Comment by vessenes
14 hours ago
That’s not untrue. But it’s also a misstatement of mathematical history. Many leading mathematicians historically have been highly competitive — Gauss comes to mind. Woe betide the lesser intellect that sent Gauss some ideas. The Newton Leibniz controversy was very serious business at the time in the UK and the continent. It was considered at the least a sin to reveal that sqrt(2) was irrational to those outside Pythagoras circle.
Mathematics has always been highly competitive.
The mathematical community was very competitive in its early years, but in the last 70 to 100 years, it has been generally less competitive and very collegial. The community was in a good place, and progress has been very good. In a few cases when competitiveness was ramped up, it lead to bad behaviour and destructive fights. Few would like to return to those competitive years.
The overarching geopolitics have always taken precedent over the preferences of the mathematical community as far as I can tell
Same is true for any area/topic though. Countries at war stop playing friendly football games against each other, as a very basic example.
Maybe it’s a dynamic equilibrium? We will become competitive for a while, then run out of questions, which in turn rewards pockets of collaboration?
The story of the cubic equations is another great example: https://en.wikipedia.org/wiki/Cubic_equation
Dudes straight up used to hoard solutions to equations and use them in math battles.
Right, the point is we're trying to avoid reverting back to such practices.
Showing my ignorance, but the only thing I can picture when I hear 'math battles' is akin to the 'street Countdown' scene from the IT Crowd
My computer contains the prime factorization of probably several dozen (if not more) large integers, and I refuse to share them with anyone!
(Because they are my private RSA keys)
Why are you still using RSA?
Where else are you going to keep your large primes?
It is easier to trust what you can understand.
Andrew Wiles was also careful about communicating progress on his Fermat's Theorem proof during the years in his attic. So yes I take the point.
I read the Mastodon thread as more about the 'flattening' and 'rawness' of the proofs these systems and their operators are producing. I mean what is the cultural significance of a lean proof that is half a million lines long or something? And what tools can be extracted for further work from such a construction?
The late William Thurston wrote about the culture of mathematics in that sense.
Also Andrew Wiles working in secret for 7 years out of fear of someone scooping him.
Partially; but also in order to be able to focus, as stated by himself in https://www.pbs.org/wgbh/nova/transcripts/2414proof.html:
"But I realized after a while that talking to people casually about Fermat was impossible, because it just generates too much interest, and you can't really focus yourself for years unless you have this kind of undivided concentration, which too many spectators would have destroyed."
But yes; him reaping the benefits of himself having the idea first was part of it too; as far as I am aware.
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Which is still something completely different than some anonymous organisation keeping mathematical research secret because it is better for hype reasons. One is competition between individuals or groups within a field; the other is boring and sometimes borderline nihilistic generating of mathematical knowledge as an marketing asset.
I've always found the story of A. Wiles sad and frustrating. He worked in secret for 7 years. He submitted a (incorrect) proof at year 4 or so. Reviewers found a problem, but he decided kept all secret for many years after. He didn't even proof the last theorem of Fermat directly, he proved some conjeture that someone else before him, proved that it implied Fermat last theorem...
I found this behavior against healthy science practices and only driven by ego. Unfortunately, I find this too often at work (working in academia). Most probably I'm too naive...
What you’re talking about is his proof of (a specialised version) of the Taniyama-Shimura-Weil conjecture[1] which had been proven to imply Fermat’s Last Theorem. The technique he used to prove this was adopted by his students to prove the conjecture in full generality so it now known as the modularity theorem. Given its importance to the Langlands programme it may be that when history looks back on this it will consider this a more important contribution than the fact that it proved FLT even though that is obviously the thing that grabs the headlines, but there’s nothing at all wrong with proving something that implies your goal rather than proving the goal directly. There’s a reason the words “it suffices to show” often turn up in proofs.
[1] https://mathworld.wolfram.com/Taniyama-ShimuraConjecture.htm...
> He didn't even proof the last theorem of Fermat directly, he proved some conjeture that someone else before him, proved that it implied Fermat last theorem...
That's how maths works yes...
Obviously, he was trying to avoid being labeled as a crank for working on a famous problem like that for so long.
While I can sympathize with this perspective, I don’t think it’s right to call it driven by “ego.” Sometimes one just wants to go at a problem without being second guessed on approaches or led astray with suggestions by others.
> He didn't even proof the last theorem of Fermat directly, he proved some conjeture that someone else before him, proved that it implied Fermat last theorem...
I think that was Ken Ribet?
Grigori Perelman and the Poincaré Conjecture is more interesting. IIRC he turned down Millennium and was decidedly not all about the Fields Medal - mostly because Richard Hamilton didn't get credit? Anyway, I am grateful I had the opportunity to learn about Poincaré in college taking a few classes from a professor who was a key contributor to the conjecture and got a Fulbright for it when I was there
Eh, it seems like it's pretty necessary for success on such a problem (but obviously not sufficient). These problems gain a reputation, and you either get judged for it or get too much attention for it.
perhaps a 2 sin 45?
Yeah but a highly productive last two decades of math research from https://en.wikipedia.org/wiki/Polymath_Project has come from collaboration.
