AI isn’t outthinking mathematicians, it’s out-remembering them

Even in a debate, if somebody just has the ability to remember tons of facts and figures, the other person will seem unintelligent by comparison, even if the other person is correct

I’ve met different people throughout my career whose intelligence came in 1 specific area. For instance, my friend is extremely good at trivia, he clearly has a lot of storage and can access it easily. I think I’ve only met one person who was excellent in all three areas of intelligence.

Obviously this is a simplification, but it’s how I like to illustrate my ideas on intelligence at parties and first dates.

Understanding especially in the context of unknowns is what intelligence is.

I know people that got to post grad math without understanding a thing but they could remember a lot easily, while many of those that understood but had a harder time remembering every last variation of everything got penalized.

There’s no such thing as a truly original idea. It is all just combining A+B!

If a time traveler went back to 1600 and started spouting off about differential equations everyone would think them quite mad.

But “remembering” is only the beginning (you have to remember first!), after that, elements like understanding relations, connecting dots and remixing, timing, etc will truly make one shine.

In one way, it’s like the current “LLM + Harness” setup for agents. LLM is how well it remembers, but different harness techniques really matters, at times even a worse model mixed with great harness can outperform great model with bad harness

I have fairly poor memory, but I can solve problems that people with better memory can't.

Really, compress. Compression is a useful proxy for intelligence.

To compress data efficiently, an entity must discover the underlying patterns, structures, and regularities within that data. Identifying these structures is precisely what we define as "understanding" or "intelligence.

Anything we put into writing can be thought of as a string of data. The Kolmogorov complexity of a string of data is the length of the shortest possible computer program that can output that string.

A good compressor doesn't just memorize (yes, there's a reason LLM writing features the "it's not just X, it's Y pattern"), it builds a generalized model to handle unseen, similar data. LLMs are trained by compressing the internet into weights, they use this to write new code.

When an LLM is trained, it compresses ...bytes of human text into a few ...bytes of neural network weights. When you ask it a question, it uses that compressed model to generate (decompress) a coherent response.

Consider the dichotomy:

LLMs display emergent reasoning, coding capabilities, and logic because they were forced to compress human language so tightly.

LLMs often suffer from "hallucinations" and struggle with novel logic puzzles because compression encourages mimicking statistical patterns rather than building a true causal model of the physical world.

An algorithm can achieve massive compression if it utilizes a highly complex, slow, or massive decompression engine. If a system takes 10 000 years to decompress a file, or requires an infinite amount of computing power (like Kolmogorov complexity), it cannot function as a practical intelligence. Real intelligence must operate within tight time and energy constraints.

> Maybe there's true brilliance out there

Brilliance is a trope. pg's 2009 "The Anatomy of Determination" deals with it: https://www.paulgraham.com/determination.html

In it, he argues that determination is the single most important predictor of success, surpassing even raw intelligence.

While popular culture loves the narrative of the "brilliant genius," Graham's experience backing hundreds of startups taught him that a highly intelligent person without drive quickly becomes ineffectual. Conversely, a profoundly determined individual can find pathways to success even in ordinary or unglamorous industries. Over time, determination actually starts to look like talent.

I'm going to commit a faux pas (a breach of HN etiquette) and give an AI summary in 3 bullet points because, well, the article deals with 3 points, so it's a good fit and I doubt I can compress it better than AI:

1. Willfulness. The desire to get your way regardless of obstacles or consequences.Being fearless about consequences and pushing through "hellacious" moments when others quit.

2. Discipline. The ability to force yourself to do what you planned to do. Being "hard on yourself" and turning off the background analysis in your mind that questions the plan.

3. Ambition. The grandness of your destination and what you want to achieve. Choosing the right target; if determination is a vector, ambition is the direction it points.

