University of David
As we have seen, the question of what “really” exists pervades the sciences and human thought in general. The belief that the infinite does not really exist goes back at least to Aristotle. Parrnenides even questioned the reality of plurality and change. (Einstein’s vision has much in common with Parmenides). Towards the end of the nineteenth century an acrimonious exchange took place between Kronecker and Cantor regarding the reality of the actual (as opposed to potential) infinite. Kronecker claimed that only the finite integers really exist and all else is merely the work of man. Cantor countered that the essence of mathematics was its freedom and that he had attained a larger vision than Kronecker had who could not see the infinite. Most mathematicians have followed Cantor and found his paradise a more beautiful and alluring universe. Hilbert accepted Kronecker’s viewpoint for his metalanguage, but tried to recapture Cantor’s paradise in a formal language. Hilbert was a formal pluralist in feeling that each mathematical discipline was entitled to its own formalization. Russell was a logical monist and felt that all of mathematics should be constructed within a single formal system. He put a great deal of labor into his program and looked askance at Hilbert. He felt that Hilbert’s approach had all the advantages of theft over honest toil. What he did not realize was that in intellectual affairs, as in economic affairs, great fortunes are rarely ever accumulated through honest toil. What is needed is the intellectual leap. Russel’s program led to much interesting mathematics, but even if in principle it could be carried out, in practice the result would be computationally intractable. One would be translating simple, clear ideas into the fog of Principia Mathematica. Russell’s program has as much relevance to complex analysis as von Neumann’s game theory has to chess. The understanding and appreciation of mathematics has very little to do with formal logic. For example, the following footnote occurs at the beginning of Wall’s (1970) book Surgery On Compact Manifolds. Recent results of Kirby, Siebenmann and Lees have now (1966) provided such a technique. All our methods now extend to the topological case, with only trivial alteration. See (K8),(K9), (L10). All the experts could see the truth of this footnote. But this seeing is not explained by modus ponens. In his beautiful book Proofs and Refutations, Lakatos (1976) has shown that the mathematical process itself is dialectical and not Euclidean. At all times our ideas are formally inconsistent. But inconsistency, while still recognized as a pathology, is no longer seen to be a fatal disease. If we come across a contradiction, we localize it, isolate it, and try to cure it. But we have to get over our neurotic phobias concerning this disease and recognize it as inseparable from life itself. Hilbert’s program collapsed with the startling work of. Godel. Mathematical logic and the study of formal systems have become a branch of mathematics instead of its foundation. Moreover, Cantor’s paradise has been raised into the metalanguage in order to prove deep theorems concerning formal systems as well as to provide a semantics for such systems. A. Robinson even defined non-standard formal systems which contain infinite formulas. One thus has a large plurality of different approaches to mathematics. Most mathematicians live in Cantor’s paradise in spite of Russell’s paradox; they simply learn to avoid making certain moves which have been shown to lead to contradictions.