Some background about the author of this MathOverflow answer:
> William Paul Thurston (October 30, 1946 – August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds.
and where he writes "I think of mathematics as having a large component of psychology, because of its strong dependence on human minds. Dehumanized mathematics would be more like computer code, which is very different" he's just blowing smoke. smh so dumb.
I especially liked this comment that proposed that this answer be called Thurston's Paean:
> This seems like an ideal counterpoint to Hardy's Lament. I'm calling it Thurston's Paean :). Seems poignant now that he has passed. --
Suresh Venkat (Aug 22, 2012)
I agree. I have now dedicated a page to Thurston's answer at https://susam.net/maze/thurstons-paean.html hoping that it'll make the term Thurston's Paean slightly more popular.
Hardy's lament is a short popular essay about how the famous professional mathematician Hardy experiences math as art with the curiosity of a child, and yet the average school math education evokes a prison of rote exercise. It was written in 1940 and I think at some point he says something about number theory being pure math with no applications. Ironically I'm submitting this comment over https so he was wrong about that.
I really like this view of academic research in general. The field does not exist as a means unto itself, but as a tool for helping and contributing to humanity. A service mindset. That's how I have always viewed both research and teaching.
The most frustrating part of academia, and one of the main reasons I left, was the sense that everyone had completely lost their service mindset. The result is a set of sclerotic institutions -- protected from competition by rules and norms motivated by an ostensibly noble purpose -- wrecking havoc on society.
In research, in particular, it's now rare to find researchers who really give a damn about anyone except themselves. Their PhD students are tools for increasing faculty influence. The motivation section of their grants are polite lies and everyone knows it. The whole thing is a big inside joke that people laugh about in elevator bays, backed by billions of taxpayer dollars. The selfishness and complete disinterest in stewardship of public funds is nauseating.
But things are worst in undergraduate education. You occasionally meet someone worried about the educational mission, but no one was ever interested in the overall well-being of students. Financial well-being, in particular, is a verboten topic. The question of whether a particular degree program might be doing irreversible and crippling financial damage to nearly every student is considered rude. The question of whether requiring a certain course might be a waste of a customer's money is derided ("they aren't customers!", despite paying mid 5 figures per year for shockingly little face-to-face time.)
Academia has completely lost its historically service-oriented pursuit. The faculty are motivated by a combination of laziness, entitlement, and ego. The administration is motivated by the same incentives as any other industry. But huge quantities of tax dollars are locked up in this broken and corrupt industry.
As a paper, I'd send it back for revision - remove the hyperbole and 99% of the emotion, and focus on facts and strong reasoning that could persuade me.
I know people in academia and they and their experiences don't fit this description. We get out in the real world and discover each individual has many motivations, both good and bad. It's messy and complicated. IMHO the insight is to realize that I am not the heroic protagonist, but instead I'm just as flawed, and then to try to bring out the best in myself and in them.
> wrecking havoc on society
Not even AIs or the online propapganda machines are doing that - though I kinda wish they would!
PhD student and faculty (quite briefly). Serving on grant panels also informed this view, but looking back, I think I was already getting disgusted by the culture of academia around my second year of grad school.
My immediate family spent/spends much more time in academia than I, including the person I live with who has built their career in academia.
My own experience of academia was actually quite positive -- I enjoyed my PhD years, was well-supported, received many tenure-track offers from elite institutions immediately out of my PhD, and I left on better than good terms with coworkers and former colleagues.
My assessment isn't a result of personal bitterness or a failure to "win the game", if that's what you're getting at. Quite the opposite, in fact -- doing extremely well in the early career game afforded me the personal confidence to leave the track to which I had previously committed.
(FWIW, the fact that I'll retire many decades earlier than I would've otherwise was not immaterial to my decision making)
I scrolled up the page and saw the following banner, which (in hindsight) couldn't be more wrong.
> This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.
It's a longtime trope of these "stack" sites: the best questions with the best answers are those that are marked as offtopic, closed, deleted, downvoted or otherwise rejected. It's a marker of good quality at this point.
Marker of good quality would be too far (and biased), as the every growing mass of genuinely hard to salvage closed question certainly outnumbers the interesting ones
But it sure looks to be bimodal. Marker of the really bad, and the so-good-it's-closed
I would guess that this bimodality correlates with one in the personalities of those who become moderators - on the one hand, those who are willing to put a lot of effort into keeping the community relevant, and on the other, those who abhor any perceived irregularity.
