[HN Gopher] Reimagining mathematics in a world of reasoning mach...
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Reimagining mathematics in a world of reasoning machines [video]
Author : kentricon
Score : 22 points
Date : 2024-12-19 21:34 UTC (3 days ago)
(HTM) web link (www.youtube.com)
(TXT) w3m dump (www.youtube.com)
| revskill wrote:
| My wish is for math to remove all the gatekeeping layers and one
| day, everyone can do maths, just like with programming.
| godelski wrote:
| I'm curious, what do you see as "gatekeeping layers" in math?
| Books? Material? Pedagogy? Nomenclature? Something else?
| wslh wrote:
| 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.
| Al0neStar wrote:
| Not really gatekeeping, but as someone who likes to self-
| study, not having solutions is very annoying.
| godelski wrote:
| I totally get that. I was fortunate to get properly
| educated up through essentially the level of an
| undergraduate math degree (minus maybe typology), but then
| continued learning a lot on my own. It's common to hear
| that the struggle is part of the learning process. The more
| I've advanced the more I find this to be true. It's that
| struggle that makes you pay attention to the small details
| that are so critical. I was also fortunate to have a
| professor who would pester me and he later told me he
| wanted me to be confident in my results, because eventually
| I would have no one to double check (and he was right). The
| struggle really helps with this.
|
| I think the main difference between learning provisioning
| and math is a compiler. To learn either you can only learn
| by doing. Reading and lectures aren't enough. What is hard
| to learn in math is to be the compiler yourself. To be able
| to verify "programs" (do I even need quotes here?). This is
| a very powerful tool to add to your tool belt and one I
| think even helps in programming.
|
| I hope others can add advice here and words of
| encouragement. The struggle is real, but it is part of the
| process, for better or for worse.
| nicf wrote:
| 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 textbooks with solutions to the exercises would
| probably help a lot with this, especially if you used the
| solutions judiciously. I think the fact that this isn't
| more common sadly has a lot to do with their role in
| undergraduate teaching: every exercise that has a solution
| in the back of the book is one that college students can
| very easily cheat on. I definitely agree that it's
| frustrating that the product has to be made worse for
| everyone else just because some people would misuse the
| better version. Far from the only such case in the world!
| random3 wrote:
| Programming should remove all gatekeeping one day so everyone
| can do programming
| Nevermark wrote:
| And gatekeeping, so many of us want to gatekeep but there
| too, other gatekeepers stand in our way
| olddustytrail wrote:
| Yes, like the Fields Medal. Everyone should get a Fields Medal,
| that way we're all winners.
|
| Or the Riemann Hypothesis! Why not call it the American
| Hypothesis? That way everyone could think it.
| Tainnor wrote:
| Everyone can do maths. All you need is pencil and paper. Or
| download Lean, if that's your preference.
|
| Of course it's possible that instead of "no gatekeeping" what
| you mean is "free tutoring".
| wakawaka28 wrote:
| 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.
| m3kw9 wrote:
| Just waiting on what Terrance Tao thinks of the new o3 intern
| pegasus wrote:
| When can infer somewhat from this section in the FrontierMath
| paper [1]:
|
| "Assessment of FrontierMath difficulty
|
| All four mathematicians characterized the research problems in
| the FrontierMath benchmark as exceptionally challenging, noting
| that the most difficult questions require deep domain expertise
| and significant time investment. For example, referring to a
| selection of several questions from the dataset, Tao remarked,
| "These are extremely challenging. I think that in the near term
| basically the only way to solve them, short of having a real
| domain expert in the area, is by a combination of a semi-expert
| like a graduate student in a related field, maybe paired with
| some combination of a modern AI and lots of other algebra
| packages..."
|
| However, some mathematicians pointed out that the numerical
| format of the questions feels somewhat contrived. Borcherds, in
| particular, mentioned that the benchmark problems "aren't quite
| the same as coming up with original proofs."
|
| It sounds like it will be able to crack some hard math
| problems, but not actually _do mathematics_. Which makes sense
| to me, given how these oN models are being trained. Synthetic
| data is bound to be in some (more or less obvious) way
| contrieved, since it 's not like we can just generate tons of
| new/natural/original mathematical results to train the model
| on.
|
| [1] https://arxiv.org/pdf/2411.04872
| versteegen wrote:
| > 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 shouldn't o3 be able to produce novel
| mathematical proofs? It's got plenty of maths textbooks and
| papers to train on, the same thing mathematicians train on.
| pegasus wrote:
| 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 math book.
|
| Doesn't mean they won't be useful to mathematicians, they
| will, just like computers in general are useful in doing
| maths today.
| pegasus wrote:
| Look at the sample problems and it will be obvious. For
| example, look at all the arbitrary constants here:
|
| "Let an for n [?] Z be the sequence of integers satisfying
| the recurrence formula1 an = (1.981 x 1011)an-1 + (3.549 x
| 1011)an-2 - (4.277 x 1011)an-3 + (3.706 x 108 )an-4 with
| initial conditions ai = i for 0 <= i <= 3. Find the
| smallest prime p [?] 4 mod 7 for which the function Z - Z
| given by n 7- an can be extended to a continuous function
| on Zp."
| sleno wrote:
| nothing very interesting was said
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