arXiv:math/9404236math.HO
In response to Jaffe and Quinn [math.HO/9307227], the author discusses forms of progress in mathematics that are not captured by formal proofs of theorems, especially in his own work in the theory of foliations and geometrization of 3-manifolds and dynamical systems.
Linked by 37 identifiable commenters in 41 comments across 36 threads, from Nov 2009 to Oct 2026. Found in 9 HN submissions from Dec 2009 to Sep 2026. Open submission discussions.
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
> 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…
> 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…
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…
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…
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…
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
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…
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…
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…
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…
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
Mentioned in a comment of that MO question is Thurston's "On proof and progress in mathematics" https://arxiv.org/abs/math/9404236
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.…
> 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…
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…
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
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…
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…
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.
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…
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…
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…
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
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…
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.
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…
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…
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
Source: http://arxiv.org/pdf/math.ho/9404236.pdf
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…
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…
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…
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…
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…
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…
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…
Bill Thurston describes math as the theory of formal patterns. He has a nice article here: http://arxiv.org/abs/math.HO/9404236
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 .
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…
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…
Connected through shared HN threads, not inferred similarity or a recommendation.
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…
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