On occasion, when I get invitations to participate in moderator elections, I look to see if I recognize any of the candidates. Once or twice, I have looked through the candidates' resumes, and one thing that struck me was that some of them did a lot more down-voting than up-voting. Make of that what you will.
They explain why they deleted the most appreciated question ever in Stack Overflow, and how they magnanimously let you look at a few scraps of the answers (but not to all, these are DELETED!), and why we can't, after all, have nice things.
> Does this effect have a name?
If I wasn't against ad-hominems, and to avoid giving undue credit, I'd propose to call this effect "Jeff Atwooding". As in, the cool answer by Thurston on contributing to mathematics has been jeff atwooded recently.
But it should be noted that the question in the blog post—share a bit of programming jargon you’ve made up—is not a good question for a Q/A site. It is a “fun to play with” social media engagement question, more suited for Reddit.
The question on the original post seems more grey-area. It isn’t really a math question that can be answered in a math-y way, but it is the sort of thing that some folks might benefit from seeing anyway…
We could call it the Thurston effect - a mistaken belief that site like SO exists solely to give solutions to abstract computer problems and not as a communal mechanism maintained by the practitioners to disseminate and further understanding of the craft.
The "I'm angry at the moderators enforcing 'keep off the grass' because I want to be on the grass and I'm special so the rules shouldn't apply to me" effect?
StackOverflow advertises itself as a place for precise questions and answers, and explicitly not for open ended questions and answers. It tells people who sign up not to post open ended questions and answers, and encourages participants to flag same, and moderators to close same. To then be bitter and rude because that happened exactly as the prophesies told you it would, is unreasonable.
These moderators act as angry old men booing people out of their lawn. People who were having innocent and wholesome fun, and overall improving the neighborhood.
It's the same kind of guys who call the police on kids selling lemonade.
It's the same kind of guys who call the police on festival goers trampling their land, ruining nice things, and using "you must be fun at parties" as if it was some kind of justification.
You can't go to a vegetarian cook-out and start cooking burgers because "I want burgers" and then mock the hosts for being unhappy at you. Well you can, but you're the jerk in that situation. You can't go to Wikipedia and start posting questions and answers and saying "I wish it was StackOverflow you angry OLD MAN horrible moderators".
> "People who were having innocent and wholesome fun, and overall improving the neighborhood."
Misusing someone else's site, in a way the creators and existing users don't want it used, in a way you agreed not to use it when you signed up, is not "improving the neighborhood".
> Misusing someone else's site, in a way the creators and existing users don't want it used, in a way you agreed not to use it when you signed up, is not "improving the neighborhood".
No one cares about "the creators", and MOST ESPECIALLY NOT what Jeff Atwood thinks about their questions, answers and comments on stackexchange sites. Atwood is gone anyway, he cashed-out and moved on.
The existing users are many and varied (I am one) but for whatever reason the gamification attracts a high level of zeal from certain personalities that tend towards petty authoritarianism and persnickety behaviors. As useful as these sites are (at least up to now), this is a serious problem.
Seriously, though, respond to the point - would you be happy with people turning Wikipedia into AskReddit where every page was "what's the sexiest sex you ever sexed?" and all the page content was memes about narwhals, bacon, broken arms, and poop knives? I wouldn't be happy with that. Most open ended questions are vague and bad. Most answers to them are bad. The existence of a few good ones does not justify "all sites should become Quora". Open ended questions are very easy to write, have no clear criterion for "answer is useful" and if they were just allowed, they would drown out everything else on the stackoverflows while being a bad interface - witness the Meta overflows which are really really poor attempts to have a chat forum in the QA interface, tons and tons of comments all over, people replying to other people with answers which get voted out of order.
You can tell because you don't want to go to Quora or Reddit for your open ended questions; allowing unfettered open ended questions generally makes sites become overrun with low quality memes which drives the more hardcore users away. You recognise somehting about high quality at StackOverflow which is why you want to go there, but then you want to bring the low quality stuff with you. And when that's ruined StackOverflow you'll chase the next site which has narrowly scoped content and demand to ruin that too. Stop.
> "No one cares about "the creators", and MOST ESPECIALLY NOT what Jeff Atwood thinks about their questions, answers and comments on stackexchange sites. Atwood is gone anyway, he cashed-out and moved on."
