[qarl2]

> However, it seems the proof is extremely concise so it seems that it is exploiting a clever trick that somehow all the experts missed.

Why is that a "however"? My reading is that it found a genuinely new solution that is both elegant and previously missed.

Seems like exactly the kind of result a human mathematician would aspire to.

[Garlef]

> a human mathematician would aspire to

Some do. But there's also the notion that a clever trick is a bad explanation.

[qarl2]

Hmmm... seems to me that if you can find a solution without creating the desired explanation - then that's a problem with the original question - not the solution itself.

And discovering a bad question leads to the correct question. No?

[Garlef]

> then that's a problem with the original question - not the solution itself

I think there's a good counterexample to this:

Atiyah/MacDonald proove the Nullstellensatz ultimately by using some trick involving determinants.

They give a very nice theoretical treatment of the content and context of the theorem. But the proof at one crucial point uses techniques that live conceptually outside of this context: While its possible to see that the argument is sound, it does not give a good explanation of _why_ it's true within the context of the theorem.

(You could of course argue that they did not give enough context ... but that's exactly my point: the trick makes the proof work but hides the explanation)

[wbl]

I was baffled enough by this comment to take my copy and look. The Nullstellensatz is an exercise late in the book long after Noetherian rings are introduced and they don't even do the Rabinowitz trick in the hints as they have enough theory to hit it the hard way. Determinants are nowhere to be found.

[Garlef]

It's been a few years (~ 15) since I read it.

The determinant trick I'm referring to is used in prop 2.4 (in the Version I found via Google).

They use the determinant of the adjugate matrix if I remember correctly... But they just hit the reader with this without any motivation or even naming it.

[wbl]

For Nakayama lemma (but they give a cleaner proof later). Note that the book was from the 1960's and from lectures at Oxford: matrix manipulations like that were a lot more commonplace.

[coldtea]

>(You could of course argue that they did not give enough context ... but that's exactly my point: the trick makes the proof work but hides the explanation)

Can't one see it in another way: that the trick illuminates a deeper explanation, connection the theorem's context and the stuff that's conceptually outside of that context. And that the problem is we don't know why the two domains (the context and the conceptually outside of it one) are related and cooperating in this way.

[qarl2]

Ah... is this just the difference between a constructive and non-constructive proof? Is that the distinction you're making?

[fnordpiglet]

But a proof isn’t an explanation it’s a proof. Proof by assuming the opposite is true and demonstrating a contradiction is very indirect and not at all directly explanatory yet it’s a proof non the less. The goal of proofs is to demonstrate something to be provably true, not expository knowledge gathering.

In fact most mathematicians (myself included!) think the more clever the trick the better the proof! The trick itself being clever is interesting because it often yields a new way of tackling or thinking about your own proofs. A bland explanatory proof that elicits some conceptually “why” is only preferable if it has a reason for doing so - does understanding why yield a new avenue of research? Often then the “why” is quite a clever trick too.

I think it’s a bit the opposite of programming. There you want your solutions to demonstrably not be clever and the code be its own documentation. It’s a different discipline.

[qarl2]

Thank you for the validation. I only dabble in mathematics - so it's reassuring to hear what I remember - a clever proof is still considered a positive thing.

I do (of course) understand the issue that non-constructive proofs can be frustrating - and a constructive proof of the same theorem is valuable tool in seeing exactly why a truth is true.

But I guess I've never seen a non-constructive proof (in a space where no constructive proof is available) considered a failure.

[calf]

Is this what the prompt means by proof strength gap, or is that something else entirely? (Sorry not a mathematician.)

[ak_111]

clever tricks has value for sure. But the main way progress is done in mathematics is by building new theory, the proof of Fermat's Last Theorem is much more important because of the math it created to solve the problem, rather than actually solving the problem.

[qarl2]

Right. I think I understand - this question was expected to produce a new theory and the clever solution avoided that.

Like I said below, I think this is a fantastic result. It discovered that this question really wasn't asking the right question. That's a determination that has eluded the humans examining the problem - and a real step forward - albeit not the hoped-for step.

No?

[pfdietz]

That depends on the branch of math. Combinatorics is trick-oriented. Maybe this is why AI has done well in that.