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In Gian-Carlo Rota's Indiscrete Thoughts, there a list of mathematical gossip among which one reads:

[...] What would have happened [...] if Grothendieck had known the theory of distributive lattices?

So... What would have happened? (and/or: what did Rota think would have happened?)

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This strikes me as being pretty speculative. Is there any way tighten it into a question that admits some kind of reasonable answer? –  Andy Putman Sep 18 '10 at 23:36
If Grothendieck had known the theory of distributive lattices then he would not have had to waste a couple of hours reinventing it. Gian-Carlo Rota got overexcited about some routine lattice theory Grothendieck developed in EGA; apparently Grothendieck either didnt know or didnt care that it was already known. –  Richard Borcherds Sep 19 '10 at 0:06
@Richard: Do you know what part of the theory of distributive lattices did Grothendieck reinvent? There are $>3200$ papers about distributive lattices according to MathSci. I am not sure Grothendieck could invent all that in 2 hours. I do think that the question is appropriate for MO, and would like to see an answer from somebody well informed. –  Mark Sapir Sep 19 '10 at 2:15
What would have happened if X had known Y questions are quite speculative and easy to generate. IMO without more background data on why this would be of interest to anyone, I don't think this is the right forum for the question. –  Ryan Budney Sep 19 '10 at 2:31
There's only one way to find out: Go and find Grothendieck and teach him about distributive lattices. –  David Corwin Oct 10 '10 at 6:13
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4 Answers

Gian-Carlo Rota made a related comment in his article "The Many Lives of Lattice Theory" (Notices of the AMS, Volume 44, Number 11, December 1997, p. 1442):

"Dedekind outlined the program of studying the ideals of a commutative ring by lattice-theoretic methods, but the relevance of lattice theory in commutative algebra was not appreciated by algebraists until the sixties, when Grothendieck demanded that the prime ideals of a ring should be granted equal rights with maximal ideals. Those mathematicians who knew some lattice theory watched with amazement as the algebraic geometers of the Grothendieck school clumsily reinvented the rudiments of lattice theory in their own language. To this day lattice theory has not made much of a dent in the sect of algebraic geometers; if ever it does, it will contribute new insights. One elementary instance: the Chinese remainder theorem. Necessary and sufficient conditions on a commutative ring are known that insure the validity of the Chinese remainder theorem. There is, however, one necessary and sufficient condition that places the theorem in proper perspective. It states that the Chinese remainder theorem holds in a commutative ring if and only if the lattice of ideals of the ring is distributive."

I think this sheds some light on the question "What did Rota think would happen?" as Ben Webster interpreted the question.

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What a nice answer! I am glad the question had not been closed prematurely. –  Victor Protsak Sep 19 '10 at 10:54
What I'd be really interested in hearing is the response of an algebraic geometer to the above statement. –  Harry Gindi Sep 19 '10 at 13:25
What annoys me is: if Rota knowingly asserts that lattice theory will contribute new insights if only algebraic geometers would pay more attention, then why doesn't he just spit it out? Given his stature, algebraic geometers would have paid attention. Why cloak it all in mystery and coyness? –  Todd Trimble Sep 19 '10 at 15:03
I wonder what version of CRT Rota had in mind? The one I think is most standard -- see e.g. p. 34 of math.uga.edu/~pete/integral.pdf -- holds in any commutative ring. –  Pete L. Clark Sep 19 '10 at 23:28
@Mark: perhaps the "emperor's clothes" went a bit too far (although I have to say that IMO quite a lot of Indiscrete Thoughts, which I have read and reread a number of times, is similarly both exasperatingly vague and polemical at the same time). To put this in balance: I am a huge admirer of Rota when he is behaving like a professional mathematician; he writes mathematics beautifully and most thought-provokingly. But in this particular case, nobody here seems to know what he's talking about exactly, and I'd honestly like to know if there's anything there. Can one of his students explain? –  Todd Trimble Sep 20 '10 at 3:25
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First, an answer to Pete Clark's comment on the Chinese remainder theorem can be found here in an article by Floris Ernst on multiplicative ideal theory. Prüfer domains are exactly those integral domains for which the ideal lattice is distributive, and he indicates that these are exactly those integral domains "satisfying CRT": see section 3.3 for an explanation of what he means.

