Gowers's Weblog

Since there are a lot of comments here from people who were interested in PCM but not sure about buying it, I'd like to remind everyone that, if you have access to a university library, then it'll definitely have it; and your public library might not have the book, but can probably get it by ILL (a badly undervalued resource, in my experience). A lot of contributions are also available from their individual authors online. Its editor, Timothy Gowers, also has a very nice blog https://gowers.wordpress.com which is worth reading by specialists and non-specialists alike. (However, as brilliant as Gowers is, I think neither he nor anyone else could have dropped the book in your lap two decades ago, before it was compiled. :-) )

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…

Hi Richard, thanks and good luck to you! A couple of thoughts on things I would love to see. First of all, if research papers could be associated with discussions that would be great. The arXiv is great for math, but you can't ask questions other than e-mailing the author. Imagine that someone posted a paper, and you asked some question "Doesn't XYZ also relate to ABC"?, etc., etc. There is some of this on people's blogs I know but it would be great if there was a lot of this in one place. You might also read http://meta.mathoverflow.net/ -- MathOverflow (mathoverflow.net) is great, but I feel there are a lot of pain points you'll notice there, and there is room to do things which are outside of MO's scope. Also, check out Tim Gowers's blog -- http://gowers.wordpress.com/ -- even if you don't know abstract math, it's well worth reading. His math posts lead to the kinds of discussion I would love a more systematic venue for that is not run by a single person. I can more or less speak for my discipline (AMA), but I understand different disciplines have very different cultures! Even CS is quite different from math from what I've heard. Again, good luck to you!

This viewpoint is curdling rapidly. The definition of "reasoning" and "intelligence" will be debated for ages by philosophers and cognitive scientists, but whatever type of logical/critical thinking is going on in the heads of software engineers and mathematicians, frontier LLMs can now emulate to a very high degree. For mathematics in particular, examples like the following will become common place: https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatg…

There are very cheap arXiv overlay journals where the processing fee is about $10 per submission.[1] And that’s with a for profit entity providing the platform. Reviewing is already unpaid, editing is often unpaid even with traditional journals. Your “journals die as a result” assumption is faulty. Not to mention for quite a few fields you can already find every remotely worthwhile paper on arXiv. Those fields didn’t collapse. [1] https://gowers.wordpress.com/2015/09/10/discrete-analysis-an-arxiv-ove…

Every time I encounter anything from Gowers I'm impressed by his humility and real concern for people's struggles with learning mathematics - particularly at the undergraduate level. He did a great blog when teaching Real Analysis to Cambridge first years. https://gowers.wordpress.com/2014/01/11/introduction-to-cambridge-ia-a…

Tao's Solving Mathematical Problems: A Personal Perspective (https://www.amazon.com/Solving-Mathematical-Problems-Personal-Perspect… mentioned here is an excellent book that provides keys to mathematical approach to problems and is now a classic ranking there with Polya's How to Solve It. One thing the article doesn't touch on is Tao's prominence in massively collaborative math research through his blog (latest such work from there was discussed on HN a day ago https://news.ycombinator.com/item?id=21542054). This new approach, first proposed by Tom Gowers (https://gowers.wordpress.com/2009/01/27/is-massively-collaborative-mat… in 2009 has successfully solved a number of open problems in the past ten years (https://polymathprojects.org/2019/02/03/ten-years-of-polymath/).

Yep. Discrete Analysis is perhaps one of the best known of those. https://gowers.wordpress.com/2015/09/10/discrete-analysis-an-arxiv-ove… https://discreteanalysisjournal.com/

Even in pure mathematics, a good rule is to give examples (if simple) right after the definitions. Examples always help. Here's a post by the Fields medallist Tim Gowers: https://gowers.wordpress.com/2007/10/19/my-favourite-pedagogical-princ…

I've always liked Tim Gowers' welcome to incoming students to the Cambridge Mathematical Tripos program. Lots of great tips on learning/studying math, asking intelligent questions, etc. https://gowers.wordpress.com/2011/09/23/welcome-to-the-cambridge-mathe…

