## November 19, 2009

### Mathematical Emotion

#### Posted by David Corfield

Continuing the season of Collingwood on mathematics, here is an extract from The Principles of Art (1938):

A symbol is language and yet not language. A mathematical or logical or any other kind of symbol is invented to serve a purpose purely scientific; it is supposed to have no emotional expressiveness whatever. But when once a particular symbolism has been taken into use and mastered, it reacquires the emotional expressiveness of language proper. Every mathematician knows this. At the same time, the emotions which mathematicians find expressed in their symbols are not emotions in general, they are the peculiar emotions belonging to mathematical thinking. (p. 268)

[‘Symbol’ is used here to mean “something arrived at by agreement and accepted by the parties to the agreement as valid for a certain purpose” (p. 225).]

Anyone who doubts such ‘emotions belonging to mathematical thinking’ exist need only read an edition or two of This Week’s Finds. They are dripping with emotion, as the modern phrase has it.

Collingwood continues:

The same applies to technical terms. These are invented solely to serve the purpose of a particular scientific theory; but as they begin to pass current in the scientist’s speech or writing they express to him and to those who understand him the peculiar emotions which that theory yields. Often, when invented by a man of literary ability, thay are chosen from the first with an eye to expressing these emotions as directly and obviously as possible. Thus, a logician may use a term like ‘atomic propositions’ as part of his technical vocabulary. The word ‘atomic’ is a technical term, that is a word borrowed from elsewhere and turned into a symbol by undergoing precise definition in terms of the theory. Sentences in which it occurs can be subjected to homolingual translation. But, as we find it occurring in the logician’s discourse, it is full of emotional expressiveness. It conveys to the reader, and is meant to convey, a warning and a threat, a hope and a promise. ‘Do not try to analyse these; renounce the dream of analysing to infinity; that way delusion lies, and the ridicule of people like myself. Walk boldy, trusting in the solida simplicitas of these propositions; if you use them confidently as bricks out of which to build your logical constructions, they will never betray you.’ (pp. 268-269)

Now who has the best example of a mathematical term for which we can construct a similarly intricate account of its emotional expressiveness? Candidates include imaginary numbers, sober spaces, pseudofunctor, and the fundamental theorem of algebra. But we shouldn’t overlook less obvious terms such as inhabited set which express a decidedly constructivist emotion. It is interesting also to see how the emotion may drain from a term as it ages. Perhaps people these days tend to play it safe to avoid their patently loaded name later looking ridiculous.

I have the next two paragraphs by Collingwood here, ending with

The progressive intellectualization of language, its progressive conversion by the work of grammar and logic into a scientific symbolism, thus represents not a progressive drying-up of emotion, but its progressive articulation and specialization. We are not getting away from an emotional atmosphere into a dry, rational atmosphere; we are acquiring new emotions and new means of expressing them. (p. 269)

If so, we could understand that commonly met thought that learning mathematics is good for the soul in terms of its capacity to shape the emotions for the better.

Posted at November 19, 2009 8:59 AM UTC

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## 263 Comments & 0 Trackbacks

### Re: Mathematical Emotion

I’ll go off on a slight tangent and mention not emotion (which word probably means a different range of phenomena for me than for Collingwood, rather than necessarily disagreeing about the actual phenomena) but instead talk about terms with connotations. It’s always fascinated me how different types of number have English names with postive or negative connotations: there are natural numbers, rational numbers, irrational numbers, real numbers, imaginary numbers, transcendental numbers. Here it’s difficult for me to tell if the common meaning actually matches current day mathematical view, eg, are irrational numbers worse than rational numbers?(There are also ones where there English common usage is clearly a much closer match with the mathematical properties, eg, prime numbers – which are arguably “prime” numbers – and perfect numbers.) It would be very interesting to know if current (eg, within the last 20 years) mathematicians still have the confidence to use common words with positive or negative strong connotations for newly discovered entities. (Off the top of my head, the last ones I can think of are “catastrophe theory” and “chaos theory” which are both over 40 years old.)

Posted by: bane on November 19, 2009 10:42 AM | Permalink | Reply to this

### Re: Mathematical Emotion

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.

Posted by: James on November 19, 2009 12:07 PM | Permalink | Reply to this

### Re: Mathematical Emotion

The history relating ratio in the proportional sense and in the sense of reason must surely be very intricate. Here’s a online Latin dictionary entry for ratio, which mainly concerns the latter sense.

What does Euclid use for ratio in the proportional sense? Presumably there’s some Pythogorean emotion behind the connection.

Posted by: David Corfield on November 19, 2009 1:29 PM | Permalink | Reply to this

### Re: Mathematical Emotion

To answer my own question - What does Euclid use for ratio in the proportional sense? - ‘Logos’, of course.

Now that was a loaded choice.

Posted by: David Corfield on November 20, 2009 9:53 AM | Permalink | Reply to this

### Re: Mathematical Emotion

What an excellent etymological revelation. As a corollary, I think we should acknowledge that irrational human behaviour is neither bad nor something to be eliminated, any more than irrational numbers are bad things that should be eliminated.

Posted by: Eugenia Cheng on November 28, 2009 4:03 PM | Permalink | Reply to this

### Spectral model not even wrong; Re: Mathematical Emotion

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.

Posted by: Jonathan Vos Post on November 20, 2009 4:24 PM | Permalink | Reply to this

### Re: Spectral model not even wrong; Re: Mathematical Emotion

Indeed! Later he writes

To put things in terms of practice: we have got into the habit of thinking that a writer must belong to one of two classes. Either he is a ‘pure’ writer, concerned to write as well as he can, in which case he is a literary man; or he is an ‘applied’ writer (to adapt the old distinction between pure and applied science), concerned to express certain definite thoughts, and anxious to write only well enough to make his thoughts clear, and no better. This distinction must go. Each of these ideals, if there is to be any future for literature, must fertilize the other. The scientist and historian and philosopher must go to school with the man of letters, and study to write as well as writing can be done. The literary man must go to school with the scientist and his likes, and study to expound a subject instead of merely exhibiting a style. Subject without style is barbarism; style without substance is dilettantism. Art is the two together. (pp. 298-299)

Posted by: David Corfield on November 20, 2009 4:38 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Grothendieck is said to have been appalled at the coining of the term perverse sheaf, with its all-too-clear emotional or moral implications. How could you saddle a naturally-occurring mathematical concept with a pejorative name like that?

Some names invite you to look down on the concept to which they are attached. Pseudofunctor is a case in point. (Surely it can’t be as good as the real thing?) Peter Johnstone, in his book Stone Spaces, makes the following remark on p.166:

for an odd offshoot from this notion, see [Nieminen 1977], whose author has enriched the English language with the term ‘weakly ultra-pseudocompact’.

I’m sure David doesn’t want this to become a thread on bad terminology. I just wanted to point out that some terms (including at least one I’ve coined myself) immediately point the reader’s emotions in a downward direction.

Posted by: Tom Leinster on November 19, 2009 12:48 PM | Permalink | Reply to this

### Re: Mathematical Emotion

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

Posted by: Walt on November 21, 2009 10:10 PM | Permalink | Reply to this

### In love with their own theories and terminology

I’m sure everybody has come across somebody so in love with their on theory and terminology that they are incapable of seeing any alternative.

This is common among crackpots, but there are many stories of famous scientists/mathematicians for which this also holds.

Posted by: RodMcGuire on November 19, 2009 12:52 PM | Permalink | Reply to this

### Re: Mathematical Emotion

A confession: for me, some terminology and notation inspires fear and insecurity. 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. Then I’ll feel even worse

The only thing that will save me is if I can’t tell what the letter is. Then I can feel a sense of peevish yet justified irritation.

Fortunately, my browser isn’t set up to show gothic letters on the $n$-Café.

Posted by: Tom Leinster on November 19, 2009 12:57 PM | Permalink | Reply to this

### Re: Mathematical Emotion

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.

Posted by: Todd Trimble on November 19, 2009 1:43 PM | Permalink | Reply to this

### Re: Mathematical Emotion

I have noticed on occasion a tendency for some people to use the biggest fanciest fonts for the biggest fanciest concepts they have.

I try to make my fonts mean something.

In my PhD thesis (see page 77ff), I used uppercase Latin letters for objects, lowercase Latin letters for morphisms, and lowercase Greek letters for $2$-morphisms. Since much of the point was categorifying a situation from a category to a $2$-category, I used italic letters in the category and calligraphic (uppercase) and fraktur (lowercase) letters in the $2$-category. Maybe this was overkill, but it kept things clear for me.

In more normal situations, I'll use italic lowercase Latin letters for unstructured elements of abstract sets, italic uppercase Latin letters for sets (including subsets) of unstructured elements, calligraphic uppercase Latin letters for sets of sets, etc. I also try not to repeat letters, even if the font has changed. So I might have $a$ an element of $X$ or $x$ an element of $A$, but probably not $x$ an element of $X$. Especially to be avoided is to have $x$ and $y$ both elements of $X$ when there is also a set $Y$ around.

Posted by: Toby Bartels on November 19, 2009 9:52 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Judging from the tone, you seem to be very proud of this set of decisions. To each his own! The thing you think is “especially to be avoided” wouldn’t bother me one little bit.

Posted by: Todd Trimble on November 20, 2009 12:40 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Judging from the tone, you seem to be very proud of this set of decisions.

Proud? No, not proud.

I really just hope to justify my use of fancy fonts to you.

To each his own!

I guess that's enough.

As for not repeating letters, that's really supposed to go the other way. That is, there's not so much danger in having variant fonts if one doesn't repeat letters, since the readers can then ignore the fonts if they wish. It's mostly Jim Dolan who (unintentionally) made me start doing this.

Posted by: Toby Bartels on November 20, 2009 3:54 AM | Permalink | Reply to this

### Re: Mathematical Emotion

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.

Posted by: Tim Silverman on November 19, 2009 1:50 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Posted by: The Fraktur Redaktur on November 19, 2009 2:39 PM | Permalink | Reply to this

### Re: Mathematical Emotion

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!

Posted by: Arnold Neumaier on November 24, 2009 3:57 PM | Permalink | Reply to this

### Re: Mathematical Emotion

I imagine:

Abstract: In this article we develop Lie theory without using fraktur letters. We demonstrate that it is possible to find proofs of all standard results in Lie theory independent of fraktur fonts.

In the second part of the article we prove by example that without fraktur fonts for Lie algebras it is easy to confuse Lie algebras with elements of Lie groups, and based on this we are able to state previously unimaginable results quite easily.

Posted by: Urs Schreiber on November 24, 2009 7:58 PM | Permalink | Reply to this

### Re: Mathematical Emotion

It has always seemed strange to me that it never became standard to denote Lie algebras generically with letters like $A$, and to denote the Lie algebra of a Lie group $G$ by something like $A(G)$ or $A_G$.
Posted by: Mark Meckes on December 2, 2009 2:33 PM | Permalink | Reply to this

### Re: Mathematical Emotion

It has always seemed strange to me that it never became standard to denote Lie algebras generically with letters like $A$, and to denote the Lie algebra of a Lie group $G$ by something like $A(G)$ or $A_G$.

I have sometimes accepted $\mathfrak{g}$ (that's ‘g’ in Fraktur if you can't see it) as the notation for a Lie algebra associated to a Lie group but then insisted that such a notation must refer to the operation rather than to its output.

So we have $\mathfrak{g}(G)$, $\mathfrak{g}(H)$, $\mathfrak{g}(G \times H)$, etc. You never have to learn any other Fraktur letters, nor do you have to give $G \times H$ a new name before you can write down its Lie algebra.

Posted by: Toby Bartels on December 4, 2009 1:40 AM | Permalink | Reply to this

### Re: Mathematical Emotion

I love Lie algebras and have thus learned to read and write a few lower-case Gothic letters. It’s tragic if there are people out there who have avoided this subject merely because they don’t like the letters. Knowledge of Lie algebras will confer magic powers upon you!

I always tell my students that Lie was a Goth.

Posted by: John Baez on December 2, 2009 4:50 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Let me try to articulate a bit more the “fear and insecurity” that I mentioned previously.

Most mathematicians, I guess, are familiar with the fear that you can feel on approaching a subject that you know nothing about. Perhaps fear is too strong a word: trepidation, at least. Here’s something from John’s old Oz and the Wizard notes:

‘So,’ continued the Wizard, ‘we will need to learn, and understand, the following formula for the Riemann tensor in terms of the Christoffel symbols.’ And he turned the page, revealing the following formula, written, it seemed, in blood: $R^a_{bcd} = C^a_{bd,c} - C^a_{cd, b} + C^e_{bd} C^a_{ec} - C^e_{cd} C^a_{eb}$ Oz’s hair stood on end. He backed away, slowly. ‘No,’ he said. ‘No. There is no way I am EVER going to understand THAT.’

The Wizard smiled. ‘That’s what most people say when they first see it. I felt that way myself.’ But somehow this did not reassure Oz […] ‘NO!’ cried Oz. ‘I will NOT understand it! NEVER!’

And here’s something from an article by William Thurston on mathematical education:

It is very intimidating to hear others casually toss around words and phrases as if any educated person should know them, when you haven’t the foggiest idea what they’re talking about […] I remember many occasions when I felt intimidated by mathematical words and concepts before I understood them: negative, decimal, long division, infinity, algebra, variable, equation, calculus, integration, differentiation, manifold, vector, tensor, sheaf, spectrum, etc. It took me a long time before I caught on to the pattern and developed some immunity.

You’d think we all — grown-up full-time mathematicians — would have developed total immunity by now. But speaking for myself, I know that’s not always the case.

There’s a spectrum of feelings between fear and distaste, and when you find yourself reluctant to learn a new subject, it’s not always easy to discern what feeling is behind that reluctance. Is it because I’m afraid I won’t be able to do it? Because I think I won’t like it? Because I think the whole theory is misguided?

Little things can be important. When you first approach a subject, you usually can’t see its deep, important aspects: you only see what’s superficial. If a subject has lots of nitpicking terminology (“quasicoherent”, “weak$*$ topology”, “pseudofunctor”) then you might be disproportionately suspicious of it. Barrages of baffling notation can do the same thing: I’m sure many people are put off logic for that reason.

These are really superficial things! It’s preposterous that any sensible adult would let themselves be influenced by them. But I think it happens.

So, back to gothic letters.

It’s true that they contribute, slightly, to my irrational aversion to Lie theory. I have trouble writing gothic letters, on the rare occasions that I try. I think I’m quite flexible about notation in general, but I have trouble thinking of something denoted by a lowercase ‘g’ as an algebraic structure (even if it’s in a different font or underlined). To me, ‘g’ is usually a map, and things like groups, vector spaces etc. get capital letters. It’s like using $\varepsilon$ for a natural number or $n$ for a small positive real. I could do it if I tried, but it’s pointless extra effort. And I have trouble reading some of the uppercase gothic letters — I genuinely don’t know what letter they’re supposed to be. I can make an effort and find out, but again it’s a little barrier — a frustrating failure of communication .

