Not only is John Armstrong a failed crackpot, he is wrong about statistics. Statistics is, from the mathematical point of view, a perfectly interesting subject; this fact is carefully concealed from us by statisticians. For example, most mathematicians know the central limit theorem, which says that the sum of large numbers of independent, identically distributed (iid) random variables tend to be normally distributed. This even has an elegant proof in terms of Fourier analysis, where addition of random variables because multiplication of Fourier transforms.
What mathematicians don’t know is that almost every other statistic ever defined also satisfies the central limit theorem. The median of a large number of iid random variables? Normally distributed. The mode of a large number of iid random variables (where the underlying distribution has a single mode)? Normally distributed. The cosine of the seventeenth percentile? Normally distributed. The simplest explanation for this cavalcade of normality involves the Gâteaux derivative in functional analysis.
So now please explain how this universal CLT works.
Oh come now. Can’t you see begrudging respect in the nickname from someone who uses “DrMathochist” as his handle in damn-near every online forum he frequents?
Still, I am first and foremost not-an-analyst, and… that field… is close enough to analysis that it just doesn’t mesh with my poor, category-stricken brain. Don’t get me wrong: I’m glad that someone does it. I’m just even more glad that that someone isn’t me.
I’ve forgotten most of the probability theory people tried to teach me, but there’s been some interesting-sounding work (Barron? Johnson?) on getting CLT-like statements out of the fact that the normal distribution maximizes entropy relative to certain constraints…
Walt: is this some kind of concentration of measure phenomenon?
hellblazer,
just for the sake of completeness, the “certain constraints” are that the normal distribution maximizes entropy over all distributions with a fixed mean and variance. So they’re not something particularly exotic.
While having a finite mean and variance might seem pretty easy-going constraints on a probability distribution, there are distributions which don’t satisfy even these. The Cauchy distribution, for example, has an infinite variance. Not a problem, except to those of us studying (eg) financial markets, where variables of interest often seem to be rather fat-tailed, and best modeled as having Cauchy distributions . . . .
Peter,
sure, there are plenty of reasonable distributions (Cauchy being the prototypical example) that don’t have finite variance. The point I was trying to make is that when I hear “certain constraints”, I often think that the constraints must be hard to explain, because otherwise wouldn’t the speaker or writer just come out and say what they were? I wanted to make it clear that the constraints in question are at least simple to state and not artificial-sounding to the uninitated.
Isabel,
Thanks — that’s what I thought, but at 3am it’s hard to remember things you read as a not-quite-graduate student… I just didn’t want to misquote the result!
I really should go back and read those papers…
Still, I am first and foremost not-an-analyst, and… that field… is close enough to analysis that it just doesn’t mesh with my poor, category-stricken brain.
How/why is analysis incompatible with categories?
It’s not incompatible (not that John was implying it was!). But, many (most?) algebra- or logic-oriented people, including category theorists, have little taste for analysis, and vice-versa. It’s more a personality thing.
I’m a categorist, and definitely not an analyst!, but I don’t mind analysis if I think I can get a grip on the conceptual point. But the kind of analysis which involves lots of grungy hard-core estimates — yuck. Not my thing.
It’s not incompatible, per se, but it’s clearly a vastly different style of work. You may as well ask why neo-classicism is incompatible with cubism.
thw: I’ll see if I can come up with a snappy formulation.
hellblazer: In a literal sense, the measure of a statistic is concentrated around its asymptotic limit, but I don’t know if this can be translated into the framework here.
At the risk of going too far off-topic: perhaps one factor is that in analysis one often makes lots of arbitrary choices — truncate your function at some height K, chop it up into places where one norm is small and another is big, take epsilon less than 0.83, and so on. This seems to me (a non-categorist) to go against the categorical way of thinking, which is more inclined to go with as much canonical or functorial information as you can.
Out of interest, how do posters above find Terence Tao’s post
here
I think that’s a nice example of rationalizing what might otherwise seem a bunch of tricks.
IMHO, the divide between algebra and analysis is something of a self-fulfilling prophecy, the real difference seems to be between people’s preferences for the way they do maths. Only a suggestion.
Aargh, sorry, mangled URL. Can you fix that Walt, or shall I repost?
Also: having thought a bit more, I remember reading somewhere (in the introduction to some paper on free probability) that the CLT can be interpreted as some kind of fixed point theorem, and there was some mention of derivatives of a non-linear operator on aninfinite-dimensional space … is that closer to what you meant?