Tim Gowers is an amazing mathematician and a person of great integrity, but this is not exactly news given that the boycott has gone on for so long now (see the date on the article). More recently, in March 2016 he founded a new ultra-low-cost journal (Discrete Analysis) which publishes all articles on arXiv. https://gowers.wordpress.com/2016/03/01/discrete-analysis-launched/

The system works smoothly for them. Not much incentive to change it unless they are prepared to think a bit more widely. One great exception is Timothy Gowers. "it should in principle be possible to run a journal for almost nothing. The legacy publishers (as they are sometimes called) frequently call people naive for suggesting this, so it is important to have actual examples to prove it, and Discrete Analysis is set up to be one such example." https://gowers.wordpress.com/2016/03/01/discrete-analysis-launched/

Well, Gowers went at great length explaining its motivations [0] (see the following quote, from [1]). So saying that Gowers started attacking Elsevier "by accident" and "was surprised that it grew as a movement" is very disrespectful to Gowers, to truth, and also a bit to the reader. >"Once I did hear about Elsevier’s behaviour, I made a conscious decision not to publish in Elsevier journals and I started to feel bad about cooperating with them in any way. [...] Now, however, I have decided that my previous quiet approach was not enough." [0] https://gowers.wordpress.com/category/elsevier/ [1] https://gowers.wordpress.com/2012/01/21/elsevier-my-part-in-its-downfa…

[Cannot edit the parent, sorry] There is a somewhat embarrassing (to my taste) discussion here http://gowers.wordpress.com/2013/03/20/pierre-deligne-wins-the-2013-ab… in which they claim that it is policy of the Abel Foundation not to bestow the prize on people who would not attend the ceremony. This strikes me as surprsing at the very least but you can never tell with foundations. I am thinking in three centuries' time people will say: "Of course, they gave Grothendieck one of the first Abel prizes" and receiving a sorry answer "well, I say, no... it was a question of policy". Kings are not kings if they do not survive someone rejecting their bounties.

It's worth noting that Cosma Shalizi has been doing this for about six years now, and it's becoming increasingly popular in certain areas of academia. Background: http://masi.cscs.lsa.umich.edu/~crshalizi/weblog/442.html http://masi.cscs.lsa.umich.edu/~crshalizi/weblog/864.html http://gowers.wordpress.com/2012/01/21/elsevier-my-part-in-its-downfal…

The blog is incredibly open. seeing it all go up with the notes around the creative process is quite fascinating even if you cant follow all the math [and I guess only 50 people in the world can]. Having the math inline works really well [wordpress renders latex formulas as pics with the latex as alt text] - you don't have to jump into a pdf viewer while parsing the main text. Its one way to get started with latex, ie. by example. Terry and Timothy Gowers are experimenting with an approach to creating math with lots of small involvements by many contributors online - see http://gowers.wordpress.com/2009/03/24/can-polymath-be-scaled-up/

I used to read Tim's expository writings religiously. The old ones can be found at: http://www.dpmms.cam.ac.uk/~wtg10/mathsindex.html and the newer stuff is on his blog: http://gowers.wordpress.com/

These are most of those I follow... as you'll see, it's very difficult to find good material on maths and physics blogs, or at least I've had difficulties on finding it. Maths: http://www.arsmathematica.net/ http://scienceblogs.com/goodmath/ http://lambda-the-ultimate.org/ (not exactly maths but...) http://plus.maths.org/blog/ http://terrytao.wordpress.com/ http://golem.ph.utexas.edu/category/ http://blog.plover.com/ (sometimes) http://blog.wolfram.com/ http://gowers.wordpress.com/ Physics: http://backreaction.blogspot.com/ http://www.math.columbia.edu/~woit/wordpress (very biased) http://angryphysicist.wordpress.com/ http://scottaaronson.com/blog http://motls.blogspot.com/ (very biased)

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…

> 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-act…

* https://gowers.wordpress.com/ His surname is 'Gowers'!

