Gate Nine: Against Analyticity — Third Move

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).

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Against Analyticity: Third Move

This gate contains five chapters:

1. Chapter One: Decidability and Completeness of Systems of Thought
2. Chapter Two: Within the System and Outside It
3. Chapter Three: Gödel’s Theorem: Provability and Truth
4. Chapter Four: Within the System and Outside It — Again
5. Chapter Five: A Historical Note

Introduction

This gate is the third and final one in the series of attacks on the analytic position conducted in the third part of the book.

The first attack, in Gate Seven, was carried out on the philosophical-epistemological plane. The second, in Gate Eight, was carried out on the logical plane, mainly in the domain of the single logical argument, which derives a specific conclusion from specific premises. After that, we expanded the discussion to axiomatic systems, which are the more general framework within which the process of particular logical argument takes place. In this gate we will continue the discussion on the broader logical plane, but we will expand it beyond one axiomatic system or another to human thought as a whole. We will examine the question whether all human thought can be described by an analytic model, and we will conclude that this is impossible. Most surprising of all is that the proof of this is mathematical, though it is not clear whether it is analytic.

Part of the discussion here will be somewhat mathematical, and the reader may skip it, as I will note in the course of the discussion. I found it appropriate to present the formal side of the issue because it presents, in the sharpest and clearest way, the core of the claim about the importance of stepping outside the system, which stands at the center of the present attack.

Here we will encounter one of the most essential features of analyticity: the inability to step outside the system of discussion. By its nature, this phenomenon is the cutting edge of the argument against analyticity. Therefore, in the next part, as a point of departure for developing the synthetic alternative, we will present the synthetic position as one that advocates the possibility and legitimacy of stepping outside the field of discussion.

Chapter One: Decidability and Completeness of Systems of Thought

Decidability

In this gate we will demonstrate more clearly the weakness of the analytic-axiomatic form of thought, and from that we will begin to hint at the direction that may allow us to overcome this weakness.

In the course of the discussion we have already encountered several times the concept of an axiomatic system. In mathematics, one asks of various axiomatic systems whether they are decidable. A decidable system is a system of premises within whose framework all questions that arise can be resolved analytically and unambiguously, positively or negatively — that is, by mathematical proof. In twentieth-century thought and science, this question takes similar forms in mathematical logic, where it was treated by the German mathematician Kurt Gödel, who formulated the incompleteness theorems that bear his name, and in computer science, where it is known as the halting problem in computability theory, whose treatment began with the British mathematician Alan Turing.

We will begin the discussion of this question specifically from a common assertion in the world of traditional Torah study: that the Torah contains all the solutions to all the problems of humanity and the world, as the verse says:

“The Torah of the Lord is perfect” — complete.

In a similar way, in general moral theory, and also in legal systems, there are approaches that maintain that there is one uniquely correct solution to every problem, whether moral or legal.

In mathematical language, this claim can be formulated as follows: the Torah, or the moral theory under discussion, is decidable if it has a defined position regarding every situation or every problem.¹ At the beginning of this discussion I will try to examine the claim that the Torah is decidable by considering absurd boundary cases. We will ask ourselves what exactly counts as a “solution” to a problem, and we will distinguish between solutions found within the framework of discussion and solutions that lie outside it. Later we will move from the Torah discussion to a broader discussion of the inherent weakness of axiomatic-analytic systems. Such systems look for solutions only “within the system,” and therefore they find themselves compelled to distinguish between “provability” and “truth,” in the terminology of Gödel’s theorem in mathematical logic. We will return to this below.

Completeness and Decidability of Systems

Let us begin with halakha (Jewish law). In halakha there are many clashes between values. One well-known halakhic ruling states that saving a life overrides the Sabbath. Behind this ruling lies the assumption that there are two values here, both positive and important: the preservation of life, and Sabbath observance. Yet despite this assumption, it turns out that there are situations in which these two values conflict with one another. In such situations we must decide which of them is to be preferred.

When a critically ill person must travel to the hospital on the Sabbath in order to save his life, the trip is indeed a violation of the Sabbath, but if he does not do so he places himself in mortal danger. Here halakha rules that saving life overrides the Sabbath — that is, the value of life takes precedence.² Another example of the halakhic scale of values is the rule that in a conflict between a positive commandment and a prohibition, the positive commandment prevails; in halakhic language, a positive commandment overrides a prohibition. There are, of course, many other rules of conduct for various situations of conflict between halakhic values.

