We need to pivot math towards intuition and accessibility.
Math has built a gated, inaccessible institution which - by design or not - served as a moat.
I believe math has little to do with mathematical notation - intuition is much more important. One can have intuition but not be able to “read math” - much as many musicians don’t “read music”.
Reading math however does not guarantee ideas or intuition. And those are what math needs.
I think LLMs are the single best things that were invented to make mathematics accessible.
I loved mathematics when I was young and wanted to be good at it, but I got burnt so many times by teachers who could not explain even if their life depended on it.
Asking an LLM and discovering how different theories belong together and what are genuinely open questions, along with different philosophical interpretations has been a wonderful gift.
I have long accepted that Mathematics is a language where you must go "all in" because for the most parts institutions are fundamentally unable to explain it in an intuitive way. There are people who "just get it" and are blind to the struggle of people who need a different approach in learning it. There's something about the field where a generational survivorship bias has created an environment where math is mostly taught to likeminded people, who then don't see why anybody could not see the difficulties learning it.
I know this is not true everywhere, but the only reason I could finish my studies was because there are some genuinely excellent online lectures on youtube, along with massive support from friends who have already been through the ordeal.
If I had had access to an LLM, I could have actually taught myself from scratch and be more playful in learning it. And no, I am not willing to learn lots of math "for fun", I can think of better "for fun" things that actually help my profession and cater to my actual interests and not just some academic ideal.
99%. Humans break into three categories: originators, amplifiers, mimics. The 99% are mimics, they won't use LLMs for anything other than direct applications to existing life tasks.
See: what they did with personal computers and smartphones. They have access to Fable and Astra right now, and they're not interested what-so-ever despite what these systems can now do. Instead they'll promptly go vaporize $50k on a giant pile of metal garbage (autos) that will rapidly depreciate. They have no practical use for Fable as is.
For the same reason people didn't rush off to their local public library to teach themselves JavaScript to acquire a better life for the cost of renting library books. They can't easily go against their restraints, their biological potential. To do so is extraordinarily taxing.
Originators generate most of the extreme wealth, and extreme outcomes. Mimics can be exceptionally successful however, most doctors are mimics for example, they won't ever originate anything.
Amplifiers are the teachers, they - ideally - shuttle what works to the mimics for reproduction across society. The mimics are largely just trying to keep up with the Joneses. They go out of their way to not risk anything most of the time. They're ideally following pre-set paths to biological success. It's why mass, rapid de-industrialization for one example is such a wipeout, the mimics don't adapt very quickly, they followed the path they were told would work and then they feel betrayed, they become like abandoned programs and don't know what to do next.
As a system you don't want too much of your population in the originator category: it's very expensive and highly prone to spectacular failure. You want the extreme majority of your biological entities following successful paths that have already been proven to work.
Humans aren't special, we're animals. Apply the patterns accordingly.
You are not wrong, but I was just saying that the possibility is there. And especially for people who are despairing over bad math education I see this as a game changer. Seeing equations used to make me angry and powerless for over a decade because I felt absolutely hopeless during my studies.
LLMs are decent at explaining up to undergrad mathematics but that is because they are parroting / plagiarizing the hundreds of human-written books that have been written for the express purpose of teaching undergrads. LLMs are actually fairly terrible at explaining higher mathematics; humans are much better.
Sure. But it was a bitch to get access to a good maths textbook- either you forked out a lot of money, got a library loan (we had to renew ours every 7 days), etc. With an LLM there's good breadth of knowledge so if can quickly hop across topics rather than follow the progression proscribed by the author. There's also no element of judgement/ego etc etc.
They are definitely doing more than parroting. When learning mathematics, it's not uncommon to have some little misunderstanding that just isn't covered by a written explanation because it's "obvious" unless you happen to have misunderstood it that particular way. When that happens, you could look up the topic in 5 books on the subject and not make progress (never mind that you have to search through them to get to roughly the right point). LLMs are genuinely good at challenges like this - they make mistakes, but they can still identify and correct your misunderstanding with a bit of back and forth.
A good human on the subject will do better, and I'm not claiming the LLM has "understanding" in any philosophical sense. But it's more useful than something that just regurgitates material.
> it's not uncommon to have some little misunderstanding that just isn't covered by a written explanation because it's "obvious" unless you happen to have misunderstood it that particular way... LLMs are genuinely good at challenges like this
For undergrad math, sure, and again, I think it's often because on some random corner of the internet someone else had a very similar misunderstanding. On the other hand, when I ask for clarifications on niche topics that have few examples on the internet LLMs (even Opus 5) often confidently cite irrelevant papers / results or simply hallucinate.
I think you're desperate to think that LLMs cannot make any sort of deduction by themselves: they must have just remembered someone else doing it. But I've definitely had conversations where they've had insights that can't be explained by that. Yes, I've had plenty of hallucinations too, but it's worth it, for me at least, overall.
Having said that, I wouldn't attempt to learn a whole topic with an LLM, that would be frustrating.
I am not desperate; I just have enough expertise and work in a niche-enough field to recognize that LLMs just are bullshit machines.
See <a href="https://claude.ai/share/6431ccd9-f4f8-4052-8996-b046d1ea1f76" rel="nofollow">https://claude.ai/share/6431ccd9-f4f8-4052-8996-b046d1ea1f76 for an example that I ran into just now.
