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Reimagining mathematics in a world of reasoning machines [video]

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Re: Reimagining mathematics in a world of reasoning machines [video]

#21

Post by godelski »

nicf wrote: post #18
Al0neStar wrote: post #10

Earlier quoted context omitted.

Not really gatekeeping, but as someone who likes to self-study, not having solutions is very annoying.

I work as a private tutor for proof-based math, and I have a lot of students who've spent some time self-studying before coming to me. The comment by godelski matches my experience: the biggest obstacle seems to be the fact that it's hard to learn how to check your own proofs if no one has ever taught you how. I see a lot of variation in how well people have managed to develop that skill on their own. Having more tex…

Yeah it’s a catch-22. I’m in grad school and occasionally I get to be instructor. When I am I focus far less on tests and more on homeworks and projects (I do ML so it’s well suited for that style). The homeworks are made to be “play around” and the project is to be very self driven (with plenty of help, but they are juniors or seniors so they be fairly self reliant) and to find passion.

The reason I do this is because grades matter so much to students that even if they care to learn material they are incentivized to cheat (and subsequently cheat themselves). I think a lot of academics still don’t get this and are resistant to change (it is a lot of work to create a class but not to much once you worked everything out).

I think this confidence thing is also something that needs to be learned in every subject. Even in CS the compiler, type checking, and even unit tests aren’t enough (though they are extremely useful).

I should also say, one unfortunate thing I find in academic teaching of coding is we often don’t look at code. There’s not enough time. But to me this feels like trying to grade a math proof by looking only at the first and last lines. I think this builds lots of bad habits and over confidence

Re: Reimagining mathematics in a world of reasoning machines [video]

#22

Post by versteegen »

Terence Tao wrote his comments on this talk here:

https://mathstodon.xyz/@tao/113601397729489852

> Akshay Venktesh gave a thoughtful and accessible talk recently entitled "(Re)imagining mathematics in a world of reasoning machines". I particularly liked his highlighting of Davis and Hersh's pithy definition of mathematics (which I had not been previously aware of) as "the study of mental objects with reproducible properties"; it is a nice way to abstract out the most essential features of the diverse universe of mathematical objects (numbers, shapes, functions, etc.) that mathematicians actually study.

> (My own personal definition of mathematics is more descriptive than prescriptive: mathematics is what mathematicians do, and mathematicians are the people who study mathmatics. This sounds like a tautologically circular definition, but I think of it more as describing the equations of motion of a stochastic dynamical system, in which breakthroughs in mathematics attract more attention by mathematicians, which in turn redefines the relative weighting of different fields of mathematics.)

Re: Reimagining mathematics in a world of reasoning machines [video]

#23

Post by versteegen »

pegasus wrote: post #15
versteegen wrote: post #13

Earlier quoted context omitted.

> It sounds like it will be able to crack some hard math problems, but not actually do mathematics. What, do you say that because format of the FrontierMath problems is a bit contrived? I think I must misunderstanding you; a benchmark can't rule out something it doesn't test. And saying solving these problems requiring "deep domain expertise" isn't real mathematics sure sounds like a No True Scotsman argument. Why sh…

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 going to print those crossed-out scrawls in a textbox either! But then the student learns from the chain-of-reasoning which arrived at the result. They learn what steps they should have followed one after the other so they can apply it effortlessly next time, what mathematical tool A on problem type B results in, and so forth. They distill some insights from it. Well LLMs can also learn many of these same things. They certainly don't learn as well or as much as humans (just look at their appalling data requirements), but my point is synthetic training data really does work. So expect training compute used on frontier models to keep scaling.

Re: Reimagining mathematics in a world of reasoning machines [video]

#24

Post by isotypic »

versteegen wrote: post #23
pegasus wrote: post #15

Earlier quoted context omitted.

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…

> 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, maybe even not-even-wrong style "proofs". And then how can training on this yield anything?

Re: Reimagining mathematics in a world of reasoning machines [video]

#25

Post by globalnode »

godelski wrote: post #3
revskill wrote: post #2

My wish is for math to remove all the gatekeeping layers and one day, everyone can do maths, just like with programming.