Of these 3, it's quite obvious at this point that LLMs are superior at 1) and 2), as LLM alignment is quite good, and it's ability to spin up sub-agents that can work for over 24 hrs is better than humans. Human limit is basically 1 hackathon's worth of cognitive work. Beyond that, health becomes a concern, and how many people can produce useful work under sleep deprivation? Codex can work for 24-ish hours, as evidenced by today's "Auto-research with codex: How I achieved a 232x Faster Kernel": https://news.ycombinator.com/item?id=49309549

So what's left for mere mortals? Apparently, nourishing your ambition:

"So here in sum is how determination seems to work: it consists of willfulness balanced with discipline, aimed by ambition. And fortunately at least two of these three qualities can be cultivated. You may be able to increase your strength of will somewhat; you can definitely learn self-discipline; and almost everyone is practically malnourished when it comes to ambition."

I wonder practically how useful it is. It should be very useful, right? Say you're a hiring manager, and now you can hire thousands of students who have almost eidetic memory across pretty much all the subjects. The students can produce amazing work if you give them specific instructions. They are also very good at mechanical pattern matching. That is, they are the uttermost crammers. They are like the students who don't really get STEM yet they could, through memorizing all the problem-solving patterns, ace any national college entrance examinations, IMOs, PHOs, Putnams, you name it, and all the way to solving hard problems like Sendov Conjecture - they even found out elegant elementary solutions! Mind you, Terrance Tao proved a weaker conclusion when studying the Sendov Conjecture and got to publish his work on The Big Four. That said, the students won't take initiatives, and they sometimes make very silly or even impossible mistakes, to the point that it requires your supervision and review. Would you hire them to do your company's work? I'd assume that answer is a resounding yes.

I can’t remember the lyrics to any songs, or the lines from any movies, or what I’m doing this week. I don’t think it’s a strong hypothesis.

My wife has a nearly eidetic memory, so marrying her was a game changer for counteracting mine :)

Doing an angular -> react refactor with Opus 5 at work.

Amazed at how good it is but also amazed at how dumb it is after I told it I want a very basic and specific code review:do the file names match the contents? is one functionality/concern split in multiple files? do the variable names match what they do? does a method read nicely?

Anyway, I was very surprised at how bad it can be at this.

I suspect it's because the LLM simply does not focus well enough on a single method.

My gut modelling is that the critical files are quite small, some 4k LOC and the LLM tries to read them all in one go and my requirement is surprisingly difficult but the LLM does not realize it, loads all the input into context and simply cannot find a good signal per token/group of tokens.

I guess I can do it because I literally cannot fit 4k LOC in my head and I actually read only tens of lines but I can reeealy dig deep in those lines.

The LLM can remember lots, it has 1M context window, was trained with needle in the haystack problems, but it can severely underestimate how much computation is required with TASTE/universal computation type tasks.

Yes, LLMs are incredible when it comes to delivering well defined outcomes, it's evident they were RL trained very well. But they are incredibly bad at understanding nuance in a text.

It's incredible how agentic coding benchmaxing seems to correlate with writing good poetry but at the same time finding good names for files and variables and wiring beautiful code does not.

Same with getting basic details of who said what in an "adversarial" conversation between 2 humans -- LLMs are incredibly bad at it, with all their "almost discovering new maths" skills

If you want an example of someone at the near peak of human ability, check out Jon Von Neumann.

Then there is an ability to peer deeply into complex problems and somehow find the simplest truths that make sense of it all. Think of Einstein.

Both are incredibly intelligent, but in different ways. I'd say Von Neumann's memory was far greater than Einstein's though.

One can flawlessly ponder anything known to man, and the other could ponder completely original ideas (to an extent)

Some people, through practice and experience are better than others at breaking down foreign information and making sense of it, or jumping to an assumption.

I also think there’s an element of just remembering what my final conclusion was on an opinion, vs all my old opinions and all the work to get there. It’s easier to just store current belief and re-work-it out how I got there later.

Being smart, at least in the context of the workplace, is about being checked in to whatever you’re doing, and drawing connections across your experiences.

They’re positively correlated. Presumably because people who have good plastic intelligence often end up learning more throughout their lifetime. The only way to form knowledge in our minds is by learning it.

