On proof and progress in mathematics · HackerPapers discussions

On proof and progress in mathematics — mentioned by martincsweiss

Fri, 23 Oct 2026 12:56:36 GMT

I'm working on a piece inspired in large part by William Thurston's "On proof and progress in mathematics"[1]. Should publish it next week. Part of why math is effective is that its extraordinarily compressible. And you can take those ideas and apply them elsewhere (composition works well). The natural world has a lot of detail and is therefore hard to compress. That's why it's a good source of new ideas. Consider the relation between the McCulloch–Pitts neuron (MCP lol)[2], developed to model biological spiking neural networks, and an inspiration for modern artificial neural networks. [1]…

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On proof and progress in mathematics — mentioned by sanxiyn

Thu, 08 Oct 2026 06:25:41 GMT

This is basically On proof and progress in mathematics by Thurston restated. When Thurston wrote it in 1994, many people didn't understand what he is talking about. https://arxiv.org/abs/math/9404236

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On proof and progress in mathematics — HN submission

Fri, 25 Sep 2026 10:35:35 GMT

Proof and Progress in Mathematics (1994)

3 points; 1 discussion reply on HN. Paper and source history. A submission is not an endorsement.

On proof and progress in mathematics — mentioned by bwfan123

Tue, 15 Sep 2026 15:37:08 GMT

> Understanding the concrete problem mathematicians are upset about can help us better understand the impact of AI on our own fields, imo, The author of this essay does not understand the concrete problem that mathematicians are upset about. There is an idea that math [1] and coding [2] are human activities whose purpose is to achieve a certain kind of insight or mental clarity of things. The simplest description of this is by Feyman [3]. AI generated proofs short-circuit human understanding and therefore goes against the primary purpose. The declaration is calling this out loudly to…

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On proof and progress in mathematics — mentioned by curt15

Sun, 13 Sep 2026 15:33:41 GMT

> He’s also always argued that mathematics is fundamentally about human understanding. It's also worth pointing out he didn't come up with this viewpoint just to "cope" with the headlines. Thurston articulated this way back in the 90s (https://arxiv.org/pdf/math/9404236) and also more recently (https://mathoverflow.net/questions/43690/whats-a-mathematici...): > The product of mathematics is clarity and understanding. Not theorems, by themselves. Is there, for example any real reason that even such famous results as Fermat's Last Theorem, or the Poincaré conjecture, really matter? Their real…

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On proof and progress in mathematics — mentioned by lhd1

Sat, 12 Sep 2026 07:52:55 GMT

If you consider this event in isolation it is cause for celebration. But the controversy around this isn't so much about how the proof was obtained but what this means for the practice of mathematics going forward. It seems we can probably expect more and more results of this nature being dumped into the community. It's happened before that one person, Bill Thurston, was so successful in his field, proving theorem after theorem, that he inadvertently killed his field. People hesitated to enter his field, knowing that they could be scooped at any moment. It took years before his field…

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On proof and progress in mathematics — mentioned by fultonn

Wed, 19 Aug 2026 21:08:42 GMT

People say similar things about automation of software engineering. Different, but similar. I'm deeply suspicious. I do not yet have a concise statement for why, but a lot of literature on the sociology of knowledge work sort of points at my thoughts. Section 5 of the Thurston article cited by Tao touches the elephant. Raduchel's article on the economics of software [2] also touches it. I've tried to put words to this for a few years. I think I'm just going to start writing versions of it as see if that helps me shape the thought into something more concise. So, in the spirit of this…

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On proof and progress in mathematics — mentioned by BeetleB

Wed, 19 Aug 2026 20:41:43 GMT

The essay On Proof and Progress in Mathematics by a Field's medalist is worth reading: https://arxiv.org/abs/math/9404236 He wrote it in 1994. He writes about how he almost "destroyed" a subdiscipline in mathematics by becoming so good at it that he outclassed everyone. PhD students were advised to stay away from the whole field. When he discovered this, he realized his error was that he was focusing on producing results, and not focusing on explaining his thought process. It's that thought process that is valuable in advancing the frontier - results alone won't do it. It didn't matter how…

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On proof and progress in mathematics — mentioned by woopwoop

