In so far as theories of mathematics speak about reality, they are not certain, and in so far as they are certain, they do not speak about reality. - William Rowan Hamilton

Mathematicians have constructed a very large number of different systems of geometry, Euclidean or non-Euclidean, of one, two, three, or any number of dimensions. All these systems are of complete and equal validity. They embody the results of mathematicians' observations of their reality, a reality far more intense and far more rigid than the dubious and elusive reality of physics. The old-fashioned geometry of Euclid, the entertaining seven-point geometry of Veblen, the space-times of Minkowski and Einstein, are all absolutely and equally real. ...There may be three dimensions in this room and five next door. As a professional mathematician, I have no idea; I can only ask some competent physicist to instruct me in the facts.
The function of a mathematician, then, is simply to observe the facts about his own intricate system of reality, that astonishingly beautiful complex of logical relations which forms the subject-matter of his science, as if he were an explorer looking at a distant range of mountains, and to record the results of his observations in a series of maps, each of which is a branch of pure mathematics. ...Among them there perhaps none quite so fascinating, with quite the astonishing contrasts of sharp outline and shade, as that which constitutes the theory of numbers.

The Devil: Okay, boys...tonight's homework. Algebra. Xn + Yn = Zn. You're never gonna use that, are you? Imperialism and the First World War. What was done is done. No point thinking about it now. German, French, Spanish. Ja, ja, oui, oui, s, s. It's nonsense. Everyone speaks English anyway. And if they don't, they ought to. So, no homework tonight. But I want you to watch a lot of TV, don't neglect your video games...and I'll see you in the morning. Shall we say 10:00, 10:30? No point in getting up too early.

There is no mathematical substitute for philosophy.