Syntax and Semantics

The year is 2002 and here we are at a symposium on Foundations and the Ontological Quest. The first thing to say is how bleak the present situation is. In foundational studies of mathematics and physics we have been stuck for seventy years; despite numerous books, articles, and meetings, there has been no real progress. Seventy years ago Gödel established his incompleteness theorem [1], destroying the foundational program of David Hilbert [2]. Just a year or two later, Gödel showed [3] that Brouwer’s intuitionism [4], advanced as a correct constructive alternative to an allegedly incorrect classical mathematics, was actually an extension of classical mathematics rather than an alternative to it. In short order Gödel dashed the hopes of the formalists and the intuitionists, seemingly leaving no choice other than blind Platonic belief. I shall say more about this later. Just over seventy years ago, Heisenberg [5] and Schrödinger [6] created quantum mechanics. There have been innumerable attempts to develop a rational understanding of this eminently successful physical theory, but not one of them has met with general acceptance. The foundations of cosmology seem to change radically every few years. It is a field in which, more than in any other science, the problem of separating observational fact from theoretical assumption is refractory. I look forward to learning about foundational problems of cognitive science. I hope there is progress there. But in mathematics and physics there has been little or none in seventy long years.