Even in situations where there is no clear determination regarding the hierarchy between two conflicting values, halakha still leaves us with a way out that allows us to decide how to act. In such a case there are laws governing doubt: ruling leniently, ruling stringently, “it is preferable to refrain from acting,” and so forth.³

There are cases in which the hierarchy of values is not transitive — that is, value A is preferable to B, B is preferable to C, and C is preferable to A.⁴ Problems of this sort do not, at first glance, seem fundamental, since even in non-halakhic fields there are many relations that are not transitive. Take, for example, the relation “being the father of.” If this relation exists between Isaac and Jacob, and also between Jacob and Reuben, that does not mean that it also exists between Isaac and Reuben.

It is, however, quite clear that the relation “preferable to,” or “more important than,” with which we are dealing here, does appear to be transitive. Therefore it would still seem that there is a problem on the level of the hierarchy of values. The theoretical problem can be solved by saying that the preference is in different respects, and indeed that is the case in most of the examples.⁵

The problem becomes more acute in situations that are not merely theoretical but practical. If each pair of values comes before us separately, no practical problem is created, even if on the theoretical plane one does indeed exist. In each such case we will decide in favor of the value that takes precedence over the other. But there is a situation in which a practical conflict arises among three such values at once — that is, two pairs, with one value shared by both. Let us now take an example in order to clarify the matter.⁶

Suppose a person sees his own lost object and the lost object of his teacher, or of his fellow, both being swept away in the river, while at the same time there is an obligation to attend to his father’s honor, for example by feeding him or giving him drink. That person cannot perform all the actions required in such a situation. He cannot save both lost objects and also attend to his father, but can perform only one of these actions.⁷ The question then arises: what should he do first? The halakhic rules governing such a case are the following:

Rule A: His own lost object takes precedence over the lost object of his teacher, meaning that he is entitled to deal with his own loss and neglect that of his teacher.

Rule B: Honoring his father takes precedence over his own lost object, according to Rabbi Yehuda.

Rule C: The lost object of his teacher takes precedence over honoring his father.

Every situation in which two of these actions meet is decidable; it can be resolved halakhically. But the problem of all three actions together creates a situation that is seemingly without escape, since the three values involved are non-transitive. If he wishes to prefer saving his own lost object, he cannot, since honoring his father takes precedence. If he wishes to save his teacher’s lost object, again he is not obligated to do so,⁸ since his own lost object takes precedence. And if he wishes to occupy himself with honoring his father, a problem again arises, since his teacher’s lost object takes precedence over attending to his father’s honor.⁹

If we formulate this more formally and generally, we may put it as follows. We are given a system with three possible courses of action in the situation — saving his own lost object, saving his teacher’s lost object, and attending to his father’s honor — which we will call A, B, and C. There are three guiding principles for the case of conflict, that is, three override rules, the rules set out above:

1. B < A — value A takes precedence over value B.
2. C < B — value B takes precedence over value C.
3. A < C — value C takes precedence over value A.

The problem is what to do when a three-way conflict occurs — that is, when all three actions are called for, but only one of them can be performed.

Let us spell this out more fully. If I choose action A, I satisfy Rule 1 and violate Rule 3. If I choose action B, I satisfy Rule 2 but violate Rule 1. And if I choose action C, I satisfy Rule 3 and violate Rule 2. This is a necessary result of the absence of transitivity among the values. At first glance this seems to be a problem with no solution. In light of this, can one still say that the Torah is “perfect”? Apparently there are problems for which we have no halakhic solution, and therefore, in the terms used above, we must conclude that the halakhic system is not decidable.

In order nonetheless to try to solve this dilemma, we must formulate a three-way decision rule — that is, a rule that will help us decide the triple dilemma directly. The problem in creating such a rule is that it cannot be derived directly from the two-way decision rules, since those lead us into contradiction.¹⁰

This problem can be generalized to many systems of thought, and we may see that although they appear complete and well-founded, they may nonetheless not be decidable.¹¹ Later we will see, to our surprise, that this phenomenon appears even in mathematical systems.

Chapter Two: Within the System and Outside It

In the previous chapter we saw that within different systems of thought there can arise situations in which the system cannot decide what answer it itself prescribes for the situation under discussion.

One possible direction¹² for making a decision in such a case is to assign weights to the principles formulated above. For example, the “weight” assigned to observing principle no. 1 is X, to observing principle 2 is Y, and to principle 3 is Z. The “cost” of violating a principle can be chosen, for example, as the opposite of the “weight” of observing that very principle. Thus, violating principle 1 carries a cost of -X, and so on.¹³ One can then construct a table of costs for each chosen action. If we choose action A, the cost will be Z - X. For action B: X - Y. And for action C: Y - Z. We may then decide on the desired action by comparing the costs of the three alternatives.¹⁴

Of course, theoretically the same situation may recur even on this level. The three “weights” themselves may also stand in non-transitive relations to one another, in which case the cost differences above will be equal, or at least two of them will be equal, and we will once again not have escaped the tangle. In that case we must repeat the process again, and again, until we arrive at a transitive system that permits a decision.

