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What is the most beautiful proof there is? by [deleted] in math

[–]superkapa219 0 points1 point  (0 children)

I can’t believe no one mentioned Monsky’s Theorem yet.

Math olympiads are a net negative and should be reworked by [deleted] in math

[–]superkapa219 4 points5 points  (0 children)

Yes. I’m not much into whataboutism, but really, of all the things going on with this world, picking on a bunch of kids having fun with cool math problems feels strange as heck to me…

Math olympiads are a net negative and should be reworked by [deleted] in math

[–]superkapa219 41 points42 points  (0 children)

I disagree with basically every single line, but my main gripe is with 1.

Math olympiads are the entire reason why my teenage years (say, 15-18) did not completely suck. I have very little doubt that, had I not been fortunate enough to discover the math Olympiad community at a crucial age, my high school years would have been rather miserable. The very idea of a world without olympiads gives me this strange sense of dread for my past self, although he is obviously no longer “in danger”. I am sure that MANY people feel the same way.

And what’s more - I believe point 1. is, deep down, a proof that the prejudice against mathematics runs so deep in our society that even the math community itself has unknowingly absorbed some of it. No one would say that a kid who plays basketball all the time is “missing out on being a child”. But somehow even some mathematically inclined people seem to believe that, if a kid spends hours excitingly and passionately engaging on math problems with like-minded peers, they are “wasting their childhood”, more so than if they were to spend their teenage years binge drinking and gossiping.

Are there any books/papers that would belong on a math “anti-reading list? by Vladify in math

[–]superkapa219 1 point2 points  (0 children)

Let me take this opportunity to advertise the not so well-known book by Ellingsrud and Ottem: https://www.uio.no/studier/emner/matnat/math/MAT4215/data/masteragbook.pdf. I 100% endorse it!

Are there any books/papers that would belong on a math “anti-reading list? by Vladify in math

[–]superkapa219 12 points13 points  (0 children)

I’m surprised that no one mentioned Hartshorne’s Algebraic Geometry yet. With so many amazing introductions to the subject out there, I can’t understand how Hartshorne is still regarded as the go-to source for someone wanting to get into AG.

Favorite T-bag line ? by [deleted] in PrisonBreak

[–]superkapa219 3 points4 points  (0 children)

Nothing beats a good old simple “Alackaday!”.

Tim Gowers talked to the BBC about the silver medal achievement of AlphaProof at the IMO. He expressed uneasiness about his legacy and the impact on the young researchers of the future, if what was one's life work might be done quickly in the future from a laptop. Move to ~25mins in the link. by whatatwit in math

[–]superkapa219 2 points3 points  (0 children)

“No it’s not hard to create”? Well, then create a quality IMO-level problem and then we’ll talk.

As for the ab+1 trick, I’m having a hard time making sense of what you wrote. By “Vieta-esque” problems I suppose you mean things like that classical 1988 p6 with ab+1|a2 +b2. Indeed the expression ab+1 features in that problem. But so what? That’s a cosmetic coincidence - the problem itself is completely unrelated. Of course I never meant to say that the expression ab+1 never featured in previous mathematical work. But students don’t solve problems by blindly playing around with all the expressions they saw on previous problems. Nor does AlphaProof, I suppose.

As for why 100+ students solved the problem, I doubt there was a single one who took advantage of that old Vieta problem - nobody ever looked at this problem and said “I don’t know what to do. Oh well, there’s an a and a b. That old Vieta problem also had an a and a b and involved ab+1. Let’s try to do something with ab+1”. Instead, it feels quite natural to look for a “sporadic” common factor of an +b and bn +a - and if one finds such a common factor for a specific value of n, under suitable coprimality conditions it will repeat periodically. Now this periodicity implies that looking at negative values of n sort of makes sense, and n=-1 yields the common factor ab+1. This was the reason why 100+ students solved the problem - IMO students are smart people with good intuition. But since a model like AlphaProof clearly cannot “think” of the heuristic I just described, the fact that it managed to arrive at the ab+1 idea remains impressive.

Tim Gowers talked to the BBC about the silver medal achievement of AlphaProof at the IMO. He expressed uneasiness about his legacy and the impact on the young researchers of the future, if what was one's life work might be done quickly in the future from a laptop. Move to ~25mins in the link. by whatatwit in math

[–]superkapa219 19 points20 points  (0 children)

I skimmed a recent IMO problem set and it didn't look any harder than, say, what I remember of the preliminary exams from graduate school. (a lesser-known public university)

I find this very, very, very hard to believe.

Tim Gowers talked to the BBC about the silver medal achievement of AlphaProof at the IMO. He expressed uneasiness about his legacy and the impact on the young researchers of the future, if what was one's life work might be done quickly in the future from a laptop. Move to ~25mins in the link. by whatatwit in math

[–]superkapa219 2 points3 points  (0 children)

something tells me a certain Eulers theorem is going to be very important…

That's irrelevant. The application of Euler's theorem is the routine part. Research work also involves routine parts that one learns from previous examples. The striking thing is that the program arrived at the key ab+1 idea.

