> Every Mathematician Has Only a Few Tricks
>
> A long time ago an older and well-known number theorist made some disparaging remarks about Paul Erdös’s work.
> You admire Erdös’s contributions to mathematics as much as I do,
> and I felt annoyed when the older mathematician flatly and definitively stated
> that all of Erdös’s work could be “reduced” to a few tricks which Erdös repeatedly relied on in his proofs.
> What the number theorist did not realize is that other mathematicians, even the very best,
> also rely on a few tricks which they use over and over.
> Take Hilbert. The second volume of Hilbert’s collected papers contains Hilbert’s papers in invariant theory.
> I have made a point of reading some of these papers with care.
> It is sad to note that some of Hilbert’s beautiful results have been completely forgotten.
> But on reading the proofs of Hilbert’s striking and deep theorems in invariant theory,
> it was surprising to verify that Hilbert’s proofs relied on the same few tricks.
> Even Hilbert had only a few tricks!
>
> - Gian-Carlo Rota - "Ten Lessons I Wish I Had Been Taught"
We may have collectively filled libraries full of books, and created yottabytes of digital data, but in the end to create something novel somebody has to read and understand all of this stuff. Obviously this is not possible. Read one book per day from birth to death and you still only get to consume like 80*365=29200 books in the best case, from the millions upon millions of books that have been written.
So these "few tricks" are the accumulation of a lifetime of mathematical training, the culmination of the slice of knowledge that the respective mathematician immersed themselves into. To discover new math and become famous you need both the talent and skill to apply your knowledge in novel ways, but also be lucky that you picked a field of math that has novel things with interesting applications to discover plus you picked up the right tools and right mental model that allows you to discover these things.
This does not go for math only, but also for pretty much all other non-trivial fields. There is a reason why history repeats.
And it's actually a compelling argument why AI is still a big deal even though it's at its core a parrot. It's a parrot yes, but compared to a human, it actually was able to ingest the entirety of human knowledge.
Even this, though, is not useful, to us.
It remains true that, a life without struggle, and acheivement, is not really worth living...
So, it is nice that there is something that could possibly ingest the whole of human knowledge, but that is still not useful, to us.
People are still making a hullabaloo about "using AI" in companies, and there was some nonsense about there will be only two types of companies, AI ones and defunct ones, but in truth, there will simply be no companies...
Anyways I'm sure I will get down voted by the sightless lemmings on here...
The combinatorial nature of trying things randomly means that it would take millennia or longer for light-speed monkeys typing at a keyboard, or GPUs, to solve such a problem without direction.
By now, people should stop dismissing RL-trained reasoning LLMs as stupid, aimless text predictors or combiners. They wouldn’t say the same thing about high-achieving, but non-creative, college students who can only solve hard conventional problems.
Yes, current LLMs likely still lack some major aspects of intelligence. They probably wouldn’t be able to come up with general relativity on their own with only training data up to 1905.
Neither did the vast majority of physicists back then.
Indeed, and so do current humans! And just like LLMs, humans are bad at keeping this fact in view.
On a more serious note, we're going to have a hard time until we can psychologically decouple the concepts of intelligence and consciousness. Like, an existentially hard time.
I've been using LLMs for much the same purpose: solving problems within my field of expertise where the limiting factor is not intelligence per se, but the ability to connect the right dots from among a vast corpus of knowledge that I would never realistically be able to imbibe and remember over the course of a lifetime.
Once the dots are connected, I can verify the solutions and/or extend them in creative ways with comparatively little effort.
It really is incredible what otherwise intractable problems have become solvable as a result.
I don’t know what this claim is supposed to mean.
If it isn’t supposed to have a precise technical meaning, why is it using the word “interpolate”?
and homo sapiens, glancing at the clock when it happens to be right, may conjure an entire zodiac to explain it.
A broken clock can be broken in ways which result in it never being correct.
They are not great at playing chess as well - computational as well as analytic.
Further evidence for the faultiness of your claim, if you don't want to take me up on that: I had problems off to GPT5 to check my own answers. None of the dumb mistakes I make or missed opportunities for simplification are in the book, and, again: it's flawless at pointing out those problems, despite being primed with a prompt suggesting I'm pretty sure I have the right answers.
I found and fixed bugs I wrote into the formulas and spreadsheets, and the LLMs were not my sole reference, but once the LLM mentioned the names of concepts and functions, I used Wikipedia for the general gist of things, and I appreciated the LLMs' relevant explanations that connected these disciplines together.
I did this on March 14, 2026
That's one way to waste a ton of tuition money to just have a clanker do your learning for you.
Unless you're teaching it, in which case I hope your salary is cut by whatever percentage your clanker reduces your workload.
80 hours! 80 hours of just trying shit!
That is not nothing, no matter how much you hate AI.