Ravi Vakil, Anton Geraschenko and I are writing an opinion piece for the Notices about Math Overflow. Following in the fine tradition that John Baez started with his opinion piece about mathematical blogging, we’d like to post our draft here, and ask for suggestions and criticism!

Note that we’re working inside a fairly strict ~800 word limit for an opinion piece, so if you tell us to add a whole new section, expect to be disappointed. (On the other hand, we are listening for ideas about things that you feel should be covered in a longer article, as we’re writing one of those too!) Particularly helpful would be advice from anyone who’s standing a little bit further away from Math Overflow than we are, and can point out any background or context we’re implicitly assuming of the reader.

Here’s what we’ve got:

The internet has changed the way mathematics is done. Online journals, MathSciNet, and the arXiv have existed for some time. More recently a number of blogs have become home to rather sophisticated mathematical discussion [1,2,3,4]. There are also mathematical wikis combining exposition and research [5] and several massively collaborative online “Polymath” projects [6].

On the new site Math Overflow, users ask and answer focused mathematical questions. They also vote on questions and answers, making it easy to sort out the good stuff. The majority of active participants are graduate students and faculty, including many leading researchers, along with a few advanced undergraduates and non-professional mathematicians. On a typical day, the site receives about 30 new questions and over 30,000 page views from around 2500 different users.

Math Overflow is a natural tool since doing mathematics is itself largely driven by asking questions. We often formulate a problem as a series of small focused questions and then try to answer them. Sometimes the answers are surprising and lead to more questions, and off you go on an adventure. Other times you get stuck, in which case you might try looking online. Math Overflow is heavily indexed by search engines, so you’re likely to find your question being discussed there. Since many mathematicians first visit the site in this way, the system is designed to make it easy to jump right in. You can start posting questions and answers without registering, and \LaTeX\ “just works.”

The site has a personal feel to it. It’s very satisfying to know that you’ve helped someone with their problem or that somebody has taken the time to help with yours. It’s like a global math department tea. You interact with excellent mathematicians of all ages and you get to do some really fun math. We know of cases where someone was stuck on a problem in their research, asked it on Math Overflow (why not?), and got a satisfying answer within an hour, or even minutes [7]! The site launched in October of 2009, and already questions have led to research papers with the asker and answerer as coauthors: [8] was a direct result of [9].

Upon first visiting Math Overflow, people are often surprised by the high quality of the content and the sophistication of the mathematics. This is made possible by features which allow the community to moderate itself, keeping the site from becoming anarchic. For example, established members of the community have increased power to make decisions. This ability is measured by a point system, somewhat unfortunately called “reputation.” Votes for something you’ve written increase your reputation. As your reputation grows, you gain abilities that allow you to organize and improve the site, so moderation duties are efficiently shared amongst the most active users. At first you can only ask and answer questions, but soon you can vote (one good question will get you this far), then you can create new tags, and eventually you can edit other people’s posts and vote to close bad questions.

A “bad question” is one which is either irrelevant to mathematicians or one where it is unclear what constitutes an answer. Homework questions, on the rare occasion that they occur, are closed and the asker is politely directed to one of many websites where such questions are appropriate. Blogs often host discussions, and wikis compile expository material; Math Overflow fills a different niche. It is optimized for focused questions which (could) have clear answers. As a result, it is awkward to use it for broad or open-ended questions, and such questions are often closed. Community norms dictate that you should use Math Overflow when it is the right tool for the job. The best questions include motivation and background, and have a clear goal.

The underlying software is StackExchange, the engine behind the extraordinarily successful programming site Stack Overflow. All contributed content is under the same Creative Commons license used by Wikipedia, and database dumps are regularly made available for analysis, reproduction, and future-proofing. [10]

It is hard to explain Math Overflow without showing it to you, so please visit http://mathoverflow.net and poke around! If you have a question that’s been bothering you that you are pretty sure \emph{someone} must able to answer, try asking it. If you find the main page overwhelming, go to http://mathoverflow.net/tags and click on a tag corresponding to your specialty. Currently some areas of math are disproportionately represented, but just as with the arXiv, it is gradually “filling out” as it matures. We’d love to have mathematicians from more areas actively involved, and already you’re sure to find something you’re interested in.

