> Mathematical Platonism is modern form is no more than a belief that > the natural world is governed by "laws", which are discovered by human beings but exist independently of them and can be expressed in > mathematical form.
The natural world is, of course, the domain of science, not mathematics. The imagined world of Platonic mathematics is most definitely *not* the natural world, as it is inaccessible to the methods of science. But mathematics as a product of human thought is most definitely accessible to cognitive science.
> Like all metaphysics worth its salt, this belief can neither be > validated nor refuted. Anybody who thinks that it can be > "comprehensively demolished" is either using rhetorics more fitting to a political than a philosophical dispute or else should catch up on > his Hume.
A hypothesis that won't stand up to test deserves little respect. (my Bayesian colleagues can even argue this mathematically). But mathematics is a human practice, occurring in the real world, accessible to experiment. Thus, to a scientist, you are in fact demolishing your view by insisting that it cannot be demolished.
> Philosophically I am close to Quine, and so I > believe that ontologically there is no fundamental difference between > the objects studied by mathematicians, such as groups or sets, and the ones studied by physicists such as atoms or electrons.
There are absolutely fundamental differences. Physical objects are accessible to experiment. The properties of groups result entirely from the definition of "group". But no amount of reasoning can tell you much about the properties of atoms given only the definition of "atom".
> They are all human posits which we use to "explain" the sense data > which arise from some independent reality. But as the the actual > nature of this reality we can only speculate and in doing so we can > rely on nothing more then our aesthetic judgement.
No. I agree completely that aesthetic judgement directs mathematics, but science is directed by evidence. We often see that aesthetic judgement undisciplined by real world evidence is wrong.
> So what > exactly is the evolutionary path from a near "laboratory animal" to > Riemann or Perelman?
I think it's similar to the evolutionary path from laboratory animal to elite athelete. Sports like ice skating are not really like anything humans evolved to do, but involve physical and cognitive "modules" evolved for other purposes, combined in novel ways. Scientific understanding of this has made it possible to teach atheletes to perform feats once thought impossible, like "quadruple jumps". I see no reason this shouldn't also apply to mathematics.