Mon, 30 Mar 2026 14:40:28 GMT

Rigor is not the whole point of math. Understanding is. Rigor is a tool for producing understanding. For a further articulation of this point, see https://arxiv.org/abs/math/9404236

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On proof and progress in mathematics — mentioned by physidev

Sat, 30 Aug 2025 20:06:35 GMT

I think the viewpoint articulated in this post fits quite well with the one expressed in the often-shared "Programming as Theory-building" article (I think it was shared here just a few days ago). Scientists, mathematicians, and software engineers are all really doing similar things: they want to understand something, be it a physical system, an abstract mathematical object, or a computer program. Then, they use some sort of language to describe that understanding, be it casual speech, formal mathematical rigor, scientific jargon -- or even code. In fact, thinking about it, the code…

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On proof and progress in mathematics — HN submission

Sat, 23 Aug 2025 08:47:53 GMT

Proof and Progress in Mathematics

2 points; 0 discussion replies on HN. Paper and source history. A submission is not an endorsement.

On proof and progress in mathematics — mentioned by enugu

Thu, 12 Dec 2024 17:28:50 GMT

Don't think you should be intimidated just by reading the article itself. It is using several domain specific terms. But you would encounter that in many other contexts - for instance a group discussing an intricate board game you are seeing for the first time. However, unlike board games where the concepts can be explained to you in a few minutes (usually it becomes clear only when you play), a lot mathematics especially algebraic geometry/number-theory can have many layers of dependencies which takes years to absorb. It would be interesting to compare it to understanding a large open source…

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On proof and progress in mathematics — HN submission

Wed, 19 Jun 2024 14:05:24 GMT

Proof and Progress in Mathematics (1994)

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On proof and progress in mathematics — mentioned by svat

Mon, 20 Nov 2023 00:53:42 GMT

TLDR it's 286,386,577,668,298,411,128,469,151,667,598,498,812,366 — there, saved you a click. [The above is a parody of a kind of HN comment common on long articles (that wants an article to simply "answer" the headline), but this case—the numerical answer being not at all illuminating in comparison to actually reading the article—is also a clear illustration of this point from the Fields medalist William Thurston's very insightful "On Proof and Progress in Mathematics" (https://arxiv.org/abs/math/9404236): > They might print out a table of the first 10,000 primes, only to find that their…

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On proof and progress in mathematics — mentioned by oblmov

Fri, 10 Nov 2023 04:24:29 GMT

All of the mathematicians I mentioned believed philosophy was relevant to their mathematical work, and that period of work on the foundations of mathematics was accompanied by extensive discussion of the work of Frege, Russell, Wittgenstein, etc. Even if we pretend philosophy never involves rigorous proofs, mathematical theorems do not spring out of thin air, and saying "anything that isn't a formal, rigorous proof is useless to mathematics" is like saying "anything that isn't a finished house is useless to the process of building a house". https://arxiv.org/pdf/math/9404236.pdf discusses…

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On proof and progress in mathematics — mentioned by joewferrara

Tue, 18 Jul 2023 20:43:04 GMT

This essay by the same person who answered the mathoverflow post greatly expands on the ideas in the post. https://arxiv.org/pdf/math/9404236.pdf It’s a great read for young people interested in mathematics and influenced me while in undergrad deciding to persue a PhD in math. Thurston, the author, was one of the best mathematicians of his time.

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On proof and progress in mathematics — mentioned by bjornlouser

Tue, 18 Jul 2023 16:21:27 GMT

Mentioned in a comment of that MO question is Thurston's "On proof and progress in mathematics" https://arxiv.org/abs/math/9404236

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On proof and progress in mathematics — mentioned by jonsen

Tue, 18 Jul 2023 10:26:25 GMT

From a comment there: > Thematically this is similar to a (much longer) essay Dr. Thurston wrote several years ago, which I highly recommend: https://arxiv.org/abs/math/9404236

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On proof and progress in mathematics — HN submission

Mon, 26 Jun 2023 12:27:22 GMT

Ask HN: What are you farovite essay-papers?

