Mathematics

Abstract

In the early years of the twenty-rst century, one might well look back over the previous 100 years and come to the conclusion that the notion of human progress - intellectual, political, and moral - is at best ambiguous and equivocal. Indeed some philosophers (for example Thomas Kuhn, Paul Feyerabend, and Richard Rorty) have written in recent years as if no such notion could be made out and they have seriously challenged the idea that standards of rational scientic progress exist. However, there is one area of autonomous, human, scientic endeavor where the idea of, and achievement of, real progress, the discovery of ever deeper and more general theorems, is unambiguous and pellucidly clear; it is mathematics. In 1900, in a famous address to the Second International Congress of mathematicians in Paris, David Hilbert listed some twenty-three open problems of then outstanding signicance. In the intervening period many of those problems have been denitively solved, or shown to be insoluble, culminating most recently, in 1994, with the proof of Fermat’s Last Theorem by Andrew Wiles. Along with enormous progress in the disciplines of pure and applied mathematics there has also come real insight into the methods of mathematics (both classical and constructive), and into the nature of proof and its relation to mathematical truth. Considerable progress has also been made in meta-mathematics (that is the mathematical study of such key notions as demonstrability, denability, predicativity, and truth), in areas just hinted at in the nineteenth century like computability and information theory, and in foundational issues with which this essay will be primarily concerned. One of the most notable foundational achievements has been the reduction of the corpus of mathematics to Zermelo-Fraenkel set theory (with the Axiom of Choice) and the proofs of the consistency of various branches of mathematics relative to it. It is also remarkable how much interesting mathematics has actually been produced in the pursuit of philosophical claims about the objects of mathematics and the nature of mathematical truth. Witness Frege’s Theorem (see below) on the one hand and the major results of intuitionistic analysis on the other, and how much philosophical insight has been gained by the interpretation of certain very deep theorems indeed of mathematics proper, the Gödel Incompleteness Theorems and the Paris-Harrington Theorem, to give but one generic example. The Paris-Harrington Theorem is especially interesting in that it provides a clear example of a statement of obvious combinatorial arithmetic content (the Modified Finite Ramsey Theorem) which is independent of the rst-order Peano Axioms for arithmetic. This is one of a number of results of clear arithmetic content used in everyday mathematics that have been shown to be independent of the Peano Axioms, thus adding to the purely meta-mathematical signicance of Gödel’s original theorem.