A lot of odd and downright false assertions ITT. Anti-Brexit people are too often mistakenly conflating Cummings/Johnson and Farage as if everyone in favour of Brexit has precisely the same views. In reality they ran two different campaigns in very different styles and in fact one of Vote Leave's main aims was to keep Farage off the TV as much as possible because he was (and is) a turn off to the majority and presumably most swing voters. I voted Remain at the time, but stumbling across Cummings' blogs and subsequently reading about Tetlock [0] + David Deutsch's arguments [1] has left me in favour of Leave (although I don't have strongly nailed down views on this). This project is a great idea to improve the effectiveness of government by creating tools that will help ministers and officials make better decisions in the face of complex systems. Will it work? I'm optimistic but there will surely be unknown unknowns that could derail progress in addition to plain old politics. [0] - Superforecasting is a great book: https://www.amazon.co.uk/Superforecasting-Science-Prediction-Philip-Te… [1] - https://www.youtube.com/watch?v=xdtssXITXuE Also, Timothy Gowers (Fields Medallist) has an interesting piece in favour Remain [2], although he overlooks the fact that differences in institutional design of UK vs EU mean that sovereignty has significant implications beyond just sovereignty for sovereignty's sake. [2] -…

Indeed. And this applies to mathematics too. Here is Ralph Boas, in Can We Make Mathematics Intelligible? following his own advice: > Suppose you want to teach the 'cat' concept to a very young child. Do you explain that a cat is a relatively small, primarily carnivorous mammal with retractable claws, a distinctive sonic output, etc.? I'll bet not. You probably show the kid a lot of different cats saying 'kitty' each time until it gets the idea. To put it more generally, generalizations are best made by abstraction from experience. (See also Tim (Sir Timothy) Gowers on Examples first: https://gowers.wordpress.com/2007/10/19/my-favourite-pedagogical-princ… and https://gowers.wordpress.com/2007/10/24/examples-first-ii/) (I frankly think this applies to the accursed phenomenon of monad tutorials in Haskell too (https://stackoverflow.com/questions/3261729/monad-in-non-programming-t… but some disagree.)

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…

Indeed the typical wisdom in math is that the best way to understand a group (and the reason we usually care about one) is its actions. There's some quote to the effect of: "Just as with men, a group is known by its actions." This was the only reference I could find off the top of my head https://gowers.wordpress.com/2011/11/06/group-actions-i/, but I've definitely heard the sentiment repeated in a couple different texts. The sentiment expressed in this article might useful to hear if you're solely interested in learning how to use monad transformers or whatever in Haskell, eg, I know a fair amount of math, but very little category theory, and manage use monads just fine without any deep intuition as to why they're an important formalism. But if you want to develop your problem-solving skills, for example, I think "quiet contemplation" of axioms is pretty much always worse than working through several examples and absorbing some good exposition. Here's a recent article that I found very though-provoking: http://cognitivemedium.com/invisible_visible/invisible_visible.html. They quote some mathematicians discussing their visual intuition for one property of groups.

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…

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

I am a PhD student in mathematical statistics. The people I have spoken too think this is very exciting and cool. There is also a sense of unease about what this will mean in terms of being a mathematician, and what effects it will have on our future employment. I am also a little worried about what it means for your training as a junior PhD. Often you would try and solve a problem your advisor thinks is doable that they assign to you as a learning exercise. It may be more and more difficult to find problems that a junior PhD can solve but that AI can not. Tim Gowers has written about that here: https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatg…

Link to source blog post: https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatg…

A very interesting comment from Baez, I'll just quote part of it. > Where does the value of thinking and having deep ideas come from? We need to think about this now. If it comes primarily from their scarcity – the fact that having certain ideas is hard – then indeed this value may drop precipitously when the manufacture of ideas can be automated. But if the value comes from the utility of the ideas – the benefit that the idea brings – then the story changes: perhaps creating more good ideas is actually better, not worse. Here I’m using “utility” in a broad sense, not just in the sense of what people often call applied mathematics. > In other words, mathematicians may need to adjust to a transformation from a scarcity economy to an abundance economy. https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatg…