John Sidles writes on Scott Aaronson’s blog:

The ongoing acceleration of global mathematical culture has produced a surplus of Grothendiecks and a paucity of Dieudonnés; in consequence “accelerated” intelligences *already* are walking among us; it is natural for STEM workers (young and old alike) to be simultaneously exhilarated and frightened by this reality.

The Stacks project, “maintained” by my office neighbor Johan de Jong, is a collective and strikingly successful solution to this problem that tonight is celebrating its 5000’th page.

The eight IHES volumes of *EGA*, in contrast, come to a total of 1814 pages; printed, however, on high quality paper.

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KANTORcommon reply on Scott Aoronson’s blog – I hate blogs –

The ” speedy gonzales ” type of science working imposed to us through blogs and all that is not a positive evolution in science sociology.I refer to Michel BLAY’s remarkable little book on Penser NOT cliquer that our friend in the US should meditate.

Apart from that I am very happy to be invited to the Stack’s party and will bring my friend Mike,my friend Joe M. and my friend Alexander (Sorik ) G. with me .

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mathematicswithoutapologiesPost authorMichael Blay’s book is not readily available in the US. Could you summarize the argument?

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KantorI will ask Michel to send copies to various libraries and Universities in the US.

I regret that the american intellectual world is so closed to european ( french /CNRS ) ideas !

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mathematicswithoutapologiesPost authorIs the argument really that difficult to summarize?

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mathematicswithoutapologiesPost authorBy clicking and thinking simultaneously I was able to find the CNRS website for Blay’s book. Depressingly enough, given its topic, the book even has a Facebook page! But it seems obvious enough to me from the book’s description that this blog would have no disagreement with Blay’s thesis; I would even submit that this blog offers another way to make the same point.

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David RobertsnLab: 11 640 web pages (some short, some very long)

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mathematicswithoutapologiesPost authorThe differences between the Stacks Project and nLab actually did come up during the party. A point no one bothered making, because it is too obvious, is that nLab is not meant to serve as a reference; it contains many assertions but few proofs (maybe none at all). Algebraic geometers have for several decades referred to EGA or SGA for claims made in the course of their arguments; now, increasingly, they are referring to the Stacks Project. I don’t think nLab can be used in that way, and this is consistent with the official stated purpose of nLab:

The purpose of the nLab is to provide a public place where people can make notes about stuff.Less obvious points were raised. On the one hand, some colleagues have complained that the Stacks Project can’t be considered completely reliable because its pages have not been verified by referees. The obvious rejoinder is that EGA, SGA, and for that matter Bourbaki’s Éléments de Mathématique were not peer-reviewed either, at least not as this is conventionally understood. On the other hand, the Stacks Project is not merely a settled body of knowledge to be cited as necessary; it is profoundly interactive, a quality it has in common with nLab. More on this in a subsequent post.

I have already mentioned several times that I have not actually found nLab useful for learning anything, whereas I was not only able to learn but even to teach the foundations of algebraic stacks after spending a few hours with the Stacks Project. This experience was pretty much universal among the people at the party, and the reason for this is not hard to find: the dependency graphs for theorems in the Stacks Project are

trees; it’s fair to say that this is a necessary condition for them to be considered theorems. nLab, in contrast, is far from simply connected. If nLab provided dependency graphs they would demonstrate that many, if not most, (if not all) of the definitions are circular.LikeLike

David RobertsThe benefit one gets out of the nLab is a more complicated affair. It’s a handy place to record obscure references to things that one needs to refer back to, to record observations that are folklore, or are scattered through the literature and so on. It really didn’t start out as a reference, but as a way of recording results of mathematical blog discussions on current research that were otherwise lost in comment threads. Given that there are things in the nLab that are recorded, as far as I can tell, nowhere else, it’s a very different beast to the SP. You could think of the nLab more like Pursuing Stacks or Les Derivateurs than EGA: it’s a living document recording progress and observations. That people came along later and used it as a reference is coincidental. (Although clearly AG was writing Les Derivateurs in a less discursive style, but it is unclear how much he meant it to be read by the mathematical public, seeing as his disappearance in 1991 happened shortly after writing it). Another comparison, even if one may hesitate to make it, is Quillen’s now-digitised private notebooks. No one would treat them as a polished product like EGA, but they are a fantastic resource if approached correctly.

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Attila SmithnLab is a content free, pretentious site whose main activity seems to forcefully translate well-known soft results (all much better explained in Wikipedia, of course) into n-category language. The site offers no insights and, more importantly, no calculations. A comparison with The Stacks Project would be insulting if it were not utterly grotesque.

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mathematicswithoutapologiesPost authorThat seems a bit harsh to me, but if you and nLab’s devotees want to open a debate, I’m sure many of us will find it enlightening — not least the Ambitious Young Historians who (I hope) are still with us.

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David RobertsThe following is just some thoughts, take them as you will. As a ‘steering committee’ member (we don’t run the nLab, just fight fires if and when necessary), I should probably say a word.

They are not really comparable, I just mentioned it for contrast. Also, since it was an offshoot of the n-category café, most of the people who regularly contribute work in higher category theory, and it’s a public notebook rather than a paper, it shouldn’t be surprising that it focusses on n-categories and leaves detailed calculations to the references it links to.

The table of (unstable) homotopy groups of the orthogonal groups, for instance, I had to dig out of old papers of Mimura and Toda, and otherwise sources not available electronically. I wanted to be able to refer to this in an easier fashion that digging through multiple paper or scanned sources, so I added more rows than was already there, giving reference to the original sources. (This table is not on Wikipedia.)

Note that there is also original research (in higher categories, in foundations etc) on the nLab in its less public corners that has never been published elsewhere (disallowed on Wikipedia, of course). There are also syntheses of existing scattered work into more conceptual unity (a recent example is https://ncatlab.org/nlab/show/prime+ideal+theorem, fleshed out by Todd Trimble who delved into the literature to find these results – no n-categories here! – and write up proofs), at a level not frequently seen on WIkipedia.

The nLab is as useful as you make it. For some it won’t be useful unless they start using it in an active fashion (i.e. not treating it as an encyclopedia, or even Wikipedia).

I can’t exactly say the nLab is useful for number theory, since none of the regulars are number theorists. The algebraic geometry content is very low, as again none of the regulars are algebraic geometers. But there’s a growing body of homotopy theory, for instance, as people are looking at that for their own research.

I’m not looking for a debate, since, sure, there are few of what mathematicians would call calculations on there. But there are references to papers with calculations. Sadly, the technology for diagrams is not up to scratch to do a lot of those, so that’s one strike against the software.

“Pretentious”? No less than quoting continental philosophers and literary theorists to explain why it’s ok to be a mathematician, surely. 🙂

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mathematicswithoutapologiesPost authorFirst of all, I’m very happy to encourage the airing of differences, if they are in any danger of getting stale and brittle. But for once I don’t feel inclined to take sides. I have explained that I have found the nLab frustrating and not helpful, but that hardly means it’s not helpful for topologists. In fact, I conjecture that nLab’s dependency graph becomes simply connected (more precisely, each of its connected components is a tree) if the nodes are removed that belong to a professional algebraic topologist’s standard training. Topologists using nLab may not perceive its circularity.

I had already been thinking of writing a post about whether literary theorists or mathematicians qualify as more obscure. I have no idea why Attila Smith called nLab pretentious. My guess is that David Roberts is using the word to mean “obscure,” in which case he would have a hard time convincing literary theorists that they are more obscure than mathematicians.

But perhaps he is using the word to mean “referring to bodies of knowledge he probably doesn’t understand, and in any case has no business trying to understand.” That would be an instance of the anti-intellectual posturing that is all too common in our profession, and that I personally find extremely irritating. Why do I find it irritating? It’s not because I believe every mathematician needs to undergo a crash course in Great Books — though I do believe a reviewer who boasts that he has not read (say) Max Weber, rather than regretting his ignorance, is simply being boorish. Nor is it because mathematicians like Riemann, Hilbert, and Hausdorff, as well as the founders of 20th century physics, were deep readers of continental philosophers — they were continental people, after all — and it’s simply impossible to understand the history of our chosen profession without being aware of this. No, it’s for a different reason. But that will have to wait for another discussion.

Meanwhile, the literary theorists I know may agree that some of their colleagues are pretentious, but they certainly don’t see themselves that way. And they, after all, are a large component of the target audience.

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David RobertsMy remark was entirely sarcastic and tongue-in-cheek; please don’t take it seriously. (Perhaps I’m unfamiliar with the latest Parisian ideas on Oulipism or what-have-you, but then I’m not a world-famous number theorist who chats to actresses at parties in New York about mathematics, and may very well be temporarily working as an adjunct lecturer. I know my position in the pecking order, and hence my total lack of charisma as discussed in MWA.) I can only point to the definition Google gives “attempting to impress by affecting greater importance or merit than is actually possessed.”, which is what I took Attila Smith to mean. And, I guess what is meant is the so-called n-Point Of View (https://ncatlab.org/nlab/show/nPOV), from which page I quote (the version at the nLab is hyperlinked):

“Around the nLab it is believed that category theory and higher category theory provide a point of view on Mathematics, Physics and Philosophy which is a valuable unifying point of view for the understanding of the concepts involved.”

“So at the nLab, we don’t care so much about being neutral. Although we don’t want to offend people unnecessarily, we are also not ashamed about writing from this particular point of view. There are certainly other valid points of view on mathematics, but describing them and being neutral towards them is not the purpose of the nLab. Rather, the nLab starts from the premise that category theory and higher category theory are a true and useful point of view, and one of its aims is to expose this point of view generally and in a multitude of examples, and thereby accumulate evidence for it.”

“If you feel skeptical about the n-point of view, you may want to ignore the nLab. Or you may want to take its content as a contribution to a discussion on what is behind the claim that category theory is the right language to describe the world, or at least the world of mathematical ideas.”

I can also recommend https://ncatlab.org/nlab/show/About, in particular the section “What the nLab is not”:

“Most importantly the nLab is…not complete and not meant to be complete. Neither its general structure nor each single entry are meant to be optimal in their current state. Many existing entries, possibly all of them, deserve to be and are meant to be eventually improved and expanded on.”

“Notice: an entry being in a pitiful state is usually more a sign of nobody having spared the time and energy to work on it, than of our joint incompetence to write a decent entry if we were being paid for doing it. So if you find your eyebrows raised by some entry, don’t turn away to be the next one not to work on it. Instead, improve it. We all do this voluntarily. We all have other duties to look after. So don’t be annoyed with “us”, help us.”

In this respect, it is most definitely the total opposite to the Stacks Project.

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David RobertsWell, that last comment I made was a bit harsh. I do agree with Michael’s thesis in MWA, and sarcasm aside, I do appreciate and see the value of his viewpoint. It was probably my inner man on the street talking, rather than my European-infused cultural side.

Like Michael’s warnings that MWA isn’t a scholalry work, that it isn’t autobiographical and so on, perhaps the nLab’s protestations and disclaimers on not being authoritative, on merely being a collection of personal notes on a scattering of topics etc will be missed, or misconstrued.

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KantorI am sending the cover and the last words from Blay’s booklet so that anyone can get an idea.

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