What must characterize such a situation in which a decision can be reached? First, it is a situation in which the “weights” are expressed in a uniform system of units. If the weights can be written as numbers, in any units whatever, provided that the same system of units can be used for all three decisions, then a non-transitive situation will not arise. Almost always there will be one number that is larger than the other two. Here we see that the impossibility of deciding among the different courses of action stemmed from the fact that their relative priority was not measured in the same units. This is what we earlier called “preferences in different respects.”¹⁵

The second characteristic of the “weighting” process described here is, in fact, that it involves stepping outside the system of principles themselves and evaluating them in a meta-system. Within a system of principles, if the decision rule or weighting is not itself formulated as an independent principle, as in the halakhic examples brought above, there is no ability at all to decide priority between two different principles. The paradox we presented has a resolution only if there is a meta-system in which all the actions and principles can be examined in the same “currency,” and a hierarchy can be established among them. One cannot establish hierarchy, or priority, between two principles that are assessed by different measures, that is, principles lacking a common measure — incommensurable principles.

We thus see that in situations where there is a dilemma that appears impossible to resolve, that is necessarily true only within the system. Exiting the system can make it possible to solve even a dilemma that at first glance appears impossible, of this sort. Of course, there need not necessarily be a suitable meta-level of principles¹⁶ in which such a solution is indeed to be found. Theoretically, it is possible that the problem of transitivity will accompany us on every plane we reach. In such a case there will be no decision, or no solution, at least not of this type, to the paradox.

In light of this analysis, one can reformulate the claim about the decidability of halakha and say that the assertion is that for every problem there is a meta-systematic level at which an unambiguous decision can be reached. The promise that such a level exists does not appear obvious at first glance, since we have seen that it is possible that on every level we may encounter the same kind of contradiction among principles. This claim is connected to the Torah conception that the whole world, with all its complexity and problems, emanated from a single root. The problematic and paradoxical nature of halakhic and real-life situations stems from the multiplicity present in creation. On the highest level there is an unambiguous resolution to all problems, since the root of all values is the one God. This is a logical-axiological aspect of belief in the oneness of God.

Even without any direct connection to one theology or another, which is brought here only as an illustration, it seems that the solution to many problems that appear on their face to be insoluble is thinking on a higher plane, or freeing oneself from a given system of principles or terminology. The problematic nature of this kind usually, and perhaps always, arises because one looks at the problem from within a closed system of principles, and within that narrow domain there is no solution to the dilemma.

We should note that within the analytic framework this is the only context in which one can relate to value principles, and therefore also to value problems. We examine the principles according to which we act only within our own system of assumptions. This is the essence of the analytic approach. According to this approach, there is no non-formal context for examining principles, or any claims whatsoever. Put differently: one cannot step outside the system of assumptions within which one operates. In the next chapter we will see a surprising implication of this analytic inability.

Chapter Three: Gödel’s Theorem: Provability and Truth¹⁷

The Meaning and Importance of Gödel’s Theorem

One of the well-known theorems in mathematical logic is the theorem of the German mathematician Kurt Gödel, known as the incompleteness theorem. There are several such theorems. This theorem provides an example, within mathematics itself, of the claim with which the previous chapter ended. The stronger incompleteness theorem states that in every axiomatic system equivalent, in known senses, to the mathematical theory that deals with the whole numbers, that is, number theory, there is at least one statement that is necessarily true and yet at the same time cannot be proved.

The central philosophical consequence of this surprising theorem is the distinction between a “provable statement” and a “true statement.” The meaning of this distinction is that there is truth even without proof — or, alternatively, that proof is not the only ultimate path to truths. In other words, the analytic position is rejected here outright.

I should note that although the theorem was proved only for certain mathematical systems, namely those equivalent to number theory, the distinction that arises from it between provability and truth is, of course, general. It is enough that there exists even one system in which a difference arises between provability and truth in order to claim — and perhaps even to prove — that such a difference does in fact exist, that is, that “true” is not necessarily equivalent to “provable.”

As I hinted, astonishingly enough there is a proof of this theorem itself. Moreover, the proof in question is what mathematicians call a constructive proof. This means that within the proof one actually constructs, inside the system under discussion, some statement, and then shows that it is true on the one hand and at the same time unprovable on the other. In order to conduct the discussion more intelligently, we will now briefly present a version, admittedly very simplistic and highly incomplete, of a proof of Gödel’s theorem.¹⁸ This will also serve as a relatively simple example of an axiomatic system and of the analytic mode of operation in mathematics, and finally also of the idea of stepping outside the system as the main characteristic of the synthetic position. Anyone who is uncomfortable with mathematical symbolism, and for whom the description that follows is too difficult, may skip this section, or merely skim it.

