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.
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.
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.
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.
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.
Sounds like someone was cruel to you at one point. The parent likely means the opposite of what you're suggesting.
Most people consider mathematical intuition something that only develops from deep study. What the parent is referring to is being able to anticipate how a system will behave before you actually do the calculation.
Most people who use mathematics in an industry job rely on their intuition rather than rigorous proofs. And most industry mathematicians will acknowledge that they're not as rigorous, nor as academically gifted, as their friends who stayed in academia.
Just like a programmer with 20 years of experience can look at a bug and anticipate the cause.
This reads well enough as a platitude, but what do you actually mean?
How is mathematics gated and inaccessible? How do you specify mathematics without notation or some language? What intuition do you mean, as many mathematical ideas are non-intuitive?
You would be surprised at the number of people (who study math formally) who just memorize proofs and having no clue what it all fundamentally is about.
I am a mathematician, though I haven't been in academia for many years, and I would indeed be surprised if there were a lot of mathematicians who "just memorize proofs". Do you really mean that, and if so what's your evidence for it? (And: how do you suppose it is possible for someone who merely memorizes proofs without caring what they are about to prove new things in their field?)
For the majority of math history, people wrote proofs using natural language. Formalized notation is relatively recent. I have read both, and the notation is (on the whole) so much easier to read. But anybody is free to learn math beyond an undergraduate level with natural language texts. If you want to be traditional, start with Euclid's "Elements."
Plus, in the good articles I've read, the structure is usually: dissection of subproblem, intuitive description of solution, converting that solution into notation, proceed to next subproblem. If you see an unfamiliar symbol in an article, look it up. If you don't understand its definition, you don't yet have an intuitive grasp of the concept. Keep going down that rabbit hole until you have the intuitive building blocks to understand the article.
Frankly, I love how much math we can teach before a kid graduates high school. Thousands of years of mathematical ideas, up to calculus invented less than 400 years ago. All before they're 18 years old and probably don't care much about math.
>The power to understand and predict the quantities of the world should not be restricted to those with a freakish knack for manipulating abstract symbols.
With powerful tools of the type Bret envisions, humans, in aggregate, might have a chance of staying ahead of LLMs at even frontier math..
(But maybe that wouldn't fare so well against future world-models, who knows)
here might be something to point those tools at--- if one isn't fixated on "predictions", or "real world"
https://youtu.be/fec8qSBiM4k
> One can have intuition but not be able to “read math” - much as many musicians don’t “read music”.
Is it really that common for musicians to not be able to read music? I'm not a musician so I wouldn't know, but it seems far-fetched. Like the story that Einstein failed math. I likewise doubt that there are many people who would be good mathematicians but unable to learn the language.
Sure notation is arbitrary and not the actual "truth" of math, but we need some way to talk about it, so why not just learn the accepted standards?
There might be at least one awesome author out there who never learned to write or type, but they're probably extremely rare, because becoming great at something usually entails a lot of practice, and that's hard to get when you're not making the effort to learn the basics.
In any case, LLMs are probably a bad way to "pivot math towards intuition and accessibility" because they, as language models, are great at generating a bunch of jargon that is hard for somebody with no training to tell apart. See the recent story [0]. Here the author (perhaps) succeeded with his proof, but was it because of superior intuition? Or just throwing compute at the wall to see what sticks?
I've played music for twenty years and can't read sheet music to save my life. Pretty easy to learn the shapes and patterns on a keyboard or guitar without reading sheet music (easier on guitar) and i figure with drums like, while i might Google the chords of a song for guitar, I've never known any drummer that would learn a cover song via notation
Most musicians in history didn't know how to read music, yeah. Less so in the present day but just about every folk song has spread by the word of mouth, and even in pop music many genres have a strong oral tradition (jazz, blues, rock, you name it), they play by ear and not by sheet.
Well, that's not really what I mean. Obviously they had some concept of communicating music, somewhat standardized instruments, etc. Pythagoras didn't phrase his teachings using algebraic symbols either. I suppose what I mean is that
> Reading math however does not guarantee ideas or intuition.
is true, but the converse, having the mind for good mathematical ideas or intuition, almost surely means that you'll have very little trouble picking up mathematical notation (not just notation, but the reasoning and ability to write a proof, be it in natural language or some formal proof assistant language).