Second, I'd just like to underscore what is perhaps the heart of the article mentioned by Marko Amnell: what Rota calls "linear lattices" (lattices of commuting equivalence relations), the most important class of modular lattices. These include lattices of ideals, lattices of normal subgroups, and much more generally, lattices of algebraic congruence relations that arise for algebras of a Mal'cev theory, much studied by universal algebraists.

The point is that such lattices (and I repeat that ideal lattices are examples) have an incredibly rich structure: they are not just modular lattices, they are also Desarguesian (satisfy an axiom which generalizes the Desarguesian axiom for projective planes), and in fact satisfy a battery of equational identities incapable of being finitely axiomatized. I'm mentioning all this because I think it's highly likely that linear lattices were very much on Rota's mind at the time of writing Indiscrete Thoughts. See for example this paper from Google books, by Finberg, Mainetti, and Rota, which gives a kind of natural deduction calculus for linear lattices.

If it is indeed the case that Grothendieck was rediscovering some of the identities known to occur in lattices of commuting equivalence relations, then perhaps Rota would have a point there. But I'm not sure about that: for example, Rota credits Schützenberger with the discovery of the Desarguesian identity, and I think this might have occurred post 1965. I haven't looked into the history (nor do I have any but the first volume of EGA at hand; treillis wasn't in the index there).

I can't resist adding, for anyone who might be interested, that the "logic" of linear lattices developed by Rota and his collaborators smells very similar to the graphical calculus outlined in Categories, Allegories, by Freyd and Scedrov (section 2.158), for deciding which equational identities expressible in the language of allegories holds in the allegory of relations between sets. Yes, Freyd and Scedrov indicate that this theory is decidable (which seems related to concerns expressed by Rota in that AMS Notices article), and the decision algorithm is based on graphs which admit a "parallel-series" decomposition in the sense of circuit theory, which looks on the face of it very similar to the graphical calculus given by Finberg, Mainetti, and Rota. I don't know whether categorical logicians have been talking much with lattice theorists, even though they were some of both at the conference where Finberg, Mainetti, and Rota read their paper.

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@Todd: Thanks! We still don't have a response from algebraic geometers. Perhaps it is because somebody removed (IMHO unwisely) the alg.geometry tag from the question, or perhaps because there is nothing for them to say. I thought that the connection between lattices and Grothendieck would be related to the etale cohomology and Grothendieck topology which generalized Zariski and Stone topologies. Chinese Remainder Theorem does not need anything outside the ordinary commutative algebra. But perhaps one can apply lattice theory to some generalizations of CRT like the strong approximation? –  Mark Sapir Sep 21 '10 at 18:51
Thanks, Todd -- this answers my question very nicely. I am (moderately) familiar with Prufer domains and their characterization via lattice-theoretic properties of their ideals. What I was missing was the precise formulation of CRT as given in the notes you linked to. (I will add though that I have used the "comaximal CRT" many, many times in a variety of contexts, whereas I think I have never needed the Prufer CRT, not even in the special case of Dedekind domains, which I have thought about a fair amount.) –  Pete L. Clark Oct 10 '10 at 9:20
You're welcome, Pete -- I was puzzled too, not knowing what Rota could have meant by "necessary and sufficient conditions" for CRT, since as you had pointed out, the more familiar version of CRT is a very simple and general thing. So I googled it, and this was one of the results. But I'm still puzzled as to why Rota seems to think that this result is something Grothendieck would have especially benefited from, or whether there's more to the story that he's not telling us. –  Todd Trimble Oct 10 '10 at 12:23
I think it's possible to feel sure that if algebraic geometers knew more lattice theory they could do good things, without knowing what these good things would be. I feel things like this all the time. Perhaps it takes a certain chutzpah to put these feeling into print, though. So I'd call Rota's sin "chutzpah" rather than coyness. You're only being coy if you're holding something back. –  John Baez Dec 4 '10 at 2:57
No fair: you and Timothy Chow actually knew Rota! Seriously, that sounds like a good possible explanation, but I keep coming back to this: "Those mathematicians who knew some lattice theory watched with amazement as the algebraic geometers of the Grothendieck school clumsily reinvented the rudiments of lattice theory in their own language." That sounds a lot more like he knows something and is holding back than it does intimations of vague feelings. (By the way, I love the word chutzpah. Did you know that if you play it in a triple-triple line in Scrabble, you can get 392 points?) –  Todd Trimble Dec 4 '10 at 11:07
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It's a tendentious question, certainly. It might mean, if Bourbaki, let us say, had had more of an interest in lattice theory, that the French word for "lattice" of this kind would be more familiar at least to me (it's treillis, which is not overloaded in the same way that the English word lattice is in mathematics).