[1] John Baez, Notices of the AMS, Febuary 2010 Volume 57 Issue 03 http://www.ams.org/notices/201003/rtx100300333p.pdf

[2] Secret blogging seminar, https://sbseminar.wordpress.com/

[3] Terry Tao’s blog, http://terrytao.wordpress.com/

[4] Gowers’ blog, http://gowers.wordpress.com/

[5] The nLab, http://ncatlab.org/nlab/show/HomePage

[6] Polymath project, http://polymathprojects.org/

[7] Shtetl-Optimized, “Prove my lemma, get acknowledged in a paper!”, http://scottaaronson.com/blog/?p=432

[8] F. Calegari, S. Morrison, N. Snyder, “Cyclotomic integers, fusion categories, and subfactors”, to appear.

[9] N. Snyder, “Number theoretic spectral properties of random graphs”, http://mathoverflow.net/questions/5994/.

[10] Mathoverflow database dumps, http://dumps.mathoverflow.net/

We’ll be listening in the comments!

Quite nice.

We’ve also posted this draft at meta.mathoverflow.net, if you’d prefer to comment or follow there.

I put this up, with full attribution and link, on my Facebook page. Many of my roughly 700 facebook friends are Math, Science, or Publications sophisticated. I’m happy to repost any good comments to you here. Looks like a great first draft to me! No, the exclamation point was not a factorial…

I would leave the citations to particular blogs out at the end of paragraph 1. The citation to John Baez’s opinion piece is sufficient, and there’s no way to pick a list of what to reference without people feeling hurt/coming off as too self-promoting.

A sentence about the origins would perhaps be nice. That, it is modelled on stackoverflow, arose out of discussions between Berkeley grad students, etc.

I agree with comment 2.

“Other times you get stuck, in which case you might try looking online. “(…) “someone was stuck”.

I find the repetition stylistically unpleasant (like, didn’t I already read this?).

I think you might want to stress that the idea, if you’re stuck on a math problem, of looking online instead of asking a person is very new, and one with a potentially high impact in the sociology, or geography, of mathematics. Especially if you add the possibility to listen in to what everyone else is asking.

In the initial paragraph where you list the many ways Internet has affected mathematics you may want to mention, without links, the possibility of “attending” lectures and conferences without physically moving. Unless you’re afraid that this will lead to a cutting of conference funds in the future.

You should mention that many people are using their real name. That’s quite remarkable.

+1 for what Noah says. Perhaps mention the all-important FAQ and meta.

About “reputation”: yeah that name is unfortunate, so isn’t it possible to just change it to “contributor index” or something, inside the source code?

Also, would it be hard work to set up a webpage somewhere which would simply automatically feed from the dumps, a kind of daily updated archive, to display & link to all the questions sorted by tag? Because the present system makes a few suggestions on the side to help someone asking a question spot whether it’s been already asked, but perhaps it’s not that well displayed, a more global database might help.

In any case, let’s hope for more great MO contributors!

I have a quibble with “including many leading researchers” and “advanced undergraduates”. I get the meaning, but in some ears it may sound like an appeal to elitism. (“Yeah, we got Fields medalists up in here.” “Don’t worry, ‘regular’ undergrads don’t go anywhere near it.”) If the intent is to address the (perhaps unconscious) stereotype a reader may have of Internet-based things being amateurish or not “serious”, this is already done by the rest of the piece. It is unfortunate that the point system is called “reputation,” but one does have to call it something; on the other hand, there is no need to declare some undergraduates “advanced”.

In the long term I think that Mathoverflow and mathematical blogs will serve a much more significant “passive” purpose— ie, a larger portion of these things will be just read, and not interacted with. I am thinking of large archives of answered questions, and years-old posts with comment threads that have petered out. These things have permanent value. It is fine for editorials to focus on how to get in and join the fray, but I wonder if the archival quality of Mathoverflow cannot be stressed a little more. To the extent the discussion leans toward prospective “active participants”, a large audience is left out— people who cannot envision ever joining an kind of active internet community. I am imagining an older, less technological version of myself, who might hear “it’s like going to tea” and think “but I don’t want to go to tea online, I just want to read mathematics.” Such a person might be more likely to browse now, and perhaps contribute at some later time, if it is clear that it does not require any particular commitment to a community.

“Community norms dictate that you should use Math Overflow when it is the right tool for the job.”

Do you mean “only when”?

An op-ed in the Notices is a good idea, though it’s a few

months from submission to actual publication in print.

Reference lists are not appropriate here, apart from a

link to the earlier op-ed by John Baez, as already noted.

I agree that something should instead be said about the

origins of (or people behind) Math Overflow.