An Example of an Axiomatic System and of the Analytic Mode of Operation: Number Theory

Number theory describes the properties of the whole numbers. One can write a simple version of this mathematical theory that deals only with the basic properties of the non-negative whole numbers.

This formal language has thirteen symbols:

(, 0, ∈, W, →, ¬, ), =, P, a, ', &, S

These symbols are defined as follows:

• ' — a prime mark. It is used to generate additional variables. If a is a variable, then a' and a'' are two other variables.
• P — a predicate. The statement Pa means that a has the property P.
• & — the logical operation “and.” The expression Pa & Pa' means that the property P applies both to a and to a'.
• ¬ — negation. The statement ¬Pa means that it is not true that a has the property P.
• → — implication. The statement Pa → Pa' means: if the property P characterizes a, then it also characterizes a'.
• W — the set of non-negative whole numbers.
• ∈ — membership in a set. The statement 0 ∈ W means that the number 0 belongs to the set W.
• S — successor. The symbol S0 is the number that comes after 0, which in our notation is 1.

Additional symbols may be introduced only by defining them in terms of the known symbols. For example, the symbol 1 can be defined by means of the two symbols S0.

We may now describe the list of axioms of this simple version of number theory. The formal formulation appears first, followed by the explanation in ordinary language:

1. 0 ∈ W — The number 0 belongs to the set W, the set of non-negative whole numbers.
2. (a ∈ W) → (Sa ∈ W) — If a belongs to the set, then its successor also belongs to it.
3. (Sa = Sa') → (a = a') — If the successors of two numbers are equal, then the numbers themselves are equal.
4. (a = a') → (Sa = Sa') — If two numbers are equal, then their successors are also equal.
5. ¬(Sa = 0) — There is no number in the set whose successor is 0; that is, there are no negative numbers in it.
6. (P0 & (Pa → PSa)) → Pa — The induction principle: if the property P holds of 0, and if its holding of a implies its holding of the successor of a, then one may conclude that this is a property of all non-negative whole numbers, that is, of the whole set W.

In every axiomatic system there are, in addition, rules of inference. These rules make it possible to derive theorems, that is, statements, from other theorems. In our system the rules of inference are:

1. One may substitute a constant, that is, a number, for a variable, or another variable for a variable, provided that the substitution is consistent throughout the formula.
2. The logical rule MP, that is, modus ponens, or the rule of detachment:

P → Q ; P ⇒ Q

The meaning of rule 2 is that if we are given the two theorems P and P → Q, where P → Q means “if P then Q,” then we may infer from them the theorem Q.

As an example, let us now present two proofs of statements in this system.

First, let us prove the far-reaching statement that 1, which in our notation is S0, is not equal to 0:

• From axiom 5: ¬(Sa = 0)
• By rule 1: ¬(S0 = 0)

As a second example, let us prove that 1, surprisingly enough, is a non-negative whole number:

• From axiom 2: (a ∈ W) → (Sa ∈ W)
• By rule 1: (0 ∈ W) → (S0 ∈ W)
• From axiom 1: 0 ∈ W
• By rule 2: S0 ∈ W

Gödel Numbering

Now that the language is clearer, we can move on to the stages of the proof of Gödel’s theorem itself. Gödel proposed a procedure for representing every statement in the language by means of a number, in such a way that different statements are represented by different numbers, and conversely. This mechanism is called Gödel numbering. First, for each symbol in the language, that is, for each of the thirteen symbols listed above, one defines an odd number that represents it. We will do so as follows:

• ( ↔ 1
• 0 ↔ 3
• ∈ ↔ 5
• W ↔ 7
• ) ↔ 9
• a ↔ 11
• → ↔ 13
• S ↔ 15
• = ↔ 17
• ' ↔ 19
• ¬ ↔ 21
• P ↔ 23
• & ↔ 25

As we have seen, every statement in the language is a sequence of symbols from this list. The representation that we are now proposing for statements in this language satisfies the requirement that every different statement will have a different number representing it, and conversely. We write the various symbols in order, so that in the place of each symbol there stands a prime number, in the order of the prime numbers, raised to the power of the number that represents that symbol. The number representing the entire statement will be the product of all these factors. Thus, for example, the fifth axiom will be represented by the following enormous number:

2^21 × 3^1 × 5^15 × 7^11 × 11^17 × 13^3 × 17^9

The bases of the powers in this number are the prime numbers in order, and their exponents are the numbers representing the different symbols in the order in which they appear in the formula of the statement. This number is called the Gödel number of the fifth axiom.

Because of the properties of prime numbers, one can show that every statement in our language has a unique Gödel number that represents it, and conversely. Let us note here that even meaningless strings in the language, that is, strings that are not well formed, such as aS=P, also have formal representation by means of their own Gödel numbers.

