

A first look at the two classifications presented above shows that the first is arranged along an epistemological axis, whereas the second is arranged along a logical axis. From this one may infer that every category of propositions formed by combining elements from these two axes is possible. We thus apparently obtain four possible categories of propositions in language: analytic-a priori, analytic-a posteriori, synthetic-a priori, and synthetic-a posteriori. Yet a deeper examination shows that there are relations between these classifications that may limit this fourfold scheme.¹⁶

A synthetic proposition, as noted, asserts the existence of a characteristic or property of the subject that is not present in it by definition. It would seem, at first glance, that such a proposition can be based only on experience. For beyond the process of analyzing the subject itself, which is done by the intellect, what else can be done by reason alone? If so, in order to learn a proposition of this type, we must go outside reason itself and interact with the world beyond us. Such activity is what we call “experience” or “empirical observation.”

From this consideration it follows that a synthetic proposition is always a posteriori, that is, based on experience. Without experience, how could one know that a certain ball is heavy? The property of heaviness, as stated, does not characterize the ball by virtue of its essence, and therefore in order to know it we must conduct an empirical observation of that particular ball.

In a similar but opposite way, one can see that an analytic proposition is always a priori, since there is no need to appeal to experience if all that is required in order to reach the assertion contained in an analytic proposition is rational understanding and analysis of the subject under discussion, which is already fully known to us. In the example above, there is no reason to conduct any observation in order to know that a certain ball is round. That property follows from the very definition of the concept “ball.” Similarly, there is no point in examining Reuven the bachelor, or any other bachelor, to see whether he is married. If we know that he is a bachelor, there is no doubt that he is unmarried. This conclusion follows from the very definition of the subject of the proposition, namely the concept “bachelor.”

If so, it would apparently seem that the difference between the two classifications—analytic-synthetic and a priori-a posteriori—is illusory. At first glance, we are dealing with only one classification that has two faces: the analytic is always a priori, and vice versa, while the synthetic is always a posteriori, and vice versa.

As we shall now see, Kant’s claim is that there is indeed a difference between the two classifications, meaning that they do not completely overlap. According to him, the root of Hume’s confusion lies in his failure to distinguish between them. Even after Kant, down to our own day, there remains considerable confusion in philosophical literature regarding the relation between these two classifications,¹⁷ and quite a few philosophers do not accept the Kantian distinction.¹⁸

Kant’s Formulation of Hume’s Problems: The Synthetic A Priori

Kant reformulated Hume’s problem in a different way. Hume presented a skeptical problem and declared: scientific knowledge of the world is impossible. Kant asked instead: knowledge of the world—how is it possible? In other words, he assumed that scientific progress in knowing the world is possible, and focused only on the question of how it is actually possible. Using our previous terminology, Kant posed the question as follows: scientific progress in knowledge of the world rests on synthetic propositions. Hume argued that synthetic propositions cannot be derived from experience, because of the problems of induction and causality. If such propositions nonetheless exist, as Kant assumes to be obvious from the very progress of science, then there must be some way to learn them without the aid of experience, that is, a priori. These propositions are therefore synthetic on the one hand, yet a priori on the other. Kant called them synthetic a priori propositions.

In Hume’s philosophy there is no category of synthetic a priori propositions at all. And, as Kant correctly saw, that absence is the root of Hume’s skepticism. It should be noted that Kant owes us an explanation of how one can actually arrive at, or ground, such propositions. Experience is not an acceptable route, since these are a priori propositions, that is, prior to experience. On the other hand, it is not clear how reason alone can arrive at propositions that are not analytic, since such propositions require some addition beyond what is contained in the definitions of the concepts themselves. Kant therefore asks himself: how can a synthetic a priori proposition be grounded?

Kant’s answer to this question was: by means of transcendental considerations. Kant argued that concepts or claims that serve as a necessary basis for human cognition—such as causality or induction—can be established as true a priori, even though they are synthetic determinations. Our proof of their truth is obtained in the manner Kant called “transcendental”: namely, if without them there would be no meaning to cognition, thought, or human knowledge. In other words: if the very processes of thinking, cognition, and understanding would have no meaning—or could not exist and function at all—without a certain principle, then that principle is true a priori, because it constitutes a necessary condition underlying the very concept of truth itself. This type of consideration will be discussed in detail in the seventh gate.

As stated, through these determinations Kant cancels the overlap that had been accepted before him between the two classifications. As we have seen here, according to Kant there are propositions that are synthetic and yet a priori. If so, the “analytic” no longer overlaps with the “a priori,” and the “synthetic” no longer overlaps with the “a posteriori.”

To understand Kant’s claims more clearly, one may look at the question somewhat differently.¹⁹ If science really does advance in knowledge of the world, and if this is done by relying on synthetic a priori propositions, then there must be a correspondence between pure reason—which is the source of those propositions, since they cannot come from experience, as Hume argued²⁰—and the world as such, which is the object of those propositions.

Put differently: Kant held that these propositions are not drawn from experience but from pure reason, and are therefore a priori, yet nevertheless they say something about the external world, which is why they are synthetic. These two determinations point to a correspondence between reason, or our intellect, and the objective world that exists outside us.

The Philosophical Basis for the Existence of Synthetic A Priori Propositions

Kant’s determination is that there are synthetic a priori propositions—that this category is not empty, as it appears at first glance and as Hume thought. In addition, Kant claims that these propositions can correctly describe the world, even though their source lies within our own reason. At this point Hume can of course press the question further: how can such a correspondence exist between two things that are apparently independent of each other—human reason and the world? How can products of our subjective mind correspond exactly to the state of affairs in the objective world? A proposition that is entirely a product of detached thought, without any observation of the world, would apparently seem incapable of providing a fitting and correct description of the external world.

Kant offers a new and more comprehensive formulation of the collection of problems Hume raised, by defining the four kinds of propositions mentioned above. He reposes the Humean question in the following form: how is such a correspondence between human reason and the objective world possible?

Kant does not suffice with reformulating the problems and subsuming them under one pattern. He also proposes a solution. His reformulation is intended to show more sharply the source of the problem, and thereby also to chart the way to its solution. The transcendental consideration is meant to answer this puzzle. To understand its nature, we must examine several possibilities, conveniently presented through Leibniz’s parable of the clocks.

The Clock Parable: Three Ways to Explain Correspondence Between Two Things

Leibniz writes as follows in his book The New System:²¹

Imagine two clocks or two watches that agree perfectly with one another. This can happen in three ways:
(a) by the mutual influence of one clock upon the other;
(b) by the care of a person who supervises them;
(c) by the accuracy of the clocks themselves.

In this parable Leibniz was speaking of the correspondence between body and soul, and later also among different souls, or monads. But we will adopt this division for the correspondence between reason and the world. In this context, the first possibility can be divided into two:

Human reason influences the world.
The world influences human reason.²¹

The interpretation of the third possibility is open in our case: either it is mere chance, in which case it can be dismissed, or, just as a human being planned and built the clocks in advance so that they would correspond to each other, there is some factor that planned reason and the world so that they would correspond to one another. For us, this possibility may merge with the second one, namely that there is an external factor that ensures the correspondence between our reason and the world. The only question is whether this is done at every moment, or whether the correspondence lies in the very matching structure that this factor gave to our reason and to the world from the outset.

The second possibility is called “empiricism” in philosophy, that is, the view that cognition is based on experience. According to this conception, it is the world that determines the content of cognition. This possibility cannot explain scientific progress, as Hume showed and Kant agreed, since concepts such as causality and induction cannot be drawn from experience, as we saw above.

We are thus left with two possibilities: either reason influences the world, or some third factor coordinates them. Kant and Hume rejected the latter possibility out of hand—for reasons that, in my opinion, are not entirely sufficient, and we will discuss this at the beginning of the eleventh gate.²² The main claim is that we have no valid proof of the existence of such an entity or factor. Therefore Hume, who did not recognize the first possibility at all, denied the possibility of scientific progress in knowledge of the world, whereas Kant adhered to this first possibility, which in his view was the only one left. In other words, Kant’s conclusion is that reason is what influences the world.²³ This is Kant’s solution to the problems Hume raised. The correspondence is created by a relation between reason and the world, a relation through which reason influences and shapes the world we observe.