Bourbaki wasn't very interested in logic, and hardly interested at all in combinatorics. Which is why Rota's comment is meant to sting a bit. Let's just assume, to keep a cool head, that the "Bourbaki" viewpoint from around 1965 is something of historical interest, not a description of how mathematics is or should be 45 years later. (It's an obvious remark, but needs making. We'll get back to Grothendieck in a moment, but I choose 1965 as a date at which the equation Grothendieck=Bourbakiste ultra would be at its most plausible. He was both more and less than that. Less because he came to regard writing up foundational material as a chore.)

Bourbaki claimed descent from Hilbert, but "reception theory" applied to French importation of ideas (particularly German thought, but this goes back to Newton and Locke) is never that simple. For logic, there was some joy in pushing aside transfinite induction, with Zorn's lemma, because ordinals could then be dropped. The negative part of Hilbert's legacy (lack of conscience about non-constructive arguments) was accepted, the interest in metamathematics not.

So we come down to a few questions:

1) To what extent is the lack of interest in logical infrastructure in this Bourbaki-1965 outlook a matter of sheer prejudice and arbitrary decision-making probably to be traced to early Bourbaki from the late 1930s and so to Weil?

2) To what extent, on the other hand, is it a principled approach to the overall conception that mathematics belongs in general theories, axiomatized in say ZFC, that this is Cantor's Paradise as spoken of by Hilbert, and logic fortifies Eden rather than inhabits it?

3) Is the expansion of general theories, such as is seen in SGA in an extreme form (some would say, not all though), really contingent?

The last one seems the good question for historians. To some extent the agenda for mathematics is laid down by "tradition", to some extent mathematicians update it by editing the list of traditional problems (say odd perfect numbers out, Weil conjectures in), and to some extent applications drive change. That assumes the agenda is phrased as concrete problems, not "we'd like a theory of X". But perhaps we would. We can take mathematicians' talents and motivations to be possible contingent factors; but we can't assume that those talents are so portable as to be as good for area B as area A.

By the way, the answer to Q3 seems clearly to be "some truth in this".

In the end, could Wagner have written symphonies that became part of the repertoire, as most of his operas have? How counterfactual would we have to be with his biography to make that more plausible? It wasn't that he needed to know more musical theory. Rota is wanting "alternative history" for mathematics, but like many who play with alternative history, he's making some kind of political point.

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Johnstone expressed a similar sentiment in the introduction to his book on Stone Spaces (you can read it on Amazon) where he gave a fairly detailed account of how the Stone representation theorem and the theory of continuous lattices it inspired anticipated some of the formalism of modern algebraic geometry.

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@Justin: Thank you for the reference to Johnstone book. He writes that Zariski topology is a generalization of Stone topology, that Grothendieck most probably knew about Stone's work, implicitely used it, but did not refer to Stone. That is not quite what Rota claimed, as I understand. Am I correct? –  Mark Sapir Sep 19 '10 at 16:39
@Mark: The Zariski topology was invented substantially earlier by Zariski, not Grothendieck, for algebraic varieties and algebraic sets. It's more likely that Grothendieck noticed that since irreducible closed subsets are in canonical bijection with prime ideals, we can keep track of them as actual points. The fact that we can do this follows from the observation that while maximal ideals are not preserved under preimage, prime ideals are. This gives us an "enrichment" of the classical Zariski topology by adding generic points. –  Harry Gindi Sep 19 '10 at 19:05
That is, if one wants to claim that the Zariski topology is a derivative of the Stone topology, one should look back to the papers of Zariski to understand the historical development better (and see if in fact the Zariski topology was developed from the stone topology). –  Harry Gindi Sep 19 '10 at 19:10
@Harry: I only repeated what Johnstone wrote in the introduction of his book. It seems to be different from what Rota had in mind. Did you read Johnstone's book? –  Mark Sapir Sep 19 '10 at 19:32
@Mark: Ah, okay. WRT Johnstone's book, no, I haven't. –  Harry Gindi Sep 19 '10 at 23:44
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