Platonism

Perhaps the most basic entities in mathematics are numbers. So exactly what are they, in particular are they synthetic, i.e. man-made, or objective -- existing in reality? If they are real, then where or what are they? In the Platonist view, named in honour of the Greek philosopher, all mathematical objects (like numbers, equilateral triangles, theorems, etc.) have an objective existence independent of human consciousness. But they exist not in our physical universe (after all, have you ever seen a number - not the symbol but the actual number?) but in a separate Platonist realm of mathematical entities, thus raising mathematics to the level of metaphysics. It follows that mathematics is a process of discovery not invention. Each question, provided it is phrased unambiguously, has a definite answer even if that answer is not known to us. So unsolved questions like the Goldbach conjecture¹ must be either true of false, it is just our human ignorance which stops us from knowing which.

It was Platonism that dominated right up to the twentieth century. This philosophy was a central pillar in a hierarchical view of science. It was argued that science dealt with absolute truth because each subject could be reduced to a more fundamental one (so biology could be reduced to chemistry which could be reduced to physics, etc.) and that this process of reduction rested on the rock solid foundations of pure mathematics and logic.

In the twentieth century Platonists have had to modify their ideas to some extent but still Carl Hempel argues that the basic formulas of mathematics and logic are `true a priori, which is to indicate that their truth is logically independent of, or logically prior to, any experiential evidence' [#!Hem83!#]. Or read [#!Pen90!#] which includes, among many fascinating insights, an orthodox exposition of the Platonist case.

Marxists might classify Platonism as a form of objective idealism : concepts and ideas (like numbers) are given an objective reality separate from human thought. But this objective truth in mathematics is not to be found in the nature of the physical world we inhabit, but in some other world of numbers and ideas. The question arises: if numbers and mathematics are located neither in our heads nor in the world then how do we come to have mathematical knowledge if its truth resides in a world that we can neither visit nor see? Some Platonists argue that humans have an `intuition' from which we can learn mathematics directly. But as Hilary Putnam put it

This appeal to mysterious faculties seems both unhelpful as epistemology and unpersuasive as science. What neural process, after all, could be described as the perception of a mathematical object? [#!Put83!#]