Phenomena and Noumena: The Thing in Itself

Kant now had to justify this claim, and in effect to show the mechanism by which reason influences the world. He therefore continues his argument and distinguishes between the world of things as they are in themselves and the world of appearances, or in his terminology, between noumena and phenomena. The appearances we observe are the way the world of things-in-themselves is reflected, through the mediation of the senses and reason, before human cognition. Everything received into our consciousness is processed according to the form of our faculties of perception. A human being cannot observe things themselves except through the mediation of his senses, reason, and cognition, and therefore he knows only their appearances and not the things themselves. In Kantian terminology, the world as it is reflected before our cognition is the phenomenon, whereas the world as it is in itself is the noumenon.²⁴

Note 1: The Perspective of “Object” and “Person” in Halakha (Jewish Law)²⁵

In the study hall there is a common distinction between laws pertaining to the object itself and laws pertaining to the person—that is, between laws attributed to the object itself, to the world itself, and laws attributed to the person. The source of these concepts is the Babylonian Talmud, Nedarim 2b, where the Talmud distinguishes between vows, which are a law pertaining to the object, and oaths, which are a law pertaining to the person. The point is that the prohibition created by a vow takes effect upon the object itself, whereas the prohibition created by an oath is an undertaking of the person who swears, and has no significance with respect to the object as such. When a person forbids himself by vow to eat a piece of meat, the prohibition rests upon the piece itself. It becomes forbidden food. By contrast, when that same person swears not to eat that piece, nothing has changed in the characteristics of the piece itself; rather, a prohibition has arisen that applies to the person, forbidding him to eat it.

Rabbi Chaim Soloveitchik of Brisk, who was active in Lithuania at the end of the nineteenth century and the beginning of the twentieth, greatly expanded the use of these concepts and applied them even to abstract notions. Thus, for example, with respect to prayer one may speak of laws pertaining to the “object” of prayer or to the person who prays. There is also a discussion among the early commentators in the Talmudic sugya there in Nedarim regarding whether biblical prohibitions in general—apart from vows and oaths, which are explicitly discussed there—are imposed upon the person or upon the object; that is, whether their meaning reflects spiritual realities that “rest” upon objects in the world,²⁶ or whether they are obligations imposed on the commandment-observing person. In the writings of later commentators there are discussions regarding rabbinic prohibitions, such as the prohibition of eating poultry with milk, and prohibitions dependent on time, such as leaven on Passover and eating on Yom Kippur: do they reflect a spiritual reality in the world as such, or are they obligations imposed upon the person? This is not the place to elaborate.

There are many shades to the use of these terms. The question relevant to our discussion is whether there is any relation between these halakhic-Torah concepts and their branches, on the one hand, and Kant’s concepts of phenomenon and noumenon on the other—and if so, what that relation is.

At first glance, there seems to be an identity here. A law pertaining to the person means, in its general sense, that no statement is being made about the world as such, but rather about the way it appears to us. When I say that a person is forbidden to eat pork, and not that pork is in itself forbidden for eating, I am claiming that pork as such is not the subject of halakhic determinations. Those determinations concern the person who wants to eat the pork. By contrast, a law pertaining to the object, in its various meanings, is a statement about the world itself. To say that pork is an “object-based prohibition” means that there is something—spiritual or physical—in the pork itself that forbids me to eat it. It is a clear fact that in halakha, according to all opinions, there is a division between laws stated with regard to the person and those stated with regard to the object. The disagreements concern only where the boundary line between them runs. It would seem, then, that in halakha there is no room for a Kantian conception which claims that we are entirely unable to say anything about the world as it is in itself, since halakha also deals with the world itself, with the object.

One may perhaps formulate the question of the relation between these concepts and the Kantian distinction between phenomenon and noumenon from another angle. How can one say that a certain law characterizes the world as it is in itself, when everything I can say about the world concerns only the way I see it? The assertion that I am commanded to do something, or forbidden from doing something, is apparently a statement about me, not about the world. If so, what difference is there, from the Kantian perspective, between a law pertaining to the person and a law pertaining to the object? In other words: what is added by the statement that there is a prohibition against eating a certain piece of meat, beyond the statement that a person is forbidden to eat it?

One may answer that even when speaking of a law pertaining to the object, the intention is still the way I see things—that is, phenomenon. One should note that even for Kant there must be a division between the observer and the objects he observes. That is, although the form of the objects is what appears within the cognition that observes them, and is imposed by the structure of that cognition, there are still two parts within that observing cognition: there is the observing subject, and there are the objects, or the world, being observed. According to this approach, both parts are within human cognition.²⁷

Below we shall see that Torah and halakha do indeed embody a non-Kantian conception. Later we will argue that Kant represents a primarily analytic approach, whereas the conception of halakha and Torah is synthetic. If so, it is not surprising that the Torah perspective will not accept his position.

This discussion recalls the explanation offered by Rabbi Meir Simcha of Dvinsk, author of Meshekh Chokhmah, in his commentary on the Torah portion Ki Tisa, for the sin of the Golden Calf. Rabbi Meir Simcha explains there that the essence of the sin was the attempt to attribute holiness to objects as such, rather than because of their relation to the Holy One, blessed be He. This is why Moses our Teacher decided to break the tablets in response to that sin: to show the people that even the holiest object, such as the tablets of the covenant, is not holy in itself, but only if it is related to as representing the one and only thing that can be considered holy, namely God Himself. If that relation is absent, or flawed, holiness as such has no meaning. According to Rabbi Meir Simcha, the attempt to attribute holiness to the calf, or to the tablets of the covenant, in such a way that holiness is a property of the object in itself, was the sin of the Golden Calf. The people of Israel should have understood that holiness is a feature of our relation to the object, not of the object itself.

Yeshayahu Leibowitz was very fond of quoting this explanation as part of his view that no mitzvah (commandment), and no holiness, has any significance beyond the simple fact that we were commanded concerning it. In his view, the mitzvah or holiness are not derived from reality and do not affect it. Therefore, in particular, we cannot place our hopes in the belief that fulfilling a mitzvah—even prayer—will help us in life. According to his view, everything is a law pertaining to the person, and nothing is a law pertaining to the object.²⁸

From the analysis above it follows, according to Kant, that entities or phenomena whose forms do not fit our faculties of perception will not be received by them at all, and no picture of them—that is, no phenomenon—will be formed for us. According to Kant, this is precisely the reason for the marvelous correspondence between reason and the world—for the fact that the phenomena appearing before us are subject to, and fit, the rules of reason. According to Kant, they have no choice. Anything without a matching form simply does not enter consciousness, or reason.

For this reason, Kant argued, we can formulate a priori propositions whose source lies in our reason, and yet they will still be synthetic, meaning that they will say something true about the world of appearances. As stated, our consciousness has no access to the world of things as they are in themselves, the noumena, since all our access to the world is mediated by our faculties of sensation and thought. But if that is so, then scientific assertions too deal only with the phenomenal level, not with the world of things in themselves. Therefore, determinations whose source lies within human thought can say things that fit the state of affairs in the “external world.” In other words: Kant solved the problem of synthetic a priori propositions by changing the meaning of the term “synthetic.” A synthetic proposition says nothing about the world as it is in itself, the noumenon, but only about the way it appears to us, or to our cognition, the phenomenon. This plane alone is accessible to pure reason, and therefore its conclusions can correspond to it.

Resolving the Humean Problems

If we return to the problems Hume raised, we may say that the principle of causality can be derived from the structure of human cognition and thought, precisely because it constitutes a cornerstone of the way we cognize and think. Clearly, all the phenomena of the external world appear before human cognition only after passing through the filter of the principle of causality, since it is one of the constitutive principles of our cognition. Only through it can we know and think about different things and phenomena in the world. Therefore, we may confidently assume that a synthetic a priori proposition—such as the principle of causality itself, or any of its particular implications, such as the claim that being in fire is the cause of a piece of wood’s destruction—will fit our scientific observations in the world of phenomena.

Put a bit differently, one may say that the guarantee that the insights and knowledge we accumulate about the world are indeed valid rests on the fact that we have no other way at all to attain knowledge, and perhaps even to define knowledge. One may say even more than that—we will later call this form of argument in the seventh gate an “analytic consideration”: in fact, this is the very meaning of the term “knowledge.”

At first glance such a proposition looks like begging the question, but it has a deeper content. We will understand this better when we discuss the concept of the “transcendental” in Kant’s thought in the seventh gate, and also when we discuss certain aspects of this distinction in the second gate.

Summary: The Second Copernican Revolution

Let us summarize the discussion thus far. According to Kant, the two classifications of propositions that we presented at the beginning—analytic-synthetic and a priori-a posteriori—are not identical. Analytic propositions are those whose truth follows from analyzing their subject. Such propositions are always a priori. Synthetic propositions are those whose truth follows from additional knowledge or premises beyond such analysis. Such propositions may be synthetic-a posteriori, which is the common and trivial case, or synthetic-a priori, and these are the propositions that underlie scientific progress. The way to derive them is by means of transcendental arguments. The category of analytic-a posteriori propositions does not exist even for Kant.