The next stage is the representation of an entire proof by its own Gödel number. If we look at the proofs above and ignore the explanations written beside them, we immediately notice that a “proof” in this language is a sequence of theorems, each of which is derived from previous ones by using the rules of inference or the axioms. The last line in the proof is the statement that the proof proves.

It is therefore clear that one can represent complete proofs, and not only isolated theorems, by Gödel numbers. To do so, we write the proof as the product of a sequence of prime factors, similarly to the representation of the theorems. The exponent of each prime factor will be the Gödel number of the corresponding line in the proof. For example, the Gödel number representing the first proof above, which has two lines, would be:

2^g1 × 3^g2

Here g1 and g2 are the Gödel numbers of the first and second lines of the proof, respectively, and the way they are constructed was described above.

It is important to note that one can distinguish between the Gödel number of a proof and that of an isolated theorem by checking the parity of the exponents of the prime factors. It is easy to see that in a number representing a proof, the exponents are even, whereas for a number representing a theorem, the exponents are odd. This is the reason why specifically the odd numbers were chosen to represent the symbols.

Another component of an axiomatic system is its derivation rules, or rules of inference. These too can be translated into arithmetic rules. Since such a rule of derivation is a mode of transition from one line to the next in a proof, and since each line in the proof is already represented by its own Gödel number, one can regard these rules as functions composed of arithmetic operations, usually the four basic operations, that produce Gödel numbers from other Gödel numbers. In the language of Gödel numbers, a rule of derivation is an algebraic operation that takes us from the Gödel number representing the first line to the Gödel number representing the second.

At this point one can examine by purely algebraic means whether some Gödel number represents a valid proof in our system or not. That is, one can ask whether the statement written in the last line of this proof, which is represented by the Gödel number of a theorem, is a legal or valid theorem in our system.

The “technology” that Gödel proposed for numbering expressions in the language brings us to a situation in which even a computer can examine, in an entirely mechanical way, whether a certain statement is “correct” in number theory or not.¹⁹ Up to this point we have developed, of course in a highly simplistic way, the mathematical technology required for the proof of the incompleteness theorem. From here on, the proof is quite short.

Proof of Gödel’s Theorem

If there is a Gödel number X representing a proof, one can “translate” it backward — that is, find the number representing the last line of the proof, which is the conclusion of the proof. In this way we extract from the proof the statement that has been proved. By this method one can calculate algebraically, from the Gödel number of the proof, X, the Gödel number representing the proved statement, Y. Since such an extraction is a well-defined arithmetic operation, that is, a function, there is therefore within the axiomatic system a valid statement that indicates the relation between X and Y, such that X is a proof of Y. Let us denote that statement as P(x, y). Similarly, if one is given the number x representing a valid proof, one can find the number y representing a theorem that satisfies P(x, y), that is, one for which x is a proof of y. By contrast, if one is given some y, clearly it is not always possible to find an x that will signify a good proof for it, that is, one that satisfies P(x, y). For example, for every false statement in number theory one obviously cannot find a valid proof.

Since according to the first rule of inference one may substitute any number for any variable, provided the substitution is consistent, it follows that one may in particular substitute the number Z into the theorem whose Gödel number is Z. We now define the operation Q(z, y) as follows: if you take the formula whose number is z and substitute the number z itself for the variable, you obtain a statement whose Gödel number is Y.

At the next stage of the proof we look at the following statement:

¬P(x, y) & Q(z, y)

The meaning of this statement, in non-formal language, is: “It is not the case that x is a valid proof of y,” and also, “y is the Gödel number obtained from substituting the number z into the theorem whose number is z.” Let us denote the Gödel number of the previous statement, the one composed of these two parts, by g. We now substitute, in accordance with rule of inference 1, g itself in place of z. The statement that we obtain after all these operations is:

¬P(x, y) & Q(g, y)

The meaning of this statement is as follows: it is not true that the sequence denoted by x is a proof of what y signifies, while at the same time y is obtained by substituting the number g into the variable in the statement represented by g. In other words, y is in fact the Gödel number of the last statement, since we substituted g into the statement represented by g and thus obtained a statement represented by the Gödel number y.

If we pay attention to the meaning of the statement we have now obtained, we see that in our language its meaning is that there is no proof x for the statement y, despite the fact that this was obtained lawfully within the system. In such a situation one of two possibilities must hold. Either y is true and has no proof, in which case both parts of the above statement are true; or y, which is the statement above itself, is not true, and in order for that to happen one of the two components of the statement above must be false. The right-hand component of the statement is true by definition, and therefore only the left-hand component can be false — that is, it is not true that it has no proof. Thus the second possibility is that y is not true, but does have a proof. Of these two possibilities, the logician has no choice but to choose the first, which is the less bad of the two, since otherwise he would be living with a contradiction in which a false statement has a valid proof within the system.