Most of papers deal with the technical details of experiments, but some of my favorite papers belong to a different type, these are often less technical, penned by one author and almost essay-like. Examples: By far my favorite, More is Different by P.W. Anderson: https://www.tkm.kit.edu/downloads/TKM1_2011_more_is_different_PWA.pdf Rodney Brooks' Elephants Don't Play Chess also comes to mind: https://www2.cs.sfu.ca/~vaughan/teaching/894/papers/elephants.pdf Another example is On proof and progress in mathematics, William P. Thurston: https://arxiv.org/pdf/math/9404236.pdf And if non-paper…

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On proof and progress in mathematics — mentioned by BeetleB

Thu, 15 Jun 2023 18:31:49 GMT

> Mathematical notation is a programming language. It actually isn't - the notation has lots of ambiguities that don't confuse mathematicians because they are aware of the context. On Proof and Progress in Mathematics[1][2] is a great essay by a Field's medalist. Part of the essay discusses this very topic, and why mathematicians choose not to use a formal notation for every day work. Some quotes from it: > The standard of correctness and completeness necessary to get a computer program to work at all is a couple of orders of magnitude higher than the mathematical community’s standard of…

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On proof and progress in mathematics — mentioned by throwaway81523

Mon, 26 Jul 2021 06:55:53 GMT

There is mathoverflow.net and its meta site as math researcher hangouts, plus ncatlab.org for category theory, and various sites, blogs, mailing lists etc. for other fields. You might also like Bill Thurston's famous article "On Proof and Progress in Mathematics", which says not so much that math is social, but that human mathematicians want personal understanding of mathematical truths. So black box computer-checked but human-incomprehensible proofs aren't of that much appeal. https://arxiv.org/abs/math/9404236

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On proof and progress in mathematics — mentioned by throwaway81523

Sun, 04 Jul 2021 08:46:47 GMT

Proofs are more about understanding why something is true, than verification per se. See this well known article by the illustrious Bill Thurston (RIP): https://arxiv.org/abs/math/9404236

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On proof and progress in mathematics — mentioned by jasonhong

Thu, 20 Aug 2020 12:20:41 GMT

Famed mathematician William Thurston has a good response here (from On Proof and Progress in Mathematics https://arxiv.org/pdf/math/9404236.pdf): "The rapid advance of computers has helped dramatize this point, because computers and people are very different. For instance, when Appel and Haken completed a proof of the 4-color map theorem using a massive automatic computation, it evoked much controversy. I interpret the controversy as having little to do with doubt people had as to the veracity of the theorem or the correctness of the proof. Rather, it reflected a continuing desire for human…

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On proof and progress in mathematics — mentioned by raincom

Wed, 17 Jun 2020 18:14:47 GMT

Human understanding is critical. See what Bill Thurston said:"The rapid advance of computers has helped dramatize this point, because computers and people are very different. For instance, when Appel and Haken completed a proof of the 4-color map theorem using a massive automatic computation, it evoked much controversy. I interpret the controversy as having little to do with doubt people had as to the veracity of the theorem or the correctness of the proof. Rather, it reflected a continuing desire for human understanding of a proof, in addition to knowledge that the theorem is true." From…

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On proof and progress in mathematics — mentioned by 19f191ty

Tue, 09 Jun 2020 02:23:33 GMT

Here's a longer article by Thurston on the same ideas https://arxiv.org/abs/math/9404236. Highly recommend irrespective of how involved you are with mathematics.

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On proof and progress in mathematics — mentioned by ssivark

Wed, 03 Jun 2020 19:05:35 GMT

I highly recommend Bill Thurston’s gem of an article On proof and progress in mathematics https://arxiv.org/abs/math/9404236 Talks about the human aspect of pursuing mathematical research, how they shape the attitude of the field towards a problem abs are crucial in progressing towards knowledge. Should be very readable for everyone; no formal math as such.

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On proof and progress in mathematics — HN submission

Sat, 14 Dec 2019 00:55:23 GMT

On proof and progress in mathematics (1994) [pdf]

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On proof and progress in mathematics — mentioned by svat

Sun, 15 Oct 2017 15:36:31 GMT

Wouldn't that be backwards? The purpose of notation is to aid human thought and communication. Technology should adapt to whatever notation would be most useful to mathematicians, rather than mathematicians adapting their notation for the sake of technology (like “more easily typesettable”). The blackboard-bold fonts are an interesting example of this, with computer typefaces being created for what mathematicians use in the real world. (Though Knuth and Serre and some other mathematicians don't approve of them in typesetting.) Besides, typesetting is a small part of the usage of notation…

