HNHacker News
TopNewBestAskShowJobs

xanderlewis

2,224 karma · joined November 18, 2023

$
submissionscomments
xanderlewis··on Terence Tao: Math 2.0 [pdf]
> Can't we get all the meaty goodness by simplifying and generalizing the proofs now we know they exist?

Perhaps, but there is a danger: it is difficult to un-see things. At least for now, AI-generated proofs are likely to be of a somewhat brute-force nature. As each such proof pops into existence, two things happen: (1) it is much harder to maintain an untainted mind and go on to discover alternative, perhaps deeper and more conceptual, proofs that cannot be obtained by massaging and cleaning up a more brute-force approach (think of this as getting 'stuck in a local minimum', if you like), and (2) at a societal level there is thereafter much less incentive to attempt to do so (announcing a proof of a previous unproven result is, for now, far more prestigious than announcing a much better, more elegant, proof of a known result).

One could ask why we prefer conceptual explanations to brute-force proofs; after all, a proof is a proof, isn't it? I suppose one reasonable answer is that conceptual explanations lead more readily to new questions and that there's also intrinsic beauty in such explanations -- though, of course, many outside the field will simply not care.

xanderlewis··on Terence Tao: Math 2.0 [pdf]
When it comes to mathematics, you won't find any well-informed commenters on HN. Or, at least, 99% of commenters have no idea what they're talking about. It's infuriating. I try to just ignore it now and let it wash over me.
xanderlewis··on 100+ reactions to 100+ solutions
It's an unbelievably unoriginal statement, and an irrelevant one too.

AI isn't going to stop hobbyists doing anything; no one ever suggested so. We're not talking about hobbyists.

xanderlewis··on 100+ reactions to 100+ solutions
Have you not already heard others make this analogy a million times by now? It seems like I read it multiple times a day at this point.

Mathematics can be enjoyed recreationally as a puzzle like any other, but it isn’t just an arbitrary puzzle. It’s a science where one discovers truth and seeks an understanding of, and dreams up, new phenomena. It’s not chess, or go, or StarCraft. There’s no fixed rule set and the goal isn’t to ‘win’ or beat your opponents.

Let’s stop repeating this nonsense as if it’s a profound observation.

xanderlewis··on “Math 2.0” will need to value mathematical progress more holistically
The real value is in advancing the understanding but very few outside academia understand this (it’s counterintuitive, which is fair enough as it’s somewhat unique to mathematics as a subject), so if you want to get funding you frame it as being for solving the problems.

Now that you can solve without understanding this setup is broken. Either the sources of funding will finally have to learn the difference and the value of the latter, or mathematics research ends.

xanderlewis··on “Math 2.0” will need to value mathematical progress more holistically
> even if they like doing it as a day to day job, they'd probably be doing something else if they had absolute freedom over their time.

Not the mathematicians I know. They’d happily drop the academic admin stuff, but they’d absolutely keep doing mathematics in much the same way.

xanderlewis··on Sharing AI progress in mathematics
You’re (as Claude would say) absolutely right, and I suspect the original commenter has no idea what they’re talking about.
xanderlewis··on Sharing AI progress in mathematics
As Kevin Buzzard recently said:

> In a 2020 piece in the Notices of the AMS, I asked the following question: “If one human had an understanding of all of modern pure mathematics simultaneously, how much further would they immediately be able to see?” Six years later we are beginning to understand the answer to this question.

xanderlewis··on The Future of Mathematics
It’s like people think problems are given to us by God.

We make our problems up!

> For some reason there is this virulent strain of anti-human AI rhetoric

It’s just… nasty, isn’t it? Facts aside, one does have to wonder what motivates not the beliefs themselves, but the, well, frankly aggressive, way in which they’re expressed. Bitterness and envy felt towards those who have actually done the hard work and achieved things?

xanderlewis··on The Future of Mathematics
> We've reached the point that it's even started solving Millennium problems.

Since you’re so certain, how long do you think it’ll be before all of the millennium problems are solved by AI (by leveraging the current literature)?

xanderlewis··on Sites in ChatGPT
> 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.

xanderlewis··on Everyone's Packing Up
Just don’t do drugs.

