I'm a graduate math student specializing in differential geometry (more specifically, 4-manifold theory). I think that modern geometry is really complicated, as in, if you take a look at John Lee's Introduction to Smooth Manifolds, you'll find that it has a lot of little lemmas and theorems that say very little, with somewhat elaborate proofs, but whose content seem trivial, given some thought. I feel this is due to the inadequacy of basing the foundations of mathematics on set theory, leading to a very cumbersome framework for geometry. Do you sometimes feel that there should be a way to base off mathematics from topological/geometrical considerations in a way to make geometry (and possibly other fields) less cumbersome?
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I'm a graduate math student specializing in differential geometry (more specifically, 4-manifold theory). I think that modern geometry is really complicated, as in, if you take a look at John Lee's Introduction to Smooth Manifolds, you'll find that it has a lot of little lemmas and theorems that say very little, with somewhat elaborate proofs, but whose content seem trivial, given some thought. I feel this is due to the inadequacy of basing the foundations of mathematics on set theory, leading to a very cumbersome framework for geometry. Do you sometimes feel that there should be a way to base off mathematics from topological/geometrical considerations in a way to make geometry (and possibly other fields) less cumbersome?
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