31–40 of 43 posts
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I'm curious, what do you see as "gatekeeping layers" in math? Books? Material? Pedagogy? Nomenclature? Something else?
Not really gatekeeping, but as someone who likes to self-study, not having solutions is very annoying.
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Think about what happens when a student solves some math exercises using some maths they don't fully understand, or some example they contrived to test what they read. They might make some false starts, work through some steps, make reference to other ideas that they do understand, and finally find a path to the solution. This is actually incredibly analogous to LLM-generated synthetic training data. And nobody's goi…
> and finally find a path to the solution. But how does the student, or in your case the LLM, know that it actually has the solution? For students, this is done by: a grader grading the homework, asking the professor at OH, working on problems with other peers who crosscheck as you go. I see no reason why this LLM produced synthetic data, without this correction factor, would not devolve into a mess of incorrect, may…
(In principle, it should be also be possible to get good enough at philosophy to avoid devolving into a mess of incoherence while reasoning about concepts like "knowledge", "consciousness", and "morality". I suspect some humans have achieved that, but it seems rather difficult to tell...)
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the only potential gatekeeping ive noticed in mathematics is (some) teachers that arent really teaching but more just showing off what they know. its a waste of everyone's time. i say potential because im sure its not always the teachers' fault, its likely just me being dumb.
There are definitely some people that either don't understand that their explanations are completely confusing to beginners or that don't even care. Sadly, teaching isn't always valued in academia.
Just waiting on what Terrance Tao thinks of the new o3 intern
What do you think of it?
Just waiting on what Terrance Tao thinks of the new o3 intern
Why does it matter what Terrance Tao thinks? Genuinely I find this obsession with what some specific math researcher thinks about x topic bizarre. What do you think of it?
Why wouldn't someone wish to know what he thought of this new advance?
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Those proofs are not by far enough - which is why the raw frontier models don't do that well on this test. The oN models are trained on synthetic data: so generated mathematical theorems and associated proofs. If we could generate legitimately new math we would already have accomplished the hard task, so the generated theorems are going to feel contrieved. I.e. they won't be theorems that ever make it into a future m…
Think about what happens when a student solves some math exercises using some maths they don't fully understand, or some example they contrived to test what they read. They might make some false starts, work through some steps, make reference to other ideas that they do understand, and finally find a path to the solution. This is actually incredibly analogous to LLM-generated synthetic training data. And nobody's goi…
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Not really gatekeeping, but as someone who likes to self-study, not having solutions is very annoying.
Do you have any favourite books? like maybe 2 or 3 that you'd grab in the event of ww3 or a zombie apocalypse? I'm always on the look out for good self-learning maths books.
• Real Analysis: A Long-Form Mathematics Textbook
• Proofs: A Long-Form Mathematics Textbook
• Mathematical Logic Through Python
• A Course in Calculus and Real Analysis
• A Course in Multivariable Calculus and Analysis
• Differential Equations With Applications and Historical Notes
• Probability and Random Processes (Grimmet)
• Probability Through Problems
• Fifty Challenging Problems in Probability with Solutions
• The Simple and Infinite Joy of Mathematical Statistics
• An Introduction to Error Analysis
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Do you have any favourite books? like maybe 2 or 3 that you'd grab in the event of ww3 or a zombie apocalypse? I'm always on the look out for good self-learning maths books.
These books are what i would consider the best books for self-study. (I have gone through many textbooks after i went back to school to study statistics.) • Real Analysis: A Long-Form Mathematics Textbook • Proofs: A Long-Form Mathematics Textbook • Mathematical Logic Through Python • A Course in Calculus and Real Analysis • A Course in Multivariable Calculus and Analysis • Differential Equations With Applications an…
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> and finally find a path to the solution. But how does the student, or in your case the LLM, know that it actually has the solution? For students, this is done by: a grader grading the homework, asking the professor at OH, working on problems with other peers who crosscheck as you go. I see no reason why this LLM produced synthetic data, without this correction factor, would not devolve into a mess of incorrect, may…
When you get good enough at mathematics, you can tell if your proofs are correct or not without asking a TA to grade them for you. Most mathematicians reach this level before they finish undergrad (a rare few reach it before they finish high school). While AI hasn't reached this level yet, there is no fundamental barrier stopping it from happening - and for now, researchers can use formal proof-checking software like…
This is simply not true - you can get a very good sense of when your argument is correct, yes. But having graded for (graduate, even!) courses, even advanced students make mistakes. It's not limited to students, either; tons of textbooks have significant errata, and its not as if no retraction in math has ever been issued.
These get corrected by talking with other people - if you have an LLM spew out this synthetic chain-of-reasoning data, you probably get at least some wrong proofs, and if you blindly try to scale with this I would expect it to collapse.
Even tying into a proof-checker seems non-trivial to me. If you work purely in the proof-checker, you never say anything wrong - but the presentations in proof checking language is very different from textual ones, so I would anticipate issues of the LLM leveraging knowledge from, say, textbooks in its proofs. You might also run into issues of the AI playing a game against the compiler rather than building understanding (you see elements of this in the proofs produced by AlphaProof). And if you start mixing natural language and proof checkers, you've just kicked the verification can up the road a bit, since you need some way of ensuring the natural language actually matches the statements being shown by the proof checker.
I don't think these are insurmountable challenges, but I also don't think its as simple as the "generate synthetic data and scale harder" approach the parent comment thinks. Perhaps I'm wrong - time will tell.
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When you get good enough at mathematics, you can tell if your proofs are correct or not without asking a TA to grade them for you. Most mathematicians reach this level before they finish undergrad (a rare few reach it before they finish high school). While AI hasn't reached this level yet, there is no fundamental barrier stopping it from happening - and for now, researchers can use formal proof-checking software like…
> When you get good enough at mathematics, you can tell if your proofs are correct or not without asking a TA to grade them for you. This is simply not true - you can get a very good sense of when your argument is correct, yes. But having graded for (graduate, even!) courses, even advanced students make mistakes. It's not limited to students, either; tons of textbooks have significant errata, and its not as if no ret…
If the error rate is low enough - and by simply spending a constant factor more time finding and correcting errors, you can get it below one in a million - then you do get a virtuous feedback loop even without tying in a proof-checker. That's how humans have progressed, after all. While you are right to say that the proof-checker approach certainly is not trivial, it is currently much easier than you would expect - modern LLMs are surprisingly good at converting math written in English directly to math formalized in Lean.
I do think that LLMs will struggle to learn to catch their mistakes for a while. This is mostly because the art of catching mistakes on your own is not taught well (often it is not taught at all), and the data sets that modern LLMs train on probably contain very, very few examples of people applying this art.
A tangent: how do human mathematicians reliably manage to catch mistakes in proofs? Going line-by-line through a proof and checking that each line logically follows from the previous lines is what many people believe we do, but it is actually a method of last resort - something we only do if we are completely lost and have given up on concretely understanding what is going on. What we really do is build up a collection of concrete examples and counterexamples within a given subject, and watch how the steps of the proof play out in each of these test cases. This is why humans tend to become much less reliable at catching mistakes when they leave their field of expertise - they haven't yet built up the necessary library of examples to allow them to properly interact with the proofs, and must resort to reading line by line.