The part that’s really disheartening is there’s so much knowledge out there, I wonder if this is actually efficient - maybe you can’t reasonably search the state of the art across all of human knowledge and so you have to invent it again. But that’s such a sad outcome.

There is also something to be said about recognizing that A + B can be combined.

And also in accurately judging that other combinations will be useless and avoiding them.

In order to remember something you need to know/learn it first.

    > it either came down to either having more energy than others at tackling a problem they thought was more trouble than it was worth, or just bringing back random knowledge from previous jobs or self study, and being able to apply it to the problem at hand.

There is a great episode of Inside the Actor's Studio where there is a Q&A with Mike Myers.

He was asked if going to college/getting more education was helpful to being a comedian.

He responded:

"Many times in writing comedy, you are on an 'island' and you are trying to figure out how to get to the next 'island'. Having more education and knowing more about the world can definitely help you do that"

I like the above quote because to me intelligence is a combination of remembering useful information but also being able to "connect the dots" between those facts and apply to them to the current situation.

As another example, I was watching a documentary about LIGO. One of the engineers had memorized the entire system in his head. The other engineers would come to him and ask "Will this change to component A affect anything else?". He would pause and think and then say "Yes, component F touches that so it will be affected." The other engineers mentioned how useful it was to haves someone who had the entire system in "one brain".

I've seen this in finance technology too where someone, usually a woman, has large parts of the system memorized and performs a similar function.

Yes, I feel like this is a huge part of the value that I bring to most software teams. I have enough experience, and good enough recall, to come up with solid critiques of proposals, or to think of gotchas before they bite, or to present with a plan where I've managed to consider more of the problem space than other people can do.

Certainly it's not just experience and recall; being able to apply that to the current problem is critical as well. That part often comes into play for me with debugging: I've lost track of the number of times someone came to me with a thorny problem that they'd spent hours or even days working on, but some random bit of past experience floats to the top of my consciousness and I set them on the right track within a few minutes.

(I realize after writing that this sounds like a brag or a flex, but I'm hoping it won't be taken that way.)

Your timelines are a bit unambitious. There's nobody expecting to make significant progress with a week of work.

Theres going to be this field day of low-hanging fruit that ML can round up, but after that I suspect it will be in fits and starts as a “connection maker” rather than some proof producer.

People go whole lives without being able to make it pan out.

https://en.wikipedia.org/wiki/J._Robert_Oppenheimer#:~:text=...

It's not out-thinking, it's just out-remembering

It's not out-thinking, it's just out-working

It's not out-thinking, it's just able to consider more things simultaneously

It's not creative, it's just randomly generating things and then selecting viable ones

https://news.ycombinator.com/item?id=48231974

Out-ralphing them, you might say!

https://ghuntley.com/ralph/

AGI ≈ artificial stupidity × infinite persistence

That is also approximately what people have always done to succeed.

I don't know how or why this would be trained on behavior, but no, it isn't true anymore that models don't say things like, "Ugh," or "this is going to take hours and maybe we should stop here."

The incentives are not.

The incentives are skewed towards "a magician never reveals her secrets". The results are presented as if a rabbit got pulled out of a hat, with a maximum ta-da! effect, and little backstory of how the hell did we get there.

Don't get me wrong, these things are discussed, often over beers (you better drink it you want to make a career in the field).

But not published.

The younger mathematicians are trying to change that with the blogging culture. But the professional incentives aren't there. (In corp-speak: can't put blogging on perf). They burn out.

That's why math blogs usually come from either the top dogs in the field, like Terrence Tao, who don't need to care about perf, or people outside academia.

That's one thing that I hope the disruptive/destructive effects of LLMs will force mathematicians to face.

As one of my fellow mathematicians sarcastically wrote¹, we've reached a point where we should become a cult because we're acting like one anyway.

The other possibility is, of course, that the shake-up will take us precisely into that direction.

My point here is that the real problem here is not mathematical; it's a social one: incentives and politics, organizational structures, policies, allocation of jobs and funding.

