On the Empirical Application of Mathematics A Comment

Abstract

Professor Körner claims to have discovered a gap between pure mathematics and empirical structures that has been hitherto overlooked by all the major schools in the philosophy of mathematics. This discovery is summed up in his statement that “contrary to the teachings of logicists, formalists and intuitionists, the structures are not isomorphic.” Empirical structures, so the argument runs, involve irreducible vagueness, ambiguities and border cases, and they do not necessarily constitute precisely defined classes. An empirically ordered aggregate may serve to represent a finite fragment of natural numbers in the usual way: Associate the number one with the subsequence consisting of the first member and its successor and so on. We are told that the basic arithmetical operations can be defined in such an empirical model but that, owing to the inexactness of the empirical in such an empirical model but that, owing to the inexactness of the empirical concepts, the resulting arithmetic “will be affected with certain imprecisions.”One is curious to see samples of this imprecise arithmetic.