If we translate the above statement into non-formal language, we see that its meaning is actually this:

Statement (): Statement () has no proof.

This is a kind of liar paradox in ordinary language, as we saw in the last chapter of the previous gate. For if Statement (*) is true, then it has no proof; and if it is not true, then it has a proof, which is impossible, because a false statement cannot have a proof. Therefore it is clear that it is true and has no proof.

The advance achieved here over the simpler formulation of the liar paradox is that we succeeded in formulating it as a legitimate statement within number theory, which is a clean and analytic mathematical theory. Within number theory this is a statement like any other, and it turns out to be true and yet at the same time necessarily impossible to find a proof for it. The translation into the liar paradox in our own meta-language merely helped us to see intuitively why this is the case. The proof can be presented, and it seems to me that this is usually how it is done in the study of logic, without mentioning this interpretation at all.²⁰

If we return for a moment to the computational side of the discussion, a computer, or in mathematical language a Turing machine, that is asked to check the validity of this statement will never halt. That is, the procedure will never end in finite time. This is the computational side of the paradox, or of Gödel’s theorem, which we mentioned above as the halting problem. The meaning of what we have seen in the mathematical aspect is that there exists a Turing machine that does not halt, at least when it receives a certain input.

From this, logicians conclude that there is a difference between a true statement and a provable statement, and more generally: there is a difference between truth and provability. This is a far-reaching conclusion, especially in mathematics, which is, as noted, the symbol of analyticity. It is a complete shattering of the analytic myth. Analyticity, as we have seen, identifies truth with the existence of a proof; that is, it identifies truth with provability. Gödel’s theorem strikes it squarely on the head — and does so with its own tools.²¹

Chapter Four: Within the System and Outside It — Again

Is There or Is There Not a Proof of Gödel’s Theorem?

Only now have we arrived at the really important question: why do logicians think that the very process we have gone through up to this point does not itself constitute a proof, within the system, of the Gödelian statement? That is, why is the statement ¬P(x, y) & Q(g, y) not proved by this very procedure that we have seen here, which is the proof of Gödel’s theorem? At first glance, we have proved this statement by reductio ad absurdum. We showed that the alternative to its being true is to live with the claim that it is false and at the same time has a formal proof in the axiomatic system presented here.²² If so, then apparently we do have an analytic proof, by reductio ad absurdum,²³ that this statement is true. It should be noted that in geometry, at least as it is taught in high school, where the concept of proof is not formalized, such a proof is accepted as a valid proof of a valid theorem.

The answer to this question lies in the fact that if we were to formalize geometry, we could prove Gödel’s theorem there as well, and then such a proof would not be accepted as valid. The problem arises because this proof, although it sounds logical and even necessary to laypeople, and also to those who are not laypeople, cannot be written formally in the language of the axiomatic system. It is a proof in our own meta-language, which we use to describe this axiomatic system “from the outside.” In other words, if one steps outside the axiomatic system, one can indeed say that this Gödelian statement has a proof, and one need not separate truth from provability. But a proof outside the system is a proof that cannot be written in a way that relies on the basic assumptions of the system itself, assumptions that are fixed and known in advance. If those assumptions too could be written explicitly, then we would build a new axiomatic system that included them as well, and with respect to that system we could find another Gödelian statement that would be true and yet unprovable relative to the new system.

Can “Truth” and “Provability” Be Identified in a Broader Sense

One may summarize and say that if we do not limit ourselves to analytic requirements, we can be fully convinced of the truth of statements even when they have no analytic proof. There are proofs that convince with the same degree of certainty, yet still cannot be written in analytic form.

Here we see a much stronger claim than what we have seen so far. It is not only that there are non-analytic truths; there are also truths that are proved and certain and yet are not analytic. In short, there are non-analytic “proofs.” This is an extreme example of the general claim of this book: that one can arrive at high levels of certainty even without proofs, in the analytic sense of that term. Here there is a procedure that can indeed be called a “proof.” It is characterized by a deductive level of persuasiveness, and yet it is not analytic, because it involves stepping outside the system of formal assumptions of the axiomatic theory within which we are operating.

The synthetic approach as it has been presented so far claims even more than what follows from Gödel’s theorem. The synthetic claim is that there are methods that are not certain even in this broader sense, that is, methods that step outside mathematics altogether, and not merely outside the particular axiomatic system under discussion, and yet one can still draw from them information with a significant degree of certainty. Of course, Gödel’s theorem cannot support such a claim, though it also clearly cannot refute it, since this is not a mathematical claim. This claim speaks about certainty and not about the existence of a proof, even in the softer sense that appeared above, namely, a proof from outside the system.