By means of this distinction, concerning the existence of the synthetic a priori, Kant redefined the problems Hume had raised and answered them. Both the concept of causality and the concept of scientific induction are transcendental concepts, or principles—that is, they are conditions of every possible cognition, conditions that follow from the structure of human reason and cognition. The claim concerning the existence and validity of these concepts and the principles based on them is a synthetic a priori claim. For this reason Hume rejected it, since he did not recognize the possibility of the category of synthetic a priori propositions—he identified the analytic with the a priori—a view that led him to skepticism, or at least to a skeptical posture. Hume’s approach was rejected by Kant, who, on transcendental grounds, accepted and indeed established the existence of synthetic a priori propositions.

Kant defined this distinction as a “Copernican revolution.” Copernicus challenged the then-prevailing geocentric conception according to which the sun revolved around the earth, and argued that the sun was at the center. So too Kant challenged the prevailing empiricist intuition, which sees the human being as a passive observer of nature and the world, and understands the world as standing at the center while the human being merely gazes at it. Kant argued that the human being stands at the center. He creates the world of appearances, or at least participates in creating it, and therefore can also make nontrivial claims about it—claims that tell us something new about it—and yet claims that are still true, even a priori. It is no accident that Kant is regarded as one of the pillars of modern humanism, which places the human being at the center. This is true of his moral philosophy, but, as we have seen here, also of his theory of knowledge. The human being stands at the center of the discussion.²⁹

Chapter Three: Another Aspect of Analyticity and Syntheticity: Forms of Inference and Axiomatic Systems

Deduction, Induction, and Analogy

In this chapter I will present very briefly several subjects whose full treatment would require extensive elaboration. I will focus here only on a few main points relevant to what follows.³⁰

There is a common division of types of argument and inference into three branches: deduction, induction, and analogy. Deduction is an argument directed from the general to the particular, induction is directed from the particular to the general, and analogy from one particular to another. Let us now illustrate these three kinds of argument.³¹

A classic example of a deductive argument, or syllogism, is the following:

Premise A (the major, or general, premise): All human beings are mortal.
Premise B (the minor, or particular, premise): Socrates is a human being.
Conclusion: Socrates is mortal.

This argument is necessarily valid. By this we do not mean that its conclusion is necessary in itself, but only that it follows necessarily from its premises. Whoever accepts the premises must also accept the conclusion.

This argument moves from the general, represented here by Premise A, which states a general truth, to the particular, represented here by the conclusion, which is a particular fact.

A parallel inductive argument would be the following:

Premise A: Socrates is a human being.
Premise B: Socrates is mortal.
Premise C (possible as reinforcement for the previous premises, though not necessary): All the human beings we have met so far are mortal.
Conclusion: All human beings are mortal.

This argument is not generally regarded as necessarily valid, even though each of us uses similar arguments all the time and often assumes them to be valid. An argument of this type moves from the particular, or particulars, to the general, in the opposite direction from the deductive argument.

A parallel analogical argument would be the following:

Premise A: Socrates is a human being.
Premise B: Socrates is mortal.
Premise C: Jacob is also a human being.
Conclusion: Jacob is also mortal.

This kind of argument is also very common, although of course it is not necessarily valid. It moves from one particular to another.³²

The Deductive Argument

Let us now focus on the deductive argument, which stands at the center of the field of classical logic.

The point of departure of formal logic is the observation that the deductive argument enjoys absolute validity for all of us, not by virtue of the contents that appear in it, but solely by virtue of its structure, its form. Any argument with a structure similar to the one above would receive the same degree of confidence, regardless of the content and character of the concepts appearing in it. The following formal argument is the valid structure underlying the example we gave:

Premise A: Every Q is P.
Premise B: a is Q.
Conclusion: a is P.

Any subject we place in the position of a, and any properties, or predicates, we place in the positions of Q and P, will yield a valid argument. This is a reliable recipe for producing valid deductive arguments.

Symbolic logic deals with the various formal structures underlying deductive arguments, classifies and orders them, and discusses their characteristics. This discussion, which began already in the Hellenistic period, received major impetus and accelerated development in the computer age, whose entire mode of operation rests on logical thought. In certain respects, one may define the computer, both at the hardware level and at the software level, as a deductive machine.

Once one is familiar with the basic formulas of logic, it becomes possible to ground any mathematical field on as few premises as possible, together with several logical rules of derivation. At the beginning of the twentieth century, Russell and Whitehead, in their monumental mathematical work Principia Mathematica, attempted to ground all of mathematics on logic. On this famous attempt and its unexpected results, see the ninth gate.³³

Axiomatic Systems

As a result of these achievements of logic, the conception of an “axiomatic system” became firmly established. Every high school student first encounters an example of such a system in plane geometry. This is a body of knowledge built upon four basic assumptions, or axioms—such as the assumption that two parallel lines do not meet at any finite distance—and from them, by purely deductive means, it extracts an astonishing number of geometrical truths, namely theorems. In principle, it is enough to know the four axioms in order to possess all the knowledge of plane geometry. From the axioms, and by using basic principles of reasoning, one can extract the entire remaining body of knowledge. It is, in some sense, already contained in them.

In the scientific world there is a tendency to see every body of knowledge as a potentially axiomatic system. Scientific bodies of knowledge can often be organized in the form of a system of first assumptions, or axioms, a collection of rules of derivation, and a collection of theorems derived from the axioms by means of those rules.³⁴ In the book by Carnap,³⁵ one of the great logicians of the twentieth century, one finds axiomatic systems not only for number theory, set theory, and geometry, but—more surprisingly—also for physics and biology. Einstein too, and many after him, aspired to find a unified field theory that would describe the entire physical world by means of one general law, from which all specific laws could be derived by a logical-analytic route. In the twentieth century, mathematical-analytic-axiomatic thinking increasingly took over all branches of human knowledge.

Deduction and Analyticity

At this point we leave the picture offered by Kant and continue to expand and generalize his definitions in ways that will serve us later in the discussion.

There is a clear relation between deductive thought, with which we are now dealing, and analytic thought, which was discussed in the previous chapter. Analytic thought was defined above as a mode of thought based on analyzing the subject of thought or of the proposition—analysis meaning dissection—which does not require experience in order to support its conclusions. This is the thought of pure reason, and it is, of course, also necessarily valid. The conclusion of such a line of thought cannot be refuted; one can only find an error in the process of reasoning that led to it. A valid line of reasoning in which no error has been found cannot be contradicted by any fact, empirical or otherwise.

In this sense, deductive thought too is analytic thought. In deductive thought we derive all our conclusions from an analysis of the first premises—which in effect determine the subject of thought—by means of rational principles, without need for experience or additional knowledge. The fact that Socrates is mortal was hidden in the general proposition that all human beings are mortal. Clearly, that determination was already known to us before the argument above was made. We merely used elementary logic to extract the conclusion from its hiding place within the general proposition that we already knew. This is a natural extension of what we do in a single analytic proposition, where we analyze the subject of the proposition and extract from it its properties, which have of course been hidden in it from the outset.

We can now see clearly the claim made in the introduction concerning the hot-air-balloon joke: something that is necessarily true says nothing. Deductive thought is “necessarily valid” precisely because it does not add any information beyond what we already knew beforehand. With regard to analytic propositions, in which the claim is hidden in the subject by virtue of its definition, we saw that they add nothing, in principle, beyond what we already knew in advance. A deductive argument, like an analytic proposition, merely exposes the details that were concealed within what was already known, namely the axioms. The necessity of the deductive-analytic argument stems precisely from the fact that, in principle, it tells us nothing new.

This relation finds grotesque expression in the relation between the two characteristics of the mathematician in the hot-air-balloon joke: the necessary correctness of his assertions, and the fact that they are devoid of informative value. The precision of his statements follows precisely from the fact that he tells us nothing new. This is the power of mathematics, but it is also its weakness. In the following note we will see a halakhic implication of the essential relation between the lack of novelty in a conclusion and its necessary validity.

Note 2: No Punishment Is Imposed on the Basis of Logical Derivation—Two Types of A Fortiori Inference

In Note 22 we will discuss the concept of the a fortiori inference in the world of halakha, and the question whether it is a deductive inference or not. We will see there that the a fortiori inference is not a deductive argument, but only the deductive continuation of an inductive consideration.