But mostly, I think, it’s the other way round. I have negative feelings about Lie theory, for whatever reason, so that negativity rubs off onto those poor gothic letters.

Posted by: Tom Leinster on December 2, 2009 7:42 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Excellent remarks, Tom! There could be a very interesting essay or even book about the fear we mathematicians experience when confronting new discipline, concepts or terminology. This fear is the flip side of our craving to know more. We don’t talk about it as much as we should.

I first met that formula for the Riemann tensor when I was in high school or college and got ahold of a book by Einstein on general relativity. I seem to recall that he just writes it down and leaves it unexplained. It scared the bejeezus out of me. That’s why I wrote what I did, later, when explaining this stuff to Oz.

Lie algebras are ‘infinitesimal’ versions of Lie groups. This is a wonderful thing: the miracle of group theory combined with the miracle of calculus. It’s already nice how a small ‘piece’ of a Lie group is enough to recover the whole thing, but it’s even nicer that we can let this piece shrink to infinitesimal size.

Personally, this is why I like notation where the Lie algebra of a Lie group $G$ is denoted by a smaller letter g. But a lower-case g won’t do, since that means an element of $G$! So, people break out the Gothic lower-case g… but I won’t write that here, since you don’t have Gothic fonts installed.

I have somehow managed to explain Lie algebras in This Week’s Finds without resorting to Gothic letters…

Posted by: John Baez on December 2, 2009 9:52 PM | Permalink | Reply to this

### Re: Mathematical Emotion

I have trouble to understand the importance of terminology etc. you articulate. My autodidact background should make me most sensitive to it, if you were correct in stressing those linguistic issues, but that is not the case. What one reacts on spontaneously before one really studies an article is the mental image, the sort of ideas and the image of the author’s mentality which the text transports. That’s it what makes an article jumping into one’s attention when one browses some journal or conference report.

Posted by: Thomas on December 3, 2009 12:06 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Thomas writes:

What one reacts on spontaneously before one really studies an article is the mental image, the sort of ideas and the image of the author’s mentality which the text transports.

OR: fails to do miserably, e.g. centipede tensor calculus or quoting Rainich:
the last thing you want to do is write it in coordinates - double entendre intentional.

Posted by: jim stasheff on December 3, 2009 2:52 PM | Permalink | Reply to this

### Re: Mathematical Emotion

There’s also the issue of learned as mathematical being totally divorced from their everyday meaning: case in point for me: field in algebra. It wasn’t until I encountered the Russian equivalent that the agricultural meaning struck me.

Posted by: jim stasheff on November 19, 2009 2:10 PM | Permalink | Reply to this

### Re: Mathematical Emotion

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?

Posted by: Tim vB on November 19, 2009 4:04 PM | Permalink | Reply to this

### Re: Mathematical Emotion

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

This is an interesting example, in that the term most languages use for what we call a “field” seems actually more like “corpse” than simply “body”. I’ve only found one other language whose term doesn’t denote a body (with whatever connotation).

Posted by: John Armstrong on November 19, 2009 5:31 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Oh! So that’s why we use ‘k’ to represent a field.

Posted by: Dan Piponi on November 19, 2009 6:07 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Could be, I know next to nothing about the history of math. Maybe it is just a coincidence or has a different reason, like using v for velocity, which, I was told, goes back to the Latin word velocitas.

An example that is pretty easy to track back is “eigenvalue”. The word eigen = “my own” could only be of German origin, I think, because I don’t know any other language that uses the “ei” diphthong (but then I don’t know many languages).

Posted by: Tim vB on November 19, 2009 7:01 PM | Permalink | Reply to this

### Re: Mathematical Emotion

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

This isn't just the Germans; everybody (well, everybody that I know of in western Europe) else uses a cognate of ‘corps’ to mean a field in the algebraic sense, while they use a more agricultural term to mean a field in the sense of physics.

This was annoying enough as an undegraduate mathematical physicist, where I simply had to keep the context straight; the trickiest bit was to distinguish the ‘field of scalars’ from a ‘scalar field’. But then I learned about algebraic geometry, where the same (at least in French) agricultural term is used for another meaning. Since this is too close to algebra (I guess), they had to come up with a completely different word in English, with a very different (but still agricultural) meaning; that word is ‘stack’.

Posted by: Toby Bartels on November 19, 2009 10:06 PM | Permalink | Reply to this

### Re: Mathematical Emotion

I think the worst case of conflation I have come across is in French where we can have a field of $C^*$-algebras which very closely corresponds to a $\mathbb{C}^\times$-stack, where both concepts are referred to as a champ, but are actually very different things.

Posted by: David Roberts on November 19, 2009 11:01 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Not everybody, even in western Europe. There’s one western European language other than English that goes the other way.

Posted by: John Armstrong on November 20, 2009 2:57 AM | Permalink | Reply to this

### Re: Mathematical Emotion

namely?

in the outside world where nouns can have gender, apparently the gender is not the same across language boundaries and this can influence the aesthetic evaluation of objects, cf. brdige in French versus German

Posted by: jim stasheff on November 20, 2009 1:38 PM | Permalink | Reply to this

### Re: Mathematical Emotion

To me there seems to be no pattern at all if you compare German, French and Spanish. German is annoyingly irregular in assigning a gender to a noun. German has male, female and neutral nouns, and the German word for girl (“Mädchen”) is neutral, so that even the simplest rule does not apply (assign the gender that the living being has).

But it never occured to me that the gender of the nouns could have any impact on my thinking, did you find the article about this topic that you mentioned in a previous post?

Posted by: Tim vB on November 20, 2009 5:34 PM | Permalink | Reply to this

### Re: Mathematical Emotion

See, a Mädchen is a maiden, by connotation a young girl, and not simply a Jungfrau. Mädchen and other Kinder are children as they exist before they may properly be brought into the business of begetting and bearing their own children. And I think it encourages a healthy sobriety when it comes to contemplating the young things.

Cheers!

Posted by: some guy on the street on November 20, 2009 6:05 PM | Permalink | Reply to this

### Re: Mathematical Emotion

True. That’s what I thought once, too. But then the word for boy (Junge) is male, so you end up saying things like:

“Look, there is a boy and a girl, he and it…” ?!?!

This annoys me so much that I avoid using the term “Maedchen” at all and talk about “the small one” (Kleine) instead. This is very convenient, because it can be used with any gender, even neutral. This is great if you want to talk about a baby or a toddler and don’t know if it’s a girl or a boy.

Posted by: Tim vB on November 20, 2009 7:23 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Somebody said

See, a Mädchen is a maiden, by connotation a young girl, and not simply a Jungfrau. Mädchen and other Kinder are children as they exist before they may properly be brought into the business of begetting and bearing their own children. And I think it encourages a healthy sobriety when it comes to contemplating the young things.

Tim vB replied

That’s what I thought once, too. But then the word for boy (Junge) is male,

Are we still on the $n$-Category Café, by the way?

Anyway, technically what corresponds to das Mädchen is das Bürschchen, diminutives of die Magd and der Bursche.

Except for the first, none of these are used much these days, they originate in a more ancient, more agricultural society.

This is related to the diminutive of die Frau, namely das Fräulein which was still used a lot in the 50s and then fought vehemently during the late 60s, because women pointed pout rightly that men would find it unthinkable to be officially called das Männchen. This you say when you want to insult somebody.

So in as far as these diminutives are used more for women than men says more about society using a lunguage than about the inner structure of that language. At least in this case.

But anyone around here still interested in higher categories? Small or not?