Another thought, partially in response to Todd’s adjective “grungy” (with which I agree, I’m not that kind of analyst). Often it seems that these things look worse than they are, because the style of formal presentation in a journal paper tends to frown on discursive hand-waving which would *explain* or *motivate* the hacking.
To overstate the case a little: I suspect that many analysts don’t think up their proofs the way they write their proofs. Of course, one can argue that most articles are written by specialists for specialists, and so no one wishes to seem like they’re talking down to their audience.
hellblazer, Terry Tao writes wonderful posts on his blog, but I find that one you mentioned particularly elegant (not to mention penetrating). That man is obviously a master.
Hellblazer sez
The exact same thing could be said about any field. Category theorists, algebraic topologists, universal algebraists.. they all leave out huge swaths of the detailed reasoning and motivation because it would be redundant among experts to include it.
In fact, the only place I think you’ll regularly find an exception is in a (relatively) new interdisciplinary field like quantum topology, where your audience might consist of people who are experts in only one slice of the field. There are highly-respected knot theorists with who, I still have to walk through the basic idea of a categorification every time I talk to them, reminding them that they’re not all like Khovanov homology and don’t all use the bicategory of chain complexes.
Anyhow, the point is that there really is a difference between fields of mathematics, just as in any other culture. Everyone can (roughly) tell a nerd on a short exposure. Nerds can (roughly) tell mathematicians from physicists from computer scientists (etc) on a short exposure. Mathematicians can tell analysts from algebraists from topologists… You just get an intuitive feel for it after a while.
Dr Unapol:
I wasn’t disputing that, although I am in what I’d guess is the relatively small set of people who frequently re-read bits of Rudin’s Complex Analysis and Weibel’s Intro (!) to Hom Alg…
I just worry in my more fretful moments that there’s a tendency to needless fragmentation. There’s a large number of analysts who would peg me, on the intuition you mention, as an algebraist manqué, but then again the words “charateristic p” fill me with dread, so go figure.
Oh, don’t get me wrong. There are always people who walk between the fields. In fact, I contend that mathematics as a whole is going through a synthetic phase.
That is, we’ve been multiplying fields for a long time (most of the 20th century) and splitting existing fields into new disciplines. Now we’re seeing that many of these fields are “really” just different sides of the same thing, and the big theorems are all about showing how two distinct areas are actually intimately related, even if they weren’t derived from the same parent field. Still, some fields are farther apart than others on the surface, and harder to rejoin.
Incidentally, part of why I love category theory is that it seems the natural (sorry) language for describing mathematics in such a cyclical framework. Of course, I’m biased, but still…
Coming late to the party… I do love Terry Tao’s weblog, and the post Hellblazer links two is great (I even used it in my teaching). However, “…example of rationalizing what might otherwise seem a bunch of tricks” I disagree with. It still seems like a bunch of tricks, just with some vague overall theme. I think this is just how analysis is: the same is often said of the Baire Category Theorem– Tom Korner says that it comes close to being a theorem which captures what the trick of “sliding humps” is all about, but it still doesn’t quite do that technique justice. And, in Banach space theory, I’ve given up trying to use theorems proved from Baire Category: it’s usually just easier to go back to source and us BC directly, instead of trying to use Principle of Uniform Boundedness etc.
On categories and functional analysis, an old paper which I like is by Herz, The theory of $p$-spaces with an application to convolution operators. He does some abstract stuff with curly L_p spaces, and then immediately some nice results about representations of groups drop out. To me, it really makes clear _why_ the result is true, and hides the nasty details. That said, unless p=2 and you are really in the world of Hilbert spaces, C*-algebras and so forth, if you try to take the theory much further, you run into very hard problems about the structure of L_p spaces. I think it’s just that we don’t know enough about the analysis, in most cases, to make taking a categorical viewpoint worthwhile.
I agree with Doormat and disagree with John A. Analysis is full of tricks. Algebra has many fewer. In algebra, once you find the right abstractions, the proofs frequently write themselves. In analysis, there’s always a gap between the intuition and the proof.
(As an aside, I think analysis books are getting better at placing the importance of tricks at the forefront. Older books on functional analysis tend to hide what makes proofs work. Newer books foreground proof ideas such as “sliding humps”.)