There is, however, a certain type of a fortiori inference that is exceptional, and I would like to discuss it here.³⁶ It will serve as an example of the power and weakness—dependent on one another—of deduction, and of analytic thought in general.

There is a rule in halakha that no punishment is imposed on the basis of logical derivation. That is, when the prohibition of an act is learned by means of an a fortiori inference—and perhaps by other methods as well, according to some commentators, which are also called “derivation”—a rabbinical court will not punish the person who committed it. The commentators offer several reasons and explanations for this law; see, for example, Encyclopedia Talmudit, entry “No Punishment Is Imposed on the Basis of Logical Derivation.” Some explain that since the act learned by inference is more severe than the prohibited act from which we learned it, the punishment due for this offense might also have to be more severe, and therefore one cannot simply transfer the punishment given for the source offense to the inferred offense. Others explain that an a fortiori inference can be rebutted—that is, one may find a consideration that rejects the inference and shows it to be mistaken; see Note 22—and therefore, even when we have not found such a rebuttal, this does not mean that none exists. Out of concern for this possibility, we do not punish the offender, since it may turn out that the act is not prohibited at all. The simplest explanation, in my view, is that the severity of a punishment is not necessarily connected to the severity of the offense. Hence one cannot infer the nature of the punishment appropriate to an offense learned by an a fortiori inference.³⁷

There is a discussion as to whether, in monetary law, one may impose punishment on the basis of logical derivation. The first mishnah in tractate Bava Kamma discusses the principal categories and derivatives of torts; see Note 13. Tosafot, there on 2a, under the heading “And not this and that,” infer that there is a disagreement between the Mekhilta, the tannaitic midrash on Exodus, and our Talmud on the question whether in monetary law one may punish on the basis of logical derivation. In that mishnah, we see that one may impose payment—liability for damage caused by someone’s animal—even when the mode of causing damage is learned by an a fortiori inference. It would seem from here that liability for damages is not punishment in the halakhic sense. By contrast, in the Mekhilta there is an exposition obligating a person who digs a pit in the public domain, even though his liability could have been learned through the following a fortiori inference: if one is liable for opening, is one not all the more so liable for digging? That is, if a person who removes the cover from a covered pit in the public domain is liable to pay for damage caused to someone else’s property that fell into the uncovered pit, then all the more so one who actually digs the entire pit should be liable for damages. Tosafot writes here in the name of the Mekhilta that a special scriptural derivation is needed to establish liability in such modes of damage, because one cannot impose payment on the basis of an a fortiori inference. In other words, even in monetary law, punishment is not imposed on the basis of logical derivation. The Mekhilta in fact says that this itself is what the Torah wished to teach us in this exposition: that no punishment is imposed on the basis of logical derivation even in monetary matters.

The commentators noted regarding this derivation that the a fortiori inference from opening a pit to digging one is unlike an ordinary a fortiori inference. Rabbi Yosef Teomim, the author of Pri Megadim on the Shulchan Arukh, formulates this distinction explicitly in his book Ginat Veradim, section 1, and determines that there is a difference between an a fortiori inference of the “one hundred is included within two hundred” type and an ordinary a fortiori inference. “One hundred is included within two hundred” means that the conclusion is explicitly contained in the premises, just as a sum of two hundred shekels contains within it one hundred shekels. That is, this is a case in which no preliminary inductive step is needed in order to formulate the a fortiori inference. The derivation from opening a pit to digging one is necessary, because anyone who digs a pit thereby also performs an act of opening, namely the removal of the upper layer of earth. Opening is inherently present in every act of digging. In such a case, the derivation is just an ordinary deductive operation. Just as the analysis of the major premise stating that all human beings are mortal yields the conclusion that Socrates in particular is mortal, so too the analysis of the act of digging shows that an act of opening has also been performed. For this reason, it is clear that the digger should be liable no less than someone who merely removed an existing cover.

The reasons we gave above for the rule that no punishment is imposed on the basis of logical derivation do not fit the a fortiori inference from opening a pit to digging it. Here the derivation is not an a fortiori inference of the usual type, but rather one of the “one hundred is included within two hundred” type. An a fortiori inference of this kind is ordinary deduction. Deduction admits no rebuttal, and there is no concern that the punishment is inappropriate, because an actual act of opening occurred here as part of the digging. If so, we can punish the digger at least for the act of opening that he performed.³⁸

In this type of a fortiori inference we see an example of the claim that the necessity of the analytic stems from the fact that it is already contained within the premises. In the ordinary a fortiori inference the conclusion is not contained in the premises, and a preliminary induction is required—see below, Note 22. Therefore, despite its apparently deductive appearance, a rebuttal may show that the a fortiori inference is mistaken. In this type of a fortiori inference—“one hundred is included within two hundred”—such a situation cannot arise, because the conclusion itself is explicitly contained in the premises. Such a process cannot be disputed or refuted. The necessity here derives from the fact that this derivation truly tells us nothing new. Its conclusion is already explicitly contained in its premises.³⁹

The Synthetic and the Emptiness of Deduction

We have seen that the power of deduction—its logical necessity—follows from the fact that it tells us nothing new. Another aspect of this claim will show us that deduction, like all analytic thought, does not concern itself with the truth of its conclusions, but only with their derivation from the premises. Whoever accepts the premises must accept the conclusion. Whoever does not accept the premises is obviously under no obligation to accept the conclusions. The truth of the conclusion as such is not something to which analytic thought can provide an answer. Analyticity deals only with derivation, or with formality, and not with the contents themselves. One cannot, by means of analytic thought, arrive at the conclusion that Socrates is mortal, or any other conclusion whatsoever. The only thing that can be inferred analytically is the derivation of the assertion that Socrates is mortal from the premises underlying the argument.

By contrast, synthetic thought will never be deductive. Synthetic thought does not suffice with analysis alone, but uses additional knowledge beyond analysis of the subject. This knowledge may come from experience, or from pure reason as Kant innovatively claimed, but in any case it is knowledge in addition to what is implicit in the first premises. For this reason it is also clear that synthetic thought, even if it is true, cannot be necessarily true. Therefore it is, in principle, open to refutation by various facts. Yet that is precisely the advantage of this mode of thought. It alone can enrich us with knowledge beyond what is already present in our premises.

Synthetic thought is characterized by massive use of the two other forms of inference: induction and analogy. These forms of inference are not necessarily valid, and therefore they can make substantive claims about the world. Precisely because they are exposed to refutation, they are not “empty.” This emptiness means the absence of new content in the conclusions beyond what we already possessed in the premises. As I will argue below, this emptiness characterizes analytic thought, and this is also the emptiness of the “empty wagon.” This emptiness is defined by the two characteristics of the mathematician in the hot-air-balloon joke: he says something perfectly correct, and therefore he is empty, that is, of no use to us. By contrast, the synthetic argument is not necessarily correct, and precisely for that reason it alone can add new information about the world beyond what we already know.⁴⁰

At this point it is worth noting again the difference mentioned above between mathematics and science. Mathematics is an analytic field by its very essence, and therefore it is natural to use axiomatic and analytic methods within it. The empirical sciences, by contrast—such as biology and physics—are synthetic fields by their essence, and therefore it is surprising that the knowledge accumulated in them can sometimes be organized in axiomatic form.

Even though the organization of scientific knowledge can take on an axiomatic character, it is clear that the nature of work and research in the sciences is not analytic but synthetic—the opposite of the situation in mathematics. Carnap, in the book mentioned above, argues that the knowledge accumulated thus far in certain sciences can nevertheless be organized as an axiomatic system and analytically derived from the axioms. That is merely a way of organizing the knowledge, not the way it is acquired, nor its natural structure. In mathematics, by contrast, this is indeed the natural form of both the acquisition and presentation of knowledge.

This, then, is the basis of the difference between mathematics, which is analytic in its essence, and the empirical sciences, which are synthetic in their essence. Once certain natural laws have been found, they can indeed be formulated as axioms, so that all the facts will be derived from them by rules of derivation, in a mathematical-analytic form. But it is clear that the way such laws of nature are discovered is synthetic: by means of inductions and generalizations from particular facts, usually observations. Therefore it is possible that these laws of nature will later be refuted, and it will turn out that they are not true at all, or that they will simply be displaced for other reasons. Such a thing can never happen to a mathematical theory. The only thing that can happen to a mathematical or analytic theory is that one may find an error in it from the outset; it cannot be refuted by a new fact. The assumptions of the natural sciences are open to empirical testing because they make claims about the world. The assumptions of mathematics are not open to such testing because they make no claim about the world. Pure mathematical activity is intended only to expose what is hidden within the arbitrary assumptions chosen for the mathematical field in question.

