Gossip about Grothendieck and distributive lattices 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?)
 A: 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.
A: First, an answer to Pete Clark's comment on the Chinese remainder theorem can be found in Floris Ernst's 2004 University of Otago Master's thesis Multiplicative ideal theory (pdf link). Prüfer domains are exactly those integral domains for which the ideal lattice is distributive, and Ernst indicates that these are exactly those integral domains "satisfying CRT": see section 3.3 for an explanation of what he means.
Edit: Pete has updated his notes on commutative algebra to give an account of the relevant facts (section 21). See also this discussion on CRT and distributive lattices at Mathematics StackExchange.
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,1 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.
1Finberg, David; Mainetti, Matteo; Rota, Gian-Carlo, The logic of commuting equivalence relations, Ursini, Aldo (ed.) et al., Logic and algebra. Proceedings of the international conference dedicated to the memory of Roberto Magari, April 26–30, 1994, Pontignano, Italy. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 180, 69-96 (1996). ZBL0862.03039, MR1404934.
A: 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.
A: 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.
A: Gian-Carlo Rota was perhaps thinking about Matroid theory.
His work is cited for example in the preprint
"Matroid for algebraic geometers" by Eric Katz.
If Grothendieck has known about lattices, he may
have defined a cohomology theory for lattices.
Gian-Carlo Rota actually defined homology groups for certain subsets,
called cross-cuts, of a lattice in 1964.
Whether Grothendieck could have anticipated
the numerous connections between matroid theory,
tropical geometry, enumerative geometry and algebraic geometry,
I don't know.
When it comes to the quote about
Grothendieck school clumsily reinventing the rudiments of lattice theory,
Garrett Birkhoff's book on lattices (1948 edition) has a section
about applications to algebra and algebraic geometry.
The well-known fact that
"every algebraic variety has a unique expression as an irredundant
sum of a finite number of irreducible components" is derived
from a general result about distributive lattices satisfying
the descending chain condition (ch IX 8.).
An algebraic geometer may be able to check if this specific
result about lattices applies to some category of schemes and
gives a theorem of Grothendieck.
