Comment by hackandthink
1 year ago
This is an excellent text to understand mathematical structuralism.
But it feels anticlimactic.
At the beginning there are fundamental problems with mathematical equality.
In the end, there is no great new idea but only the observation that in algebraic geometry some proofs have holes (two kinds), but they can be filled (quite) schematically.
"Some proofs" are the proof used to verify that the definition of an affine scheme makes sense, and the author motions in the direction that every single proof in algebraic geometry uses identically wrong arguments when using the language of "canonical" maps.
Thus the replication crisis of mathematics is revealed. In the words of John Conway: a map is canonical if you and your office neighbor write down the same map.
At least it's a short piece, compared to A New Kind of Science.