Posted by: Urs Schreiber on November 23, 2009 5:00 PM | Permalink | Reply to this

### Re: Mathematical Emotion

men would find it unthinkable to be officially called das Männchen.

Incidentally, German ‘Männchen’ is cognate to Dutch ‘manneken’, whence (via French) English ‘mannequin’.

But anyone around here still interested in higher categories?

This thread is about mathematics, emotion, and language.

Are there mathematical terms whose connotations are unwittingly influenced by the form of the word? We all know that calling something ‘hemidemisemi‐’ will doom it to obscurity, but you know the risks when you use that name. When have we had to rebel, like feminists in the late 1960s, against an inappropriate and (perhaps) inadvertent case?

Posted by: Toby Bartels on November 23, 2009 6:09 PM | Permalink | Reply to this

### Re: Mathematical Emotion

We all know that calling something ‘hemidemisemi‐’ will doom it to obscurity.

I’m quavering at the very thought.

Posted by: John Armstrong on November 23, 2009 7:32 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Hemidemisemiquavers are not at all obscure, although I personally prefer the term “semidemisemiquavers” as it generalises.

Posted by: Eugenia Cheng on January 1, 2010 1:22 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Happy New Year!

Of course ‘hemidemisemiquavers’ generalizes too: you could do quavers, and then semiquavers, and then demisemiquavers, and then hemidemisemiquavers, and then semihemidemisemiquavers, and demisemihemidemisemiquavers, and
hemidemisemihemidemisemiquavers, and so on; of course you could argue that this is quirky and overly complicated, but then I could argue that it’s overly complicated to do quavers and then semiquavers and then semidemisemiquavers and demisemidemisemiquavers, etc., when you can just do quavers, semiquavers, semisemiquavers, semisemisemisquavers, semisemisemisemiquavers, and so on.

I mainly posted this just for the sheer joy of writing that sentence, but here’s a question: how far up this progression would one ever need to go? You’ve seen a lot of sheet music: have you ever seen a 256th note? A 512th note? Did anyone ever write music with 1024th notes but put it in a really slow tempo, just as a kind of joke?

Posted by: John Baez on January 3, 2010 5:37 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Did anyone ever write music with 1024th notes but put it in a really slow tempo, just as a kind of joke?

Posted by: J-L Delatre on January 3, 2010 6:35 AM | Permalink | Reply to this

### Re: Mathematical Emotion

John said:

semisemisemisemiquavers […] how far up this progression would one ever need to go?

One of the more bizarre features that one observes, looking at the long term history of music, is the addition of ever shorter notes.

One starts off with Gregorian plain chant in the Dark Ages, with two note lengths: longa (“long”) and breve (“brief”).

After a couple of hundred years, apparently it becomes necessary to add a shorter note: the semi-breve. Another couple of hundred years or so, and a still shorter note is added, the minim (“shortest”). But even that’s not enough, over the course of time we find crotches, then quavers, then semi-quavers and so on.

In the meantime, the original longa and breve pretty much fall out of use.

(Apparently at one point there was even a maxima, twice as long as a longa.)

One does wonder what was going on in performance, to cause this strange progression. And whether it will continue in the future, and what will happen to the long notes if it does.

Posted by: Tim Silverman on January 3, 2010 2:34 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Tim wrote:

One of the more bizarre features that one observes, looking at the long term history of music, is the addition of ever shorter notes.

[…]

One does wonder what was going on in performance, to cause this strange progression. And whether it will continue in the future, and what will happen to the long notes if it does.

Maybe everything is speeding up!

Or, maybe it’s an effect analogous to pitch inflation: the perennial desire for musicians to want to play slightly sharp, which keeps pushing up the frequency of notes.

A similar thing happens with crickets. They tend to roughly synchronize their chirps… but each one likes to chirp slightly ahead of the rest, in a desperate attempt to stand out and get more mates. Conformists who nonetheless want to be ahead of the pack!

In all three cases, it’s possible that a kind of looping effect occurs. E.g.: it’s quite possible that music is not actually faster than it used to be — rather, new notes with shorter-sounding names keep getting introduced, while old notes with long-sounding names keep becoming obsolete. In which case what’s called a quarter note now may roughly match what was called a whole note once upon a time.

The phenomenon of Shepard tones is a great example of this ‘looping’ effect.

And then there’s monetary inflation. Things are not actually getting harder and harder for everyone to buy. But, prices keep going up.

Posted by: John Baez on January 7, 2010 6:15 AM | Permalink | Reply to this

### Re: Mathematical Emotion

John said:

it’s quite possible that music is not actually faster than it used to be

Indeed, I kind of assumed that there was some kind of competition between composers, who wanted their music to sound faster and more exciting, and performers, who just can’t move their fingers/diaphragms/larynxes/elbows/etc fast enough, so that although there might be cycles, there isn’t likely to be a secular trend (except that plain chant really is supposed to be slow and solemn).

Thanks for the other examples, too! This is an interesting phenomenon. Do you know of any mathematical treatments of this sort of thing? Something similar presumably shows up in fashion (you want your skirt/trousers to be just the same as other peoples’, except just a bit shorter/longer/wider/narrower).

I can think of two (related) examples in linguistics. One is the result of a competition between speakers and listeners. Speakers want to avoid effort, so their ideal utterance would be a disorganised laconic mumble, with the listener doing all the effort of working out what they meant. Listeners want to avoid effort, so their ideal utterance would be a perfectly articulated thought, unambiguously grammatical, spoken with crystalline clarity, so the speaker does almost all the work. But on the other hand, both sides want communication to take place. Rather than settling down in an equilibrium, this produces a cycle, which manifests in various ways. A typical way is for the work-avoidance of speakers to drive the decay of long sequences of sounds into short ones. Short grammatical words get worn down and attach themselves to neighbouring words as affixes. The affixes pile up on the word to form an “agglutinating” morphology. Certain conventional combinations of affixes become dominant, fuse together, and gradually wear down to form multi-function “inflectional” morphology. The shortening continues, with different affixes becoming more similar, losing distinctions and sometimes disappearing completely. But at this point, speakers who want to get their point across need to start using some other method to give grammatical structure to their utterances. So they start deploying whole words in ways which become increasingly grammatical. Using separate words for grammatical purposes results in an “isolating” style of morphology, and the cycle is ready to begin again.

Accompanying this is the “grammaticalization” cycle, affecting semantics. Words with ordinary (“lexical”) meanings like <go> and <want> get hijacked for grammatical purposes and get a more general, “grammatical” meaning, such as future tense. This metaphorically broadens out to cover more general meanings. This can continue, with the meaning getting weaker and weaker and more and more general and bleached. Eventually, the grammatical word (which by now may be an affix or even just an internal sound change) has almost no meaning at all, and ceases to be of grammatical use. Meanwhile, new ordinary words are being grammaticalized to take its place.

There are also cycles in the sounds themselves.

Unstable competition between growth and decay is somewhat reminiscent of diffusion-limited aggregation, too, though not quite the same.

Posted by: Tim Silverman on January 7, 2010 3:44 PM | Permalink | Reply to this

### Re: Mathematical Emotion

John B said

[H]ave you ever seen a 256th note? A 512th note?

Donald Byrd’s collection of extremes of conventional musical notation mentions Anthony Phillip Heinrich’s Toccata Grande Cromatica from The Sylviad, Set 2, m. 16 (c. 1825).

However, Byrd says that the 2048th and 1024th notes should be 1024th and 512th notes, respectively.

Aside: the use of “quarter note” and “eighth note” sounds American to my ears – does the use of “quaver” and “semidemihemiquaver” sound British to American ears?

