- Mathematics is the study of statements which can be proved.
- Science is the study of statements which can't be proved, but can be falsified.
- The humanities are the study of statements which can neither be proved nor falsified, but whose credibility can be supported or undermined by advancing evidence.
- Philosophy is the study of statements which can neither be proved nor falsified, and for which evidence cannot be advanced.
We're left with a puzzle, though: empirically, mathematics is difficult, when it ought to be easy. I'd like to suggest several reasons for this. Firstly, mathematics is very old, and has been worked on in the past by beings of otherworldly intelligence: all the easy and accessible problems were solved long ago, mostly by Euler. The git. These days, even finding a sufficiently easy problem is challenging for us mere mortals. Secondly, much of mathematics is highly abstract, and humans are not evolved for highly abstract thought: the capacity to grasp concepts with high degrees of abstraction (which is not the same thing as intelligence) seems to be quite a rare one, and requires substantial training to be brought to a useful level. Thirdly, performing experiments in mathematics was largely impractical until the invention of the computer, and even today the technology for performing mathematical experiments is at an early stage of development. This means that until recently our experience was limited to those systems which can be worked out in the head or on paper.
1 This is, of course, a slight exaggeration. For instance, the alert reader will have noticed my implicit appeal in point 2 to Karl Popper's principle of falsifiability: Popper's theories have greater credibility and explanatory power than those of the logical positivists, and thus represent an advance in the philosophy of science. But I bet you could find a philosopher who disagreed with it without too much difficulty, probably just by walking into any philosophy department common room and declaring your support for the principle in a loud voice. Philosophers are an argumentative bunch. For comparison, try finding a mathematician who doesn't agree with Cauchy's residue theorem, or a physicist who doesn't agree that general relativity represents a good approximation to reality.