Some observations about Hilbert


Hilbert's Challenge

David Hilbert was born January 23, 1862 in Konigsberg, in what was then Prussia.

Hilbert was invited to give a special address to the Second International Congress of Mathematicians at Paris in 1900. Inspired by the date, Hilbert wanted to give a talk that projected forward to speculate about developments in mathematics in the new century. He gave a now famous lecture, in which he outlined 23 problems which he thought may be decisive in the development of mathematics in coming years. The prophecy in part fulfilled itself, as young mathematicians set themselves to Hilberts challenges. Several passages from Hilbert's talk are relevant to our interests:
It remains to discuss briefly what general requirements may be justly laid down for the solution of a mathematical problem. I should say first of all, this: that it shall be possible to establish the correctness of the solution by means of a finite number of steps based upon a finite number of hypotheses... which must be exactly formulated....

In later mathematics, the question as to the impossibility of certain solutions plays a prominent part; and we perceive in this way that old and difficult problems... have finally found fully satisfactory and rigorous solutions, although in another sense from that originally intended. It is probably this remarkable fact along with other philosophical reasons that gives rise to the conviction (which every mathematician shares, but which no one has as yet supported by a proof) that every definite mathetical problem must necessarily be susceptible of an exact settlement, either in the form of an actual answer to the question asked, or by the proof of the impossibility of its solution and therewith the necessary failure of all attempts....

This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus. (From "Mathematical Problems," Bulletin of the American Mathematical Society 1902 (8): 437-479.)
This reference to ignorabimus is an attack on a popular conviction in Hilbert's time pronounced by the philosopher (now forgotten) Emil duBois-Reymond, who preached that "ignoramus et ignorabimus" -- we are ignorant and we shall remain ignorant. This apparently became something of a commonly repeated quote, justifying defeatism in intellectual endeavors. It is hard not to share Hilbert's impatience and even disgust with this kind of facile, one might well say lazy, mysticism. Later in his life, in 1930, Hilbert gave a talk in which he further asserted:
In an effort to give an example of an unsolvable problem, the philosopher Comte once said that science would never succeed in ascertaining the secret of the chemical composition of the bodies of the universe [i.e., stars]. A few years later this problem was solved....

The true reason, according to my thinking, why Comte could not find an unsolvable problem lies in the fact that there is no such thing as an unsolvable problem. (quoted in Reid 1996)

Then Hilbert asserted, "Wir mussen wissen. Wir werden wissen." We must know. We shall know. Unfortunately, Hilbert's dream would be proved wrong -- although, in defence the goal he set for us, in a rather rigorous and well defined way!

The second of Hilbert's questions was on the compatibility of the arithmetic axioms. He questioned whether we could prove that the axioms of arithmetic were consistent -- whether we could show that they could not prove a contradiction. Closely related to this concern would be the question of whether arithmetic is complete: whether all the truths of arithmetic are provable by the axioms of arithmetic. These are closely related because, if a system is inconsistent and you can find the inconsistency, then you can typically use certain tricks to prove anything. Thus, inconsistent systems are complete, but in a trivial way: they can typically prove not only all the truths, but also all the falsehoods too! In a lecture in 1929, Hilbert explicitly added the completeness of arithmetic as a challenge to future mathematicians.

One of the junior professor at Gottingen, Ernest Zermelo, pointed out to Hilbert in 1904, independently and in parallel with Russell, a paradox of set theory: "The set of all sets that are not members of themselves." This raised grave doubts about set theory as it existed then. Furthermore, the context here is important. Hilbert was adamently opposed to the position of an earlier mathematician named Kronecker who had insisted that only constructive mathematics was appropriate. This meant that we needed a clear proof that constructed a particular object in order to draw conclusions about it; we could not, Kronecker insisted, merely prove something about some kind of objects in general, or prove that some object of the kind existed which we cannot demonstrate, or prove the existence of something by showing it's inexistence led to contradictions. But these demands would eliminate much of mathematics. More recently, the Dutch mathematician Brouwer had introduced a related view called "intuitionism," which was similar to constructionism in both its views and effects. Both Kronecker and Brouwer, along with many other mathematicians, were opposed to Cantor's transfinite mathematics and much of his set theory. Hilbert granted that mathematics could be more rigorous, but insisted we could do this by using a formal axiomatic method. It was Hilbert who first explicitly suggested that proofs could be mathematical objects, themselves studied by mathematics.