If we were satisfied with the verbal statement of Gödel’s theorem, according to which there are axiomatic systems in which there are true statements that cannot be proved, it would appear that this buries the analytic approach once and for all. But from the detailed presentation of the proof of Gödel’s theorem, we have seen that the situation is more subtle, and apparently the bewildered analytic thinker still retains an honorable escape route. He can make do with a proof outside the system, in the meta-language, without giving up its degree of necessity.

But this escape route in fact means giving up the demand for the full analyticity of proof. Thus the strong version of analyticity was rejected by Gödel’s theorem in any case. Yet still, for the purposes of the present discussion, the analytic thinker may remain with a softened analytic version: only analytic proof is accepted, but it may be conducted even by means of tools outside the system, provided they are logical tools.²⁴

The Meaning of the Conclusion Drawn from Gödel’s Theorem

The conclusion that emerges from this discussion is paradoxical in a certain sense. If we had needed to decide in advance which of the two types of systems of thought is decidable, meaning capable of deciding positively or negatively and categorically every question within its domain, an axiomatic system or an intuitive system, we would have expected the axiomatic system to be more decidable, while the intuitive system would certainly not be. Here we have seen precisely the opposite: an axiomatic system is not always decidable, and it is precisely the addition of non-analytic methods, that is, stepping outside the system, that can increase its decidability.

On deeper reflection, this is not surprising. This phenomenon resembles the widespread illusion that a postmodern approach is the opposite of an analytic position, which demands precise proofs for everything. Postmodernism is perceived as a rebellion against the rule, or even the very existence, of analytic reason and necessary certainty. As we have seen, this is not correct. Postmodernism not only does not contradict the analytic position; it is in fact an almost necessary conclusion of the analytic approach. See Gate Three, Chapter One, and Gate Five for the Nietzschean analysis of modernity.

This is another aspect of the “philosophical uncertainty principle” presented in Gate One: the more necessary a statement or an inferential process is, the less effective it is — that is, the less new information can be learned from it. A statement or inference that is necessarily true contains no new information. That is what we learned in Gate One from the hot-air-balloon joke.

In the earlier parts of the book, we saw that living within an axiomatic system cannot lead to the accumulation of new information. In this gate we saw that even the weaker question, whether a conclusion follows from a set of premises, is not necessarily decidable within such a system.²⁵ It appears that not only the decision concerning the truth of statements requires synthetic assistance, but even the examination of their derivability from the premises, which is usually thought to belong exclusively to the domain of analytic thought, also requires such assistance. This was also the main claim presented in the previous gate.

Certainty can be based only on interaction with what lies outside the system. This completes the arguments presented in the previous two gates, where we also saw that the process of deriving conclusions from premises itself presupposes synthetic stages at its foundation. The deductive process requires grounding in a meta-language, that is, outside the deductive system itself.

Chapter Five: A Historical Note

At the end of the nineteenth century, the logician and philosopher Gottlob Frege attempted to build a formal language that would enable him to ground all of arithmetic in logic. Bertrand Russell ruined his plans when he showed that this enterprise could not succeed because of his famous paradox in set theory.²⁶ Russell himself solved the problem in his own analytic way, by proposing a formal form, namely axiomatic set theory. This solution, of course, did not solve the real philosophical problem, as distinct from the mathematical-analytic one, namely: what is a “set” in everyday language? It is a formal solution, in Russell’s characteristic style, which restricted the use of the concept “set” so that the paradox could not arise. In mathematics this proved to be a fruitful and appropriate solution, but with respect to the philosophical problem it is of little help.

Frege’s worldview was that mathematical constructions characterize the world, and not the human being who observes it. In this context he was what is called a Platonist. In our terms he was a synthetic essentialist. The reader is referred to the end of Gate Two, to the discussion of the problem of ostension, in order to see another aspect of his view. There too our conclusion was that Frege’s solution is satisfactory, whereas the formal solution, Russell’s, does not solve the real problem. Here we have a confrontation between the father of analyticity and a synthetic mathematician. At that stage it seemed that analyticity had won. On the mathematical plane it is indeed true that analyticity must set the rules of the game. But in philosophy, and certainly in life, that is not the case.

The continuation of the historical process is no less interesting. Russell himself, who was also a mathematician and not only a philosopher, together with the mathematician and philosopher Alfred North Whitehead, once again tried, and in a more ambitious way, to refine Frege’s approach and ground all of mathematics in logic. This attempt is presented in their monumental work Principia Mathematica. At this point Kurt Gödel arrived and did to Russell what Russell had done to Frege. He showed that this enterprise was a priori impossible. It is not possible to present all of mathematics in the form of a formal, that is, axiomatic, system, because of the essential limitations of such systems, as we demonstrated in this gate.

