
Platonism is one of the main theories in the philosophy of mathematics, and is one of the major explanations of what mathematics really is. The question it attempts to answer is whether mathematical truth has an independent existence do: mathematicians discover mathematical truths that are, in some sense, out there to be found, or do they invent or create them? The answer to this question will determine the very way in which a mathematician will look at his or her subject, and it is also part of the question of whether mathematics is a science or an art, or even possibly a game.
The Platonist position, based (as the name implies) on ideas in the works of Plato, is that mathematical truth is discovered. The idea is that mathematics consists of absolute truths, which were a priori (needing no other foundation, but being inescapable consequences of logical deduction). This position was reiterated by Kant in the 18th century, particularly with respect to Euclidean geometry. Unfortunately, the almost immediate discovery of nonEuclidean geometry put paid to any such idea, because it showed that the axioms on which Euclidean geometry was based are not all necessarily true. These discoveries, along with that of the set theory of Georg Cantor (1845  1918) led directly to the major schools in the philosophy of mathematics of the early 20th century, logicism, formalism and intuitionism.
The problem today is that all these schools have been to some extent discredited as methods by which mathematicians should work, and that many mathematicians have returned to a modified version of the Platonic viewpoint. The idea now is that once all the rules (or axioms) are fixed, then so is the truth or falsity of proposed theorems. There are theorems which are true in one system though false in another; for example, depending on the acceptance or rejection of the axiom of choice or the axioms of (say) Euclid or Nicholas Lobachevsky. A dependence on the rules of logic is also recognized. (For Platonism in philosophy, see universals and particulars.) SMcL
Further reading B. Russell, An Introduction to the Philosophy of Mathematics (1919). 
