does not mention it explicitly, but Platonism
is one of the primary streams1
thought. This is the notion that the geometric and algebraic forms
s 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
) 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.
The others are finite constructionism