dependency3

That’s as much as would fit of the force-directed dependency graph of Stacks Project Tag 07Y5, the tag for a result that I actually cited in my course at Columbia last spring:

80.17.2

The Stacks Project is now 5000 pages old, but not everyone at the project’s birthday party knows much about what mathematicians do, so it’s not surprising that within minutes of arriving I stumbled, or rather was shoved by the forceful personality of the inimitable mathbabe, “into the awkward moment every mathematician dreads” and asked to explain what the Stacks Project is about to one of the guests.   As mathbabe knows well, however, I have worked through my dread of such moments by writing some 50 pages about one of those moments in MWA; moreover, my interlocutor, whom I shall call MW (M not for mathematics), was resolved that (unlike the exchange between NT and PA in MWA) our exchange would be cooperative rather than adversarial.  Thus ensued a dialogue that went roughly like this.

NT:  You probably know that mathematics is based on rules, and mathematicians agree to follow the rules, but periodically we encounter a phenomenon to which the rules don’t apply.  Thus we need to develop a new set of rules [here there was some mumbling to forestall possible objections about the axioms of set theory] that allow us to incorporate the new phenomena.

MW:  Can you give me an example?  I don’t know much about mathematics, but maybe you can use analogies from biology.  So would this be like finding a new disease?

NT: [not quick enough to realize that the discovery of prions or viruses might actually provide a very helpful analogy] No, the rules in mathematics operate at a more basic level…

MW:  So like cells?  Or proteins?

NT:  [thinking this over]  It could be that … if we discovered something new about how cells work that can’t be accomodated by our old theories…

Bystander:  More like evolutionary biology…

NT: But I’d like to say that it’s more at the level of how life is possible at all.

MW:  So if life were discovered on another planet, but not carbon-based, we would have to revise our understanding of what life is.

NT:  That’s actually a very good analogy.  The rules that are specific to what we know about life on earth would have to be broadened to allow for the new discoveries.

MW:  And it’s something like this that explains why you number theorists need the Stacks Project?

NT:  Yes and no.  In a way the Stacks Project has made it possible to use ideas that have been around for a long time but were never clearly explained.  But — I don’t want to exaggerate the role of the technology — but the Stacks Project has also replaced its immediate predecessors, which took the form of books… [some of the younger post-docs and students look at each other quizzically]… whereas you can ask the Stacks Project what you’re looking for.  Not only that, it not only gives you the answer but it provides pictures that you can copy and paste into your own work!

[The Maintainer of the Stacks (Garde des Champs, in French, though he’s not French) has been listening with approval to the last part of the dialogue.]

MW: [between amusement and disbelief] So you can steal from the Stacks Project!

The Maintainer:  We want them to use what’s in the Stacks Project! That’s what it’s there for!

NT:  You know, it occurs to me that sometimes when I ask a question I get a list of answers… in the future it should be possible — I mean it will be possible — to talk to the Stacks Project  and get the precise answer to my question, or more likely SP will voluntarily offer the answer to the question I should have asked.

[While the Maintainer is wondering whether or not he should apologize because this option is not yet available, the dialogue continues.]

MW: And this would force the biologists to rethink their understanding of the principles of life yet again…

Had MW seen the above force-directed dependency graph she would have found the analogy even more convincing.  Later at the same party I mentioned that I had found letters I had received from one of my colleagues when we were both graduate students, and one of the post-docs blurted out, “you mean you actually wrote letters?”  while the others tried to imagine what that might entail.