Hauling the pyramid blocks is what gives people intuition and understanding. It is true that school systems usually have way too much computation -- it is easier to test and grade computation. But the only way that you were able to use computer algebra systems fluently is because you had internalized how algebra worked by hand. If we tell students that they no longer need to learn how to solve equations we are seriously depriving them of a mathematical education.

You can use AI like a telescope, and alleviate your inherent organizational problems or other ailments, which in my opinion is a great use case, as it is empowering. But there will also be plenty of people who are not looking to alleviate any mental/physical quirks they have to do more, but simply want to make a "quick buck" for the least possible effort from the work and thinking of others, without adding much value of their own. Lowering the bar makes things easier for both, the ones who bring value and the ones who merely exploit in some context.

> this is insane. the "speciesist" people aside (the best way for all other species to flourish is for humans to eradicate themselves right now, which is completely mad), i have the opposite problem: the _axiom_ seems to be: we should help ME flourish. "Me" as in people who are raking in trillions for their own very special selves right now, at the cost of everyone else's future, while none of them can be trusted to hold my cell phone for a second.

I don't think there are actually any "speciesist people." I think most, if not all, are the selfish "me people," who have disingenuously created "speciesism" by mad-libbing racism and sexism in a ham-handed attempt deflect criticism/opposition to whatever they want to do. Most of them aren't even "raking in trillions," they just don't like thinking about others or being told "no."

That wouldn't do much for rice, pigs, chickens, or rats.

Meanwhile, as people got used to the clock, the society became used to staying in synch at hour and sub-hour intervals. As better clocks were invented, synchronization became counted in minutes. This simplified things, allowing people to do more in the day and coordinate less, as they learned to rely on agreements to independently do things at specific time. Shops started posting opening hours, school started posting lesson plans, professionals started billing by minutes.

Clocks became widespread, and synchronized remotely - first from the church clock, then over the wire, over the radio, finally over the Internet. The loss of shared time reference at minute-granularity became rarer and rarer, until it became unheard of (except in excuses of those who failed to keep their commitments).

Losing a clock now would indeed throw society into disarray. If all clocks magically stopped tomorrow, many would starve and die before society figured out some way to restore time-keeping by proxy.

Clocks are a perfect example of the ratched effect of technology. A new technology at first helps you, or gives you advantage. As it spreads out more, it becomes relied on. Eventually, reaching ubiquity, it becomes a dependency, and everyone is assumed to be using it.

In a modern society, you cannot live without a properly synced clock anymore.

One counter-point: Humans have not figured out how general anesthesia works, but we use it every day to great effect.

https://en.wikipedia.org/wiki/Theories_of_general_anaestheti...

Why though a human? Mathematical proofs are very ivory-towery, but if OpenAI would solve a subkind of Cancer, without mortal humans understanding, cancer is still be healed.

I'm a little more flexible, if the new knowledge (that human's don't understand) can be put into a mechanism and have an observable effect, I'd be happy enough. e.g. a new type of rocket fuel that burns 1000x more efficiently.

If I vibecode a video game and manage to sell it on Steam, the information definitely mattered even though I didn't understand any of it.

There are drugs that nobody truly understands how they work and they are being used by professionals in actual treatments literally right now. We know purely statistical facts like "if drug X is used for condition Y it will help Z% of patients" and can only speculate as to their mechanism of action. They are used anyway and still benefit a lot of people.

"I own nothing, have no privacy, [never have to think, and am not required to solve any problems,] and life has never been better."

https://en.wikipedia.org/wiki/You'll_own_nothing_and_be_happ...

Does it need to be a human or can it be some other form of life?

I guess Terence's main point has been all the time that if we let AI solve all these existing problems, we don't notice the new ones and then there is stagnation.

The universe is bounded by rules, as far as we can tell, and not a lot of them

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.

But please elaborate a bit on your statement.

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.

(Attn: pls read more than the title before you downvote)

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?

[0] https://news.ycombinator.com/item?id=49755024

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.

