Calegari reframes solved problems as doorways, not endings
1 Nov 1 1:28 PM · 7d ago · 1 article · 2 posts · 8 comments · 3 sources · development 1 of 1
Calegari argues that solving a major open problem is not the end of a field but the beginning of exploration into new territory. He uses the periodic table's relationship to chemistry—its construction did not solve chemistry but opened new directions. He calls for mathematicians to better communicate this reality to counteract the feeling that AI has "solved" their fields.
“But even if it had solved every problem in the Langlands program, to imagine that would have 'solved number theory' would be as naive as saying that the construction of the periodic table solved chemistry.”
Frank CalegariFrank Calegari Mathematician, guest authorTerence Tao Mathematician, blog hostHugo Duminil-Copin MathematicianOpenAI AI research company
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Reported in the same hours no headline names this development itself — these 2 claims were published in its stretch
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first by What's new, 6d ago · also HN Best, HN Frontpage
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2 outlets Egoless Mathematics in the AI Era
first by HN Frontpage, 6d ago · also What's new
What people said 24 voices · best of 85 · verbatim
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The quote reads to me like “as long as you can analyze the solution after the machine solves it and make it suggest new ideas based on that, you’re almost the same as a discoverer yourself.” I’m deliberately avoiding “mathematician” here because I think Tao probably didn’t intend to take it that far. But, for many people that difference will be…
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There's a long collection of comments by mathematicians at [1]. A large number of workers in the field are rather discouraged. The main quality complaint is that the proofs are ugly, long, and incomprehensible. One mathematician comments that the first proofs of new major results are often like that. Elegance comes later. The main social complaint…
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That’s a great paper, I was tripping on these same thoughts last night. The thing that rang true was kind of getting back to these roots, the Philo-Sophia as love of knowledge, and a community that loves knowledge. It doesn’t matter if one guy has a 70 IQ and loves fables, or another is 170 and loves something I can’t comprehend, they are one…
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Watching Star Trek TNG as a kid, I vividly recall hearing Geordi or Data often say things like "run a level 3 diagnostic on the Worf Barrels" or whatever to the computer and wonder what the heck that would even be. I wondered where are all ship devops and programmers were hidden. Though I recall the one accurate part was Barclay in the Holodeck…
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> understands it thoroughly, and uses it to develop interesting further questions or directionsHe phrases it like these criteria are requirements, but they really aren't for every problem. They make sense for those theoretical problems whose solutions' main values lie in their understandability and further development, but not every problem is…
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Sure, mathematics will survive and even prosper with AI, but what about the mathematiciansIt's not a given that mathematics will survive without mathematicians. It's not even likely. The proofs OpenAI threw over the wall are unpublishable garbage, top to bottom. A big reason mathematicians are so upset is that they feel like they've been drafted…
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I know it's against the rules to be a curmudgeon, but this just reads like a lullaby for people preparing to drink the SF kool-aid. The opportunity cost of becoming an expert will remain, but the reward of pushing the frontier is mostly gone. Now, if you want to save universities as institutions(*), you'll have to offer people some other positive…
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The lens to use is the Venn diagram of: 1. What you like doing 2. What you are good at doing 3. What people will pay you to do For math, if your goal is to prove something that no one else has proven, then all three are under attack.I think the "egoless" path is taking a 50,000-foot view of mathematics that ignores the above lens. Sure…
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I guess the issue is -To advance math you need people who could be really high earners in any other field and stuck to math mainly because they find it intrinsically motivating.You change the calculus for motivation you potentially devastate a generation of talent who just decide to go to wallstreet or something since it seems there is no longer…
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> Mathematics isn't a way to keep a specific kind of intellectual happy by providing fun, meaningless puzzles to solve in isolation.Yes it is. At least in part.> It's the bedrock of all other science.Yes it is. At least in part.> The goal is to find solutions and to advance the fieldThat is only one of the goals - but it is not the goal.> we seem…
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My perception of mathematics when I was young was that it is a field of the absolute. It doesn't matter what anybody thinks, it matters what's true.To be fair math does have a great deal of truth, but it also has, surprising to me, a great deal of social cohesion. Some things are sufficiently true because the important people are pretty sure it's…
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Math envisions and communicates logic-based structures and objects that, as a side effect, may be relevant to certain real world problems decades to centuries after the initial research.I'd argue your statement is treating this discovery process as an engineering exercise: there exists an unexplored bounded space of logical axioms within which we…
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Yes, we should apply this same reasoning to AI and the companies that produce it, since they claim its the future. All that matters is truth, no valuations, no profits, not IPOs. All that matters is uncovering the truth and having the best models -- so the AI companies should just release everything for free, and not hope for notoriety as a…
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> It's the bedrock of all other science. The goal is to find solutions and to advance the field…That may be true but that may not be the feature that attracts people to the field.> As we solve lower level problems, we can unlock real world math solutions.That is an argument often made for funding mathematics research. But again it may not be the…
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> Human pursuits cannot be separated from the humans who pursuit it.But it's pretty clear you can chuck "human pursuits" in the garbage, in favor of what really matters: intensification of the market system and wealth accumulation for its own sake.Your technology leader thinks: Humanity? People? Who the fuck cares about that BS? Fuck 'em (or "I'll…
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Given the capabilities of OpenAI's unreleased model, I think the model helps mathematicians more than it hurts them. Some of the open problems that were solved were "It would be nice to know that", but many of the others represented ways in which we had been stuck for many years. There had been very little progress on analytic number theory…
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Why do nearly all the articles exploring the development of AI applied to math assume that capabilities won’t extend beyond theorem proving?> other skills will matter more: choosing which terrain is worth exploring, steering fleets of airplanes, and explaining what they find to the rest of the world.These all seem totally achievable by LLM agents…
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Mathematicians both build/formulate abstract conceptual frameworks and formulate problems within those frameworks. Solving the problems has historically been one of advanced the conceptual frameworks.Now, what has been useful for advancing physics and biology has been the abstract conceptual frameworks - using Hilbert Space for model quantum…
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The field before AI was build up so that humans can learn and understand and teach other humans how to advance these fields.AI is here 'suddenly'.Let them discuss this out, let them go through the emotions.And its unclear today IF we still need anyone doing anything in 5 years or not. In theory we could also just freeze 90% of things we do and put…
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I really struggle to understand this discussion. Mathematics isn't a way to keep a specific kind of intellectual happy by providing fun, meaningless puzzles to solve in isolation. It's the bedrock of all other science. The goal is to find solutions and to advance the field - but we seem to be treating it like some sort of occupational therapy for…
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I think that math can still be a creative endeavor, ultimately. In many cases is not so creative, and really is an exercise in searching the potentially solution space. However, math can also be about the creation (discovery?) of new language itself, which in turn enables us to ask questions that we couldn't previously ask in the first place.As a…
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As i understand it. The difference is the journey.By short circuiting the exploratory process via AI to find a solution. Some of the questions that would normally be asked during the struggle of said process will also be circumvented.Questions that would lead to other discoveries, ideas, and perhaps entire fields of study.At the end of the day AI…
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This is one of the best posts I've seen on this topic in a while.I finished a math PhD before I decided to get out of academia and into startups. Since then, the older I get the more of a Platonist I become, especially now that I've taught my daughters a bunch of math (I'm their homeschool math+science teacher). And the more of a Platonist I…
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> To me, someone who chooses a worthwhile problem, obtains a proof completely using AI, understands it thoroughly, and uses it to develop interesting further questions or directions has made a substantial mathematical contribution. I would give that person substantial credit even if they supplied no new ingredient in the original proof.Yup. This…
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