Posted by: Simon Willerton on January 4, 2010 3:20 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Simon wrote:

Aside: the use of “quarter note” and “eighth note” sounds American to my ears – does the use of “quaver” and “semidemihemiquaver” sound British to American ears?

To me they sound like cool words that specialists use, rather than the ‘ordinary’ words for these notes. Only after reading Eugenia’s comment did I go to a dictionary and discover these words were ‘chiefly British’.

Actually, ‘semidemihemiquaver’ sounds like a British mathematician trying to remember the right word and not quite succeeding. Or maybe testing us Americans to see if we notice it’s backwards?

How about ‘sześćdziesięcioczwórka’?

And how about ‘quasihemidemisemiquaver’? This gives me the impression that someone ran out of prefixes that mean ‘one half’, and hoped we wouldn’t notice.

Posted by: John Baez on January 7, 2010 4:35 PM | Permalink | Reply to this

### Re: Mathematical Emotion

some guy in the street said:

See, a Mädchen is a maiden, by connotation a young girl, and not simply a Jungfrau.

German nouns with the diminutive suffix “-chen” are automatically neuter.

Posted by: Tim Silverman on November 20, 2009 10:08 PM | Permalink | Reply to this

### Re: Mathematical Emotion

That may well be, but still, custom arose to use a word with such an ending! I’m sure they didn’t need to!

Posted by: some guy on the street on November 21, 2009 4:00 AM | Permalink | Reply to this

### Re: Mathematical Emotion

That is true, of course, but today there is no noun without this suffix (“Maed” ?), which is why I did not even recognize for a long time that Maedchen has the diminutive “chen”.
Usually you take an existing noun and append the “chen”, like ape and apechen (Affe, Aeffchen).
You can do that to the noun Junge too, and get a very condescending word (don’t call anyone Jungchen if you are not prepared for a fight).

Posted by: Tim vB on November 22, 2009 4:09 AM | Permalink | Reply to this

### Re: Mathematical Emotion

today there is no noun without this suffix (“Maed” ?)

Actually it derives from ‘Magd’ via ‘Mägdchen’. Of course, the loss of the ‘g’ just furthers the idea that it is a separate word in its own right.

Posted by: Toby Bartels on November 22, 2009 5:22 AM | Permalink | Reply to this

### ‘Mädchen’ (Was: Mathematical Emotion)

today there is no noun without this suffix (“Maed” ?)

Actually it derives from ‘Magd’ via the earlier form ‘Mägdchen’. Of course, the loss of the ‘g’ just furthers the idea that it is a separate word in its own right.

Posted by: Toby Bartels on November 22, 2009 5:26 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Tim vB said:

today there is no noun without this suffix (“Maed” ?)

That’s true. My point was that I think that, etymologically, the reason that Mädchen ended up as it did had more more to do with the diminutive semantics of the suffix than with the fact that it also made the word neuter; I suspect the latter was purely a side effect. Modern Germans may give its neuter gender more significance—that’s something I don’t know about.

Posted by: Tim Silverman on November 22, 2009 3:20 PM | Permalink | Reply to this

### Re: Mathematical Emotion

There seems to be no significance as far as I can tell, in the case of Maedchen.

But in general this is surprisingly difficult to understand: Sometimes the gender has no significance at all (function is female, vector space is male, nobody cares), sometimes it seems to play a minor role. Earth is female (in German), so you can talk about “mother earth” and that she gave birth to humanity.
But on the other hand mountain is male but can give birth too: “the mountains gave birth to a mouse” is like “much ado about nothing” (in German: “Die Berge kreissten und gebaren eine Maus.”).
The noun Person (= person) is female, so even if you talk about someone very male like me, you would say “see that person over there? She…”.

Posted by: Tim vB on November 23, 2009 7:56 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Tim vB said:

today there is no noun without this suffix (“Maed” ?)

I guess most of you already noticed, but: we have that noun in English, spelled just a bit differently. And also the corresponding diminutive ending in ‘-en’! They mean interestingly different things…

Posted by: John Baez on January 7, 2010 4:36 PM | Permalink | Reply to this

### Re: Mathematical Emotion

And I think it encourages a healthy sobriety when it comes to contemplating the young things.

It doesn't.

Posted by: Toby Bartels on November 21, 2009 8:03 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Well, isn’t that too bad… concupiscence is persistent indeed. Now, there’s a word with plenty few connotations at large today!

Posted by: some guy on the street on November 21, 2009 11:20 PM | Permalink | Reply to this

### Re: Mathematical Emotion

This is an example that does prove the point of our guy on the street rather than disproving it. An 18 year old would never ever be addressed as a “Maedchen” (this stops somewhere around the age of 15). You say “Maedchen Maedchen!” to a teenager if she did something very stupid or childish, or if you get into a fight and want to tell her that she is not a grown up (which she is at the age of 18, by the way. At this age she is allowd to buy hard liqor in every supermarket).

The movie is pretty funny, and it’s less about concupiscence and more about finding love (like “sex and the city”).

Posted by: Tim vB on November 22, 2009 4:24 AM | Permalink | Reply to this

### ‘Mädchen’ (Was: Mathematical Emotion)

This is an example that does prove the point of our guy on the street rather than disproving it.

And here I was translating the title literally as ‘Girls, girls!’ (since the singular and plural are the same when there is no article). So while a sexually available woman may be called a ‘girl’ in English (although sometimes this is considered insulting), that is not so for ‘Mädchen’ in German. Interesting!

Posted by: Toby Bartels on November 22, 2009 5:24 AM | Permalink | Reply to this

### Re: ‘Maedchen’ (Was: Mathematical Emotion)

Agreed.
Learning a language takes 6 months, learning the connotations takes a lifetime.
The word “girl” is a good example.
Imagine a 16 year old girl falling in love, Germans will tell the mother “she is not a Maedchen any more, she is a young woman!”.

Posted by: Tim vB on November 22, 2009 6:40 AM | Permalink | Reply to this

### Re: ‘Maedchen’

Imagine a 16 year old girl falling in love, Germans will tell the mother “she is not a Maedchen any more, she is a young woman!”.

You can say that in English too; it's just optional. You could just as easily say (best to her rather than to her mother) “you are still a girl, so don't go too fast!”. Either term could apply to her, depending on what connotations you want to bring out.

Posted by: Toby Bartels on November 22, 2009 6:57 AM | Permalink | Reply to this

### Poets-in-Residence at Caltech confused by “Field”; Re: Mathematical Emotion

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?

Posted by: Jonathan Vos Post on November 24, 2009 8:53 PM | Permalink | Reply to this

### Re: Mathematical Emotion

A “structure” carries the connotation of a scaffolding or a way to order other things. The boundaries within which you should think of other things. Which fits nicely with how sets became the prevailing way of thinking about everything.

“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.

Which reminds me of a type of emotional “math-rhetorics” which is perhaps more prevalent in physics but which can be found everywhere. The scientist proclaims that his mathematical construct is “a generalization of” a well-known thing or has well-known thing “as a special case”.

This caters to our expectation that if you can generalize and see the big picture, like how e.g. magnetism and electricity were shown to be only special cases of a more general electromagnetism, then that’s a good thing. “General” trumps “special case” emotionally.

We are led to believe that the scientist has achieved a new insight when in fact, he or she has only added a new term to an equation or made a constant into a variable. When the new term is zero or the variable is set equal to the previous constant, the famous theory appears as a “special case”. I’ve seen this a lot in particle physics

Posted by: Martin RB on November 19, 2009 2:44 PM | Permalink | Reply to this

### Re: Mathematical Emotion

“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.

