> I don’t open emails anymore i let it summarize draft and handle most of the work and I only talk to Claude and not read/write the emails myself.
I pity the people on the other side.
2,071 karma · joined November 18, 2023
I pity the people on the other side.
It’s hard though when Google tries to tempt you to do drugs every three seconds.
I get pissed off reading the ‘weird takes’ on mathematics from people who come from the tech world and don’t know what they’re talking about (but for some reason think they do). Not only is it just painful to read; it’s actively harmful.
Mathematicians are largely ignorant of software and are aware of and will freely admit to their own ignorance, but for some reason the software engineers here are largely ignorant of mathematics AND think they know it inside out.
It sounds like you’re actually more well informed than most. Don’t take my comments personally, in any case.
This is true of all subjects, but it’s hard to think of a subject where the gap is as stark as it is in mathematics — even intelligent outsiders have no idea what the subject is really about, and, even worse, they think they know (mainly because we all studied something also called ‘mathematics’ as children and assume that research mathematicians probably do something similar).
By the way, I agree with your original comment.
So I have my reasons to weigh his statements more heavily. I certainly don’t take anything for granted.
Ultimately, you could give a few sentences to summarise the proof of FLT as long as one accepts various concepts and their (also deep and conceptual) proofs. In the case of the four-colour theorem this isn’t true because rather than being amenable to summary in words and fancy concepts it’s just a loooooad of cases to be tediously checked.
I can imagine that AI might be able to produce reams and reams of extremely tedious ‘proofs by a million special cases’ that we’d never be able (or even have thought) to do by hand, but that’s not to say there doesn’t exist an alternate, more conceptual explanation that is compressed.
In fact one might argue that facts that are ‘inherently incompressible’ are by definition uninteresting. The profound statements in mathematics are those that appear to be incompressible, yet turn out to be provable in an elegant way by a change in perspective. In this sense, noting the use of the word ‘appear’, it becomes clear that what is and isn’t interesting (and therefore what is and isn’t to be considered trivial) is entirely subjective and hence mathematics is an inescapably human pursuit. But we’ve known this for a long time.
I don’t think it’s a stretch to say that each side’s claim isn’t equally trustworthy.
So there are some counterexamples.
He says they stole it; see for example his recent interview with Numberphile. So is he silly? Or do you know better?
> There's not enough pre-einstein text digitzed to make a fair shot at such an experiment
Then why was it possible for humans to do physics, or indeed function at all, pre Einstein? I guess humans are special after all.
OpenAI stole the work of Tristan Buckmaster to do it, and refused to explicitly deny it.
What do you think has enabled the 'staggering rate' of improvement? Scaling up? More data? It seems like, since LLMs do not solve problems from first principles and make crucial use of human work (without which they'd have nothing), once all the data has been eaten, some sort of plateau will indeed be reached. Anthropic are already reportedly buying up all of the world's second-hand books and destructively (their words) scanning them in a desperate rush for ever more training data, so I don't think this theory is unfounded.
What do you think of the Hassabis Test? Until it passes, I don't see how humans are replaced.
I like the test proposed by Demis Hassabis. Something like: train an LLM on pre-Einstein physics and see if it can rediscover what Einstein did. Despite the incessant noise online, we seem to be no closer to this. If there's material evidence that we are, I'd sincerely like to hear about it!
There's also a difference between important and hard. There are important problems that turn out to be easy, and hard problems that turn out to be useless.
As usual, I think everyone would agree that something like "curing cancer" would be both hard and important!
I think the AI labs' current obsession with showing off mathematical results to uninformed outsiders is a cheap trick. If they were really interested in 'enabling human flourishing' (rather than just wowing, by any means possible, investors with more money than sense), they'd be showing off cures to diseases rather than solving obscure problems in combinatorics only previously considered by three Russians fifty years ago and then declaring that The Singularity is here.
Mathematical progress does (quite obviously, on the whole) help advance humanity, so progress is a good thing. The problem is that defunding mathematicians and handing over control to AI and the companies that create them will cause the subject to stagnate. Sure, for a while we might get progress on existing questions using (perhaps quite novel) combinations of existing techniques, but, so far, given the character of the results we’ve seen, there’s no indication that it will continue indefinitely. Even if it did, what would be the point? Huge textbooks full of work no one can understand or benefit from?