OK, not the creators, the /owners/. Y'know, "my house, my rules", that kind of owner.
What’s the alternative? We already have quora and Reddit for less moderated questions. Stack Overflow remain the site that, when you are looking for an actual answer to a technical question, is worth clicking on at least.
> Stack Overflow remain the site that, when you are looking for an actual answer to a technical question, is worth clicking on at least.
These days I'm skipping Stack Overflow entirely and jumping straight to the docs, unless it's a common question that I can never remember properly (like some esoteric git-fu command).
The answers on SO are outdated or crappy. When you can't ask questions about an evolving framework without it getting marked as duplicate, then the answers quickly lose all value. And the people that were answering because they wanted to give back (not because they want to farm karma) are leaving.
At least in the "default" stack overflow for software developers; it's supposed to be a question and answer format about narrowly-focused programming topics. More philosophical questions are technically off-topic.
But, in the case of this post, it's pretty easy to see why Math Overflow would consider it in their best interest to make an exception! (And perhaps have a slightly different banner.)
It's easy to forget the amount of moderation that a good internet message board needs. (Just take a look at Dang's hard work on Hacker News.) Unfortunately, there are many people who just don't have the maturity, or discipline, to ask good, topical, questions.
Banners like that are a good compromise, IMO, between the need to focus on a particular topic, and allow flexibility for discussions that straddle the line between on and off topic. I think putting up a banner, but still letting the discussion continue, is a good compromise.
I feel like this comment hits at the general discussion of "Why should I learn something which I won't be the best at?" and it applies just as well to maths, science and arts: you should learn it since if you really enjoy it then you can contribute by maintaining a healthy community around it; and this can mean really working on the community or just being a person who can understand the value of discoveries of others. Having a good audience is the dream of every artist and I think this applies quite universally.
Bill Thurston! He did amazing topology - hyperbolic knot compliments, Foliations, eversion of the sphere, geometrization conjecture... A brilliant man.
I had the honor of meeting him several times here in Berkeley; he encouraged me to make Klein bottles and other topological shapes.
> Mathematical ideas, even simple ideas, are often hard to transplant from mind to mind.... Our understanding frequently deteriorates as well... The experts in a subject retire and die, or simply move on to other subjects and forget.... In short, mathematics only exists in a living community of mathematicians that spreads understanding and breaths life into ideas both old and new.
One of the great things about software is that you don't need a living community that holds every bit of every code base in their head. You can rely on abstraction boundaries, and then jump down rabbit holes only if and when necessary.
I really wonder how the work of Mathematics will change once computer-encoded proofs become mainstream. I have to imagine that will happen within the next generation or two -- as soon as the automation becomes more helpful than the hindrance of formality.
I think you're missing his point. He does acknowledge that statements of facts and their proofs can be easily communicated, but claims that the understanding their discovery was a side effect of cannot be communicated easily. This is not affected by having a better way to express facts and their proofs.
No, I get his point. I think the transmission of knowledge and insight and clarity works very differently for executable artifacts than for artifacts written in natural language.
> This is not affected by having a better way to express facts and their proofs.
But it is if those expressions are possible to interact with and use through not only conceptual abstraction but also procedural abstraction.
With good proof automation? Maybe you can interact and play with ideas through many layers of automation-enabled abstraction. Maybe you have a tactic or ATP procedure that uses a lemma in clever ways without you even knowing it exists, and only after building on that edifice for years do you bother to dig in and understand what the proof automation is even using. Mind you: not what the lemma means or how it's proven or the underlying historical development of ideas, but that it even exists. Maybe you never dig in and realize it exists.
Maybe you use these new theorems and proofs -- discovered only through extremely essential and clever use of a critical lemma you don't even know that you are using -- to do something useful in the world.
That can't happen today; not really. It would take very powerful proof automation for this world to exist.
"The world does not suffer from an oversupply of clarity and understanding (to put it mildly)" – how this can be remedied by mathematical thinking, though, is rather unclear. Take the USSR with its strong tradition of math education in secondary schools and math research. Despite producing a lot of important results and some painfully beautiful math books, like Gelfand's Linear algebra, it has more or less zero effect on the wider society or even wider intellectual discourse.
Rather than going through the list of USSR scientists & mathematicians who made titanic contributions, I'm just going to throw one unexpected name out there: Max Levchin.