Some longer perspective is desirable here, starting with

the overstated first line: “The internet has changed the way mathematics is done.” A generation ago one might have

proclaimed: “The copy machine has changed the way

mathematics is done.” Would either have changed Euler’s

work for the better? Hard to say. The current Internet/internet is both a vast repository of knowledge and a vast junkyard It’s definitely true that email, the

arXiv, online journals, and blogs have had major impact on

the working habits of mathematicians. Those who thrive

on instant communication with colleagues around the

world are well served. But “doing” mathematics still has

a solitary aspect of hard thinking about problems. Sites

like Math Overflow may be an alternative time sink for

some, like watching a hundred videos at once.

Minor stylistic criticisms of the draft are easy to state,

notably the overuse of the flabby sentence construction

“A, and B”. The sentence “Upon first visiting…” is

better written “When first visiting…” Etc.

Re: Jim Humphrey’s critique, change

“The internet has changed the way mathematics is done.”

to

“The internet has changed the way mathematics is communicated.”

Which presumably is what you mean.

@DC, we’ll certainly do something about your first comment. Until a very recent draft this had been slightly more modest, but we cut a few words. The second one (long-term value of MO) is useful, as it’s something I don’t think we pay enough attention to anyway, being too close to the fray. We’ll have to think about how to include this. (We’re also writing a longer piece for another venue, and it may be relegated there — we’re really thinking of this opinion piece more as a hook than a thorough discussion of MO.)

@Akhil, we do mean “only when”, and had in fact already argued whether it was clear from context internally. Maybe we’ll argue it again. :-)

@Noah, good idea, I think.

On DC’s first point, I think it’s worth emphasizing the scope of the userbase of the site. In particular, I think some form of “leading researchers” is necessary to get across the point that you can get good questions answered by experts in their fields. The exact phrasing could be improved, but the idea is important to get across.

On the other hand, I think DC’s right about the “advanced undergraduates” mention. The point you’re trying to make there is that although undergraduates use the site, most of the questions are at a graduate or higher level. But there’s probably a better way to make that point. For example, saying that it’s a great place for undergraduates to watch questions that come up in research get discussed and answered in realtime.

@Jim, yes, we hadn’t been sure about including the references, and have already been half-assuming that the Notices will want them out. We’re already working with the appropriate person at the Notices, but I don’t know how that will affect the publishing timeline.

I guess we’ll have to argue some about how and whether the internet has changed the way mathematics is done! It seems the comparison with a copy machine is stretching it a little, but maybe, having always had copy machines, I’m not understanding the effect they had. I couldn’t say what effect such things would have had (sometime I wish English had a more explicit counterfactual past tense) on Euler, but personally I think that it’s absolutely clear that they have had an enormous positive effect on many current mathematicians and their work, notwithstanding the continued necessity of sitting down and thinking. Perhaps those who doubt that won’t be interested in Math Overflow at all, and as such it doesn’t matter if we lose them at the first sentence?

But perhaps I’m just misinterpreting the “longer perspective” you’re suggesting, and would love to hear what sort of thing you would say in this preamble paragraph.

further, @Jim, thanks for the stylistic comments: I’ll go hunting for “A, and B”s, and think about them. We’ve already nitpicked every sentence amongst ourselves though, so explicit criticism on other stylistic points is very helpful.

Thanks for referring to my earlier AMS Notices opinion piece! I guess Steven Krantz is trying to systematically bring AMS member up to date on the growth of web-based mathematics resources. That’s great.

When I wrote my opinion piece, Krantz asked me to say a bit about the downside of blogs. I did, and I think that made it better balanced and more convincing. Your draft comes across as glowingly optimistic, a bit like an advertisement. Of course, that’s your prerogative – it’s an opinion piece, after all. And maybe it’s too soon to detect any downside to MathOverflow. But I think one sentence about problems or difficulties would make this piece feel more human.

… or maybe a question you have in your minds about the future of MathOverflow…

@math_reader: unfortunately we don’t have access to the software — it’s proprietary. We didn’t want to go into this in detail in this piece, but will in a future, longer one. There are very good reasons for putting up with not having access to the source code, and if you’re interested in talking about it more now, I’d encourage you to come across to meta.mathoverflow.net. There’s also some work going on “behind the scenes” in this respect, which Anton and I will talk about soon.

The developers are, however, very responsive, and it is likely that we will gain a lot of options to customize the software soon after it officially comes out of beta (we’re on the last beta, now, apparently). I’ve just opened a feature request for exactly this.