When one adopts a mathematical theory within some scientific field—for example, group theory, which is heavily applied in modern physics—that use itself contains an empirical claim: namely, that the segment of the world under discussion is correctly described by group theory. Therefore, if one refutes that claim by means of an observation that does not fit its predictions, that is not a refutation of the mathematical theory, but only of the claim that this mathematical theory is suitable for describing the physical world.⁴¹

Chapter Four: Forms of Thought and Positions: Analytic and Synthetic

The Relation Between Analytic Thought and Pluralism

In the previous chapter we presented a generalization of Kant’s concept of the “analytic proposition” into a more inclusive form of thought that we called “analytic thought.” Analytic thought is deductive in character, and its conclusions are necessarily valid. The single analytic proposition analyzes its subject and arrives at its assertion through that process of analysis. The deductive argument, which reflects the form of analytic thought, is a course of reasoning that includes the inference of conclusion-propositions from premise-propositions, or axioms. As with the analytic proposition, here too the conclusion is derived through analysis. We analyze the premises and arrive at the conclusion, which was hidden in them from the start.

While presenting the deductive argument in the previous chapter, I mentioned the fact that its conclusion is not necessarily true, but only necessarily follows from its premises.

It follows that this form of thought has a corresponding reverse characteristic: if we assume different premises, even opposite ones, our conclusions will also be opposite. Moreover, those conclusions too may follow necessarily from the new premises we have adopted. Mathematics, which as already noted is the canonical analytic field, offers a famous example of such a reversal: the non-Euclidean geometries discovered, or invented, in the previous century. If we assume, for example, that two parallel lines do indeed meet, contrary to the usual Euclidean assumption, we obtain a completely different geometry. In such a geometry the sum of the angles of a triangle is not 180 degrees, and many other strange results follow. On the mathematical plane these results are no less valid than the Euclidean ones. They too follow necessarily from their premises. The difference lies only in which premises one starts from. Tell me your premises, and I will tell you what your “necessarily correct” conclusions will be—or, more precisely, what conclusions necessarily follow from those premises.

It follows that within analytic thought, and especially in mathematics as the outstanding representative of that form of thought, it is true that all results are “necessarily correct.” But since the results only necessarily follow from the premises, and are not necessary in themselves, opposite premises will in exactly the same way produce opposite conclusions that are also “necessarily correct” in the same sense, that is, they necessarily follow from the premises.⁴²

Here we arrive at an important relation between analytic thought and pluralism. The demand for analyticity, which apparently stems from logical maximalism—the demand that every claim be proven—leads, by a strange reversal, to a position that advocates pluralism: “everyone has his own truth.” This is the basis for defining analyticity as a philosophical position, beyond the merely logical plane—a definition we will give immediately.

Synthetic thought is correspondingly defined as a form of argument in which the derivation of conclusions from premises is not necessary. It is generally carried out by means of analogical or inductive processes of thought. As we have seen, such a form of argument can add information about the world specifically because it is not necessary.

The Analytic and the Synthetic Position

Up to this point we have seen two forms of thought: analytic and synthetic. We now continue to generalize the Kantian definitions, and define analyticity and syntheticity as philosophical positions, a definition that will serve as the basis for everything that follows. The philosophical position we will call the “analytic position” is defined here as the view that only the analytic form of thought can lead us to valid conclusions. By contrast, the “synthetic position” accepts other forms of argument as well as adequate grounding for conclusions that may be considered true, and perhaps even certain.

In light of the foregoing, it would seem that we have reached an absurd and apparently paradoxical conclusion: the demand of those who hold the analytic position for necessary correctness—that every conclusion have an analytic justification or proof—leads us to pluralism, or more precisely, to complete intellectual nihilism. A view that accepts only absolute truths, proven to be true, leads us to total ignorance and in practice to skepticism. According to this approach, anyone who is consistent with the premises he has assumed at the basis of his discussion, regardless of what those premises are, reaches conclusions that are equally valid. We have seen that one can build several axiomatic systems that contradict one another, each of which is correct—no more, but also no less, than its counterpart. The issues that must be clarified with respect to the validity of the premises themselves are never found within the axiomatic system, and therefore do not belong at all to the domain of analytic thought.⁴³ The analytic method can be used only to examine whether conclusions follow from premises, not to examine the validity of the conclusions themselves. I hope this description reminds the reader of the schematic picture of the postmodern argument presented in the introduction to the present gate. That argument will be discussed further in the eighth gate.

This is the deeper meaning of the hot-air-balloon joke presented on the title page: the mathematician, like every analytic thinker, says things that are necessarily correct and therefore devoid of substantive content. They say nothing, or at least nothing beyond what is already known. In economics there is a law of direct proportion between risk and profit. The analytic thinker takes no risk at all of being mistaken, since he says only things that were already hidden in the premises, and therefore he cannot “profit” anything. As stated, certain propositions based on mathematical-analytic proof add no knowledge beyond what we already knew, and for precisely that reason they are “necessarily correct.”

The Emptiness of the Analytic: Emptiness, Pluralism, and Powerlessness

We may summarize the upshot of all this in the principle that analytic thought is empty. This emptiness has meaning in three aspects, which are of course connected:

The validity of the deductive argument relates only to the form or method, not to the content of the argument. The validity and truth of the content itself are subject to an examination of the first premises. This is in fact the well-known philosophical claim of “the emptiness of the analytic.”
If analyticity is the only criterion for valid claims, then there will always be parallel systems that contradict one another in content and yet are equally valid. Each of them will follow necessarily from its own premises. For this reason we will not be able, analytically, to decide which of them is the “correct” one. In fact, in a completely analytic world there is no concept of “correct” at all in the ordinary sense. This is the pluralistic characteristic of analyticity.
Analyticity is empty of content because it cannot add to our knowledge beyond what is already implicit in the premises, which we already possess. This is the phenomenon that may be called analytic powerlessness.

Below, we will sometimes refer to all three of these characteristics—emptiness, pluralism, and powerlessness—under the single heading “the emptiness of the analytic.”

One should note that induction and analogy, which characterize synthetic thought, generally do not tolerate propositions whose content opposes them. Often there is a principle of common sense that determines only one reasonable outcome for an induction or an analogy. In general, it seems that these forms of thought and argument depend more on common sense than deduction does, and we will return to this later as well.

In fact, one may say that the three forms of inference defined in the previous chapter serve different stages of thought—we will see this in greater detail in the eighth gate. Induction and analogy serve the acquisition of new truths, and here we cannot accept the opposite of those truths as equally valid. Deduction, by contrast, serves only to analyze in greater detail the knowledge we already possess. Therefore, by means of the same deductive logical tools, one may arrive at opposite conclusions if the preexisting knowledge is opposite. This phenomenon follows from the fact that deduction does not determine which knowledge is the correct one; it only analyzes what is hidden in the knowledge we already have. It is neutral with respect to the content itself, and therefore can also tolerate its opposite.⁴⁴

The Transcendental Alternative

Despite the foregoing, Kant has already pointed us to the possibility of obtaining certain truths, namely synthetic a priori propositions. He claimed that this can be done by means of transcendental arguments. This seems at first sight to be a promising direction, one that offers certain truths that are not merely formal but also have content, since they are synthetic. For that reason they are not exposed to the second defect listed above, namely pluralism, according to which one may accept the opposite of any premise with the same degree of certainty. A transcendental argument is apparently an argument without premises, and therefore it cannot be reversed by reversing premises, as in an ordinary analytic argument.

One thus gets the impression that Kant broke through the frame that binds logical necessity and forces it to be empty. It seems that we have here a type of certain proposition that is not empty, because it is not obtained by analytic-deductive methods but by transcendental methods.

This would seem to be the realization of the dream of the postmodern approaches described in the introduction to the present gate. As stated there, they accept as binding and valid only arguments that are not based on first assumptions. In the introduction I claimed that this demand is absurd, yet it would now seem that Kant offers it a foundation and a real possibility. We will later see that this is nothing but an illusion.

Analytic Philosophy: A Method of Philosophizing or a Philosophical Position

In twentieth-century philosophy there is a strong tendency in the analytic direction. A large part of philosophical research deals with questions concerning the meaning of propositions or concepts in language, or with the characteristics of different philosophical methods from the past. Very few new metaphysical systems were born in that century.⁴⁵ This phenomenon stems from the position that no philosophical assertion has greater validity than its opposite, so long as there is no mistake in deriving the conclusions from the premises. The only thing left to the philosopher who does not wish to engage in speculation and subjective ideas is the analysis of the philosophical systems that preceded him, which he himself usually classifies as dogmatic and speculative. This is precisely the relation, pointed out above, between analyticity and pluralism. Sometimes it appears as a move from analyticity to a pluralistic conception of truth, and sometimes the reverse: from pluralism to analyticity.

