The constructivist regards the natural numbers as the fundamental datum of mathematics, which neither requires nor is capable of reduction to a more basic notion, and from which all meaningful mathematics must be constructed.
The Platonist regards mathematical objects as already existing, once and for all, in some ideal and timeless (or tenseless) sense. We don’t create, we discover what’s already there, including infinites of a complexity yet to be conceived by mind of mathematician.
The formalist rejects both the restrictions of the constructivist and the theology of the Platonist. All that matters are inference rules by which he transforms one formula to another. Any meaning such formulas have is non-mathematical and beside the point.