It’s hard though when Google tries to tempt you to do drugs every three seconds.

xanderlewis··on Responsible Release of AI-Generated Mathematics
I agree, though it might not be what you mean!

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.

xanderlewis··on Responsible Release of AI-Generated Mathematics
No one’s saying you have to believe whatever the famous mathematician says — that would be stupid. They’re saying that there’s a difference between the opinion of an expert and some random outsider.

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.

xanderlewis··on Responsible Release of AI-Generated Mathematics
Listening to him, I didn’t get that impression. I get that academia at large is full of egomaniacs (or so I’ve heard and read online), but the mathematics community is a weirdly non-toxic oasis of reasonable enthusiasts — genuinely, surprising though that may be to those with no experience!

So I have my reasons to weigh his statements more heavily. I certainly don’t take anything for granted.

xanderlewis··on Everyone's Packing Up
I’ve learned a huge amount from YouTube and am very thankful for its existence. It’s the absolute insistence on showing me idiotic ‘shorts’ that made me leave. I didn’t want to.
xanderlewis··on Everyone's Packing Up
I’m just providing another data point; no need to get your knickers in a twist.
xanderlewis··on We're gonna need a lot more mathematicians
The proof(s) of FLT aren’t like that of the four-colour map theorem; despite being heavy on prerequisites, they’re conceptual. In fact FLT is a very nice example of this ‘telescoping’ effect.

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.

xanderlewis··on Responsible Release of AI-Generated Mathematics
There is an important asymmetry, though: Buckmaster is a mathematician in it for fun (as almost all mathematicians are, because there certainly isn’t much money in it), and OpenAI is a company desperately scrambling for anything they can present as evidence that the singularity is here and you must invest your money now. Buckmaster abides by the unwritten code of mathematicians (alluded to in said interview); OpenAI abides by absolutely no such code other than ‘move fast and break things’.

I don’t think it’s a stretch to say that each side’s claim isn’t equally trustworthy.

xanderlewis··on Everyone's Packing Up
I’ve left everything (Facebook, Instagram, I’ve even deleted the YouTube app from my phone, etc.) apart from HN. I occasionally look at Twitter, but only when someone sends me a link.

So there are some counterexamples.

xanderlewis··on Responsible Release of AI-Generated Mathematics
> To say they stole his work would be silly.

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.

xanderlewis··on Responsible Release of AI-Generated Mathematics
> internal models can even solve a millenium prize problem.

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.

xanderlewis··on Evan Doorbell's Phone Tapes – Brought to You by Telephone World
Yeah. It's his.
xanderlewis··on Responsible Release of AI-Generated Mathematics
That is what I'm saying. There's a gold rush to extract what can be done with LLMs, but no indication that it'll continue. Maybe it will, but the evidence for its continuation seems to be entirely hype based on the fact that we've suddenly thought of new things to try. It's not because 'the capabilities are improving at a staggering rate omg the singularity is here guys'; it's mostly because no one bothered to spend enough time and money (and currently absurd amounts are required) to give it a serious shot until recently.

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!

xanderlewis··on Responsible Release of AI-Generated Mathematics
Not all of the 'famous' open problems 'mathematicians have failed to solve for decades' are actually that famous though. In some cases they've remained open because no one cared or had even heard of them. Those outside the mathematics community seem to think that all mathematicians can, and do, work on essentially all problems (for example, most mathematicians have in mind the millennium problems as a goal), but this is very far from the case. Most serious problem statements are not even particularly understandable to most mathematicians, let alone workable on.

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.

xanderlewis··on Responsible Release of AI-Generated Mathematics
That’s a false dichotomy.

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.

xanderlewis··on Responsible Release of AI-Generated Mathematics
What (some) mathematicians are asking for, whether one agrees with it or not, is exactly the opposite of 'outlawing thinking'.
xanderlewis··on Responsible Release of AI-Generated Mathematics
> AI mathematicians have been around for a month or so.

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).

xanderlewis··on Evan Doorbell's Phone Tapes – Brought to You by Telephone World
“Show HN: I used Claude to scrape Evan Doorbell’s Phone Tapes”
xanderlewis··on Evan Doorbell's Phone Tapes – Brought to You by Telephone World
Glad to see this here. Evan Doorbell’s YouTube channel is one of my favourites ever.
Page 1 of 33Next →