Pisier’s counter-example is useful for constructing further counter-examples in the same sub-field, or an inch or two outside it, and as far as I can see, nothing in Apéry’s paper ever applied anywhere else.

So what?

For those of us who worked through Lang’s Algebra once upon a time and haven’t yet resigned ourselves to permanent residence on the other side of a bright line between the two cultures, a certain amount of husbandry is required to apply limited time and intelligence where it’s most rewarding.

In the case of Grothendieck’s conjecture, it’s more fun for me to think about why the maximal and minimal products almost always differ, and this question isn’t resolved by Pisier’s prestidigitation. I wouldn’t be surprised if this were also closer to Grothendieck’s real intention with his “conjecture”… Mathematics of the almost-true is just the sort of thing to appeal to his sideways sense of humor.

It may also interest you to know that in the other culture, where artists mix with the upper-brackets, it’s never okay to make jokes about someone’s name. некультурный, the Russians call it… “uncultured.”

I’ve seen a number of proofs, for instance, of the fundamental theorem of algebra. They range from the estimate-heavy ones you can do as soon as you’re done with Green’s theorem, to the homology-heavy ones that wrap the estimates and bounds and such up into tight little lemmas. I can follow both, and I know that they both have the exact same content (one just hides more in its lemmas). But they still feel radically different to me. The one feels like analysis and the other feels like algebraic topology.

I think it’s a mistake to cast Grothendieck in a role supporting the contention that “we don’t know enough about the analysis, in most cases, to make taking a categorical viewpoint worthwhile,” since Grothendieck’s first significant result was a very productive reformulation of Riemann-Roch

But doormat’s original post clearly refers to the specific instance of Grothendieck’s own question on a Banach space with unique Banach tensor square. He, and I, never said that just because something seems like category theory won’t help, that that supposition is accurate.

Likewise your (tongue-in-cheek) implication that just because I link to a crap film as light relief, I might be the kind of person who

“believe that some branches of mathematics are too old or too new to produce significant progress on contemporary problems, but the history of mathematics is full of counter-examples, where “exhausted” branches produce surprises like Apéry’s entirely classical proof of the irrationality of zeta(3) in 1978”

Yes yes, it’s not like I haven’t heard of the zeta(3) result, or the Ball-Rivoal follow-up, all right? I was simply using the following heuristic:

1) Grothendieck, after introducing beautiful ideas (tensor norms) and making penetrating observations [his inequality for bilinear forms on C(K) is a gem, in all its proofs] makes a natural conjecture motivated by a categorical viewpoint [he's already shown, I think, that every nuclear Banach space is finite-dimensional].

2) Grothendieck never writes or claims a solution to his own conjecture, and instead proceeds to (re-)invent algebraic geometry

I claimed that this “indicates that Grothendieck left his conjecture unresolved precisely because the categorical viewpoint could not at that time be applied to greatest effect.” What part of this don’t you agree with, Jacob? I’m not saying he couldn’t have solved it himself — perhaps he just wasn’t bovvered enough, I have no way of knowing.

You seem to take my comment as espousing a view that mathematics is limited by its arbitrary self-division into disciplines, and counter that the work of geniuses (or one-off ingenious work of us PC Plods) transcends such limits. *I’m not disputing that*. I do think that if a lot of clever people have tried to use their techniques to solve something, and failed, then a better bet is to look at new techniques for that problem, or new versions of the problem. If someone comes along and solves it using classical or “post-modern” technqiues, I’ll be just as happy as the next person. That doesn’t mean I read any of the arXiv proofs of the Riemann hypothesis in any detail…

Really, you seem to be inferring too much from my post on someone else’s post on a very particular problem and a very particular paper by Pisier (have you gone and had a look at it?) In fact, what I was trying to say seems close to your rejoinders: Pisier brilliantly exploited some recent but standard results of cotype and Fourier analysis and applied them, with *implicitly categorical ideas*, to resolve a question everyone thought intractable. So how can I possibly be saying that category theory is too new or old to help? I think Pisier’s paper is in fact valuable ammunition if you want to assemble a case that categorical methods are at present under-employed in parts of analysis, which is something I thought several posters in this thread might agree with.

Walt:
Isn’t that a little much to ask of anyone, that they carefully judge every mathematical utterance on how it will affect theoretical precocious high-school students?

Again, I just want to raise people’s awareness. Perhaps John’s utterance isn’t likely to do much damage in the scheme of things, but it did seemingly invoke an unnecessary contrast between categories and analysis. Why not avoid that sort of thing?

Todd:
I second thw’s request to see how this Gateaux derivative stuff explains the ‘cavalcade of normality’.

I third!