Like yeah, before the written language there were great storytellers who couldn't write. But now that humanity does have standardized languages, it's way more rare to come across such a person.
I've never met someone who believed this who was actually able to solve simple math or physics questions given to them. Attitudes like this always feel like cope to me.
> Math has built a gated, inaccessible institution which - by design or not - served as a moat.
This has always been my problem with math going all the way back to college. It was obvious to me that the concepts were far less hard than the combination of notation and esoteric jargon with liberal use of symbols and weird letters made them seem. Math felt (and still does feel) "encrypted."
The impression math gave off is of an arcane discipline that uses its arcane-ness as a gatekeeping tactic, intentionally or not, and makes itself intentionally hard for newcomers to learn without being hand-held by members of the guild.
Of course I can say the same about a lot of computing, and I'm old enough to remember efforts to make computing more approachable like easier to learn languages and GUIs being mocked and scoffed at by "real programmers."
I think this is a pretty typical human group behavior.
I think this underlies a lot of AI hate from these communities today. AI makes it easy for outsiders to bash their way into the field with the help of an LLM. Yes, this often results in low-effort "slop," but if used correctly it can also help people climb the learning curve really fast. I've had great luck having an LLM make me some passable "slop" and then explain it and go around with me as we fix and refine it, explaining each step, and as it does so it feels like we are learning together. It's very powerful and I, as the student, can control exactly how the teacher presents the material.
I was able to do this to finally start grasping the math behind LLMs themselves: attention layers, tensors, etc.
A lot of people have a powerful visceral probably instinctive reaction to large numbers of migrants entering their area. "Build the wall!" I suspect this is brain stem stuff going back to evolving under conditions of scarcity where migrants meant less food.
> the concepts were far less hard than the combination of notation and esoteric jargon with liberal use of symbols and weird letters made them seem
I really don't get the hate for mathematical notation here, do you have specific examples? Notation is a necessary tool to express complex ideas compactly so that others can understand them. It allows you to carry out technical proofs that confirm things that are "obvious". Then it uses previously introduced concepts to build new concepts on top of them. This keeps mathematics interconnected, there's value in explaining a concept in terms of already known things rather than just vibing it.
> makes itself intentionally hard
It doesn't make itself intentionally hard, it is just hard. You can simplify some expositions and make concepts easier to understand intuitively but it won't necessarily give you the skills to actually work with them in mathematical practice (which is the point of mathematics as a discipline).
Environment makes certain traits visible - eg noise and smell sensory issues, even social settings. Put me in a forest and i will be indistinguishable from a neuro-prevalent - no social pressure, no noise, no smells. Actually the neuro-prevalent might be suffering for the lack of those i suppose
Schizophrenia is known to have genetic factors that can increase its likelihood, but is also linked to environmental factors like cannabis use and stress. The group without the environmental factors has a lower incidence of schizophrenia, so it's generally believed that the onset of schizophrenia is triggered by them.
Male homosexuality is believed to have a small genetic component but is also linked to birth order. One theory is that in utero testosterone from older brothers can affect the sexuality of a younger brother.
I disagree that it makes no sense for environmental factors to lead to the onset of autism, on the basis that environmental factors can make someone gay or schizophrenic. That is to say, they can influence a personality and a mind in complicated ways that don't seem intuitive.
If you have 100k people who don't smoke cannabis and 100k who do, 1000 of the non-smokers and 1400 of the smokers will have schizophrenia.
How do you detect schizophrenia in the 400 people in the non-smoker category? The answer is that you can't. They don't have schizophrenia. This is what it means for something to have an environmental trigger.
I get that you're making a point about ableism and environment, but in order to do so you're confusing two things: the onset of autism (at a very early age or even before birth and triggered by who knows what), and the occurrence of any specific symptom of autism (at any age and triggered by things we generally understand).
Async await is a glorious fucking event loop which obscures the primitive, crude simplicity to something unrecognizable which most developers think does something it absolutely doesn't.
it could be sth along the lines of:
on(x=foo()){
//land here when x is computed
}
catch{
//sth got wrong with foo
}
Visual basic was superior to async/await crap. Not even joking.
bigstrat2003 can’t even figure out basic things. They can’t program well despite putting years of life into that effort. I'm not worried they will suddenly develop intelligence out of nowhere.
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.
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