Re “reputation”: I actually think the term is pretty good. No matter what we call it, some people will have the first impression that their worth is being reduced to a number. In my mind, the purpose of the paragraph (and the word “unfortunately”) is to communicate that you shouldn’t take your reputation score too seriously. It’s not meant to be a measure of your worth, but a rough tool for allowing the community to self-moderate.

@16 It’s impossible to write a first line, but for example you might try: “The internet has rapidly augmented the often clumsy tools available earlier for communicating or collaboratively developing mathematics.” [Maybe

too elaborate…]

Copy machines actually were a revolutionary tool, for example replacing carbon paper and mimeograph machines. It is said that at least one famous manuscript with no copies got lost in transit and enjoyed only an

underground reputation for many years. Back further,

the Selectric typewriter was quite a boon to mathematics.

Manuscript used to be meant literally. And being a

copyist was a job option.

@17 While the “A, and B” sentence is legal and sometimes

quite natural, you should ask whether the comma plus “and” might better be replaced by a semicolon or a new

sentence. The structure gets tedious, and it can often be

avoided, and it can even get annoying, and …

To edit my first line, replace “augmented” by “supplanted”

to make sense of the wording.

I clicked on the dumps link and was surprised to find a directory listing with only one dump and no information about how often one should expect dumps nor information about how to load them into something interesting. The directory also lists two favicons…I think that if you are going to point a link to somewhere, it should nicer than that…

@Yossi, as you can see from the dates, the dumps are very new! We’re not even sure yet how often we’ll make dumps. Further, as far as I know we don’t even know how to load them into something interesting. We’ve been using grep.

The favicons, at least, I’ve removed.

I agree with most of the comments made so far. In particular, this piece reads strongly as an advertisement. Perhaps this is intended and appropriate, but it should be a conscious decision.

I’m not a huge fan of the “you” in the fifth paragraph, but I don’t have any suggestions for how to fix it.

In “If you have a question that’s been bothering you that you are pretty sure \emph{someone} must able to answer, try asking it.”, I would replace “must be able to” with “knows how to”.

“It is hard to explain Math Overflow without showing it to you, so” can be completely dropped. It doesn’t contribute to the piece. Instead, start simply with “[P]lease visit”.

Drop the scare quotes around “just works” in “You can start posting questions and answers without registering, and \LaTeX\ “just works.””.

“Math Overflow is heavily indexed by search engines, so you’re likely to find your question being discussed there.” is a very awkward sentence. Most awkward: the pronominal adverb “there” doesn’t have a clear antecedent.

The false dichotomy between “The internet has changed the way mathematics is done.” and

“The internet has changed the way mathematics is communicated.” sweeps under the rug the central meta-question, which is: “To what extent is communication between Mathematics central to the practice of Mathematics?”

This gets to the heart of which metaphysical stance on Mathematics, long debated between Mathematicians and Philosophers, best fits the data on recent publications and their impact.

Whoops — that brings up the question of to what extent e or pi exist,

before we compare existences. That’s the oldest known metaphysical

debate in Mathematics, or at least on ontological status of

mathematical objects. One one side, Platonists (Transcendent Realists) who hold that 8 and pi and e and triangles and aleph-null exist, if anything, MORE than humans do.

“O my friends, what are these wonderful numbers about which you are reasoning, in which, as you say, there is a unity such as you demand, and each unit is equal, invariable, indivisible, — what would they answer?”

–Plato, “The Republic” [Jowell translation], Chapter 7.

On the second side, the Logicists, following Gottlieb Frege, who hold

that mathematics can be known a priori, but suggest that our knowledge of mathematics is just part of our knowledge of logic in general, and is thus analytic, not requiring any special faculty of mathematical intuition. In this view, logic is the proper foundation

of mathematics, and all mathematical statements are necessary logical truths. Cf. Rudolf Carnap (1931).

On the third side, the Formalists, such as David Hilbert, Rudolf

Carnap, Alfred Tarski and Haskell Curry, who hold that mathematical

statements are equivalent to statements about the consequences of

certain string manipulation rules. Some some formalists now propose

that all formal mathematical knowledge should be systematically

encoded in computer-readable formats (Cf. QED project).

On the forth side, the Intuitionists whose motto is: “there are no

non-experienced mathematical truths” [L.E.J. Brouwer].

Cf. Arend Heyting.

“God created the integers; all the rest is the work of Man.”

[Leopold Kronecker, tr. from German].

On the fifth side, Constructivists who hold that only mathematical

objects which can be finitely and explicitly constructed in a specific

sense properly belong to mathematical discourse.

