The n-Category Café

Someone once told me that the everyday word “irrational” actually came from the mathematical word, which simply means “not a ratio”. I was shocked, but after a second thought, agreed that that was the simplest explanation. We usually think it’s mathematics that takes words from everyday life, but sometimes it goes the other way. Someone should make a list of those, too.

Many texts distinguish between language of “connotation” [most extreme in Poetry] and language of “denotation” [most extreme in Mathematics]. However, this presupposes a linear order, a spectrum with Math at one end and Poetry at the other end. Collingwood was (and is) a breath of fresh air, throwing that naive model into the dustbin of history.

I love the term “perverse sheaf”. The term “middle perversity” is even better.

I’ve had conversations with various people (such as John Baez) about this sort of thing. I guess I won’t speak to gothic letters in Lie theory or in algebraic number theory (where there were strong traditions in Germany), but I have noticed on occasion a tendency for some people to use the biggest fanciest fonts for the biggest fanciest concepts they have. What this probably means, as one of my interlocutors tartly observed, is that they are not yet comfortable with their concept.

People like Ross Street react oppositely: they are quite happy to use a lower-case $e$ or something innocent-looking for what others might presume to be a scary abstract concept. I think that’s probably healthy from a psychological standpoint.

Tom Leinster said:

I can barely look at a gothic or fraktur letter—I mean the ones that Lie algebras are done in—without feeling narrow and ignorant.

That’s amusing: I always find fraktur letters exotic and thrilling, as though Lie algebras can only be reached through the back of a magic wardrobe, or will enable me to be inducted into an order of sages or something. On the other hand, pretty much everything makes me feel narrow and ignorant, so fraktur letters can’t get much of a hold there.

TL: I can barely look at a gothic or fraktur letter — I mean the ones that Lie algebras are done in — without feeling narrow and ignorant. On a bad day, I’ll start thinking about the people who’ve told me that I must love Lie theory because it’s beautiful, etc., whereas I’ve always found it irrationally repugnant.

Here are Lie algebras without Gothic letters!

Maybe you could infer from different names in different languages that the concepts were discovered and established independently, example:

ideal (English) = Ideal (German)
ring (English) = Ring (German),

but

field (English) = Körper (= body, German)

If you are German and don’t like algebra, maybe it’s just because you have a problem with your coenesthesis?

There was a mutually baffling poetry workshop at Caltech circa 1971 with that year’s Poet in Residence Robert Kelly.

The crux was a famous quotation: “poetry is a field.” Or was it “a poem is a field”?

In any case, EVERY one of the students took this as a metaphor to Electric Field, Magnetic Field, Gravitational Field, or a set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.

After several minutes of confused discourse, I asked the professor what HE meannt by a “field.”

“You know,” he said. I paraphrase loosely. “Tilling the ground and rearing cattle. A meadow of grass and other non-woody plants, with poems like beautiful flowers.”

The science students were all surprised. They’d all forgotten that there were such things as haymeadows, pastures, and non-grassland habitats such as heathland, moorland and wood pasture.

The same kind of thing happened with another poet in residence at Caltech,
Diane Wakoski.

But didn’t we have our Two Cultures discussion in another thread?

“Semi” carries connotations too that are not alway warranted mathematically. A “Semilattice” would be “like a lattice, but only a surrogate, not having all the properties of a lattice” while in mathematical fact, semilattices is a concept that include lattices as well as other things.

Depends upon what you think “Semi” means, some people think it means “similar to”, “sort of” or “not fully” but as Latin it means half. The precise lattice meaning is that while a lattice has two operators ($\wedge and \vee$) a semi-lattice has only one.

You also have to watch out for the red herring principle.

“Lattice” =French: “reseau” and “treillis”. I think it wise
to retain two words for English too.

The sublattice stabilizers of the Leech lattice form a
lattice.

I am horrified that Collingwood should stoop to introducing a bastard neologism like “homolingual” instead of “homoglottal”.

Why would you assume he introduced the term? I think its use must have been current. On p. 260 he writes about the “principle of homolingual translation”

This is an assumption about sentences corresponding to the lexicographer’s assumption about words (or, to be precise, about what I have called lexicographical units) when he ‘defines the meaning’ of a given word by equating it with that of a group of words taken together. According to the principle of homolingual translation, one sentence may have precisely the same meaning as another single sentence, or group of sentences taken together, in the same language, so that one may be substituted for the other without change of meaning.

This is one of the assumptions of a logician, he claims, not one he himself makes. It is part of a “modification of language, not a theory of it”.

While I’m defending Collingwood, last time out people objected to what they took as an assumption on his part that all mathematicians were male, based on the use of ‘men’. Since then I have read him speak about writers, using ‘men’ in just the same way, and then proceed to illustrate his point by describing some feature of Jane Austen’s work.