How can you say you disagree with me? You just described a huge difference in the culture, and that’s all I asserted to exist!
But, many (most?) algebra- or logic-oriented people… have little taste for analysis, and vice-versa.
Paul Halmos, anyone?
In general, I think the counterexamples to this stereotype are sufficiently distinguished that the stereotype doesn’t deserve to exist.
The notion of analysis and algebra being “opposing forces” amounts in my opinion to an implicit slight of huge areas of mathematics, particularly functional analysis, where they interact closely.
In algebra, once you find the right abstractions, the proofs frequently write themselves
The demigod of this style of mathematics, Alexander Grothendieck, began life as a functional analyst. The theories of nuclear spaces and topological tensor products are largely due to him. His Ph. D. advisor was Laurent Schwartz, the inventor of distributions.
I cannot help but think that, to the extent that there are such cultural divides in mathematics, they are the result of people being underinformed. (I think this is pretty close to the point of Walt’s original post!)
If you read Terry Tao’s blog, you’ll frequently see him make comments such as “I’m not particularly well-versed in this area, but…”. He will rarely if ever say anything like “Yuck!” about a branch of mathematics.
Who said anything about “opposing forces”? And it’s got nothing to do with being underinformed. When I read analysis, I can follow it fine, but it just doesn’t resonate with me like my preferred fields (knot theory, category theory, representation theory, “quantum topology”). And I’m far from the only one who finds certain fields just “feel right” to an extent that others don’t.
komponisto, I don’t think I need a lecture from you. We already agree that analysis and algebra are not incompatible; far from it. All I asserted is that many algebra-oriented people have little taste for analysis. Which happens to be true, like it or not.
Despite the isolated “yuck” (which you seem to have taken amiss), I actually like a lot of analysis, and I’ve certainly enjoyed teaching it (even at the graduate level). I’d also guess that there are areas of mathematics that Terry Tao doesn’t resonate with as well as others; should that come as any surprise?
Please don’t go slinging words like “underinformed”. You don’t even know me.
The example of Grothendieck is good though, I think. He left analysis claiming the field was “dead” (see page 8 in Notices of the AMS article (PDF). This, in hindsight, was both right and wrong. I’ve worked a bit in the theory of tensor products of Banach spaces, and it’s a beautiful area which has lots of applications to Banach space theory (and lead to Pietsch’s theory of Operator Ideals and so forth). However, one of Grothendieck’s conjectures was that for any infinite dimensional Banach space X, the maximal and minimal tensor products of X with itself always differ. It took until 1983, and someone of the power of Pisier, to find a counter-example! (Link to Acta Math paper) That is, it took many, many years for the knowledge of the analysis to catch up to the point when the more category theory type ideas of Grothendieck could be attacked.
Thanks to “Doormat” for his very stimulating comments, but 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-Roche in terms of morphisms between varieties, “at a time when no one thought of applying categories and functors to such a hard problem.” (Borel)
(taking a break from marking. Apologies in advance for long and rambling post)
Responding in reverse order of posts:
* I think Mr. Freeze (Dr. Fries?) misunderstands Doormat’s point, which as I read it is that Pisier’s counterexample needs some analytic input (due to Kislyakov or Bourgain, if I recall correctly? — corrections from DM welcome!) and so indicates that Grothendieck left his conjecture unresolved precisely because the categorical viewpoint could not at that time be applied to greatest effect.
* In fact — and thanks to Doormat for the link, I’ve lost my photocopy of that paper — there is IMHO something *very* categorically flavoured in part of Pisier’s construction — he references some observations of Kislyakov which are nothing but pushouts with norm control, and the counterexample itself is constructed as some kind of iterated pushout or similarly-flavoured colimit. (It has a construct-a-left-adjoint-by-force feel to it, for the categorists reading.)
* without wishing to inflame things further: I wouldn’t go so far as komponisto does, and I would *certainly* not bandy words like *uninformed* around (although I do see some intemperately zealous language on the n-cat cafe from time to time). However, seeing as most people on this thread believe that analysis and algebra have distinct flavours that almost all mathematicians with working taste buds could distinguish, I’d like to modify my original witterings above slightly…
Walt said:
Analysis is full of tricks. Algebra has many fewer. In algebra, once you find the right abstractions, the proofs frequently write themselves. In analysis, there’s always a gap between the intuition and the proof.