In the final analysis, the conclusion is, as stated, that even in mathematics, which is the pure analytic domain, one cannot work by analytic means alone. If that is so, then certainly the other domains of thought cannot be founded upon this form of thinking. A synthetic move outside the system is always required in order to ground and represent human thought in its fullness.

Gödel’s Theorem, Positivism, and Postmodernism

It is commonly thought that Gödel’s theorem struck a blow against positivism, that is, against the optimism that holds that analytic reason is all-powerful. On the face of it, this would seem to be an important contribution to postmodernity, which believes that reason lacks the power to discover absolute truths.

According to the picture presented in this book, postmodernism actually arises out of positivism, which expresses a pure analytic position. When one strikes such a position, one thereby strikes postmodernism at its roots as well. Gödel’s theorem does not show that there is no absolute truth, as it may at first seem and as many indeed think. Rather, its main point is that truth and certainty — which certainly do exist — are not necessarily attained by means of proofs. Gödel, as noted, separates truth from provability, and determines that there are truths that are not provable. Gödel’s theorem itself is a true theorem, and a proved one; that is, it actually presupposes that truth exists. The problem lies only in its proof. This is certainly not what postmodernism tries to infer from this theorem. It is true that the skeptical postmodernist still has the option of claiming that he does not recognize even mathematics and logic as valid tools, and by that means can reject also the proof and the claim of Gödel’s theorem itself. But he certainly cannot rely on Gödel’s theorem itself as a basis for his skeptical claims. Once again we encounter the same two claims that accompanied us at earlier stages:

1. There is a close connection between positivism and postmodernism. They are not polar opposites, as many tend to think.
2. The only alternative to the synthetic position is radical skepticism, or, in other words, intellectual nihilism.

Summary of the Discussion in This Gate

In this gate we completed the series of attacks on the analytic position. We saw that it can be proved by analytic, that is, mathematical, means that many analytic systems contain statements that are necessarily true and yet cannot be proved within the system, as shown by Gödel’s incompleteness theorem. This is the mathematical distinction between truth and provability.

This discussion led us to the insight that in order to attain completeness, we must be prepared to step outside the system and speak about it in a meta-language external to it. This is a non-analytic way, or at least a way that is not analytic in the full sense, of proving properties of a mathematical system.

We also saw the necessity of stepping outside the system from additional angles, such as the resolution of paradoxes and logical loops. There too we understood that the central problem is the inability, or unwillingness, to step outside the system. It turns out that an alternative philosophical position, one that will cope better with this problematic, will be a position that recognizes both the need and the ability to step outside the system that we are examining.

This point opens the discussion of the next part. There we will examine the relation of the synthetic approach to these questions. Before that, however, we will introduce a gate devoted to the question whether there really is such a thing as a pure analytic thinker — a question sharpened greatly in light of the untenability of pure analyticity that has been proved in the last three gates.

Footnotes

This way of presenting the dilemma and its solution is not necessary. For example, in his novellae to tractate Ketubot 15, the Hatam Sofer makes an interesting remark on this issue. The Gemara there explains that an infant whose status is uncertain—whether he is a gentile or a Jew—must be saved even at the cost of desecrating the Sabbath, at least according to Maimonides’ ruling; see Maimonides, Mishneh Torah, Laws of Forbidden Intercourse 15:26. Yet after he is saved he is not at all commanded to keep Sabbaths, and in fact may even be forbidden to keep the Sabbath, since he may be a gentile, and a gentile is forbidden to keep the Sabbath. This is only an anecdote, though a significant one, and this is not the place to elaborate.

First, I apologize for my lack of attention and thank her for the remark. Second, I correct the error and present here the correct dilemma according to Rabbi Yehuda’s view as it is presented in Tosafot.

It is entirely possible that such a donkey would indeed die of hunger. If this donkey desires life, it must enter a meta-level discussion in order to decide rationally—that is, in order to survive. Buridan’s ass would not return any lost object to its owner either, were it placed in the previous situation. The analytic thinker, in this sense, is like Buridan’s ass. The superiority of man over this donkey, as we shall see, lies in his capacity for synthetic thought, and in the terms of the present gate: in his ability to step outside the formal system of thought within which he operates.

Let us also note that one may have a situation even in the same units in which all the weights are equal—for example when Z = Y = X—and once again no decision can be made. I do not wish to enter pathological situations, since this is only an example of stepping outside a system in a way that can solve problems. It is true that stepping outside a system does not necessarily solve them.

Thus “Gödelianity” is not merely an esoteric property of meaningless propositions alone, although on the principled level it makes no difference even if no such familiar propositions had been found.

I believe that truly meaningful propositions have not in fact been found that are true and unprovable, that is, propositions expressing Gödel’s “strong” theorem, the one we proved here.