Analytic philosophy deals with questions such as: what follows from a certain philosophical system? What may one who holds it allow himself to think? What would render him inconsistent? In essence, the main content of analytic philosophy is the examination of consistency and inconsistency among different claims. It is an empty philosophy, in the same sense that analyticity is empty. It does not try to say anything about the world; it only guides those who do wish to say something about the world to do so consistently and in correct language, not to fall into errors, and to present a philosophy of proper structure. It does not examine the first premises of the philosophical system under discussion, but only the consistency of the conclusions with those premises.

As stated, analytic philosophy is apparently only a method of philosophizing and not itself a philosophical position, and therefore analytic philosophers themselves were not all holders of an analytic position. The description here fits mainly the earliest figures of this movement, and also many who came after them, such as Carnap. Yet these motivations characterize to a large extent others who traveled the analytic path as well, even those who do not advocate it as a full-fledged philosophical position. The analytic position defined above is the one that regards analytic philosophy, in its narrow sense, as the only relevant method for clarifying truth. This is an approach that assumes that certain truth can be attained only by analytic means, and therefore, as we have seen, within pure analytic philosophy truth and certainty do not exist at all.⁴⁶

In the terms proposed at the beginning of the book and in its title, analytic philosophy is the philosophy of the hot-air balloon: a philosophy that is perfectly precise and yet of no use to us. This is also, according to the interpretation offered here, the emptiness of the analytic “wagon.” This is an emptiness of content and of certainty that is built into the analytic conception. In light of what has been described here, the expression “empty wagon” is not a term of contempt, but a description of the character of the wagon. Understanding the meaning of the hot-air-balloon joke sheds light on the emptiness of the wagon. After we understand in the coming gates the relation between secularism and analyticity, it will become clear that this is also an interpretation of the emptiness of the “secular wagon.”

The philosophy of the wagon, the full one, as distinct from the philosophy of the hot-air balloon, is synthetic philosophy. It claims the superiority of one system of thought over its rivals, or at least the possibility of such superiority. Such a wagon can collide with another wagon unlike it, unlike the empty wagon, which is a structure without content. Analytic philosophy, being a structure without content, cannot collide with any other philosophy,⁴⁷ since if their premises differ they cannot share a common language, and therefore they also have no way of examining which of them is correct and which is mistaken. The metaphor of the hot-air balloon, which has served us in describing this approach, thus acquires an added significance.

By contrast, synthetic philosophy asserts the truth of specific contents. It is a philosophy that can and wants to debate, and sometimes also to clash, with its opponents. The full wagon bears a large sign proclaiming: there is truth in the world. It does not suffice with examining consistency alone; it insists on addressing and examining the contents as well.

The Expanded Definition of the Analytic Position

We now broaden the definition of the analytic position beyond what one might expect it to include. We will define an analytic philosophical position as one that recognizes as true not only claims proven by reason, but also claims grounded empirically. It is hard to imagine any worldview that does not admit that what we see or sense indeed exists and is real.⁴⁸ This is the highest level of certainty according to all opinions. We should note here that this expansion is by no means self-evident on the philosophical plane, even if in practice analytic thinkers do usually acknowledge the existence of an external world. At present we are proposing it ad hoc; later we will return to discuss it.

At first glance, this expansion seems to save the analytic from its emptiness. There is now a basis of facts on which analytic machinery can operate. If we know something as a result of sense data, then we seemingly have a certain base of knowledge, axioms, on which one can build knowledge by purely analytic means. This would appear to be the synthetic component of knowledge that was missing from the analytic picture as drawn thus far.

One might think that this addition to the analytic position rescues it from emptiness, but that is not the case. Such an approach is merely a return to naive pre-Humean empiricism. In light of the problems Hume raised—especially the problem of induction, and also the problem of causality—it is clear that this is a meaningless expansion, and it cannot save the analytic from its emptiness. General propositions will not be able to receive confirmation, even in this expanded sense of analyticity, because no general scientific law can be inferred from empirical data through a purely analytic process. We will always need inferential processes that contain a synthetic dimension.

Summary

Let us summarize by saying that in this chapter we distinguished between forms of thought and positions. An analytic position is an approach that demands certainty akin to mathematical certainty—sense data or analytic proof—in order to recognize any assertion as true. We saw that, absurdly enough, specifically this approach leads to the impossibility of certainty altogether. By contrast, the synthetic position recognizes additional ways, beyond the analytic one, of arriving at certainties, and therefore it can also make contentful claims and relate to them as binding truths. Obviously, the holder of the synthetic position does not reject the tools of analytic thought, and does not use only synthetic thought. The holder of the synthetic position disputes not the validity of analytic thought, but only its exclusivity.

Put differently, the analytic position leads its adherents to a relative view of both their own principles and those of others. It is a kind of conventionalism, which holds that each person adopts a system of beliefs and principles by way of agreement. That is: whoever agrees to my basic system will also agree to my conclusions. With others I have nothing to discuss at all. As noted, I also cannot claim that they are more right or less right than I am. In the next gate we will see that analyticity is generally accompanied by a conventionalist position with respect to concepts. In the fourth gate we will see that such a pluralistic position also does not allow constructive exchange of views or openness.

Two Remarks

We will conclude with two remarks.

(a) Kant himself also demands certainty in order to accept a claim as true. As stated, his principal innovation is that some synthetic propositions also possess the required certainty. In light of this, the term “analytic,” which we will sometimes use from now on, should in Kantian terms sometimes be understood as “a priori,” that is, prior to all experience. I prefer the term “analytic” in the broader contexts with which we will be dealing. We saw above that even proponents of the analytic method in philosophy, who also hold an analytic position, call their field “analytic philosophy,” not “a priori philosophy,” which might perhaps have been more fitting. Additional reasons for this terminology will become clear in the third part, where we will challenge the very Kantian distinction between the analytic and the a priori, and show that Kant too belongs to the analytic, that is, a priori, side of the philosophical map.

(b) An analytic position that accepts as true only what has been proven analytically may stem from two different philosophical positions. One is the position that there is no truth at all, but only derivation from premises, which are themselves arbitrary. The second is that there is indeed theoretical truth, but that we have no way of reaching it beyond what can be reached by analytic means. These positions may seem completely different in essence, but this difference has no practical significance and, in my opinion, no philosophical significance either. Therefore, in what follows I will refer to both approaches under one general name: the analytic position, or approach.

Summary of the Discussion in This Gate

First, the essential problems inherent in the process of cognition and scientific generalization, as they emerge from David Hume’s thought, were presented. These involve the rejection of the principle of induction and the principle of causality. Following this, the concepts of the “synthetic” and the “analytic” were defined in accordance with Kant’s thought, and his manner of approaching and formulating the problems Hume raised was presented, culminating in the “Copernican revolution” that Kant proposed as their solution.

Next, three types of argument were presented: deduction, induction, and analogy. “Analytic thought” was defined as deductive, analytical thought, a generalization of the analytic proposition. Following this, the “analytic position” was defined as a philosophical position that uses only that form of thought and the senses. The “synthetic position” was defined, by contrast with the analytic one, as a position willing to rely also on synthetic thinking—induction and analogy—as a tool for acquiring truths and certainties, in addition to analytic tools of thought.

After that, it was explained that analytic thought is empty in three senses: it deals only with analyzing premises and adds no new information; it deals only with derivation, that is, with forms of argument, and not with content; and it allows the parallel existence of different, and sometimes contradictory, systems possessing equal validity.

At the end of the gate, an interpretation was proposed for the emptiness of the “secular wagon” in terms of the emptiness of the analytic. This interpretation naturally depends on a fuller grounding of the identification it presupposes between secularism and analytic positions, a grounding that will be discussed in greater detail in the second and third parts.