On the sixth side, Fictionalists, such as Hartry Field published

“Science Without Numbers” (1980), rejecting or reversing Quine’s

argument on indispensability.

On the seventh side, Embodied Mind Theorists who hold that

mathematical thought is a natural outgrowth of the evolved human

cognitive machinery embedded in our physical universe; hence Math is not universal and does not really exist, except in human brains, which construct (not discover) mathematical objects, in efficacious ways that benefit Darwinian fitness.

Cf. “Where Mathematics Comes From” — George Lakoff and Rafael E.

Núñez; Keith Devlin’s “Math Instinct.”

On the eighth side, Social Constructivists (Social Realists) such as

Imre Lakatos, Thomas Tymoczko, Reuben Hersh, Philip J. Davis, and Paul Ernest, who hold that mathematics is a social/cultural construct, akin to English Common Law, or a Black Queen in Chess. Mathematical objects come from an empirical endeavor dictated by the fashions of the social group performing it, and/or by the needs of the society or power elite financing it, and are ultimately a political struggle of mathematicians seeking sex, money, or power.

Or should I classify Lakatos as a Quasi-Empiricist along with Popper

and Hilary Putnam?

There are also Linguistic theorists and Aesthetic theorists of the

ontology of Mathematics.

And I’m probably missing whole schools of thought.

One more thing. you don’t seem to state your involvement with MO…don’t you think you should be upfront about it?

@Yossi, hrm, interesting point. I guess we’d been thinking it was obvious from context that we’re some of the people making Math Overflow happen. I’ll talk with Anton and Ravi about this.

@GS, about origins, that is a good point, and interesting. Key reasons for MO being able to get off the ground so successfully are Berkeley-specific: there is a (large) critical mass of graduate students interested in a wide number of areas, who have a culture of being broad and talking across all of mathematics. This is big enough that I wonder if it better belongs in the longer piece (that I won’t be an author on).

@DC, you make two points I wanted to address/mention. Definitely the passive archival point is important. (Anton and Scott — how to easily include it?) And about “leading researchers”: that was there for the reason you suspect: to get people who are not used to this sort of thing (most of our colleagues) to realize that this isn’t like the comment section to nytimes.com, and that they will be in serious company. Your point about “advanced undergraduates” is well-taken, and we should be sure to phrase this well (if it is still kept — at this point, I think it is out).

@Akhil Mathew: yes, that’s what I think we meant (although I’m not sure if Anton and Scott agree).

@Jim Humphreys: I’m very intrigued by what you write, and some of the follow-up comments from others implicitly agreeing with you. I would have thought that it was clear that the internet has changed the way mathematics has been done, and I now realize that a case needs to be made (that you and others may not buy). I’ll stick by that sentence (in an opinion piece), and begin to make the case here. First of all, I *do* think there have been innovations that have changed the way mathematics has actually been done (not just communicated), including some subset of the following: the copy machine, the telephone, the advent of the mathematical journal, the research library, LaTeX, the arXiv preprint server. In each case, it isn’t that all mathematicians changed how they did math; most individual mathematicians I suspect stayed fairly stable. But newer mathematicians developed different habits than their older colleagues. (It may be easier to see in another field: the way in which literature gets composed has been substantially changed by the advent of word processing programs.) I would *not* claim that Euler would have been better for these changes. I potentially have more to say, but must run, so will continue later. (I find this particular discussion very interesting, and would enjoy continuing it at a departmental tea… but will likely have to settle for this electronic version… :-)

@Barbara’s point about being able to watch lectures.

Apologies about that last sentence: I’ll mention Barbara’s good point about being able to watch lectures later, and forgot to delete it in my draft.

He’s mostly talking about his novels, but he did graduate from Caltech and earn a PhD in Astrophysics first:

http://www.davidbrin.com/advice.htm

Look at the acknowledgments page at the back of every book I publish. There are at least thirty names listed, sometimes more — names of people to whom I circulated early drafts.

Yes, this is at the extreme end among writers. Many circulate manuscripts early in their careers, then stop doing so, telling themselves — “I am a professional now, so I don’t need feedback.”

Baloney! If you are a daring writer, you will always be poking away at new things, and exploring new ground. Testing your limits. That means making both wonderful discoveries and awful mistakes. So? Refine the discoveries and solve the mistakes! It helps to have more eyes — the outsider perspective — to notice thing that your own eyes will miss.