Some contemporaneous objections: Poincare on Hilbert and Logicism

In Hilbert's time, many influential mathematicians disagreed with both his project and Frege and Russell's related project. Henri Poincare, recognized in the late nineteenth century as the greatest living patriarch of mathematics, wrote critically of the new logical approach to mathematics. Poincare makes 3 central points.

First, against logicism, Poincare is concerned that highly formalistic approaches cannot capture all of mathematics: "it is certain that we cannot reduce mathematical thought to an empty form without mutilating it" (1908/2001: 464). There are severl different programs that Poincare is criticizing here, including Hilbert and Russell (for some reason he snubs Frege, and does not address his work). In a foreshadowing of the birth of the turing machine, Poincare sums up both Hilbert's program, and more generally what he objects to in the various logicist programs, by noting what Hilbert's goal, if realized, would mean:

Thus it will readily understood that, in order to demonstrate a theorem, it is not necessary or even useful to know what it means. We might replace geometry by the reasoning piano imagined by Stanley Jevons; or, if we prefer, we might imagine a machine where we should put in axioms at one end and take out theorems at the other, like that legendary machine in Chicago where pigs go in alive and come out transformed into hams and sausages. It is no more necessary for the mathematician that it is for these machines to know what he is doing. (1908/2001: 463)
(The Jevons "Piano" was a mechanical calculator, and was said to be the first to be faster at calculations than doing them by hand.) In contrast, Poincare adopts a Kantian line, and argues that you can never found mathematics on pure logic, but must always make some synthetic a priori judgments. In particular, Poincare argues that the principle of mathematical induction is always assumed by the logicists, and cannot ever be derived. Poincare's reasoning goes like this:
If... we have a system of postulates, and if we can demonstrate that these postulates involve no contradiction, we shall have the right to consider them as representing the definition of one of the notions found among them. If we cannot demonstrate this, we must admit it without demonstration, and the it will be an axiom. (1908/2001: 467)
Poincare is adopting a position weaker than constructivism: in order for a definition to identify an object, one must show that it is not contradictory to posit that object (as opposed to requiring that we construct such an object). This can be done by constructing such an object, by finding a single example, since if there is such an object, it must not be contradictory. For example, if I define a even number as a number divisible by 2, and I then show that 4, 6, and 8 are even, I have shown that there are such things as even numbers, and so my definition must not be contradictory. However, if I define an evod number as a number divisible by two and which is also odd, then I've contradicted myself, and will never be able to demonstrate an example.

But, sometimes (especially for statements that are very general), to prove that a statement does not produce a contradiction, we cannot demonstrate or construct a single example. For example, my statement may entail infinitely many different kinds of statements, so that a demonstration of each is impossible. Then, we must use the principle of mathematical induction. Recall, this is the common tool of mathematics in which we order our cases, and then we show that the 0th case has some property P, and then we show that if any nth case has our property P the n+1th case must have it. It seems obvious then that, like falling dominoes, every case must have the property P. So, we can use this tool to show that infinitely many cases of some kind have a property P.

Poincare claims that the logicist and those who are trying to reduce and clarify mathematics with logic cannot escape assuming mathematical induction is a correct principle. He believes it is correct, but he believes that since we cannot prove this without assuming mathematical induction itself, we grasp it in some other way. We see that it is correct by way of a synthetic a priori judgment.

Poincare's second objection is directed at Russell. Bertrand Russell's logicism is clear and explicit, and aims at a complete logical system that will be able to derive all of mathematics. Poincare writes

...what appears to me doubtful, is that after these appeals to intuition we shall have finished: we shall have no more to make, and we shall be able to construct the whole of mathematics without brining in a single new element. (1908/2001: 476)
About this Poincare has been proven right.

Poincare's third objection is to Cantor's set theory, and particularly his theory of the transfinite. Poincare believes that it is reasoning about infinity which has led to the "antinomies." He adopts the traditional line that we cannot reason about actual infinite wholes; "There is no actual infinity. The Cantorians forget this, and so fell into contradiction" (1908/2001: 499). Here, Poincare sees a powerful benefit to Hilbert's approach over Russell's: "belief in an actual infinity is essential in the Russellian logistic, and this is exactly what distinguishes it from the Hilbertian logistic" (1908/2001: 500). Infinity, then, is a potentiality, never a complete whole.


REFERENCES AND FURTHER READING