What was Plato Thinking?

Last week I pointed to a few discussions of mathematics I found interesting and this is my first chance to follow up.  One of them took note of the surprising persistence of a platonic view of mathematical objects, a view that inevitably introduces into our scientific culture some version of a metaphysical idea.  Paul Bernays addresses it directly in a 1935 essay. He draws attention to the difference between the way Euclid describes a line and the way Hilbert did.

we notice that Euclid speaks of  figures to be constructed whereas, for Hilbert, system of points, straight lines, and planes exist from the outset. Euclid postulates: One can join two points by a straight line; Hilbert states the axiom: Given any two points, there exists a straight line on which both
are situated. “Exists” refers here to existence in the system of straight lines.

This example shows already that the tendency of which we are speaking consists in viewing the objects as cut off  from all links with the reflecting subject.

From the point of view of mathematics using totalities, of numbers, of functions, of sequences (in other words, all of a particular class of objects) Bernays makes clear,

constructive definitions of special functions, sequences, and sets are only ways to pick out an object which exists independently of, and prior to, the construction.

Bernays gives a number of examples in which the existence of a class of objects is accepted before a particular one is constructed.  Hence he says easily, “it is not exaggeration to say that Platonism reigns today in mathematics.”  But he also indicates that the paradoxes which arose from the discussion of totalities, removed the possibility that there is some other world where these objects exist, and requires that we consider only a restricted Platonism – “an ideal projection of a domain of thought.”

A well-known contemporary debate between the neurobiologist Jean-Pierre Changeux and the prominent mathematician Alain Connes is presented in the book Conversations on Mind, Matter, and Mathematics.  Here Connes defends the truly Platonic perspective and makes the following point:

Projective representations of mathematical objects are certainly physical brain states, but reducing mathematical reality to these states would be like reducing literature to the chemical reactions of ink and paper.

An interesting review of the book by Michael Atiyah can be found here.

Raphael Nunez, the cognitive scientist, would agree but for very different reasons.  For him, mathematics develops by virtue of inference-preserving conceptual metaphors.  In other words something in our experience is mapped to an abstract idea in such a way that the abstract idea preserves the implications of the original experience. (See – Numbers and Arithmetic: Neither Hardwired Nor Out There).

But the body is built to find generalities, create ideals.  It’s part of how the tissue of which we’re made responds to the world.  This is described in an earlier post: Zeki, The Brain and the Art of Abstraction.

It seems to me that what mathematics may be providing is an opportunity to correct our usual perspective.  An unwillingness to accept a truly platonic view of the world is partly due to our materialistic tendencies, but also to our habit to distinguish ourselves from all that we are part of.  By this I mean that Changeux’s and Nunez’s views make sense because we accept the idea that human thoughts are only human or purely human.  I do not find this self-evident at all.  Human imagination grows out of the world and so must be related to it in some way.  It seems that mathematics is powerful by virtue of the fact that it moves easily, back and forth, between its role as a language for science (or for the material) and the way it relies on non-material ideals.  In other words, it is comfortable with what the senses perceive and what they don’t.  It’s neurobiological source shows its relation to life at the cellular level.  Its role in revealing the inaccessible properties of quantum mechanical worlds reveals the reach of implications it may have inherited from our more common experiences.  Side by side, these things challenge the boundary between what is human and everything we believe is not, even between what is conscious and everything we believe is not. This is why, when mathematics is the subject of the discussion, a platonic view of things persists.