The Exact Sciences and Non-Euclidean Logic

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. Parmenides even questioned the reality of plurality and change. (Einstein’s vision has much in common with Parmenides.) Toward 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.