=math

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".