Could you elaborate on the emotions coming from “atomic”? It was associating “emotion” with that term that completely lost me in the original piece. It might be that as a 35 year old atom bombs aren’t anything visceral for me, whilst as a computer scientist I read and think about “atomic operations” as “can’t cut (ie, subdivide) operations” on computers almost every day; maybe it was different when Collingwood wrote his text, although had Wheeler et al even figured out the possibility of fission in 1938? (I can honestly say that this thread is the first time I’ve mentally connected the “atomic operations” and “atomic bomb” concepts.)

Am I missing other emotional context other people have around atomic?

The antipathy extends to matroids too.

Hey! As an applied mathematician I object! It’s applied math that’s art and is guided by a sense of beauty. God knows what pure math is guided by.

John wrote

I think the discussion of ‘mathematical emotion’ becomes too restricted if we only talk about how certain terms carry emotions with them.

I hope I didn’t give the false impression that Collingwood aimed at such a restriction; he was merely finding a rapid way to exemplify emotion in a technical discipline.

I think the interesting question is how far to distinguish cases of such emotion. One end would have a different emotion for each piece of mathematical apperception. The other extreme would talk of a few general emotions, including ecstasy and frustration, experienced in different settings.

Collingwood speaks in favour of the former. This goes with the main thesis of the book which is that each work of (properly called) art is the expression of new emotion. And is he is happy to have art shaped by intellectual concerns so that Dante expresses the emotion of accepting a certain kind of Thomism, and Shelley a kind of Copernicanism.

I guess there is a “math emotion” connected with spontaneous global information processing, e.g. I sometimes “feel” a mathematical concept or text as very interesting long before I have read or understood it. The reliability of that made me increasingly follow such “emotions”. Ignoring that often turned out to be an error. The backside of that is a tendency to ignore actually very interesting things which do not instantly connect with such an emotion.

With regard to the computer-aided attack on the four-color problem, I wouldn’t say it was “unfair” – but I would say it was “brutal” (which also has an ethical ring to my ears).

I personally don’t have many thermal associations with concepts. I suppose a sufficiently rarefied notion might be thought of as “cool”, as if up on a high mathematical mountain or being icily remote. But generally I’d think of temperature as connected more with passion, as in a “hot” topic, and of course feelings about that will diverge wildly. (Temperature is given a technical meaning in the study of Conway games: a hot position is intuitively where both players are eager to make a move.)

Does anyone else agree that algebra is cool while geometry is warm?

Also, limits are white while colimits are black.

Well, in the Republic Plato doesn’t want the future philosopher-kings to study maths for more than 10 years. There’s a consistent theme of his that mathematics is a half-way house. It draws you away from the ephemeral and points you to the permanent, but ultimately it is the Forms of the Good, the Just, and the Beautiful that we should contemplate.

There was a psychologist, Margaret Donaldson I think, who argued that learning mathematics had a beneficial effect on the psychological lives of children. Perhaps in Children’s Minds. Too much of a good thing may be damaging though!

It’s all about balance isn’t it? It’s just like with food - carrots are “good for you”, but if you *only* ate carrots it would be an entirely different matter. Being a professional mathematician can be like only eating carrots.

Chess courses or workshops have been shown to make prisoners more peacefull, do math courses that too?

In response to Eugenia’s original comment, I’m surprised that you place the “dangerousness” on logical argument rather than on a tendency to take all categories (in the colloquial sense) as precisely defined and then run into problems dealing with other people when they aren’t as rigid (and sometimes even self-inconsistent) in applying categories.

Are mineralogists, chemists and embryologists equally guilty in your eyes?

Incidentally I also object to the “bra” and “ket” terminology, for similar reasons. Here there is the added irritation that the etymology is silly - why is it not “brac” and “ket”? This gives one the strong impression that someone just wanted to get the word “bra” into mathematics somehow.

Now whenever I hear a mathematician use the word ‘cleavage’, I’m going to think about toes.

Tom said:

People: I’m depressed at the way this conversation has been conducted.

Thanks for saying this, Tom. I’ve been feeling increasingly uncomfortable with this myself. I still find anon’s original remarks fairly annoying, but I could have been vastly more restrained, and I want to apologise for letting loose.

If so, the correct response is none: don’t feed the troll.

Well, no. When people take the troll (or non-troll) seriously, then you need to point out the truth. (Getting into back-and-forth arguments with a troll is pointless, but I don't think that this is what has happened here.) And the truth here is that anon is the one who has made a sexist claim: that women are incapable of participating in certain fields of mathematics. I can't just let that go by while people take it seriously.