All of this directly impacts how we do mathematics, who we do it with and teach it to, how we teach and communicate, and, of course, what math we even do and look at.

Given that, I'm neither too worried about humans vs. AI standoff, nor hyped about the Glorious New Future full of AI-assisted discoveries.

AI or not, the organizational issues in the field are still there, as are the incentive structures (including the infamous publish-or-perish).

We are doomed, yes, but by our own hands and committees. And it's up to us, not the AI, to get us out of there.

The little shove from the AI might be just the thing we need.

____

¹ https://www.mcsweeneys.net/articles/an-open-letter-to-the-ma...

In a way what they are doing, only publishing positive results is no longer good science.

Do you really think an average person can internalize complex math? Them compressing it effectively and then remembering it is a sign of (very) high intelligence.

I don't want to read another LLM documentation ever again. The information to text ratio is way too small.

That's more because a PhD (especially in this socioeconomic era where academia is also a microcosm of capitalism) does not automatically teach or focus on deep inquiry, rather narrow research programmes. Case in point, fallaciously reducing it all to a superficial theory of tricks means that by the same light, general relativity or any other major scientific result is merely just a trick. That framing ought to be absurd on the face of it but the fact is many PhD students do still graduate with very unexamined understanding of science, cognition, philosophy, etc. (Another way to see this is to note how such an argument is letting the notion of "trick"/"heuristic" do all the heavy lifting--something an advanced education should have nurtured the critical thinking skills to avoid that very metacognitive pitfall in the first place, which is ironic.) (Yet another way is to see how increasingly experts of one field or another insist that one must stay on their lane when speaking about a complex topic.) There are professors and thinkers who have written/spoken about this but they are a minority. Even Einstein himself when he complained that more and more scientists were trained/structured to miss the forest for the trees, and that was almost 100 years ago.

Also I suspect that, apart from that, the results on smaller, cleaner codebases are better. LLMs degrade when following more than N instructions (where N depends on the model) even if the context window is not full yet; I suspect they also degrade when code has too many unnecessary concepts and details

I'm lucky that for my side project (an interpreter) I've written all the code myself, so I've built up its design in my mind over the past year, and so as I mull over what I'm writing I start coming up with simpler designs.

Interestingly using Opus 5 (and LLMs in general) has made me worse at this, since I don't feel the pain of writing something over and over again. On the other hand, I don't really want to implement a whole stdlib, so I have it write more of the auxiliary code. The hardest thing is that I have to manually manage the context, which is painful when I personally remember every helper function and why. I have to remember to keep the list of helper functions updated, which is irritating.

That doesn't follow. We could still understand it just by studying it and committing it all to long-term memory, it just takes longer. And there's a hard cap on the working memory of LLMs, due to the quadratic scaling cost of the full attention layers that have proved unescapable for all SOTA LLMs.

But was it?

https://www.splcenter.org/resources/hatewatch/wikipedia-wars...

https://openpsych.net/forums/18/thread/25/?page=1#124

That's something AI companies would really want you to believe.

I agree.

> The age of humans comprehending things is coming to an end: our brains just won't have the capacity to make meaningful contributions to science, math, or technology.

I don't know if I see this being true for quite a while, if ever.

This is not to say that a human couldn't understand a streamlined version or that the AI would not be better if it made more streamlined statements to begin with.

(I am not saying that everything mathematical that an AI produces is in any sense trivial.)

Math is not magic, a proof is just a series of applications of a set of rules on some axioms. A mathematician could understand any proof given enough time to study it; the only way for AI to make proofs that a human couldn't understand is by making really, really long proofs.

As to your second point, Terry Tao already has an answer [1]: the proof isn’t the contribution, shared understanding is. This issue was already raised back when the four-colour theorem was proved. Machine proving and machine proof checking are useful tools but they don’t mean anything without the interpretative work and the communication necessary to build shared understanding.