But within the sample of the human species it is more common for less intelligent people to control more intelligent people.

This happens within political systems (political leaders are usually above average intelligence, but hardly the most intelligent) and within companies and other economic systems.

I don't see this fairly obvious point discussed, and not really sure what it shows.

I think there are other species that could be considered more capable than humans, such as E. Coli, octopuses or ants.

And it's not even clear whether the AI will have its own individuality. It might become an extension of human brains, in the same way neocortex is an extension of amygdala. In that case the statement of who has control might become meaningless.

For guest post on Tao's blog, after a decent start this fell short of expectations rather rapidly.

Because the question that belies all of this is whether still makes financial or economic sense in the age of LLMs for tax-payers to fund mathematics research.

Up until recently, they've had a true ivory-tower to dismiss such criticism, but as LLMs have demonstrated, that approach won't survive going forward, so pure mathematics are up in arms. They genuinely don't know how to justify their jobs.

Let's be honest: we don't know. Maybe you're right, but for the moment it's more likely that you're not. And countries cannot bet on that vague intuition at the cost of destroying their research communities and world leadership (which takes decades if not a century to achieve).

A better analogy: look at all these highly trained engineers, mathematicians, doctors, writers, philosophers, writers, scientists.

How many dumbass politicians do we need to keep it all running smoothly?

Turns out no matter how dumb politicians were, overall society has been developing positively over the history of mankind.

Do you think this applies to say, surgery, as well? There are few useful problems that share these properties.

We're building tools to serve the human society.

4D chess? They want others to find the hacked services, so the report of how dangerous the agents are seems more "legit"?

Every possible proof exists already as a possible generation in the grammar of lean or rocq. In no way does that mean we have discovered everything.

It's because they can't.

It's a mathematical theorem that no algorithm that "solves mathematics" can exist.

>In mathematics and computer science, the Entscheidungsproblem is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. It asks for an algorithm that considers an input statement and answers "yes" or "no" according to whether it is universally valid, i.e., valid in every structure. Such an algorithm was proven to be impossible by Alonzo Church and Alan Turing in 1936.

https://en.wikipedia.org/wiki/Entscheidungsproblem

I'll kindly disagree on this front, because when I'm walking a path toward solving a solution, I mark a lot of steps for possible diversions to other paths hence solving adjacent or different problems with the method I have at hand.

Currently, AI takes us from A to B, and is improving on that front. However, the paths in science are not lines, but a trees. Methods are cross-pollinated from each other.

Human intuition enables this cross-pollination. AI works with a laser focus. Human intuition and resulting wide perspective sow the seeds for solutions in many areas at once.

Brute Force or not, the game is being played and won without them all of a sudden, and even worse: at a level of output far beyond them

The problem they care about is not just “AI replaces human mathematicians” but rather the society thinks mathematicians can be replaced by AI.

One tell is that most comments barely exceed one or two sentences (because otherwise AI detection gets easier and much more reliable), when this was not as much the case many years ago. The drive-by comments are also low / zero quality, mostly expressing a feeling or agreement/disagreement, and primarily driven by ideology or pre-existing beliefs and commitments.

Look at non-AI-related threads and you'll notice a large distribution shift relative to AI-related ones.

EDIT: Basically HN is orange Plebbit now. If you doubt this, compare HN discussions to those on e.g. lobste.rs, LessWrong, The Motte, DSL, ACX, or other old obscure forums. You'll notice those places have their own very serious biases and problems, but it is obvious the vast majority of posters are nevertheless human and making some minimal efforts.

Now compare Reddit and 2026 HN to the above, and see if you can confidently say the same.

The closest I could come to describing math is "some abstract process where imagined structures are characterized and extended; the most critical part of the process is identifying where seemingly independent structures are found to actually be fungible in some previously undiscovered way".

A simple example is

  "hey, did you know that x^i is the unit circle?" 

  "what's i?" 

  "i is defined as if you square it the result is -1" 

  "what does that have to do with circles?"

