Webster 1913 does not mention it explicitly, but
Platonism is one of the primary streams
1 of
mathematical thought. This is the notion that the geometric and
algebraic forms that
mathematicians like to play around with exist outside of human thought.
This would seem like common sense to most people:
Physics obeys mathematical laws. (Should I add a litany of examples?)
Even the fact that objects in the physical world can only approximate these forms only add weight to the argument, as object after object approximates the same form time and time again, with different patterns of error.
But there are real objections to certain bits of mathematical Platonism.
The flimsiest criticism is that Platonism conjures notions of the ideal, say,
chair, the very paragon of
chairness. This is pretty
silly.
Less silly is the argument that if these ideal forms have to exist somewhere, and if not the real Universe (we don't even have a
Euclidean space to deal with!), then where? (Plato had his
true world, but that does have an ideal chair in it). Responses of "the human mind" only make the finite constructionist's argument for him.
The shakiest part of
Platonism seems to lie in some authors' presentation of its response to
Godel's Incompleteness Theorem: The Platonist would argue, so they say, "Ok, such-and-such a statement of arithmetic is
undecidable, but that doesn't mean it isn't
true or false!" The strict Platonist would argue that the
Axiom of Choice and the
Continuum Hypothesis are "true" independently of the rest of
set theory.
The (post-
Hilbert) Formalist would argue the creation of two new finite systems, one with the undecidable proposition as a new
axiom, and one with its negation as an axiom. This seems to me to be more in the spirit of Platonism than the preceding view.
1The others are
finite constructionism and
formalism.