Footnotes

Gadi Taub, in his book HaMered HaShafuf, HaKibbutz HaMeuchad, Tel Aviv, 1997, offers an illuminating description of this whole range of phenomena. It should be noted that HaMered HaShafuf came into my hands only after the writing of this work had been completed. I found in it parallels to some of the claims that will arise later, though in different forms and on less abstract planes. The sixth gate addresses Taub’s main arguments relevant to our concerns and places them in the perspective of my own argument. ↩
A critique of this sort of psychiatry is presented in Mordechai Rotenberg’s book Christianity and Psychiatry, in the Open University series, Ministry of Defense Publishing House, Israel, 1994. Following Max Weber’s Protestant thesis, he argues that modern psychiatry is based on Protestant assumptions. In fact, he follows the critique of Michel Foucault (especially in his History of Madness in the Age of Reason), who was among the founders of the new critique discussed here. To his credit, it should be said that he certainly does not suffice with merely pointing to the underlying assumptions of psychiatry, but examines them and even proposes an alternative. ↩
A more detailed description of modern philosophy from the beginning of the modern age to Kant may be found in the books of Professor Hugo Bergmann on this subject, and in summary form at the beginning of his book The Philosophy of Immanuel Kant, Magnes, Jerusalem, 1980. ↩
Some of the thinkers in the world of philosophy were scientists themselves. The division between the disciplines that is so sharp today was not then so clear-cut. This is a certain advantage that the thinkers of that period have over those of our own day. ↩
He does indeed distinguish between objective and subjective sensations through their relation to images of God, though this is not the place for further elaboration. Let us note that another interesting angle on Berkeley appears below in the literary intermezzo at the end of the second section. ↩
There is an additional disagreement between him and Kant regarding the classification of mathematical statements. This disagreement will be mentioned below. ↩
It may be that this is not certainty in exactly the same sense as the certainty of demonstrative propositions. Later, when we discuss Kant’s method, the concepts will be defined more clearly and the difference will become plain. ↩
The term “scientific induction” here is meant to exclude “mathematical induction.” The latter belongs, according to the understanding of most who deal with it, to the domain of mathematical principles. The intuitionist approach in mathematics disputes this claim, but here we will suffice with the fact that common sense affirms it. The use made here of what will later be called “the principle of induction” (without the addition “scientific”) is in a broader sense, including ordinary inference from the particular to the general, and not only scientific induction. ↩
Even apart from Hume’s rejection of the concept of causation as a fictive notion, as will be explained below, one may still wonder how one can determine with certainty that the tree’s being in the fire is the cause of its burning. That is, even if the concept of cause truly has the naïve meaning attacked by Hume, are we not simply hiding the problem in our claim that the reason the tree burned is only that it was in the fire? A finite number of past trials cannot suffice to eliminate other factors that may be relevant causes of the burning. For an interesting (and highly problematic) discussion of the concept of cause in general, see Yuval Steinitz’s Etz HaDa’at in the second part, and the references cited there. See also our critique of him in the appendix. ↩
The temporal precedence between events can also be classified as something whose source is not the impressions of the senses, a point that, in my opinion, Hume ignores, and which in Kant becomes a cornerstone of his method. People often fail to notice that Kant disagreed with Hume on this point as well. ↩
Hume’s basic assumptions and his errors will be discussed indirectly throughout the discussion as a whole. For a direct and detailed discussion, see especially throughout the third and fourth sections, and in the appendix. ↩
This distinction is usually attributed to Hume and, following him, to Kant, who sharpened it. In fact, Leibniz, about fifty years before Hume, had already defined and formulated the analytic-synthetic distinction more precisely. English-speaking philosophy, for some reason, tends to attribute it to Hume. Here we will follow the accepted historical narrative of the development of Kantian ideas. ↩
Kant regarded such propositions as a priori but also synthetic, assigning them to the synthetic-a priori category that was innovated in his school and will be presented below. The claim that algebraic propositions, such as this one, are synthetic has been the subject of a long-standing philosophical dispute, but that they are a priori is, so far as I know, entirely agreed upon, apart from Kripke’s challenge (who brings as an example the pedagogical argument I mentioned above in the body of the text, and in my opinion is simply speaking of a different concept of apriority). This challenge will be mentioned again below. ↩
In mathematical development, creativity and even activity that is synthetic in essence are often required, but this is only on the meta-mathematical level. For example, locating a fertile conceptual framework for mathematical development is synthetic work, whereas the development itself is entirely analytic. Sometimes one can point to even more than this—that the development itself involves synthetic effort—but that is only a shortcut that could in principle also be carried out analytically. ↩
Of course, one may wonder whether the property of a tree to burn in fire is not analytic—that is, whether it is not contained in the concept of tree itself. In a simple sense, no, because it requires in addition various laws of nature that determine which materials burn in fire. Once those laws are known, it may perhaps be possible to regard this fact as analytic with respect to the tree, but the significant scientific step is the determination of those laws themselves, and that has a clearly synthetic character. Some maintain that in the end, if all scientific knowledge were theoretically in our possession, it would be possible to arrange it so that every property would be analytic with respect to its subject. This is a hypothetical discussion that uses a not-simple definition of the concept “analytic,” and this is not the place for it. ↩
See Saul Kripke, Naming and Necessity, translated by Avishai Revah, University Publishing Enterprises, Israel, 1994, where he points out that there are indeed four such categories. His claim is that the a priori/a posteriori distinction belongs to epistemology, whereas the other distinction, analytic/synthetic, belongs to the world of things or concepts themselves, or to the world of syntax. In my opinion, his examples of analytic a posteriori statements are unconvincing. His definition of a posteriori is too extreme. See the previous note, and the next one as well. In the next gate we will discuss the differing epistemologies characteristic of those with analytic and synthetic approaches. ↩
See the table presented by Adi Tzemach, “Necessary, Analytic, A Priori,” Iyyun 36, Jerusalem, 1987, p. 168. ↩
Below I will comment on Quine’s claims, for he is one of the most prominent among those who do not accept Kant’s distinction between the analytic and the a priori. ↩
See Yuval Steinitz, LeOlam Tehe HaMetafizika, Dvir, Tel Aviv, 1990, in the introduction. ↩
I do not mean to claim that every synthetic proposition is taken from pure reason, but only that such propositions exist. Kant’s innovation was that there is at all a category of synthetic-a priori propositions, that is, that the classifications above are not identical. ↩
It is clear that the same difference between analogy and analogue exists in Leibniz’s context as well. The analogy deals with two clocks that are identical entities, and therefore the picture is symmetrical. But in the analogue there is a great difference between the influence of the soul on the body and the influence of the body on the soul. This is not the place to expand on the matter. ↩↩
This point will be discussed at greater length mainly in the eleventh gate. In fact, Kant’s main innovation is his finding of escape option A1 for one who is unwilling to believe in the existence of an external factor, yet still wants to adhere to and believe in the progress of science. The common belief that Kant refuted option B is incorrect. Kant’s argument is merely a possible way out (apparently; see the seventh and eleventh gates) for someone unwilling to accept option B, which in my opinion is self-evident. His words do not refute it in any way. It is entirely clear that the “atheistic” position (which does not believe in an external factor) is here on the defensive and not on the offensive. ↩
Some claim that in this sense Kant anticipated quantum theory, and indeed there are attempts to interpret that theory in Kantian terms. See my article, “What Is ‘Legal Effect’?”, Tzohar 2, Tel Aviv, 2000. ↩
This distinction of Kant’s will be discussed at greater length in the seventh gate. There are several interpretations of this Kantian claim; here we have chosen the simplest and most common, namely the one that sees in this phenomenon a cognitive limitation. In the second gate (and to some extent also in the seventh) I will explain why, to the best of my understanding, this approach is incorrect (not necessarily as an interpretation of Kant, but in its own substance). ↩
I was told that Rabbi Joseph Dov Soloveitchik had already discussed the connection between the two halakhic concepts of the legal object and the legal subject and the Kantian concepts of phenomenon and noumenon. See his book BeSod HaYahid VeHaYahad. ↩
On this matter, see my aforementioned article “What Is ‘Legal Effect’?” and the ongoing exchange in subsequent issues of Tzohar with Rabbi Baruch Kehat. ↩
This note is also relevant to attempts to give an interpretation of quantum theory in terms of Kant’s philosophy. In quantum theory, the principle is commonly accepted that reality is determined by the observer, which seems to point to a Kantian view (see my article “What Is ‘Legal Effect’?” mentioned above). Here too one should note that it may be only the way the observer sees reality that is in question, whereas there is also a reality in itself. I made this remark in the article mentioned below in Note 7. ↩
It is important to note that this connection is not precise. These remarks of Or Sameach do not necessarily lead to Leibowitz’s conclusion. ↩
It should be noted that Kant’s distinctions were attacked by many philosophers, one of the best known being Quine. Quine argued that we do not understand concepts “atomistically,” that is, one by one, but “molecularly.” Our understanding of concepts, in his view, takes the form of a network in which all the concepts are woven together and connected by complicated links. In such a view, the distinction between analytic and synthetic loses much of its significance. There are concepts that lie closer to the center of the network (the “analytic” part, in Kant’s terminology), and these are harder for us to change—that is, we are more “compelled” to accept them—whereas the more peripheral ones (“synthetic,” in Kantian terminology) are more easily changed. See W. V. O. Quine, “Two Dogmas of Empiricism,” in From a Logical Point of View, Cambridge: Harvard University Press, 1953, pp. 20–46. In the present discussion I am only making use of Kant’s definitions, and therefore I do not find it necessary to exhaust the discussion of the synthetic-a priori issue and the various approaches to it. My goal here is to present my own position on this issue, and I do indeed assume, unlike Quine, an “atomistic” cognition of concepts. ↩
A reader interested in further expansion may turn to almost any basic book on logic. On the subject of axiomatic systems there is also Douglas Hofstadter’s excellent book Gödel, Escher, Bach, Vintage Books, New York, 1979. It is a popular and clear book, written in a fascinating literary style, though it is very detailed and quite long. ↩
In the eighth gate we will discuss these three forms of argument, examine the differences between the various kinds, and also discuss the critique of the very separation between them. ↩
There is much room to discuss the conditions for the validity of inductive and analogical arguments—for example, the relevance of the similarity between the particulars in an analogy, or between the particular and the general in induction. We will discuss this somewhat in the eighth gate. ↩
They were preceded, though in a more limited way, by the German logician Gottlob Frege. See below in the ninth gate, and also at the end of the second gate. ↩
An axiomatic system also requires rules of formation, and various demands on its assumptions and rules of derivation, but this is not the place for such details. The interested reader is referred to the basic literature in mathematical logic. ↩
R. Carnap, Introduction to Symbolic Logic and Its Applications, Dover, New York, 1958. ↩
In my article, “A Fortiori Inference as a Syllogism,” Higayon 2, Aluma, Jerusalem, 1992 (mentioned in the note to Note 22), I briefly mentioned the difference between the two kinds of a fortiori inference that will be discussed here below. There I tried to subsume them under one formula, and I showed that this attempt does not succeed. ↩
See my article, “He Gives Evil to the Wicked According to His Wickedness,” Alon Shevut – Bogrim, vol. 9, Alon Shevut, Iyyar–Sivan 1996, p. 145. ↩
The only way to understand the Mekhilta that links this a fortiori argument to the ordinary a fortiori argument is to say that there they apparently understand the rule “one does not impose punishments on the basis of logical inference” as a scriptural decree derived from a verse (see the relevant entry in Encyclopedia Talmudit). In any event, this is strained. ↩
Further on this subject of the difference between the two kinds of a fortiori inference, see also Maharsha on Babylonian Talmud, Bava Kamma, second edition to 49b, and in Yonat Elem, section 16. See also Tosafot, s.v. “and witnesses,” Babylonian Talmud, Bava Kamma 4a, and the commentators there and there. Also Rashi on Babylonian Talmud, Chagigah 11b, regarding his daughter from rape. Also Nahmanides in his commentary to Leviticus 11:3, and the Re’em there (and verse 8). and likewise Maimonides in Sefer HaMitzvot, negative commandment 172, and Sefer HaChinukh, commandments 154 and 195. ↩
Hugo Bergmann, at the beginning of his book Introduction to Logic, Bialik Institute, Jerusalem, 3rd ed., 1975, distinguishes between contentful logic and formal logic. The term “contentful logic” did not take root in the philosophical world of our time. This too stems from an analytic approach that sees only the analytic domain as a domain of discussion that can count as logic. True, today there are some attempts to build inductive logics and various kinds of fuzzy logics, but they are not at the center of logical discussion. In fact, these are attempts—sometimes paradoxical—to create a system that is “logical” in the accepted senses even for those synthetic domains. ↩
For example, the “addition,” or composition, of forces in mechanics is not correctly described by the rules of arithmetic addition. This in no way refutes the rules of addition in elementary arithmetic. At most, it refutes the claim that one can describe the addition of mechanical forces by means of ordinary arithmetic addition. In fact, the addition of forces in mechanics is done by the mathematical operation called “vector addition.” The mathematical part is not subject to refutation. If anything is refuted, it is only the claim that the mathematical theory can be applied within empirical science. We will discuss this at length in the next book of our trilogy. ↩
At the end of chapter 3 of the eighth gate we will see that non-Euclidean geometries (as well as the theory of relativity) are a clear example against the relativity of truth, contrary to what is commonly believed. ↩
This subject will be discussed at length in the eighth gate. ↩
Whoever understands this distinction will see the problematic nature of attempts to create inductive logics. This is the application of a method to a domain to which it seemingly does not belong at all. In one of the previous notes we pointed out that, because of an analytic conception, there is no willingness to recognize “contentful logic.” Here we see another side of the same coin: one who is already willing to deal with domains of content also tries to subject them to formal rules. Often this is done for the same very reason: an unwillingness to recognize the legitimacy of non-mathematical modes of thought. ↩
I mean philosophical methods and not mystical ones. This classification is somewhat formally problematic, and certainly in the eyes of analytic thinkers, who classify every metaphysical method in some sense as mysticism. Even so, in practice it is usually quite clear who is a philosopher and who is a mystic. The renewed flourishing of mysticism, and its significance, will be discussed in the second section. ↩
A more detailed discussion of analytic philosophy will come in the second and third sections. There we will also discuss the common approaches to analytic philosophy that see it specifically as a conservative stance. ↩
The only possible clash for analytic philosophy is a clash with synthetic philosophy, or with a “full wagon,” regardless of the content of the “fullness.” The analytic approach cannot and will not accept the existence of groups that claim the truth of contents. Any claim to the correctness of some content will immediately come into conflict with the conception of the empty wagon. Seemingly, this is the only contentful claim that the analytic approach must accept—and indeed it accepts it and fights for it with all its strength: there is no certainty, only consistency. This issue itself is not in dispute. The connection between analyticity and skepticism, and the question whether skepticism itself is certain in the eyes of the skeptic, will be discussed below. There also arises here the question of analytic tolerance toward positions that are not analytic, a subject to be discussed in the fourth gate in the chapter on tolerance. ↩
Philosophical idealism is such a position. Today it is hardly found at all, and one cannot seriously claim that it underlies Western thought. Nowadays it is specifically seen as more connected to the East. Still, as we saw above in the description of Berkeley’s philosophy, even idealism has empiricist roots; that is, even there skepticism (in the sense of non-recognition of the existence of an external world) stems from excessive demands on cognition. This is a parallel phenomenon to the one I pointed to here, namely that skepticism concerning the concepts of certainty and truth stems from excessive demands placed upon them. ↩