[1] https://news.ycombinator.com/item?id=49056620

Humans can’t compete with AIs on vastness of material they are familiar with, or the depth of effort they are willing and able to throw at a problem.

But scale isn’t the only aspect of difficult scientific endeavours. There’s also theory. And advancements sometimes come through hard graft of knotting together many things. And sometimes they come through the revelation of a deeper truth, or a new framework, a fundamental insight.

AI might help us reach the next level. But that doesn’t mean we won’t understand anything. It could be we have periods of vast intricacy we cannot follow, punctuated by profound elegance we (or at least experts) relatively easily can. And then the scaffolding we needed to get there falls away.

I think perfect rationality doesn't exist, because it is rational to reject something that you don't understand. So rationality of a given physical system will always be bounded.

That sentiment makes me cringe. If you understand how LLMs work, you'd know it'll never be possible without a fundamental change in how these work.

We're also supposed to be reaching that point, somehow, without the LLMs ever being intelligent (in the dictionary definition sense, not the "high reasoning model" marketing sense).

Based on observations, the ones who are fooled by the supposed emergent properties, are just that, fools. Any sufficiently unintelligent agent will perceive transformer based LLM text predictors as possessing high intelligence.

Ask yourself: is that really the world you want to live in? It's a world where people, all people, are sidelined.

I think the happy ending of that path is something like Idiocracy. And the more likely ending is something like "automated capitalist economy without the people, because the people couldn't compete."

They don't know and can't know. Without an external source of input that corrects them, their output can never be verified.

The thing is, some number of advanced proofs start out "too intricate for most mathematicians to understand" but many of these get rephrase and reframed until they're accessible to undergraduates. Hopefully, AI math can be guided to do that sort of reframing to increase the level of accessible math as well as extend the border of math.

I'm sure humans will comprehend some things and not others just as now. They may be smarter than me but the Einsteins and the like already were. It may be that the cutting edge gets pushed by AI more than humans at some point though.

Why would you want to replace your lawyer with a set of tensors that does not actually think and makes mistakes like this? Lawyers tend to get hired in high stakes situations. Why wouldn't you instead say that this would be a great tool for lawyers to use judiciously in researching precedents, etc?

I don't understand what people are doing with models that makes them assign agency or intelligence to them. When I manage to forget the financial fuckery of the AI buildout and its implications, when I manage to forget scaremongering by loathsome CEOs, I still have the same fascination and excitement at the idea of LLMs as I did when I was playing with the GPT API prior to the release of ChatGPT.

LLMs are, to me, truly amazing tech. It's so fascinating to me that they now DO have emergent properties that look at face value like reasoning and intelligence. But every day that I work with them, I am repeatedly clobbered over the head with the fact that they do NOT reason and are NOT intelligent.

Why can't we be fascinated by emergent properties of intelligence without immediately jumping 10 steps into the future and, like a limit in calculus, assume that "this is it-- we're on the cusp of AGI"? To me, the fact that LLMs can combine existing ideas that people hadn't thought of combining in solving a novel problem is extremely cool. But my first thought is-- this is an amazing new tool for mathematicians and researchers. Instead, most everyone seems to jump the gun to the "humans are obsolete next year" conclusion.

Law and medicine are fundamentally harder fields to obtain decent training data for, and LLM results are therefore expected to be less powerful. Also, making mistakes in these fields is costly, but perhaps you were alluding to that already.

[1] https://en.wikipedia.org/wiki/Mathematical_proof

I think the real scandal is that we are almost 3-4 years into this (I think the release of GPT 3.5 is a good marker of when this public frenzy started) and all we’ve seen is OpenAI and the other major AI frontier companies constantly retracting their preposterous claims every time. We appear to have reach a local maxima in that it has some value in places that tend to be a little easier to scope and limit (computer programming, mathematical proofs). So, given the actual useful economic value this has provided, does this justify the investments? I think we are approaching 1 trillion in CapEx for AI [0]. For context, I believe the annual GDP of Norway is $600 billion.

[0] https://www.fool.com/research/ai-companies-spending-on-data-...