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On proof and progress in mathematics — HN submission

Wed, 28 Jun 2017 15:17:43 GMT

On proof and progress in mathematics (1994)

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On proof and progress in mathematics — mentioned by mrkgnao

Thu, 23 Mar 2017 15:45:52 GMT

You're not stupid. At least, there is very little chance that the math you're trying to work on right now requires super-normal cognitive abilities of any sort: it's more about learning to be comfortable with the ambiguity or lack of comprehension that the process of learning entails. Here's a grab-bag of stuff that have helped me (and still do) lift myself out of "damn I'm dumb" situations such as the one you seem to find yourself in. "Does math have to be ambiguous or cryptically written? That doesn't sound right." Well, no. Sometimes, the solution is just "get a better book" (although, see…

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On proof and progress in mathematics — mentioned by posterboy

Sat, 13 Aug 2016 10:13:28 GMT

Site's down. Google is yielding a link to the arxiv. I didn't know the site was that old. https://arxiv.org/abs/math/9404236

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On proof and progress in mathematics — mentioned by mrkgnao

Sat, 21 May 2016 11:59:17 GMT

If I remember correctly, you could forget the "inverses" bit and define a group action with only the monoid corresponding to the group. So not all the "groupiness" is captured by actions. Also, Bill Thurston's beautiful "On Proof and Progress in Mathematics" [1] contains several ways to understand the derivative, in addition to it being a look into what a very distinguished mathematician thought of the nature of the mathematical pursuit. Pretty similar ideas, explored in depth. Edit: Apparently this is linked to in TFA :) [1]: https://arxiv.org/abs/math/9404236

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On proof and progress in mathematics — mentioned by erje

Fri, 10 Jul 2015 10:16:37 GMT

If you haven't already, take a look at Bill Thurston's paper "On proof and progress in mathematics" http://arxiv.org/abs/math/9404236 Your comment could very well be its abstract.

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On proof and progress in mathematics — HN submission

Fri, 29 May 2015 12:30:04 GMT

On proof and progress in mathematics (1994)

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On proof and progress in mathematics — mentioned by davmre

Sat, 07 Mar 2015 18:19:30 GMT

The great thing about math research, or really research in any field, is that it's highly parallelizeable. There are a few geniuses like Terry Tao exploring specific directions of thought, but there are so many things to think about and problems to solve that even the best researchers can only focus on a very small fraction. For every genius there are hundreds of other researchers who are only, say, 10% as productive (this is maybe your typical professor at a top-20 school), but through virtue of numbers they can explore a much wider range of problems and have a large aggregate impact.…

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On proof and progress in mathematics — mentioned by cmrx64

Sun, 30 Nov 2014 02:44:50 GMT

William P. Thurston 1994, "On proof and progress in mathematics" Gives a good amount of insight into how academia works for mathematics, and gives a good contrast with how CS works. Don't be scared by the abstract, it's a completely non-technical paper. The academic/research culture can be more important than the results. http://arxiv.org/abs/math.HO/9404236

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On proof and progress in mathematics — mentioned by bkcooper

Thu, 04 Sep 2014 03:07:13 GMT

That's an excellent response. In one of the comments on it, there is a link to a paper[1] by Thurston that is well worth your time. I found it an inspiring read. [1] http://arxiv.org/abs/math/9404236

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On proof and progress in mathematics — mentioned by murbard2

Mon, 05 May 2014 13:34:15 GMT

Source: http://arxiv.org/pdf/math.ho/9404236.pdf

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On proof and progress in mathematics — mentioned by gjm11

Sun, 12 Jan 2014 13:05:33 GMT

No disrespect, but dkural's idea of what mathematics is about is no more irrelevant than sillysaurus2's or yours, and dkural's view is quite well represented among research mathematicians, first-rate ones included. For a famous example, see William Thurston's "On proof and progress in mathematics" at http://arxiv.org/pdf/math/9404236v1.pdf .

