Two more platonisms – and their opponents

Platonism, as understood by our mathematicians of Chapter 6, holds that there is a mathematical reality, independent of the human mind, which human beings investigate, explore, discover, very much as we gradually learn more and more about the material world in which we live. Alain Connes, self-avowed Platonist, has a deep commitment to the existence of an archaic reality constituted by the series of numbers. The anti-Platonists, Alain Lichnerowicz and Tim Gowers, deny that this ‘reality’ makes sense. They do not think of themselves as exploring a reality and discovering the truth, but as discovering what can be proven. All three describe their views more as attitudes than as systematic doctrines. These attitudes express their visions of the mathematical life, of what they are doing, as mathematicians.

Now we turn to philosophical doctrines. They run parallel to the mathematicians, but arise from different interests. The platonistic side in the ensuing debates is best understood by what opposes it. I shall distinguish two different types of counter-platonism. The first, introduced in Part A below, is connected with intuitionist and constructivist tendencies in mathematics. It is what led Paul Bernays to introduce the word ‘platonism’ in connection with mathematics. The second, in Part B, arises against a background of denotational semantics, and is the primary focus of today’s platonism/nominalism debates among analytic philosophers.