Posted by: RodMcGuire on November 19, 2009 4:40 PM | Permalink | Reply to this

### Re: Mathematical Emotion

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

Posted by: Toby Bartels on November 19, 2009 10:08 PM | Permalink | Reply to this

### Re: Mathematical Emotion

“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.

Posted by: John McKay on November 23, 2009 6:16 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Isn’t the name category a prank that Saunders McLane played on philosophers because it as a very specific (and different) meaning in the works of Kant?

Posted by: Tim vB on November 19, 2009 4:08 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Acquaintance with the Greek alphabet has always been a prerequisite for elementary particle physics. Rather more than that is desirable for understanding where much of mathematical terminology comes from. I am horrified that Collingwood should stoop to introducing a bastard neologism like “homolingual” instead of “homoglottal”. OK, I know that English is full of horrors like “television” but there is no need to multiply them needlessly.
Sammy Eilenberg once boasted that he had been the author of more mathematical terminology than anybody else in the twentieth century. He was particularly proud of “exact”; he pointed out that invented terminology is more useful if
1) it can be declined or incorporated (“exactness”, “left-exact”, …) and 2) it makes historical sense (“exact differential”).
Invention of terminology is a serious matter.

Posted by: Gavin Wraith on November 19, 2009 5:50 PM | Permalink | Reply to this

### Re: Mathematical Emotion

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.

Posted by: David Corfield on November 19, 2009 9:15 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Collingwood made a good choice with his decision to elaborate the emotional expressiveness of ‘atomic’ as “a warning and a threat, a hope and a promise”. Of course, such expressions need not be conveyed by single technical words. For example, here we find Mark Hovey issue a warning/threat:

One thing I will say here; if I am called to referee a paper on elliptic cohomology that does not deal with the Hopkins viewpoint on elliptic spectra, I will almost surely reject it. The time is gone when one could write papers about the Landweber-Ravenel-Stong elliptic cohomology based on the Jacobi quartic–we now understand that that is only one of many different elliptic cohomology theories, and all papers on elliptic cohomology should now accept that and deal with it.

Posted by: David Corfield on November 20, 2009 9:33 AM | Permalink | Reply to this

### Re: Mathematical Emotion

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?

Posted by: bane on November 20, 2009 3:29 PM | Permalink | Reply to this

### Re: Mathematical Emotion

I wonder when ‘atomic proposition’ was first coined? One tends to think of Russell and Wittgenstein. Let’s see, Wikipedia has:

An Atomic sentence (or possibly the meaning of an atomic sentence) is called an elementary proposition by Wittgenstein and an atomic proposition by Russell:

* 4.2 The sense of a proposition is its agreement and disagreement with possibilities of existence and non-existence of states of affairs. 4.21 The simplest kind of proposition, an elementary proposition, asserts the existence of a state of affairs.: Wittgenstein, Tractatus Logico-Philosophicus,[1].

* A proposition (true or false) asserting an atomic fact is called an atomic proposition.: Russell, Introduction to Tractatus Logico-Philosophicus, [2]

Perhaps Russell used it earlier. In fact, in Principia Mathematica you see atomic propositions being defined in contrast to molecular propositions on p. xv. Hmm, but this in the introduction to the second edition published in 1927.

Anyway, even had Russell known that physical atoms were not fundamental particles, he could still claim that by describing a certain kind of proposition, one which was not decomposable, he was drawing on the etymology of ‘atom’ as un-cuttable, and expressing the hope, as Collingwood describes, of their being building blocks.

Whether Russell or Wittgenstein were successful in showing us atomic facts or propositions is debatable. The Principia examples are “this is red” and “this is earlier than that”.

Posted by: David Corfield on November 20, 2009 4:33 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Maybe I didn’t answer your question. I don’t think you’re missing anything about the word ‘atomic’. The point is that sometimes terms are chosen which express their author’s emotional response to a thought. Other times, this is not expressed through the term but by other means.

We might run through as a second example what the ‘natural’ in ‘natural transformation’ is expressing. I gave this a go in chapter 9 of my book. At the very least it expresses a preference for something which doesn’t rely on arbitrary choices.

Posted by: David Corfield on November 20, 2009 4:46 PM | Permalink | Reply to this

### Re: Mathematical Emotion

I think I was more perplexed why atomic is a good example and why it particularly embodies hope and fear, but as mentioned it might just be CS “conditioning”: atomic is just a workaday word that describes things that are indivisible, and is of the same class as word like round or heavy, and did genuinely think I was missing something.

Incidentally, I’m not arguing against emotion in mathematics, it’s just that most often it emerges out of the melting pot of terminology connotations, visualisation, expectations, level of complexity, fit with current tasks, and lots of other stuff in the mathematics at hand at the “top level of my experience”. If anything, visualisation tends to be much stronger “lone” emotional invoker for me.

Posted by: bane on November 21, 2009 11:16 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Alain Connes wrote an interesting remark on groupoids in his classic “Noncommutative Geometry” (which you can download for free on his webside!), on p.13:

“It is fashionable among mathematicians to despise groupoids and to consider that only groups have an authentic mathematical status, probably because of the pejorative suffix oid.”

I remember that I was a little deterred when I first encountered groupoids, probably for exact that reason.

Posted by: Tim vB on November 20, 2009 5:51 PM | Permalink | Reply to this

### Re: Mathematical Emotion

The antipathy extends to matroids too.

Posted by: David Corfield on November 20, 2009 6:00 PM | Permalink | Reply to this

### Re: Mathematical Emotion

I think more often it’s: I know what an —- is but what’s a —-oid?

Posted by: jim stasheff on November 21, 2009 1:11 PM | Permalink | Reply to this

### Re: Mathematical Emotion

I think the discussion of ‘mathematical emotion’ becomes too restricted if we only talk about how certain terms carry emotions with them. The role of emotion in mathematics is incredibly important and incredibly neglected. Though I’m sure it’s not the case, one can at least imagine dry and computer-like applied mathematicians who solve whatever practical problems they are paid to solve. But pure mathematics, like music or art, is a subject where we have no preordained task to accomplish. Instead, we let ourselves be guided by mysterious emotions — like a ‘sense of elegance’ — to choose what tasks to accomplish, and how to accomplish them. So emotion is paramount here — but of course tightly constrained by the rules of the game we choose to play.

Posted by: John Baez on November 20, 2009 6:34 PM | Permalink | Reply to this

### Re: Mathematical Emotion

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.

Posted by: Eugene Lerman on November 20, 2009 7:15 PM | Permalink | Reply to this

### Re: Mathematical Emotion

So you’re a theist, eh? Must be the result of too much Platonism.

Posted by: John Baez on November 20, 2009 7:32 PM | Permalink | Reply to this

### Re: Mathematical Emotion

I agree that too much Platonism is bad for you.

I suspect pure math is actually guided by a bunch of bloggers.

Posted by: Eugene Lerman on November 20, 2009 7:41 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Well, they are dual to each other:

applied and unemployed math or

pure and impure math,

depending on your point of view.

Pure mathematicians are applied mathematicians that apply their theorems to prove other theorems. Applied mathematicians are pure mathematicians that apply their theorems to prove other theorems that weren’t recognized to belong to the realm of mathematics before that. So the first ones are the workers, and the second ones are the coworkers, or the other way round, in any case they probably follow dual rules.

Posted by: Tim vB on November 20, 2009 7:38 PM | Permalink | Reply to this

### Re: Mathematical Emotion

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.

