mathematical intuition




I was asked:

Why are "completed" mathematical proofs by good mathematicians usually correct even when there's a flaw in the proof?

If a proof could only be complete or wrong, then mathematicians wouldn't be able to progress towards a proof in an intentional way. If an incomplete proof wasn't probably right, it wouldn't be possible for mathematicians to create large proofs.

So the question is really, how do mathematicians do that? How can mathematics progress towards a goal?

Sometimes, it's useful to convert such a question into an imperative form. For example, "Why does anything exist?" doesn't go anywhere, but if you convert it into instructions:

1) Follow a causal chain backwards until you find an event with no cause.
2) What is the cause of that event?

then the problem becomes clear.

Instead of asking "Why are mathematicians correct even when proofs are flawed?" we could say:
"Design a hypothetical system that proves mathematical theorems in such a way that proofs are probably correct before completion."

Suppose we mainly use artificial neural networks for that. Neural networks represent items as vectors which can be interpolated between, which can be considered points in a high-dimensional space which is approximately Euclidean.

Mathematicians will sometimes describe concepts as being "close" to each other. Proofs have sometimes been compared to finding a connection between two nearby points that have a "wall" between them. Sometimes new techniques have been likened to finding a "door" in such a wall.

It's also common to use neural network ensembles, in which case there are multiple spaces in which an item is represented as a point. Mathematicians sometimes describe mathematical concepts as being "close in some ways, but far apart in other ways". So we could suppose that mathematicians represent concepts using an ensemble of multiple spaces.

If we're supposing all that, then how would mathematical concepts be organized within such a space?

Luckily, mathematicians have helpfully made a very large collection of names for mathematical concepts. Names like "symplectic manifold" and "smooth fiber bundle" - whatever those are - presumably represent distinct and noteworthy high-level concepts.

Development of mathematical intuition, then, would involve organization of such concepts in abstract mental spaces in useful ways. If a mathematician can make good new proofs, then they probably must have a good mathematical intuition to find routes to them, so their proofs are probably right even if there's a flaw somewhere, because a gap in the proof would only be a small "distance".


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