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On proof and progress in mathematics — mentioned by jordigh

Thu, 24 Oct 2013 18:57:19 GMT

I find it funny how you think of differentiation in terms of "frequency domain", "bandlimit" and "filter". Very signal-processy, very engineery. A mathematician would speak about unbounded operators in a Banach space. :-) Edit: Reminds me of the list on page 3 here: http://arxiv.org/pdf/math/9404236v1

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On proof and progress in mathematics — mentioned by proofschool

Sun, 25 Aug 2013 15:42:12 GMT

Quick response from Proof School co-founder: Can't comment on all the critiques here, but thank you for them, especially those intended to be constructive--the board of directors and advisory board members will consider some of points made here. If you haven't already, I encourage you to check out who's supporting Proof School. We are fortunate to have their backing and advice. First, to user interface... the pitch deck format of the current website is meant more as a landing page to pique curiosity and to solicit interest--it will change to a more familiar UX soon. Apologies if it's slow or…

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On proof and progress in mathematics — mentioned by calibraxis

Mon, 19 Aug 2013 11:43:26 GMT

Interesting argument, though I currently think correctness is more important in programming. Their proofs are hand-wavy compared to our programs. Thurston expands, "It’s just that the reliability does not primarily come from mathematicians formally checking formal arguments; it comes from mathematicians thinking carefully and critically about mathematical ideas." (http://arxiv.org/abs/math/9404236) But I could also accept the claim that mathematics is more intellectually gratifying, if they're not so much at the mercy of standardization and other boring fiddly issues. I don't know.

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On proof and progress in mathematics — HN submission

Thu, 23 Aug 2012 19:02:34 GMT

What do Mathematicians Do?

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On proof and progress in mathematics — mentioned by noblethrasher

Wed, 22 Aug 2012 23:14:44 GMT

It's been a while since I've done serious mathematics so I'm almost certainly wrong about this, but the "logical" definition for a derivative looks incorrect under the section "How do People Understand Mathematics?" (pg 3). Shouldn't the distance between the quotient and d be less than ɛ? http://arxiv.org/pdf/math/9404236v1.pdf

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On proof and progress in mathematics — mentioned by gruseom

Wed, 22 Aug 2012 22:01:34 GMT

I agree with everyone who recommends the stellar "On Proof and Progress in Mathematics". It a gem that could only have been written by a lovely man. http://arxiv.org/pdf/math/9404236v1.pdf The parts about how communication works inside mathematical specialties and about the difference between math and computer programming are worth their weight in gold. Really, everyone, just do yourself a favor and read it – all the way to its astonishing final section and beautiful ending. It doesn't require you to remember any math.

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On proof and progress in mathematics — mentioned by tokenadult

Wed, 22 Aug 2012 21:25:30 GMT

Fields medalist Terence Tao writes about earlier Fields medalist William Thurston, "In addition to his direct mathematical research contributions, Thurston was also an amazing mathematical expositor, having the rare knack of being able to describe the process of mathematical thinking in addition to the results of that process and the intuition underlying it. His wonderful essay “On proof and progress in mathematics“, which I highly recommend, is the quintessential instance of this; more recent examples include his many insightful questions and answers on MathOverflow." Besides "On proof and…

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On proof and progress in mathematics — mentioned by sz

Sat, 21 Aug 2010 20:11:40 GMT

Bill Thurston describes math as the theory of formal patterns. He has a nice article here: http://arxiv.org/abs/math.HO/9404236

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On proof and progress in mathematics — HN submission

Mon, 07 Dec 2009 22:17:51 GMT

On proof and progress in mathematics (1994)

25 points; 5 discussion replies on HN. Paper and source history. A submission is not an endorsement.

On proof and progress in mathematics — mentioned by mreid

Mon, 07 Dec 2009 22:11:18 GMT

Great points, especially the one on multiple perspectives. If you haven't read it already I highly recommend Thurston's essay "On proofs and progress in mathematics" - http://arxiv.org/abs/math/9404236 .

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On proof and progress in mathematics — mentioned by btilly

Tue, 10 Nov 2009 20:42:34 GMT

I accept your correction that there is a difference between "promoting progress" and "coming up with ideas". (By coincidence I just reread http://arxiv.org/abs/math.HO/9404236 which argues, among other things, that progress in mathematics can be helped by not proving facts too quickly.) I agree with you on the intent of the design of the patent system. However in a world filled with overbroad junk patents, where accidental infringement is much better than willful infringement, there are real incentives for innovative people to _not_ read patents. Because reading them opens you up to liability…

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