Posted by: David Corfield on November 20, 2009 8:03 PM | Permalink | Reply to this

### Re: Mathematical Emotion

David wrote:

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.

Good. But somehow the ensuing conversation seemed to revolve around mathematical terminology, rather than the far broader and (to my mind) more interesting issue of the overall role that emotion plays in mathematics. Maybe it’s because mathematicians are very used to talking about terminology — they love doing it (see, another emotion!) — but not very used to talking about emotion. Expression of emotion in discourse about mathematics is carefully limited. And that would be an interesting subject in itself!

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.

That’s good. Emotion is not a simple thing; it’s incredibly subtle!

To take a random example, when Weyl talks about ‘angel of topology and the devil of abstract algebra fighting for the soul of every individual discipline of mathematics’, he’s referring to some very interesting sort of inner struggle. I feel it in myself — though I tend to pose geometry and algebra as the opposites — and most mathematicians I know well have felt it as well. What’s going on here?

I could list further examples endlessly…

Posted by: John Baez on November 20, 2009 9:48 PM | Permalink | Reply to this

### Dirac; Re: Mathematical Emotion

P.A.M. Dirac, the “theorist’s theorist”, long disdaining Philosophy, late in his life, famously lecturing on Beauty in Mathematical Physics. See the major article on him in this month’s Physics Today.

Posted by: Jonathan Vos Post on November 21, 2009 12:38 AM | Permalink | Reply to this

### Rainich Re: Dirac;

This was even more pronounced with Yuri Rainich (yes, that Rainich) who worked on math until the last couple of weeks of his life when he finally told his emanuensis - OK, now bring the philosophy books.

Posted by: jim stasheff on November 21, 2009 1:21 PM | Permalink | Reply to this

### Re: Mathematical Emotion

John said

Expression of emotion in discourse about mathematics is carefully limited.

Yes, it’s hard to think of anyone writing about their view of mathematics as Darwin does about his view of nature:

It is interesting to contemplate an entangled bank, clothed with many plants of many kinds, with birds singing on the bushes, with various insects flitting about, and with worms crawling through the damp earth, and to reflect that these elaborately constructed forms, so different from each other, and dependent on each other in so complex a manner, have all been produced by laws acting around us. These laws, taken in the largest sense, being Growth with Reproduction; inheritance which is almost implied by reproduction; Variability from the indirect and direct action of the external conditions of life, and from use and disuse; a Ratio of Increase so high as to lead to a Struggle for Life, and as a consequence to Natural Selection, entailing Divergence of Character and the Extinction of less-improved forms. Thus, from the war of nature, from famine and death, the most exalted object which we are capable of conceiving, namely, the production of the higher animals, directly follows. There is grandeur in this view of life, with its several powers, having been originally breathed into a few forms or into one; and that, whilst this planet has gone cycling on according to the fixed law of gravity, from so simple a beginning endless forms most beautiful and most wonderful have been, and are being, evolved.

I dare say it is harder to write so well about mathematics, but then all the more reason to include the mathematician here:

The scientist and historian and philosopher must go to school with the man of letters, and study to write as well as writing can be done.

Posted by: David Corfield on November 21, 2009 10:43 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Thanks for that excerpt from Darwin. Sometimes it is good to go back to the prophet himself.

Posted by: jim stasheff on November 21, 2009 1:18 PM | Permalink | Reply to this

### Re: Mathematical Emotion

it’s hard to think of anyone writing about their view of mathematics as Darwin does about his view of nature

But biologists today also remark that Darwin wrote in much less staid a style than they write in now. (I should be able to find a good quotation from Steven Jay Gould about this, but I don't know how to search for it.) And there are some good quotations from (say) Riemann in the other direction.

I think that it's the times more than the subject matter.

Posted by: Toby Bartels on November 21, 2009 7:58 PM | Permalink | Reply to this

### Re: Mathematical Emotion

But does Gould lament the change or merely observe it? I’m sure that’s right that nineteenth century mathematicians were more expressive – Hamilton, Cayley, Sylvester, Clifford, etc.

But then there are Weil, Weyl, Mac Lane,… and let us not forget Grothendieck.

Something admirable about the Darwin extract is that it comes at the end of his book. I like the idea that this form of expression is seen as fitting to express the sentiments of one who made the cognitive journey through the text.

Posted by: David Corfield on November 22, 2009 12:53 AM | Permalink | Reply to this

### Re: Mathematical Emotion

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.

Posted by: Thomas on November 21, 2009 1:46 AM | Permalink | Reply to this

### Re: Mathematical Emotion

I think this discussion of emotions in mathematics should be expanded to include the ethics of mathematics, a closely related subject. For example, there is a link between intellectualized emotions and intellectual virtues. Surely, we can say that a mathematician was courageous (Wiles working away on Fermat) or that a particular problem was solved unfairly (the four color problem). Collingwoods remark that mathematical symbolism is a refinement of our ordinary human capacity for emotion is telling.

On another note, perhaps there is a bodily basis for the emotional tone of mathematical concepts? Just as certain colours are warm and others are cool, perhaps some kinds of mathematics are hotter or colder than others. Do people have converging intuitions about the emotional temperature of mathematical structures and theories?

Posted by: RajeshKasturirangan on November 23, 2009 2:44 AM | Permalink | Reply to this

### Re: Mathematical 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.)

Posted by: Todd Trimble on November 23, 2009 3:19 AM | Permalink | Reply to this

### Re: Mathematical Emotion

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

Also, limits are white while colimits are black.

Posted by: Toby Bartels on November 23, 2009 6:28 AM | Permalink | Reply to this

### Re: Mathematical Emotion

No strong thermal distinction between algebra and geometry for me (although I’d lean toward that direction and not the other way), and never heard or felt that about limits/colimits. Would you consider yourself a synaesthete?

Posted by: Todd Trimble on November 23, 2009 8:17 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Would you consider yourself a synaesthete?

I don't think so.

The (co)limit colouring is strongest for (co)terminal objects and probably ultimately derives from a vision of 0 as black and 1 as white; the unit interval consists of linearly varying shades of gray. While I don't remember where that came from, it has a pretty direct interpretation, especially if you think of white as presence of light (rather than black as presence of pigmentation, which would be backwards).

I'm pretty sure that I didn't sit down one day and decide that 0 would be black from now on, but it's still hard to square this Wikipedia's description of ‘automatic, involuntary experiences’. It's clearly a deliberate act of imagination on my part, and I can just as easily do it the other way (although it feels less familiar). As a category theorist, of course, it's important for me to be able to switch directions like this!

The pictures of the number line on Wikipedia's page seem particularly inapt to me. Except for small numbers, they're very linear and clearly base 10. For me, numbers have personalities (at least when I'm familiar with them), but these have much more to do with their multiplicative structure than with their additive structure. I see a much greater difference between 47 and 48 or 47 and 49 than between 47 and 53, for example.

Thomas wrote:

Brain hyperconnectivity may not only relate to synaesthesia, but to bipolar personality

As some of you know, I have bipolar disorder. (The term ‘bipolar personality’ seems to be a misnomer to me. But in some cases, the term ‘personality’ changes what you're talking about; compare obsessive-compulsive personality disorder and obsessive-compulsive disorder.) I found it a bit harder to think about mathematics while I was taking medication (although the sacrifice was worth it).

Posted by: Toby Bartels on November 23, 2009 6:39 PM | Permalink | Reply to this

### Re: Mathematical Emotion

When I was an undergraduate and got an introduction to functional analysis, the professor once took the other way round: He said that good names for mathematical gadgets should connect emotionally to the audience and be something easy to visualize, which is why he took some forgotten concept from Stefan Banach and called it “red”. The punchline is that I have forgotten what is was about, of course. A topological vector space is red iff…? But I think he was not serious anywaw, but was making fun of all the students that complained about the abstractness of his lectures.