He is arguably the brains, and the irreducible core, of the movement that led ultimately to PayPal, the PayPal mafia, and every other startup & cultural influence that resulted from that. He has described having a strong math & science background as being important to him, and it was obviously at the very core of PayPal - like I said, the one irreducible piece of that was Levchin's math-based contributions. Arguably without Levchin there is no Elon Musk, no Peter Thiel, no Reid Hoffman, not as names that we recognize, the list goes on...
And if that doesn't count as influence in the larger society, I don't know what does.
but you can think it in the other direction.. Even in a sociey so closed and constrained (to simplify a lot), math and art, literature, ideas, lived, fighted and sometimes flourished, history time still flowing
This is such a completely US-centric comment. Go talk to folks in India or China or Vietnam or Europe or Africa, and you'll find that the USSR had much larger cultural and societal impact.
For the record I was born in USSR and never lived in the US. More to the point, my comment was about the USSR-internal discourse and how it was (not) shaped by mathematicians in any meaningful way – not a pronouncement on what good Soviet engineers did abroad or, more largely, on the practical usefulness of mathematics.
> Despite producing a lot of important results and some painfully beautiful math books, like Gelfand's Linear algebra, it has more or less zero effect on the wider society or even wider intellectual discourse.
In Computer Science we have Adelson-Velsky and Landis's self-balanced binary trees, and many other results, as well as other results in math and the sciences.
In engineering and otherwise the USSR launched the first satellite, person in space, sent the first probes to the moon and other planets, and pushed the US into space.
The civil rights moment in the US was often accused of being communist because - a lot of it was communist, and supported by the USSR, at least in the days of Paul Robeson and the Scottsboro Boys and Martin Luther King at the Highland Folk school (billboards through the South had a picture of King there "King at communist training school"). Actually if you watch the Nightline of Mandela coming to the US, half the speakers accused him of being a USSR-aligned communist. The USSR contributed to the anti-colonial struggles around the world - from China to Cuba to Vietnam, and contributed to it in other countries like India.
Plus it did most of the fighting in WWII for the US and UK to defeat Germany (of course nowadays the US and UK flipped it to Germany against Russia).
The USSR also helped push a blue collar working class proletariat into a kind of middle class. This began being rolled back even when the USSR still existed but was weak, and is still being rolled back.
> Plus it did most of the fighting in WWII for the US and UK to defeat Germany (of course nowadays the US and UK flipped it to Germany against Russia).
That's quite a spin on history. The USSR (already an imperialist power) and germany secretly divvied up eastern europe between themselves before Germany started the war, and it was only Germany's betrayal of that deal which led to the USSR actually joining the fight against Germany, with the US and UK finding it a convenient way to wear down a shared foe to provide them with some of their industrial capacity. You make it sound like this was some kind of puppet mastery by the US and UK (which to be clear, are not saints either, but decidedly on the right side in WWII) as opposed to two imperialist powers seeking to conquer others by force fighting amongst themselves.
That's some spin of the UK side of things. When the USSR was looking in vain for an alliance with France or the UK, France and the UK instead made a pact with Germany, handing Czechoslovakia to Germany.
Insofar as the UK and imperialist powers seeking to conquer others by force - Scotland, Ireland, India, Hong Kong and the Opium Wars, Australia, Iraq, Africa from Egypt down to South Africa, I can go on...
Good point about understanding.
Lets say there is a catastrophe, world goes to hell, understanding is lost.
Then people go into a library and start opening books.
The arcane symbols of Math would be indistinguishable from books of magic.
Without the understanding, fresh minds would be hard pressed to distinguish math formula from magic runes.
The understanding of the math comes from some community of people passing it along from generation to generation more than from the symbols.
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.
Just want to point out how cool it is that a Field Medal winner is on MO. Don't know if any Turing Award winners or someone like Jeff Dean is on StackOverflow.
This is a comment I think about a lot lately when it comes to discussions of AI and art too — if there’s a hazard in “dehumanized mathematics” and value in mathematics as community and reflection of human understanding then surely there’s something similar in art as communal conversation and reflection of human experience.