← Introduction to Part OneContentsGate Two: Matter and Form — The Thing in Itself and Its Appearances →

Basic Concepts: The Analytic and the Synthetic

Introduction

Chapter One: Pre-Kantian Dogmatism and the Challenge to It3

Historical Background

David Hume: The Problem of Induction and the Problem of Causality

Chapter Two: Kant’s Solution: The Synthetic A Priori

Two Classifications of Propositions

The Importance of Synthetic Propositions

The Relation Between the Two Classifications

Kant’s Formulation of Hume’s Problems: The Synthetic A Priori

The Philosophical Basis for the Existence of Synthetic A Priori Propositions

The Clock Parable: Three Ways to Explain Correspondence Between Two Things

Phenomena and Noumena: The Thing in Itself

Note 1: The Perspective of “Object” and “Person” in Halakha (Jewish Law)25

Resolving the Humean Problems

Summary: The Second Copernican Revolution

Chapter Three: Another Aspect of Analyticity and Syntheticity: Forms of Inference and Axiomatic Systems

Deduction, Induction, and Analogy

The Deductive Argument

Axiomatic Systems

Deduction and Analyticity

Note 2: No Punishment Is Imposed on the Basis of Logical Derivation—Two Types of A Fortiori Inference

The Synthetic and the Emptiness of Deduction

Chapter Four: Forms of Thought and Positions: Analytic and Synthetic

The Relation Between Analytic Thought and Pluralism

The Analytic and the Synthetic Position

The Emptiness of the Analytic: Emptiness, Pluralism, and Powerlessness

The Transcendental Alternative

Analytic Philosophy: A Method of Philosophizing or a Philosophical Position

The Expanded Definition of the Analytic Position

Summary

Two Remarks

Summary of the Discussion in This Gate

Footnotes

שתף

Chapter One: Pre-Kantian Dogmatism and the Challenge to It³

Note 1: The Perspective of “Object” and “Person” in Halakha (Jewish Law)²⁵