We all run into hallucinations like in your example, but that doesn't mean that the AI cannot make connections between different areas of knowledge it has, and combine them usefully to something new that wasn't there before. Like Navier Stokes, for example. I've had it make plenty of such connections in my own research, giving me new results I didn't have before.
They don't parrot if you know what to ask for. You can name for examples, name your own thoughts and observations and these things will often quote something related or the fact that a famous mathematician has also made this observation some point in time which then lead to the study of something else.
I am not going to study a whole book just so I can find the answer to a couple things I might never need. I like to focus on a few areas, learn everything I need to understand it, know if there are any dependencies and see what is there to discover and then move on.
I know many people get joy from learning knowledge that is not directly applicable and might never be applicable. And I know that the academic spirit would be to learn these things. But my time is seriously finite and I am not using math in my life in any fashion whatsoever other than occasional curiosity.
strideashort · · focus · HN ↗
Math has built a gated, inaccessible institution which - by design or not - served as a moat.
I believe math has little to do with mathematical notation - intuition is much more important. One can have intuition but not be able to “read math” - much as many musicians don’t “read music”.
Reading math however does not guarantee ideas or intuition. And those are what math needs.
DarkNova6 · · focus · HN ↗
I loved mathematics when I was young and wanted to be good at it, but I got burnt so many times by teachers who could not explain even if their life depended on it.
Asking an LLM and discovering how different theories belong together and what are genuinely open questions, along with different philosophical interpretations has been a wonderful gift.
I have long accepted that Mathematics is a language where you must go "all in" because for the most parts institutions are fundamentally unable to explain it in an intuitive way. There are people who "just get it" and are blind to the struggle of people who need a different approach in learning it. There's something about the field where a generational survivorship bias has created an environment where math is mostly taught to likeminded people, who then don't see why anybody could not see the difficulties learning it.
I know this is not true everywhere, but the only reason I could finish my studies was because there are some genuinely excellent online lectures on youtube, along with massive support from friends who have already been through the ordeal.
If I had had access to an LLM, I could have actually taught myself from scratch and be more playful in learning it. And no, I am not willing to learn lots of math "for fun", I can think of better "for fun" things that actually help my profession and cater to my actual interests and not just some academic ideal.
SecretDreams · · focus · HN ↗
For every person with this mentality, how many people do you think there are that aren't learning from the LLM at all?
adventured · · focus · HN ↗
See: what they did with personal computers and smartphones. They have access to Fable and Astra right now, and they're not interested what-so-ever despite what these systems can now do. Instead they'll promptly go vaporize $50k on a giant pile of metal garbage (autos) that will rapidly depreciate. They have no practical use for Fable as is.
For the same reason people didn't rush off to their local public library to teach themselves JavaScript to acquire a better life for the cost of renting library books. They can't easily go against their restraints, their biological potential. To do so is extraordinarily taxing.
Originators generate most of the extreme wealth, and extreme outcomes. Mimics can be exceptionally successful however, most doctors are mimics for example, they won't ever originate anything.
Amplifiers are the teachers, they - ideally - shuttle what works to the mimics for reproduction across society. The mimics are largely just trying to keep up with the Joneses. They go out of their way to not risk anything most of the time. They're ideally following pre-set paths to biological success. It's why mass, rapid de-industrialization for one example is such a wipeout, the mimics don't adapt very quickly, they followed the path they were told would work and then they feel betrayed, they become like abandoned programs and don't know what to do next.
As a system you don't want too much of your population in the originator category: it's very expensive and highly prone to spectacular failure. You want the extreme majority of your biological entities following successful paths that have already been proven to work.
Humans aren't special, we're animals. Apply the patterns accordingly.
DarkNova6 · · focus · HN ↗
adventured · · focus · HN ↗
Nobody will hate AI more than the con-artists in academia pumping out fake papers by the zillions that are all about to be wiped out.
queenkjuul · · focus · HN ↗
tylerhou · · focus · HN ↗
alansaber · · focus · HN ↗
quietbritishjim · · focus · HN ↗
A good human on the subject will do better, and I'm not claiming the LLM has "understanding" in any philosophical sense. But it's more useful than something that just regurgitates material.
tylerhou · · focus · HN ↗
For undergrad math, sure, and again, I think it's often because on some random corner of the internet someone else had a very similar misunderstanding. On the other hand, when I ask for clarifications on niche topics that have few examples on the internet LLMs (even Opus 5) often confidently cite irrelevant papers / results or simply hallucinate.
quietbritishjim · · focus · HN ↗
Having said that, I wouldn't attempt to learn a whole topic with an LLM, that would be frustrating.
tylerhou · · focus · HN ↗
See <a href="https://claude.ai/share/6431ccd9-f4f8-4052-8996-b046d1ea1f76" rel="nofollow">https://claude.ai/share/6431ccd9-f4f8-4052-8996-b046d1ea1f76 for an example that I ran into just now.
auggierose · · focus · HN ↗
DarkNova6 · · focus · HN ↗
I am not going to study a whole book just so I can find the answer to a couple things I might never need. I like to focus on a few areas, learn everything I need to understand it, know if there are any dependencies and see what is there to discover and then move on.
I know many people get joy from learning knowledge that is not directly applicable and might never be applicable. And I know that the academic spirit would be to learn these things. But my time is seriously finite and I am not using math in my life in any fashion whatsoever other than occasional curiosity.