> Mathematical ideas, even simple ideas, are often hard to transplant from mind to mind.
this is a problematic way to frame the issue which I agree is very difficult.
by framing it like this, I think, it sets up this task to be impossible. I'm also considering a subtle distinction between 'teaching' and 'education'
he's using a teaching framework. to put the idea in the head of the student.
but education is more about guiding the student to find the answer on their own; if teaching is difficult, education may seem freaking impossible.
with a 'teaching framework' simple ideas are more difficult (and complex ones turn impossible)
but with the 'education framework' (which in my direct experience is not available publicly) simple ideas may be trickier, but the hope is that sophisticated ideas will become easier to communicate.... dunno
I'm my mind echoed something like... "don't ask yourself what you can do for mathematics, but what mathematics can do for you.."
in my opinion the 1st/biggest contribution to math (and any science, discipline, art, and work) is to study/pratice it, is the best contribution to a living body of knowledge and wisdom, new ideas and discovery are side effect, welcomed, but not mandatory.. but is the discipline involved that change the single and later the world. The sea is made of drops..
the true goal is the/any-single way in praticing, not the end/highest point (if exist).
So, what's so special about this answer? Seems like a pasta, replace "mathematics" by {SUBJECTNAME} and you get yet another popular idea of the day. Startup founders, for one, having been claiming they are "contributing to humanity" for ages now.
People often think of mathematics as uniquely objective, and as standing apart from the social processes of collaboration and invention. And mathematics kind of is special in some of those ways.
But mathematicians know, while laypeople might not, that at its core, mathematics remains a social endeavor just like the rest of the sciences. That's the piece which is a really valuable reminder imo.
We must distinguish between theoria, praxis, and poiesis. Praxis, or the practical, has as its end action, something other than itself. When I clean my house, I don't do this for its own sake, I do this for the sake of having a clean house to live in. Similarly, poiesis, or production, also has something other than itself as an end, some thing. When I cook dinner, I don't cook it for its own sake, I cook it to feed myself and others. On the other hand, theoria, or theory, has as its end truth. The truth has practical consequences, of course, but there are truths that are valuable knowing for their own sake.
Ends, of course, need a terminus. While a practical or productive act terminates in something other than itself, you must now ask what the end itself is for. Is it for itself, or something else further still? When we follow practical and productive ends, they must ultimately terminate at something that is not for the sake of anything else, but is valuable in itself. Indeed, when I said the truth has practical consequences, it was an understatement; the entire regime of practical and productive activity presupposes an ultimate end for the sake of which it exists, that justifies it, that it is in service of.
I will omit discussion of the ultimate here intentionally, because my purpose here is to frame the problem first and predispose the mind toward the answer; giving the answer straight away, or an answer, is more likely to trigger reflexive prejudices here, I think. Once framed this way, people are liberated to ask "whom or what do I ultimately serve? why do I live? why do I act?". The vista of ends and order is thrown open for examination. This clarity enables one to leave Plato's "cave" and makes it difficult to obfuscate and settle for cheap answers as it can corner the conscience into pestering us to reject dishonest rationalizations. Of course, people will often try to do so, but doing so is more likely to gnaw at the soul, something that can prompt people to keep searching instead of holing up in an intellectual ghetto and persist in mediocrity.
> We must distinguish between theoria, praxis, and poiesis.
Must we?
> When I clean my house, I don't do this for its own sake, I do this for the sake of having a clean house to live in.
Why not both? A clean house isn’t necessary for survival. Why do you enjoy a clean house? For its own sake perhaps? Many would argue that we do these things for the sake of them as much as we do these things out of necessity.
Reframed slightly, we do many things for the sake of doing them, and we seem to enjoy bringing this mindset to the necessities of life.
Mihaly Csikszentmihalyi explores this extensively in Flow, and it’s a recurrent theme in many contemplative traditions.
> When I cook dinner, I don't cook it for its own sake, I cook it to feed myself and others.
Again, why not both? If the only value gained from cooking is sustenance, why do we enjoy dessert? Why do we enjoy it more when other people are with us?
I’m not saying we don’t do things for reasons or that it’s uninteresting to assign labels to those reasons and explore them, but such intellectual/philosophical framings are ultimately just reductions of unfolding experience into a low resolution explanation of reality relative to our immersion in it.
Should we explore these ideas? Yes.
Must they be the foundation from which we explore other ideas? I don’t think so. Sometimes, sure.
> William Paul Thurston (October 30, 1946 – August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds.
Source: https://en.wikipedia.org/wiki/William_Thurston