Gowers's Weblog

I want to preface my response by saying that I don't buy most of what the AI labs say. I don't think that LLMs will replace most white collar labor for example. I also find many of the practices of these labs to be abhorrent. However, all of these opinions are orthogonal to the fact that LLMs have gotten extremely good at mathematics.

> Who says no one was making progress?

Let's look at the Jacobian conjecture, since that was the open math problem I was most familiar with prior to its solution. Yitang Zhang, one of the worlds most renown mathematicians (famous for his lower bound on the twin prime conjecture) spent 8 years working on this problem with his advisor (who himself is a renown mathematician) and turned up completely empty handed. His advisor described it as a "waste [of] 7 years of his own life and my time" [1]. Of course, these two were not the only ones working on this problem for the almost 100 years its been open, but they should have sufficient credentials to show that they were not fools or amateurs.

And in a single afternoon an LLM disproved the conjecture. How is that not an extraordinary feat of technology?

> Who? And doing what?

A close friend is studying differential geometry in a PhD program. Sadly I doubt anything I say on his work will convince you, so I will instead offer two anecdotes:

Terrence Tao (widely considered the worlds greatest living mathematician) has said AI is precipitating "a crisis in the foundations of mathematical values and…

It's important to distinguish between researchers, reviewers/editors and publishers.

The academics who write a paper or make the decision to include it in a journal typically aren't paid. They benefit from the prestige signal generated by a famous journal (editors gain some control of research directions in their field and young academics get a signal validating the quality of their work). Publishers do the typesetting, hosting, distribution, marketing and other admin responsibilities and get grossly overpaid for this task [1].

The publishers clearly want to maintain the status-quo. However, I think both authors and editors would prefer lower journal subscription fees/greater open access for the papers. It would lead to lower costs for universities and broader audiences. However, coordinating away to this outcome is tricky, since (as you say) upstart journals are less prestigious so would offer lower initial influence and prestige to defectors. This is basically the classic path dependence problem from economics[2].

There are a couple of possible solutions to the problems:

(1) Incumbent publishers could be breaching European and American antitrust law. Competition authorities could prosecute them for charging excessive prices, bundling practices or placing unreasonable restrictions on academics wishing to make their research open access? Governments could also address this with specific policies.

(2) High profile academics, newly emerging fields or those particularly…

In the quest to prove P != NP a great number of techniques have been tried, and many of these techniques have attempted to use circuit complexity. The circuit complexity of a function with n inputs is, roughly, the size of the smallest circuit that computes it. This is a function depending on n, and so we can talk about polynomial size circuits, etc. The important fact is that if we can find a specific problem (a problem in NP) which doesn't have a polynomial size circuit, then P is not NP.

Unfortunately many of the elementary techniques that try to show this don't work, and for a long time nobody knew why. Then in the 90's Alexander Razborov and Steven Rudich proved a general theorem that these elementary techniques could never prove P != NP.

What Gowers explains in his blog post (and the follow-up) is the Razborov-Rudich theorem, called natural proofs. Roughly, it says that all of the attempted techniques use a "natural" property P of boolean functions. A natural property P asserts one specific function needs large circuits, while all small circuits cannot satisfy the property (and P is efficiently computable, in a sense).

The insight is that all of these properties also apply to TOO MANY other functions. Indeed, if P is natural for polynomial size circuits, and it applies to too many functions, then it can be used to violate another well known conjecture in cryptography, a conjecture on the existence of a certain kind of secure pseudorandom number generator. This is…

HN has always been a great place to get news not being reported elsewhere

I'll let other users here check my user profile, including its extensive list of submissions, and then let me know how I fit into the community here by how I submit articles. On my part, I agree that HN has long been a place where one can find stories that aren't reported elsewhere, and that can be both a feature and a bug. HN is an especially good place to find breaking news on its core topics (?) of start-up business and new applications of software technology. It is also a good place to find thoughtful discussion on stories from a variety of sources about the impact of technology on society, education policy in light of technological changes (what drew me here), and the personal development of hackers (what keeps me here, as the parent of four children who may all become hackers). I love the thoughtful comments here, even on metadiscussion topics like the issue of "What's on topic on HN?"

That said, I don't find it bad to find story sources here that are "mainstream," or sources one could find elsewhere. We still won't see the same kinds of comments on those stories elsewhere that we can find here on HN. Indeed, compared to many of the blogs that are submitted as sources on HN, I would MUCH rather see more stories from the Wall Street Journal, the New York Times, National Public Radio, Time magazine, and The Atlantic, simply because those are all professionally edited sources that often (not…

It depends entirely on how many people you're teaching. If you're sitting in a lecture hall full of 250 students, asking continual questions is just not going to get any response. No-one wants to embarass themselves in front of an audience that large. You'll just slow down the lecture while you pause and listen to a stony 10 second silence every couple of minutes. I've had lecturers do this, and it's extremely frustrating.

Personally, I don't mind lecturers "telling" the answers straight away. Obviously give motivation and lead into it rather than just delivering it with no introduction, but I've found a lot of problems are simply too large or too non-obvious to take a straw poll from the class.

The best teachers I've had have been:

(i) Really bloody good at their subject. My probability lecturer (the Field's medallist Tim Gowers), when presenting the proof of Stirling's approximation, said something along the lines of "now, the normal method for proving this bit is a bit complicated, and on my cycle in this morning I thought 'There must be an easier way', when the following struck me. Now, I've never written this down before, so hopefully it works...", and of course laid out a simplification of a famous proof that worked perfectly. Inspirational. (See http://gowers.wordpress.com/2008/02/01/removing-the-magic-from-stirlings-formula/)

(ii) Incredible elucidators. The ability to take a complex concept and pick exactly the right words or metaphors to explain it is a…

> Naturally, they don't explain all the ways they didn't solve the problem when they finally do solve a problem.

I do think something related to this is less obvious than you're suggesting. A proof generally presents a train of thought gets you from point A to point B. It's almost never a representation of the train of thought that the author actually followed to get from point A to point B. This is obvious to people with experience in mathematics, but it's much less obvious before you have that experience. Even if you know it's true in principle, appreciating it viscerally and adapting the way you interact with mathematics to account for it is basically a lifelong task.

A big component of good mathematical communication is explaining what motivates a solution to a problem (or even what motivates the problems in the first place). Sure, you don't have to do that every time, but it's fair criticize a broad lack of this. A lot of mathematical communication is... not good in this sense. Notably, the problem/solution in the OP isn't really motivated by the author. A skilled reader will think about it themselves, but they need to learn that skill somewhere.

More or less on topic, here's an example where an experienced mathematician tries to reproduce one of Sylow's theorems without looking it up, and writes down a more-honest account of the thought process involved: https://gowers.wordpress.com/2011/12/10/group-actions-iv-intrinsic-actions/