Modern Logic: From Frege to Gödel: Hilbert and Formalism

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MODERN LOGIC: FROM FREGE TO GÖDEL: HILBERT AND FORMALISM

The leading exponent of the formalist philosophy of mathematics was David Hilbert (1862–1943), who pioneered in a development of logic known as proof theory or metamathematics. From the time of his first papers on the foundations of mathematics, Hilbert stressed the importance of the axiomatic method and its superiority over the genetic approach, by which concepts are extended piecemeal as the need arises. Once a theory is axiomatized, however, it invites a number of general questions concerning the logical relations holding between its propositions, and Hilbert was soon to consider as central among such questions the problem of establishing consistency, or freedom from contradiction. Hilbert did not himself think that there was any support for the allegations of inconsistency in analysis, as made by Hermann Weyl. Nevertheless, he wished to consolidate once and for all the foundations of mathematics and to give them such clarity that the axiom of choice would be as perspicuous as the simplest arithmetical truth. To this end he needed to devise consistency proofs. He had, in 1899, shown the consistency of Euclidean geometry relative to the theory of real numbers, but proofs of this form do no more than shift the problem of consistency to the system to which the original theory has been reduced. Some new, more direct method seemed to be called for.

Despite his confidence in the consistency of classical mathematics, Hilbert contended that operating in an abstract way with mathematical concepts had proved insecure, and his remedy was to interpret number theory as relating to the observable domain of such signs as 1, 11, 111. Elementary number theory is thereby assured of a concrete interpretation—"3 > 2," for example, can be understood as asserting that the concatenation of three strokes extends beyond the concatenation of two strokes. However, the possibility of such an interpretation does not extend to all branches of classical mathematics, for such entities as transfinite cardinals do not allow of representation as sequences of strokes.

Hilbert's solution to this difficulty was to treat such numbers as "ideal" elements. Thus, appealing to Kant, he argued that one precondition for the application of logical laws is a domain of extralogical concrete objects, given in actual perception and capable of being exhaustively surveyed. Nowhere in nature is an actual infinity to be found; therefore, whereas for finite numbers a perceptually given basis could be given, transfinite numbers had a place in mathematics only as ideal elements, much like the ideal factors introduced to preserve the simple laws of divisibility for algebraic whole numbers. Such a reduction was, Hilbert claimed, a natural extension of the work of Weierstrass, who had shown that reference to infinity in the context of calculus involved merely a façon de parler, replaceable by a theory of limits requiring a potential infinite rather than an actual one. Similarly, the infinities introduced by Cantor, though apparently irreducible, had to be shown to be indispensable, and arguments proceeding via the infinite had to be replaced by finite methods that achieve the same goal. Again, since the transfinite enters with the use of unbounded quantifiers, statements containing these had to be regarded as ideal statements.

With this approach Hilbert hoped to partially vindicate classical mathematics against the attacks of the intuitionists. Complete vindication, however, required a proof of consistency, and the method that Hilbert proposed for obtaining such a proof is closely related to his method for providing elementary number theory with a sound basis. That is, just as he had considered numbers as sequences of strokes, so he now regarded formulas and proofs as sequences of uninterpreted signs. In this way he provided a concrete subject matter for a proof of consistency, a proof that was to invoke only logical principles whose security and perspicuity are equal to the security and perspicuity of the perceptually given domain on which they are to operate.

Thus, the consistency of some given formalization of a branch of mathematics could be unquestionably established if it could be shown by finite combinatorial methods that no manipulation of the symbols which represents a passage from axioms to theorems could result in the derivation of the expression "0 = 1" or of some other concatenation of symbols which, when interpreted, is seen to be an absurdity. The theory itself might contain symbols for transfinite cardinals and other ideal elements, but this would be no obstacle to a consistency proof, since in such a proof we are required only to treat these symbols as perceptually given objects and to show that they will never figure in a formula whose negation is also provable. On the other hand, Hilbert believed that although nonfinitary concepts are allowable within mathematics proper, they are not to be countenanced in the theory of proof that is to ensure consistency.

The formalist school, which included Wilhelm Ackermann, Paul Bernays, and John von Neumann, succeeded in establishing a number of metamathematical results of considerable significance, but without completing Hilbert's original program, for although successively stronger systems of arithmetic were proved consistent, no proof was forthcoming for the full system required by classical number theory. And, indeed, results obtained by Kurt Gödel in 1931 indicate that no finitary consistency proof is possible, since any proof of consistency must make an appeal to principles which are more general than those provided by the system and accordingly are as much open to question as those principles whose consistency we wish to establish. Attempts were subsequently made to prove consistency by means which were as close to being finitary as possible, notably by Gerhard Gentzen in 1936, but even if "finitary" were thought to apply to the methods used—in this case an application of transfinite induction—it would not follow that classical mathematics had been vindicated against the intuitionists, since to their way of thinking the mere consistency of mathematics would not suffice to confer a clear meaning on the crucial concepts of classical mathematics.