Hilbert's Statement of His Program

We must remember that in Hilbert's time all mathematicians were excited about the foundations of mathematics. Intense controversy centered around the problem of the legitimacy of abstract objects. Weierstrass had greatly clarified the role of the infinite in calculus. Cantor's set theory promised to raise mathematics to new heights of generality, clarity and rigor. But Frege's attempt to base mathematics on a general theory of properties led to an embarrassing contradiction. Great mathematicians such as Kronecker, Poincaré and Brouwer challenged the validity of all infinitistic reasoning. Hilbert vowed to defend the Cantorian paradise. The fires of controversy were fueled by revolutionary developments in mathematical physics. There was a stormy climate of debate and criticism. The contrast with today's foggy atmosphere of intellectual exhaustion and compartmentalization could not be more striking.

As the leading mathematician of his time, Hilbert considered it his personal duty to defend mathematics against all attackers and skeptics. This task was especially urgent in view of contemporary scientific developments. According to Hilbert, the most vulnerable point in the fortress of mathematics was the infinite. In order to defend the foundations of mathematics, it was above all necessary to clarify and justify the mathematician's use of the infinite [13].

Actually Hilbert saw the issue as having supramathematical significance. Mathematics is not only the most logical and rigorous of the sciences but also the most spectacular example of the power of ``unaided'' human reason. If mathematics fails, then so does the human spirit. I was deeply moved by the following passage [13], pp. 370-371. ``The definitive clarification of the nature of the infinite has become necessary, not merely for the special interests of the individual sciences but for the honor of human understanding itself.''

Hilbert begins with the following question. To what if anything in reality does the mathematician's use of the infinite correspond? (In my opinion Hilbert's discussion of this point would have profited from an examination of Aristotle's distinction between actual and potential infinity. According to Aristotle, there is no actual infinity, but potential infinity exists and first manifests itself to us in the continuous, via infinite divisibility. See also Lear [18].)

Hilbert accepts the picture of the world which is presented by contemporary physics. The atomic theory tells us that matter is not infinitely divisible. The quantum theory tells us that energy is likewise not infinitely divisible. And relativity theory tells us that space and time are unbounded but probably not infinite. Hilbert concludes that the mathematician's infinity does not correspond to anything in the physical world. (Consequently, the problem of justifying the mathematician's use of the infinite is even more urgent and difficult for Hilbert than it would have been for Aristotle.)

Despite this uncomfortable conclusion, Hilbert boldly asserts that infinitistic mathematics can be fully validated. This is to be accomplished by means of a three step program.

2.1. The first step is to isolate the unproblematic, ``finitistic'' portion of mathematics. This part of mathematics is indispensable for all scientific reasoning and therefore needs no special validation. Hilbert does not spell out a precise definition of finitism, but he does give some hints. Finitistic mathematics must dispense completely with infinite totalities. This means that even ordinary logical operations such as negation are suspect when applied to formulas which contain a quantifier ranging over an infinite domain. In particular, the nesting of such quantifiers is illegal. Nevertheless, finitistic mathematics is to be adequate for elementary number theoretic reasoning and for elementary reasoning about the manipulation of finite strings of symbols.

2.2. The second step is to reconstitute infinitistic mathematics as a big, elaborate formal system. This big system (more fully described in Hilbert [14]) contains unrestricted classical logic, infinite sets galore, and special variables ranging over natural numbers, functions from natural numbers to natural numbers, countable ordinals, etc. The formulas of the big system are strings of symbols which, according to Hilbert, are meaningless in themselves but can be manipulated finitistically.

2.3. The last step of Hilbert's Program is to give a finitistically correct consistency proof for the big system. It would then follow that any $\Pi^0_1$ sentence provable in the big system is finitistically true. (For an explanation of the role of $\Pi^0_1$sentences in Hilbert's Program, see Kitcher [16] and Tait [25].) Thus the big system as a whole would be finitistically justified. The infinite objects of the big system would find meaning as valid auxiliary devices used to prove theorems about physically meaningful, finitistic objects. Hilbert viewed this as a new manifestation of the method of ideal elements. That method had already served mathematics well in many other contexts.

Such was Hilbert's inspiring vision and program for the foundations of mathematics.

I have only one negative comment. With hindsight, we can see that Hilbert's proposal in step 2.2 to view infinitistic formulas as meaningless led to an unnecessary intellectual disaster. Namely, it left Hilbert wide open to Brouwer's accusation of ``empty formalism.'' Brouwer's accusation was clearly without merit. A balanced reading shows that Hilbert's overall intention was not to divest infinitistic formulas of meaning, but rather to invest them with meaning by reference to finitistic mathematics, the meaning of which is unproblematic. Nevertheless, this part of Hilbert's formulation was confusing and made it easy for Brouwer to step in and pin Hilbert with a false label. The whole drama had the bad effect of lending undeserved respectability to empty formalism. We are still paying the price of Hilbert's rhetorical flourish.