Foundations of Mathematics

Intuitionists and Formalists.

This defect in the system of Principia Mathematica has prevented its obtaining any general acceptance, and several of the greatest mathematicians now ap proach the problem on entirely different lines. The Intuitionist school, led by Brouwer and Weyl, proposes to abandon many generally accepted parts of mathematics, and to retain only such propositions as they can prove without using the Law of Excluded Middle, which says that either all numbers have a certain property or at least one number does not have it. They think that it is wrong to say that there is a number with a certain property, e.g., satisfying a given equation, unless we have a definite construction for finding one.

On the other hand, the Formalist school, who follow Hilbert, hope to put an end to this disastrous scepticism, by taking an alto gether different view of what mathematics is. They regard it as merely the manipulation of meaningless symbols according to fixed rules. We start with certain rows of symbols, called axioms; from these we then derive others by substituting certain symbols called constants for others called variables, and by proceeding from the pair of formulae p, if p then q, to the formula q. Mathematics is thus regarded as a sort of game, played with meaningless marks on paper, rather like noughts and crosses; but besides mathe matics there is, according to Hilbert, another subject called meta mathematics which is not meaningless, but consists of real asser tions about mathematics, telling us that this or that formula can or cannot be obtained from the axioms according to the rules of de duction. The most important theorem of metamathematics is that it is not possible to deduce a contradiction from the axioms, where by a contradiction is meant a formula with a certain kind of shape which can be taken to be o o. Although no complete proof of this theorem has yet been published, it is supposed that it can be proved, and that scepticism arising from the fear of contradiction will then be finally disposed of.

Since, whatever else a mathematician is doing, he is certainly making marks on paper, it must be granted that the formalist view consists of nothing but the truth; but it is hard to suppose it the whole truth, as our interest in the symbolic game surely arises from the possibility of giving meaning to some, at least, of the marks we make, and the hope that with the meaning so given they will represent knowledge and not error.

Wittgenstein.

We see, then, that these eminent mathemati cians, great as are the differences between them, are yet agreed that mathematical analysis as ordinarily taught cannot be re garded as a body of truth, but is either false or at best a mean ingless game; and if classical mathematics is to be defended, some answer must be found to their criticisms. Such an answer might be based on the logical and important work of Wittgenstein.

If we take the view that mathematics can be reduced to formal logic, there is one problem of fundamental importance which Wittgenstein claims to have solved. He has clearly defined the peculiar characteristic of logical propositions. It was formerly supposed that any true proposition which mentioned no particular thing or relation, and so could be stated in purely logical terms, was a proposition of logic or mathematics. But such a view is evidently mistaken, as such a statement as any two things differ in at least 3o ways" can be stated in purely logical terms and may well be true, but even so it would not be a logical or mathematical truth. Logic and mathematics have a further char acteristic, which you can call either necessity or tautology accord ing to your philosophy, and which Wittgenstein has precisely analysed. Further, Ramsey claims to have shown that using Wittgenstein's work the system of Principia Mathematica can be reconstructed so that the unsatisfactory Axiom of Reducibility is no longer required. Thus classical mathematics, interpreted as one with formal logic, may yet be rehabilitated.