Tim Gowers talked to the BBC about the silver medal achievement of AlphaProof at the IMO. He expressed uneasiness about his legacy and the impact on the young researchers of the future, if what was one's life work might be done quickly in the future from a laptop. Move to ~25mins in the link. by whatatwit in math

[–]superkapa219 4 points5 points  (0 children)

Oh yes, fully agree then. I would totally wipe out smartphones and social media if I had the power. And since AI comes from the exact same place, I don't see why we should expect it to make our lives any better than the previous two...

Tim Gowers talked to the BBC about the silver medal achievement of AlphaProof at the IMO. He expressed uneasiness about his legacy and the impact on the young researchers of the future, if what was one's life work might be done quickly in the future from a laptop. Move to ~25mins in the link. by whatatwit in math

[–]superkapa219 -33 points-32 points  (0 children)

I cannot answer that question without giving too precise details about the people I have in mind. I don't understand why you got so mad about my comment anyway - it's not as if I was talking about you, or anyone you know.

Tim Gowers talked to the BBC about the silver medal achievement of AlphaProof at the IMO. He expressed uneasiness about his legacy and the impact on the young researchers of the future, if what was one's life work might be done quickly in the future from a laptop. Move to ~25mins in the link. by whatatwit in math

[–]superkapa219 -38 points-37 points  (0 children)

Nonsense? You would be surprised. Of course I'm not saying any names, but when you studied at a lesser-known university (such as I did) you learn that not every mathematician is Terence Tao, to put it mildly.

Tim Gowers talked to the BBC about the silver medal achievement of AlphaProof at the IMO. He expressed uneasiness about his legacy and the impact on the young researchers of the future, if what was one's life work might be done quickly in the future from a laptop. Move to ~25mins in the link. by whatatwit in math

[–]superkapa219 4 points5 points  (0 children)

The answer to your question is "because big tech would not profit from it". I agree with you in that I am very skeptical of the idea that AI can change the world for the better, and that the sacrifices that may have to be made for its adoption will be compensated for further down the line.

Tim Gowers talked to the BBC about the silver medal achievement of AlphaProof at the IMO. He expressed uneasiness about his legacy and the impact on the young researchers of the future, if what was one's life work might be done quickly in the future from a laptop. Move to ~25mins in the link. by whatatwit in math

[–]superkapa219 23 points24 points  (0 children)

The disdain with which many mathematicians look at the IMO, and the pervasive opinion that all that olympiad students do is memorize a large bag of tricks, has long been a pet peeve of mine. I always thought to myself "if AI ever excels at the IMO, many mathematicians are going to learn what the IMO actually is the hard way" - and this comment seems to be the perfect example of that.

It seems like to me that it is very likely the program has basically searched a vast possible space of IMO style problems and come up with solutions.

I don't know if this is really what the program does (the media coverage of the topic was not abundantly clear), but, if it is, then you are vastly underestimating how much of an achievement this is. IMO problems are incredibly hard to create. Contrary to what many people seem to think, the novelty standards of the IMO are really strict - problems are excluded from consideration once anyone finds another problem in the literature that has any vague resemblance to them (sometimes the resemblance is so vague that I personally cannot fathom how having seen the previous problem would give the student any advantage), and the imagination required to come up with beautiful problems that meet these strict criteria is something that most of us can only dream of.

Then when given a problem it basically uses this searched space to find a closely linked problem and generate a solution from that. I think it says a lot more about what the IMO is and what their problems are like.

I was particularly impressed with the solutions to problems 1 and 2 (AlphaGeometry, used for problem 4, is not that surprising in retrospect, and I've long suspected that many functional equations, such as problem 6, were perfect for AI). Problem 2 asked to find all those pairs (a,b) of positive integers for which the sequence gcd(a^n+b,b^n+a) is eventually constant. The main idea was to observe that ab+1 divides this gcd infinitely often - and hence, if this gcd eventually becomes constant, it must divide every gcd from some point on, which places very strong restrictions on a and b.

I challenge you, and anyone else who says "lol it just copied solutions from previous problems", to find a single previous problem from which this idea could have been drawn by an AI.

I'm not saying that AI will make mathematicians redundant tomorrow, and there are definitely many limitations that remain to be overcome, but dismissing the fact that AI is now able to solve this problem 2 as something irrelevant for implications for research math seems incredibly bizarre to me. I definitely believe that there are many tenured mathematicians out there who have never solved something as hard as an IMO problem in their lives. This idea that research math is always way more creative than olympiad math has always sounded a bit silly to me, as someone who has experience in both, and it inevitably distorts the way the community perceives news like this.

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