You miss the point. Humans don't mind competing with others. I love competition, but I don't want to compete with you and your machine. I love to play chess, I don't care if you are grand master, whoop my ass. But not if you are going to pair up with stockfish. I don't even care if you are a newbie that started playing yesterday with an ELO rating of 900. If I wanted to play the damn computer I'll do it myself. Likewise, mathematicians will not mind sharing and competing with other fellows, but if another has a billion dollars worth of GPU and you don't? Then you best be carefully what you say.
Could you tell the difference between a grandmaster and stockfish if playing them online? If not, why would you care which one you are playing against?
Of course you can. Stockfish plays very different compared to a human, and it never ever blunders or makes mistakes.
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because we value competence
this is not untrue but it's a pendulum swinging back to ancient times man
Between people.
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Do you have a real argument to make rather than just appealing to authority?
It is a strong argument in this case though, because Terence Taos expertise is directly linked to his ability to not misstate the history of mathematics.
Also note how the quote by Tao is in all likelyhood not meant as an absolute; rather than a statement of a trend - a handfull of counterexamples do I no way change anything about the truth value of Tao's quote.
On the other heand; consider how absurd it would be if "... in the direction of no longer sharing any promising research directions with the broader community, which would reverse centuries of traditions of open science ..." would indeed be a misstatement; which would imply that far more promising research directions were not shared with the broader community (i.e.: published). I wonder what different reading of that counterfactual there could be other than secret societies that kept their discoveries and research directions to themselves - which we just learned about (since we would otherwise not be refering to the secret societies and their supposed promising research directions).
All pretty straightforward, I would say - both that "misstatement" is hopefully based an overly strict reading of Tao's quote, and that mentioning Tao's background as one of the fields leading practitioners is relevant as well. Again; to make sure: A few counterexamples achieves nothing here. It would need to reach a certain threshold of such counterexamples before we will have to write the history of mathematics; and before Tao actually made a misstatement here.
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And the other an appeal to tradition. There was only one Gauss to scoop and focusing on him misses the larger math culture which Terry might be aware of, where most trust others to not scoop.
And if any mathematician's AI usage on a problem leads to scooping, the volume of agents involved gives them a huge advantage which could prompt mathematicians to not use LLMs.
Though you can say Terry's claim is a slippery slope.
So let me get this straight: you're saying that Terrence Tao, one of the most prominent mathematicians alive today, doesn't know math history? And me pointing this out is merely an appeal to authority?
Get outta here.
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Surprised to see someone on HN arguing against open science. Seems like the opposite of the lessons we should learn from Newton and Gauss, actually, hoarding results for decades at the expense of progress.
(the Pythagorean thing isn't really competition either, is ahistorical, and from what we actually do know it's again people hoarding results instead of sharing them).
FWIW, your post comes off as a middlebrow dismissal, surface level and not actually engaging with the substance of the comment. It's also just wrong. You claim "it’s also a misstatement of mathematical history", but don't specify which part. That there's "centuries of traditions of open science"? But your examples are from centuries (and millennia) ago, and there was never any claim that these traditions are universal.
But more fundamentally, competition doesn't mean you can't also have open science. And the very long, damaging events like the Leibniz/Newton feud are exactly what make many mathematicians work to maintain a spirit of collaboration and attribution even when they're competing on approaches.
Nothing in their comment reads to me as "arguing against"
Reads like nothing but historical context
> Nothing in their comment reads to me as "arguing against"
If competition is somehow the opposite of "centuries of traditions of open science", and "mathematics has always been highly competitive", then open science is neither sufficient or necessary for the future of mathematics. Their clear implication is that we don't need to worry about it, though, because it's always been that way.
> Reads like nothing but historical context
They literally accuse Tao of "a misstatement of mathematical history".
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They aren't arguing against open science, they are trying to educate you on the history of science. It's always been this way.
Also, your third paragraph is highly ironic.
> They aren't arguing against open science, they are trying to educate you on the history of science. It's always been this way.
Always been what way? And how does that contrast to what Tao said (since it was apparently "a misstatement of mathematical history")?
> Also, your third paragraph is highly ironic.
You'll have to be more specific, since I engaged with my parent's argument, while they waved away Tao's quote by suggesting he was wrong because of exactly the kind of events that helped lead to the norms and mores working mathematicians have today.
Not arguing against open science - it's super valuable. I'm saying that pearl clutching by people reading Tao isn't useful, because it misses some long history which tells us that this kind of science has been seen as fundamentally competitive for millennia.
Should it be competitive? Is it more useful to be collaborative? How collaborative can it be when it's fundamentally competitive? Is it only fundamentally competitive because of some common 'quirks' of math types, or are there deeper forces pressuring it to be competitive?
These are all questions that I think are worth discussing, as is the note that the pendulum seems to be swinging away from cooperation in the face of competing for $trillion+ valuations (and a real enthusiasm for proving cool math stuff). The alternative, tweeting complaints on twitter without some context, is mostly a waste of space. I mentioned the history in hopes we could get informed complaints on twitter.