I remember the quotation that “God cannot do pure mathematics” from somewhere but unfortunately Google doesn’t, so I can’t provide a reference. The idea, of course, is that once the axioms of any system are written down, an omniscient intellect would immediately see all the consequences. The reason we need to do pure mathematics is that, with our limited intellects, we get the fun of solving the puzzles.
There is, of course, an underlying assumption that may rest on an incorrect idea of the way mathematics is done. We teach children in high-school science that a scientist uses what is known as the “scientific method”: the scientist creates a hypothesis, devises experiments to test that hypothesis, carries out the experiments and then accepts or rejects the hypothesis. Tell that to Kuhn and Popper! Similarly we teach (taught? I may be out of date here) children that pure mathematics is done by laying down axioms:
1. Things which are equal to the same thing are equal to each other.
2. If equals be added to equals, the wholes are equal.
down to the more worrisome
12. If a straight line meet two straight lines so as to make the interior angles on one side of it together less than two right angles, these straight lines will meet if continually produced on the side on which the angles which are together less than two right angles.
and then deducing things therefrom. This has always seemed a process as unlikely as that of the “scientific method” but it took section 7.5 of David Corfield’s Towards a Philosophy of Real Mathematics to give me a structure in which I could understand why.
I’ve been wondering about applying the scientific method to the question of whether pure mathematics is tautological. It is said (although not by me) that over the past centuries scientists have made progress while philosophers are still discussing, without much success, the same problems that Plato wrote about 2500 years ago. No scientist spends much time today worrying about the number of epicycles in the Ptolemaic theory of the solar system but many philosophers are still engaged in questions of free will, the reality of the material world, etc. So there must be something in that scientific method after all. Mustn’t there?
So, we need to devise a deterministic experiment to test whether mathematics is tautological: whether, for example, Pythagoras’ Theorem was discovered or invented. When we’ve carried out the experiment, we could devise another one to determine the question of free will. An inability to devise such experiments could be taken by some people to argue that the very question is invalid.