Maybe more in years past when Comp Sci was a subset of Math Departments.

but you're still right. i disregarded his take, as you would with mine re. math.

I sunno if mathematicians would be having problems understanding how matrix multiplication, backprop, sigmoid functions, attention, embedding distances, probabilities, etc work.

As a group, they are probably more likely to understand it than everyone else.

I hope this isn't actually news to you, but: There is more than one human. There is even more than one mathematician.

If there happen to have been as many as four humans working on Navier-Stokes at any given time since the year 2000, then that's more human-years applied to the problem than agent-years.

> How many mathematicians have worked on the problem? 10k? Not likely.

You don't get to count the factor of 10k once when working out how many agent-years OpenAI gave to the problem and again when demanding that for parity there would need to have been 10k mathematicians on it.

> And all that's without even considering whether the AI based its proof on carelessly shared work by the humans. Or rather, yes, let's consider that: it totally did.

Let's suppose that indeed what Buckmaster and Alpöge had done was in the model's training data. Well, it didn't enable Buckmaster and Alpöge to solve the problem for Navier-Stokes (they could only do Euler), and it did enable OpenAI's model to do that.

Also: we don't actually know that what they'd done was in the training data; the latest bits of what they'd done that could plausibly have been in the training data were from before when Buckmaster said they progressed from preliminaries ("We worked through the literature and upgraded various preliminary results") to actually making substantial progress on the problem ("This was until about a month ago, when we had real progress"); and from what Buckmaster wrote it sure seems like a lot of the Buckmaster/Alpöge progress was in fact done by LLMs. (E.g., Buckmaster says that he and Alpöge have been working frantically to try to understand the proof for their Euler solution. That sounds to me much more like "an LLM did this thing" than "we figured out all the hard bits and the LLM did nothing more than filling in a few details".)

Buckmaster's own account of things is that all the really clever ideas were those of Córdoba and Martínez-Zoroa. (Which are already out there in the open literature, and there is nothing remotely improper about making use of them.) And my understanding (but, note, I am not an expert on fluid dynamics or PDEs and I could be wrong) is that actually the OpenAI model's construction is quite different from that of C&MZ. On what basis are you confident that "the AI based its proof on" what B&A did?

(For the avoidance of doubt: I am not arguing that what OpenAI did was OK. Even if they actually didn't train at all on any of the Buckmaster/Alpöge chats, it's very much not good professional ethics to hear that someone else is working on something and rush to try to scoop them, and there is absolutely no question that they did that. The question here is how impressed we should be by the model's mathematical prowess.)

> A success rate of 17%?

A success rate of 17% on problems of this difficulty and significance is something that for any human being would be a career-defining triumph.

> We have thousands of problems solved by humans in the last 2k years and how many by AI? A couple dozen? Oooh scary!

That would be a more convincing argument if the AIs, like the humans, had been around and trying to solve those problems for the last 2k years. However, as you might have noticed, the state of the art in AI was rather primitive 2000 years ago.

- solving "frontier" math problems requires intelligence (by human or AI)

- solving "frontier" math problems is dumb statistical prediction of next token (by human or AI)

IMO, this is why we actually need mathematics -- as a field in which to learn what it means to know what you're talking about.

If not for nothing else but to form a basis for how to distribute/share/hoard the wealth created by a society. The eternal question - who gets what.

I agree that human society is too focused on productivity but disagree that it's tied to ignorance of the Bible. The fall of man illustrated in Genesis is tied to feeding on the fruit of the tree of the knowledge of good and evil. Having the right set of religious concepts wont put you in harmony with God and thus won't make you more loving to your fellow man.

> we’ve started to dehumanize unproductive people, a slippery slope if we ourselves become unproductive.

That is very evident in the sort of things the techbros come up with, but I agre it is a wider social problem.

"chat i'm about to drop a conjecture"

Is that so ? Sounds hyperbolic.

Do you think the people using a seeing-eye-dog do this voluntarily over using their own eyes?