Cleaning Up the Mess

Minhyong Kim has an interesting post on how mathematical theories can develop messily, but then the mess gets cleaned up. Minhyong wonders what’s lost in the transition to neatness.

Class field theory is an example of a subject that has been polished to a high gloss, but its very neatness makes it hard to understand what it’s about, which is why I prefer expositions that talk about the history and start with inspirational examples. I suspect that a streamlined but mostly-historical introduction to class field theory would be incredibly interesting.

Algebraic geometry is another good example. Linear systems are basically an elementary idea (the Wikipedia page doesn’t get across how elementary), but in practice these get coded up in terms of ample line bundles, which seem a long way away from the original idea.