Towards an epistemological classification of scholarly disciplines

• 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.

This classification suggests some further ideas. Firstly, mathematics is in some sense the easiest branch of scholarship: in fact, mathematics is precisely "that which is easy", for the appropriate definition of "easy". Secondly, philosophy is really, really hard. This accounts for the almost total lack of progress in philosophy in the last 2,500 years. Philosophers are still debating problems posed by Thales of Miletus, and defending (or attacking) positions advanced by Plato; pretty much all we've achieved is to clarify our statements of the problems¹. Is there a single statement whose truth would be agreed-on by all philosophers? I'd love to be corrected, but I don't think there is. Compare the progress achieved in philosophy to the progress achieved in mathematics over the same time period, or with the progress achieved in science in a mere 500 years, and you'll see what I mean. As far as I can see, all progress in philosophy has come by re-stating philosophical questions as scientific or mathematical ones. And this despite philosophy attracting some of the best and brightest minds of every generation. Hell, even the humanities people are arguing about different books now. Thirdly, mathematics is neither a science nor a branch of philosophy, though it has things in common with both.

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.