Anon, I'm sorry for saying that you made it up; I have no way of knowing that. In any case, your opinion, that you don't like terminology that may have suggestive doubles entendres, is a valid one, just as much as (for example) Tom's opinion that Fraktur letters are ugly, and you should be able to express it. But if you're really not able to participate in a field of endeavour where words like ‘cleavage’ and ‘ball’ are used metaphorically to refer to their ordinary nonsexual meanings, or where abbreviations like ‘bra’ and ‘Ass’ are used without regard to their ordinary sexual meanings in some languages, then it's not because you're a woman.

Maybe I’ve missed, ignored, or been cluelessly oblivious about the most objectionable parts of this discussion, but I think

Male mathematicians line up to heap ridicule on her.

is an overly harsh characterization. Nevertheless, I can’t disagree with

it confirms part of anon’s point: that mathematicians can be socially insensitive

and I add my own apology to Tim’s.

Neither ‘bra’ nor ‘cleavage’ as used in mathematics or physics gave me much pause for thought before, but after (re)observing my own reaction to $Ass$, I can definitely better appreciate anon’s annoyance, without going so far as to declare any of these terms deal-breakers. I think it’s good though to bring this out in the open. ‘Pseudo-splitting’ seems to me to be worth considering for use in the nLab. And ‘brac’ now strikes me as better or at least less needlessly provocative in hindsight, whatever Dirac’s reasons might have been.

By the way, while it’s quite likely that anon is a woman, there are some men out there who feel very strongly about the same set of issues. It’s probably better not to presume.

Richard, I’m shocked that you would so casually throw around a word like “loaded” when there may be recovering alcoholics reading this comment thread who may be insulted by your wholly unintended insinuation. Please try to be more careful in the future.

Much as I’d rather not engage in serious argumentation, this comment deserves at least some censure. I don’t know where you live, but you’re doing a pretty good job of conforming to the (mostly unjust) caricature of Americans as a people who believe their culture to be the one everyone should ‘adapt to’ and ‘exist in’. English is now Lingua Franca in large parts of the world and is indeed evolving dynamically in the different dialects. As I mentioned above, this blog is accessed by an enormous range of people.

Furthermore, it’s quite odd to read a pretty aggressive series of assertions that conclude with the oxymoronic combination

‘Get over it…respect others.’

a thousand ships (well, several) are launched by the males in the group to smother and kill that complaint with linguistic, philosophical, and sociological arguments

That’s a strange take on the situation. It seems to me people are just giving their honest reactions and having a discussion in broad daylight. There is no intent to smother, and no evil designs to kill.

finally culminating in the ridiculous complaint by someone that they had actually been insulted by anon

What culminating complaint are you referring to, specifically?

Face it folks, the word bra has only one real meaning in the english language – something that has something to do with women’s breasts – and while the word cleavage actually has a technical meaning that transcends reference to women’s breasts, it is now an extremely LOADED word in (at least) American culture. Words shift in meaning, reference and weight with time. Get over it. Understand and adapt to the culture you exist in and respect others.

I don’t think anyone here has denied that ‘bra’ has only one nontechnical meaning in English, and I think some of us have already said that in retrospect it was an unfortunate choice for a technical term, although it is very debatable that it was originally introduced with sinister anti-woman motives. As for ‘cleavage’, several participants including myself have said that the usage in mathematics carried no breast-related associations for them before this discussion took place (I think the word was probably a translation from French, via Grothendieck and his school), but now that it has, we can’t help but see them. We hardly need to have it pointed out again, or to be told that language changes – everyone is very aware of that.

So there’s nothing particularly to get over – I don’t think any one of the participants is in any denial about these words. I also think it’s good that anon raised her voice. But turning the discussion to more productive directions, now that the point has been raised, the question is what to do about it. And we’ve talked about that as well.

So, Richard, what do you propose we do about it? Do you have something productive to say, besides delivering a little sermon to us males?

How about ‘accessible category’? An accessible category is a (potentially) large category that can nevertheless be understood through small reasoning.

Indeed a good idea. I probably should also have avoided contributing so much distraction, and will attempt to refrain once again. But since segueways of a political flavor do unavoidably come up, I do urge everyone to remember that we really are writing for a worldwide readership. Since I occasionally come here, I’ve received positive comments about the blog from as far and wide as India, Japan, Iran, and South Africa. I’m sure others, especially the hosts, have similar experiences. In the end, it’s a real privilege to live in an age when such rich communication is possible on a daily basis. It would be a pity to squander such global privilege on the futile exercise of trading local cliches. (As usual, I’ll simultaneously apologize in parentheses for that last harsh description of our exchange. Also to Richard for my sharp criticism.)

Good example, schlicht is either “yeah, it’s really that simple!” or (if you say “(s)he is schlicht”) “that person is dumb, take care!”.
That’s why I was very surprised that schlicht functions are such a rich and interesting topic in complex analysis.