Mathematical Empiricism and the Mathematization of Chance: Comment on Gillies and Schneider

Abstract

Professor Gillies is a mathematical empiricist. According to him, natural systems instantiate mathematical concepts or properties; consequently, mathematical knowledge is generated and justified in a manner not essentially different from empirical knowledge (“decimal arithmetic is a very well-confirmed theory”). Those branches of mathematics that (when conjoined with suitable auxiliary hypotheses) have implications for the physical world are ultimately empirical. Though they are not falsifiable since the results of a negative test can be blamed on failure of an auxiliary hypothesis they are confirmable (and often in practice confirmed) by the empirical predictions that they support. Much of arithmetic, geometry, analysis and set theory is, for Gillies, empirical in this way. Gillies further holds that this style of mathematical empiricism has implications for the history of mathematics: “the growth of mathematics should exhibit the same patterns of development as any other theoretical branch of natural science.” To the extent that we find supporting case histories, Gillies claims, his view will be confirmed; mutatis mutandis for refuting counterexamples.