I'm curious, what do you see as "gatekeeping layers" in math? Books? Material? Pedagogy? Nomenclature? Something else?

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.

Re: Reimagining mathematics in a world of reasoning machines [video]

#26

Post by zxvkhkxvdvbdxz »

revskill wrote: post #2

My wish is for math to remove all the gatekeeping layers and one day, everyone can do maths, just like with programming.

> one day, everyone can do maths, just like with programming.

Everyone can absolutely not do programming with an AI.

What is true tho is that everyone who cannot write program code, also cannot evaluate if the code the AI produces is correct.

And we who do write alot of code are not all that impressed.

Re: Reimagining mathematics in a world of reasoning machines [video]

#27

Post by zxvkhkxvdvbdxz »

wakawaka28 wrote: post #8
revskill wrote: post #2

My wish is for math to remove all the gatekeeping layers and one day, everyone can do maths, just like with programming.

What gatekeeping? Anyone is free to do math to the extent of their abilities. Math is known for accepting contributions from other fields. You don't need special credentials in math to publish a book or paper about math. It helps but it is not essential. Some academic positions are very difficult to get without a proper degree in the subject but that applies to every field, and for good reason.

Not OP, but I read it like they meant gatekeeping == lack of skill, because OP seems to think that everybody knows programming now thanks to AI.

I hope I am misreading this.

Re: Reimagining mathematics in a world of reasoning machines [video]

#28

Post by wslh »

godelski wrote: post #20
wslh wrote: post #5

Earlier quoted context omitted.

Not OP, but I'd like to add the term “learning complexity reduction” which relates closely to “pedagogy”. Reflecting on my experience as a CS graduate with a strong interest in math (though not very advanced knowledge), I realize I would have benefited from a more intuitive, black-box approach when first engaging with complex topics, rather than diving straight into their intricacies.

I think this is kinda like the C++ vs Python as your first language question. C++ is better if you want to really learn programming but it’s harder to get to the magic of your first real program. Python gets people making programs sooner but it’s harder to build good habits and get that deep understanding. It depends what your goals are. There’s stuff like this for math but it’s kinda like trying to learn math from 3…

I'm not suggesting that you can fully learn math in this way. My point is that the approach you take can boost your confidence and motivate you to progress further. Of course, some people don’t need that kind of push or entry point.

I've seen other great examples of this from good professors. For instance, in probability and statistics courses, we would validate our results by combining theoretical concepts with simple Monte Carlo simulations to ensure they matched.

Re: Reimagining mathematics in a world of reasoning machines [video]

#29

Post by Tainnor »

globalnode wrote: post #25
godelski wrote: post #3

Earlier quoted context omitted.

I'm curious, what do you see as "gatekeeping layers" in math? Books? Material? Pedagogy? Nomenclature? Something else?

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.

Re: Reimagining mathematics in a world of reasoning machines [video]

#30

Post by godelski »

wslh wrote: post #28
godelski wrote: post #20

Earlier quoted context omitted.

I think this is kinda like the C++ vs Python as your first language question. C++ is better if you want to really learn programming but it’s harder to get to the magic of your first real program. Python gets people making programs sooner but it’s harder to build good habits and get that deep understanding. It depends what your goals are. There’s stuff like this for math but it’s kinda like trying to learn math from 3…

I'm not suggesting that you can fully learn math in this way. My point is that the approach you take can boost your confidence and motivate you to progress further. Of course, some people don’t need that kind of push or entry point. I've seen other great examples of this from good professors. For instance, in probability and statistics courses, we would validate our results by combining theoretical concepts with simp…

Sure, I don't disagree. But there's a lot of trash and not a lot of gold. It is hard to find the gold and to make the problem even worse, what's gold is trash to others. Worse than that, what may be trash to you at one point in time may be gold at another, and vise versa. It's unfortunate, but I think this is true in any learning process. There's lots of great programming articles on Medium, but you're more likely to learn garbage, and even more likely to not know you are.
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