OK, that’s certainly true (though I think that’s a function of how we do analysis at the moment, and the accepted mores of how one writes up the argument for maximum kudos). The point of my Life of Brian link earlier was to suggest, obliquely, that the differences within the Two Tribes can be as great as those between them. Or, to be more geeky, perhaps the Hausdorff distance between them isn’t that much bigger, and may even be smaller, than their diameter.
F’rinstance: when I started my PhD I thought algebra, and by extension, your generic algebraist, was all about functors and pullbacks and taking the graded ring and all that stuff I was too coarse to appreciate. Then, I ended up going to a lot of talks in geometric group theory, and it seemed much more reminiscent of combinatorial analysis (with a real hacky case by case flavour at times). Sure, there’s a conceptual underpinning (did I hear Ronnie Brown shout “groupoid” just then?) but I saw very little category theory (for instance) in these talks, whereas I ended up using it all the time . Trying to ask my fellow grad students about, say group representations, was a non-starter! and that’s before we mention the derived category of modules in characteristic p
Related anecdote: apparently in some working seminars it was observed with slight labour that the centralizer of the centralizer of the centralizer of a set was the centralizer of the same set. At which point the *analyst* in the audience reportedly shouted “Galois connection” …
(I guess this is my real source of mild irritation: the labels Algebra and Analysis are so wide as to be misleading. I’m more partial to Tim Gowers’ floated suggestion of two cultures in mathematics myself, though I don’t at all suggest it is The One And Only Truth, and to be fair neither does he.)
Todd and John, it was not my intention to suggest that you personally are underinformed; nor does my comment amount to a “lecture”. All I am doing is stating a point of view that I think is insufficiently voiced.
It is the community in general that is underinformed. The result is that certain attitudes are created in the culture, and these have a tendency to infect even the well-informed. I have run into too many algebra-type people who express disdain for analysis in such a way as to seem unaware that “soft analysis” even exists. Conversely, I know of at least one reknowned analyst who, as editor of a journal, would automatically reject a paper if it discussed “topological vector spaces”. I contend that this situation is lamentable.
If I have any problem with the discussion here, it is not that people are pointing out the existence of this divide –who could fail to notice it? — but rather the sort of c’est-la-vie complacency there seems to be about it. Furthermore, and contrary to John Armstrong, I don’t think it should be regarded as analogous to neoclassicism vs. cubism. Areas of mathematics have specific logical relations to each other. Banach spaces form a category; operators studied in PDE are elements of C*-algebras; and so on. Opposing these things to each other is like saying you love the integers but can’t stand the rational numbers. People do say this kind of thing, but I think it’s silly, and they ought not to.
Well, nor was it my intention to perpetuate stereotypes, or suggest a notion of “opposing forces”.
Nor do I think it’s right to infer, from what I wrote, that my attitude is one of complacency. Your original question, “How/why is analysis incompatible with categories?”, innocent as it sounded to me at the time, seemed to invite a simple explanation that it isn’t, but that some people’s minds work differently from others. Then somehow a spotlight was thrown on my admittedly very casual response, which I now begin to regret. Sheesh.
Anyway (once again), I think we are in fundamental agreement: I too like to see bridges built across disciplines, and I am so not into turf wars. But firmly entrenched minds (like that of the renowned analyst you mention) are not likely to be changed by anything except dramatic results which draw from across disciplines. In other words, yes, sure, I agree with you, but then what?
hellblazer, you’re not contradicting anything here (not that you’re trying). I’d say that geometric group theory studies groups — which, yes, originated as algebraic objects of study — in an analytic style with analytic tools. You seem to think I’m ascribing certain techniques for certain objects of study, when all I’m doing is describing technique and style.
And the exact same thing goes for komponisto. I’m not saying that there are these things that we study with those tools, and so we parse up all of mathematics.
What I’m saying is that some mathematicians feel more comfortable with one style and one set of tools, and sometimes we find ways to apply our tools to objects that other people have traditionally used their tools on, but in the end I just don’t think in terms of estimates and bounds and all this messy convergence without a lot of effort, and I’m far from the only one. On the other hand, I’ve known analyst after analyst answer, “can’t you see that as a functor?” with “yes, but why would you want to?”
It’s not about what we study, it’s about how we study it. Stop putting words in my mouth.
Crikey. I wish I’d stuck to the CLT now… Rather than dig myself further into trouble, I think I’d best bow out with one of my favourite homilies:
“I ain’t too certain about where people stand. P’raps what matters is which way you face.”
