Mathematical Platonism

It is a general truism that all forms of cognition are produced jointly by the world and the mind. Mathematics is a form of cognition, and thus no exception. The scientist does not circumvent this truism by claiming objectivity, nor does the mathematician by embracing mathematical Platonism. Yet, this does not prevent some mathematicians from holding that mathematics has a reality independent both of nature and of human minds. Mathematical Platonism is the belief that mathematical truths exist apart from mathematicians and their particular constructs in any given era. It is discovered as a pre-existing realm, not created by mere mortals. The totality of mathematical truth or logical possibility is a finished and timeless structure. It is not a question of empirical experience, but something transcendent, eternal, objective, fixed. While Gödel evidently believed in such a thing, ironically he showed that any constructed part of mathematics (of a certain complexity) cannot map the whole. A given mathematical construct is a product of definition. As such, it may be considered “timeless.” Yet someone constructs it at a particular time and place. The corpus of mathematics too has a history and is a work in progress. On the one hand, mathematics is a matter of formal definition and of logic, to which time seems irrelevant. Yet even logic is not fixed for all time if logic is a product of human evolution. Its very game-like nature, as a voluntarily adopted set of rules for manipulating a set of symbols, means it can be altered at will. That is, as a complete abstraction mathematics is an arbitrary convention. Its relation to physical reality, on the other hand, is a function of the evolution of human cognition. This relation too could change. To insist that it is fixed for all time and that mathematics has nothing to do with cognition is simply naïve.