@John and Theo, This is an opinion piece rather than reporting, and like many opinion pieces, this is intended to advocate a certain point of view. I hope we don’t come across sounding like everyone will use MO in a few years. John, your point that the piece can be made stronger by adding some interesting questions about the future is sensible. But I don’t really have any such questions other than the rather vague: “What will this turn into?”

I respectfully disagree with #8: the word “advanced” in “advanced undergraduates” is important. It is intended to convey (isn’t it?) that MO is a site which is firmly beyond standard undergraduate level mathematics. This is one of the most important features of MO — its (usually!) unapologetically advanced level compared to most extant internet math sites.

When I first read this passage, I had the exact opposite reaction: this one word “advanced” is being overloaded. I thought that the true meaning is that we entertain people who are able and willing to ask and answer questions at the level of a serious PhD student in mathematics or a research mathematician. An “advanced” undergraduate sounds like a junior/senior in college who is studying from Munkres or Rudin or Dummit and Foote — MO does not in fact entertain questions at that level. Really, we entertain questions from those who can participate in the discourse _as if they were_ at least serious PhD students of mathematics, whether they be amateurs, undergraduates or (in at least one case!) high school students.

I respectfully disagree with #36: MO is a very nice tool for the just curious general people like me, where one can ask questions arising from mathematical bedside reading, even when one is unsure if they are ok. “Real students” react and one finds hints to interesting texts in MO, which one would have not found, or whose interest/readability one would have not detected, by oneself so fast.

With all due respect, Thomas, I find it hard to characterize you as “just curious general people”. (Of course you’re very welcome on mathoverflow, and my apologies if the brevity of this piece forced us to omit mentioning some parts of the population.)

I’d just like to add my voice to those who are defending the assertion that the internet has changed the way that mathematics is done. It may perhaps be an exaggeration, and there is probably a significant proportion of mathematicians to whom it does not apply at all, but the internet has already changed the working practices of many mathematicians and has the potential to change them much more in the future. Whether there is a pithy way of expressing that I’m not sure.

Re #39, I think “The internet has changed the way mathematics is done.” is a perfectly good pithy way of expressing that.

Although you can’t perfectly defend against misinterpretation, I don’t think that a reasonable reader should read the opening sentence as written as implying that “The internet has completely changed the way all mathematics is done by everyone.”

Also, there are lots of mathematicians whose working practices weren’t changed per se by the internet, because they were formed in the context of the internet. I personally have no idea how I would go about doing research without email, MathSciNet, the arXiv, and online journals; just as many previous generations have been unable to imagine working without research libraries.

I’ve also grown up with email, mathscinet, the arxiv, etc., and in addition I’ve seen changes in how I “do mathematics” even more recently.

Subversion (or equivalents) has really made collaboration less painful. I remember one moment a few weeks ago when four emails arrived in my inbox in the space of an hour, informing me that four different collaborators had just committed changes to the repositories of four different projects!

Wave is also affecting how some people do mathematics — already for some work I prefer it to instant messenging (I realise that many mathematician have never used IM to talk about maths, either!), and from what I see some of the people who hang out here are using much more intensely than I have. (Actually, this draft was written entirely in Wave, and there were quite a few sessions where Anton and I were working at the same time, doing the sort of arguing-over-minutiae that is otherwise only possible in person. It’s less good for writing final mathematical text, however.)

The internet

is changingthe way that mathematics is done?Maybe that works. I’m OK with “has changed”, but “is changing” might please everyone and draw attention to the fact that the process is probably far from over.

Looking back to Jim Humphreys’s original complaint (#11, “Would [the internet] have changed Euler’s work for the better?”), I’d also add that the first sentence, as written or with Izabella’s suggested change, makes no such claim. Whether new, internet-based tools result in better mathematics or not, mathematicians are indisputably using them.

It also occurs to me that the right answer to Jim’s question “Would [the internet] have changed Euler’s work for the better?” is “Maybe not, but I bet a lot of other people would have done better mathematics if they could have read Euler’s papers on the arXiv and traded emails with him.”

I also find it a little strange that Jim would say this, when he’s sent me several helpful emails about my papers. I don’t know about Euler, but I’ve probably done slightly better mathematics because of having the ability to email with Jim.

Re #44,#45, actually I added the qualifying “Hard to say” in

speculating about Euler. Certainly better communication

would have helped mathematicians in the old days, as

witness the time Ramanujan spent rediscovering known

ideas. (By the way, I’m planning to

publish my collected emails in a hundred volumes or so,

unless someone pays me not to.)

I think you missed a BE in the last paragraph. Sorry of it’s a repost.