One possible analogy is that humans work to add new points to the space of mathematical knowledge, and AI then fleshes this out to attain the ‘convex hull’ of these points. Essentially, humans ‘invent’ the definitions and pose the questions and AI does the grunt work as well as some creative exploitation of known results and tools to bring down all the low-hanging fruit that follows (important note: what appears to be non-low-hanging fruit to us may in fact be technically low hanging once AI is involved; we saw this for example with the Jacobian conjecture). This seems to be the current situation, and to argue that humans are fully replaced it is necessary to argue that AI is adding points outside the convex hull of human mathematics. A sufficient example would be a first-principles AI proof using alien techniques, and this we haven’t seen so far.
The mathematics-chess comparison is, to put it bluntly, nonsense. I see where it comes from, but, as absolutely anyone with any research experience will tell you, mathematics is orders of magnitude (and this really isn’t strong enough) more open-ended, and doesn’t consist of a game one is seeking to ‘win’. The goal is understanding itself.
LLMs have been around for years, and they're explicitly trained on the entire history of human mathematics (without which they'd be unable to do anything).
By the way, I don’t think climate denial has (or at least should, though I know Americans like to politicise everything under the sun) anything to do with ‘conservatism’.
We were told in 2023 that there’d be no more jobs within months and the singularity was here. At some point you have to realise it’s not happening.
Let me know when the other (much harder) millennium problems get resolved, let alone explained in a satisfactory way.
The whole point of mathematics is to vastly exceed that natural ceiling by gradually building a framework for understanding. In fact it’s wrong to speak of a ceiling altogether. If there were a ceiling, we’d have hit it long ago.
AIs have already swallowed the entire history of human thought, but apparently they’re ’just getting started’. I can only assume you don’t know what mathematicians actually do.
Most pure mathematicians are no more interested in ‘producing commodities’ than any other academic is. That’s not what the subject is about — at all. The confusion arises because mathematics turns out to be extremely useful (no surprise; it’s quite useful to have a detailed understanding of the basic principles of reality).
Again, this seems to be an alarmingly common fallacy here on HN. As a commenter above observed, pure mathematics (and that is what we’re talking about here) is in important ways closer to the humanities than it is to other sciences.
Thanks for providing a (much needed!) correction.
Unfortunately, it is actually surprisingly hard to pin down, and I think mathematicians (and, as a student, I count myself as one to some degree at least) now have the task of making this a lot clearer. If we want to justify our existence in the face of new machines that can seemingly ‘do our work for us’ (so far in a restricted context), we should give a robust defence of our practice. If we can’t do this, we simply don’t deserve the funding (which, by the way, again contrary to some misguided statements here, isn’t very much anyway!). I think all of this will become clearer to outsiders as time passes, but for now it’s not easy to give a quick answer — though I can try.
Mathematics is about understanding things. Isn’t that what every subject is about? Well, I suppose so, but mathematics more specifically does something like the following:
(1) observe some phenomenon in ‘reality’.
(2) attempt to formalise that phenomenon in such a way that it can be manipulated purely symbolically.
(3) use this (perhaps fairly arbitrary; remember that we can invent as many formal systems as we like) system to deduce from our initial assumptions new facts that would otherwise have been very non-obvious.
It seems like outsiders have a decent grasp of (3) and the application of AI to it, but have very little idea about the other two steps. It seems to be widely assumed among non-mathematicians that problems are essentially god given and that the job of a mathematician is therefore to chug away on these problems, manipulating symbols and trying out tools, in the hope of learning a yes/no answer to each one.
The first two steps are by far the hardest and most important, and they’re also the parts that AI seems currently unable to help with.
NOTE: this is not a deeply insightful description of what the subject is about, and there are many better characterisations out there. I think Tao and various others have written recently about why complicated and inscrutable AI-generated proofs aren’t nearly as valuable as one might imagine. (That’s not to say there’s no value to such proofs; perhaps in time, as technology improves, mathematicians will come to accept AI as part of the process.)
If you want to understand all of this issues better, reading the recent slew of guest posts on Tao’s blog would be a very good start.
Unfortunately, mathematics (especially pure mathematics) is by its very nature very, very poorly understood by those who haven’t worked as a mathematician. Even worse, those who don’t understand are seemingly not at all aware of their misunderstanding and are entirely confident in their (very wrong) characterisation of the subject.