“hellblazer” says…
If your brain is wired for obvious associations (Mr. Freeze? Like in Batman?), then it’s probably comforting to 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, or where a new branch springs into existence before anyone can assess its unreadiness for the task at hand.
Leaving aside the most obvious examples (Galois, Poincaré, etc), I might as well seize this rare opportunity to shine a little light on Riemann’s utterly obscure 1861 non-prize-winning essay for a contest at the Paris Academy, where Riemann applied his newly invented curvature tensor to a problem about heat flow in solids, to the absolute stupefaction of the judges. Was Riemannian Geometry ready for thermodynamics in 1861? Was Category Theory ready for Riemann-Roch in 1957? Any number of similarly absurd questions can be generated from the assumption that the productivity of any branch of mathematics is limited by anything except the ingenuity of individual mathematicians.
John:
It’s not about what we study, it’s about how we study it. Stop putting words in my mouth.
Well, you did say the following:
that field… is close enough to analysis that it just doesn’t mesh with my poor, category-stricken brain.
… fairly clearly drawing a contrast between two fields (not styles), and suggesting that prolonged exposure to category theory has had a deleterious causal effect on the ability of analysis to “mesh with” your brain. (Hence the “incompatibility” I was referring to).
Look: just regard my complaint as an instance of consciousness-raising, rather than a criticism of your own personality or point of view. It would be better if people winced slightly before writing sentences like yours above. I’m not so much interested in changing the minds of those who are set in their ways as I am in preventing the spread of unhelpful prejudices to the less initiated. I’m thinking for example of a precocious undergraduate or high-school student who may be reading this, to whom it has never occurred that a fondness for categories might entail a dislike of analysis, or vice-versa. What good does it do to plant such an idea in their mind?
‘Twould be nice to get back to the mathematics. I second thw’s request to see how this Gateaux derivative stuff explains the ‘cavalcade of normality’.
komponisto: 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? John phrased his point in a self-deprecating way, as opposed to a hostile way, which I think is the most that politeness demands. If John was expressing real disdain for analysis, I might agree with you, but he’s not. (I also think that there’s some truth to what he’s saying. If you look over the broad expanse of math, you’ll see a wide range of thinking styles; so many that it’s hard to keep them all in a single human brain.)
Now you’re telling me that I said a field is different from a style. Stop it!
John: A field is different from a style. Do you deny this?
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!
OK, I know I said I was bowing out, but I want to clarify something with Jack Frost here (and I haven’t even seen that d***ed Batman film, thank goodness)
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.
Yes, komponisto, I do deny it. A field is the tools it uses, not what it uses them on. What I’m less than comfortable doing myself is not “analysis”, but analysis-style proofs.
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.
Jacob, let me remind you the house rules. Your comments about Riemann, etc., are perfectly welcome, but keep your insults to yourself. I removed my joke (which I intended to be friendly), since it seemed to offend you.
hellblazer: Sadly, or not very, I can’t understand Pisier’s paper about the Grothendieck conjecture, and even more sadly, or not, virtually nobody who can understand it could also understand Apéry’s zeta(3) monstrosity, where extremely weird formulae appear in sequences that mysteriously turn out to consist of the integers. If you can look at those oddities and see integers, you’re probably Henri Cohen.
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.”
Jacob, I thought you initially made the joke about your name, but looking back on it I see that it was hellblazer. On my side of the cultural divide (which may be population one, I’m not sure) when someone makes a joke on a subject, it is a friendly gesture to make a follow-up joke on that same subject. I’m sorry that I misunderstood, and offended you. (I do have to say though that while what artists think is uncultured counts for something to me, what the upper-brackets think is uncultured counts for absolutely nothing.)
Apéry’s proof did stimulate some interesting further developments, however. In Graham-Knuth-Patashnik’s Concrete Mathematics (p. 238), it’s mentioned that a recurrence relation announced by Apéry for certain binomial sums (now called Apéry numbers) defeated mathematicians at the time, but was later established by Don Zagier and Henri Cohen. Their proof was in turn a key clue which led Zeilberger to develop his general methods for evaluating hypergeometric sums.
Walt: The upper brackets aren’t what they used to be, and Charlestonians like me are born touchy. So I stuck a post-it that says “Chill out!” on my monitor, and hope to be milder.
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