Mathematics is not the exploration of anything. At least not literally (it is misleading to say so). It is something we create, and not more than what our mathematical calculus has been expanded into through logical inference. Mathematics is more like a symbolical machinery, a collection of algorithms.Maths is an exploration of the space of all possible forms using various representations of localised substance to construct every kind of shape that can be imagined.
We use mathematics to explore things we can imagine, and more tangible things. By postulating certain properties of concepts we can draw conclusions based on our mathematical calculus. It may very well be so that this is the 'purpose' of mathematics - if you want to put it that way (and I'll agree with you) - but it is not mathematics.