Posted by: Tim vB on November 23, 2009 8:18 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Like Todd, I have no color associations with limits and colimits, though geometry is certainly warmer than algebra.

There is a more general question about the nature of mathematics,over and beyond particular color and temperature associations. The larger debate is about the perceptual/body basis of mathematical concepts. When we say that Grothendieck style algebraic geometry is a powerful mathematical technology, it seems to me (a la Lakoff and Nunez) that a body metaphor is in use. Nothing particularly surprising here, since the power metaphor is a commentary on the reach and applicability of the high-powered technique; it doesnt go so far as describing the content of the mathematical theories. Schemes aren’t intrinsically powerful.

Collingwood seems to be claiming more. He says:

“The progressive intellectualization of language, its progressive conversion by the work of grammar and logic into a scientific symbolism, thus represents not a progressive drying-up of emotion, but its progressive articulation and specialization. We are not getting away from an emotional atmosphere into a dry, rational atmosphere; we are acquiring new emotions and new means of expressing them.”

Here, Collingwood seems to be asserting that a rarified emotion is part of the content of a mathematical concept. Are mathematical concepts intrinsically synesthetic?

Posted by: Rajesh Kasturirangan on November 23, 2009 9:19 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Like Todd, I have no color associations with limits and colimits, though geometry is certainly warmer than algebra.

I tend to think of this difference in terms of peanut butter - algebra is “crunchy” (lots of little bits) while geometry is smooth.

Words and metaphors for emotions are very often drawn from those for eating, taste, and smell.

One can talk about a satisfying theorem, or a proof that contains a disgusting hack.

Posted by: RodMcGuire on November 23, 2009 11:05 AM | Permalink | Reply to this

### Feynman’s Synaesthesia; Re: Mathematical Emotion

“When I see equations, I see the letters in colors – I don’t know why. As I’m talking, I see vague pictures of Bessel functions from Jahnke and Emde’s book, with light-tan j’s, slightly violet-bluish n’s, and dark brown x’s flying around. And I wonder what the hell it must look like to the students.”
[Feynman, Richard. 1988. What Do You Care What Other People Think? New York: Norton. P. 59]. He and I talked about this at length. My wife and I (mild synaesthetes) disagree about what color electrons are: blue or violet? Different integers have different personalities to me (and as we have seen, to John Baez et al), but not extremely as to Ramanujan.

Ramachandran, V.S., and Hubbard, E.M. (2001), “Synaesthesia: A window into perception, thought and language”, Journal of Consciousness Studies, vol. 8, no. 12, pp. 3-34.

I wonder what fraction of mathematicians have one of the several forms of Synaesthesia.

Posted by: Jonathan Vos Post on November 23, 2009 4:11 PM | Permalink | Reply to this

### Re: Mathematical Emotion

geometry is warmer than algebra

algebra is “crunchy” while geometry is smooth

I’d say algebra is mechanical (like intricate clockwork), while geometry is organic. Maybe that goes along with other cultural emotional responses to lead to the hot/cold bit.

Posted by: John Armstrong on November 23, 2009 5:22 PM | Permalink | Reply to this

### Smooth/crunchy Math; Re: Mathematical Emotion

When I make a synaesthetic map from {Algebra, Geometry, n-Category Theory} to Peanut Butter, I get a different result which I could could illustrate with a Venn diagram. I’m not really joking; these are the sense impressions that I imagine as I type this comment. Your mileage may vary.

Algebra on continua (i.e., x a real variable) maps to smooth peanut butter.

Geometry with continua (i.e. real lines, Euclidean circles) maps to smooth peanut butter.

Discrete algebra (i.e. x an integer variable) maps to smooth peanut butter.

Discrete Geometry maps to crunchy peanut butter.

Descartes introduced sandwiches with both smooth and crunchy, flavored with Algebra and geometry intrinsically intermixed.

Complex algebra and complex geometry map to peanut butter with jelly premixed in.

Topology peanut better sandwiches are on bagels.

Hyperbolic Geometry peanut better is in a spicy sate’ sauce of peanuts into which is dipped a saddle-shaped potato chip…

String Theory is that same sate’ flavor, but on long noodles.

n-Category Theory maps to {almond butter, pecan butter, macadamia nut butter, …}

Of course, we’re talking about nutritious Math with good taste.

Posted by: Jonathan Vos Post on November 23, 2009 5:37 PM | Permalink | Reply to this

### Re: Mathematical Emotion

John Armstrong said:

I’d say algebra is mechanical (like intricate clockwork), while geometry is organic.

For me, I think, geometry is like fabric, whereas algebra is like architecture. It’s analysis that is like some hideously complicated machine.

In fact, for me, differentiable manifolds are like silk, whereas algebraic varieties are, depending on how I’m thinking about them, either something much more coarse, like canvas or even rush matting, or else something more intricate, like lace.

(Now I feel as though I ought to elaborate this metaphor, comparing complex manifolds to bombazine and finite geometry to chintz or something, but I’m not quite that far gone …. Actually, finite spaces are more like corsets.)

Posted by: Tim Silverman on November 23, 2009 6:37 PM | Permalink | Reply to this

### Re: Mathematical Emotion

To belabor my metaphor, analysis would be materials science. It’s incredibly valuable for someone to do, but I’m glad I’m not that someone.

Posted by: John Armstrong on November 23, 2009 7:35 PM | Permalink | Reply to this

### Re: Mathematical Emotion

I certainly see algebra, and algebraic geometry, and rigid, and topology as flexible (not just the objects, but the idea). But in the former is more ‘spaced out’, like a plantation (though not as orderly), and the latter like an overgrown rainforest - you can find anything you like in topology, but it is surrounded by things similar but crucially different. In algebraic geometry it seems to me that one has a large number of carefully prepared ‘trees’ on which one can try out ideas, form very precise conjectures, but in topology the hapless explorer just points at the forest in about the right direction and says ‘that one there’.

Number theory is very sparse, but any structure that does exist is very powerful. It evokes more of an ‘outer space’ setting, with the numbers as stars utterly separated, but subtly interacting by unseen forces.

Posted by: David Roberts on November 25, 2009 4:03 AM | Permalink | Reply to this

### Re: Mathematical Emotion

conc. synesthesia and complex mental functions, this article could be interesting:
“(the experiment) shows that even without hyperconnectivity in the brain, you can still have synaesthesia,” says Cohen Kadosh. He says hypnosis may reactivate connections that had been suppressed by the brain. … Synaesthesia seems to underpin some savants’ enhanced memory and numerical skills.”

One can participate in an online experiment too: “synaesthetic experiences are not only triggered by a sensory experience, they can also triggered simply by thinking about things. … it’s just the predisposition to have extra pathways between areas of the brain,” says Dr Simner. “And we can see those connections.” With a newer imaging technique, called diffusion tensor imaging (DTI), Dr Simner says you can “almost count the extra pathways”. (article, online experiment)

Perhaps the way one associates visualizations to mathematical concepts has some (learned) synaesthesian aspect? E.g. I experience a concept only then as understood if I associate visualizations, but lack interest in terminologies. Some things/terminology only interest me because they induce some nice image.

Posted by: Thomas on November 23, 2009 11:54 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Brain hyperconnectivity may not only relate to synaesthesia, but to bipolar personality (acc. to this anonymous blog a relevant, but rarely open discussed issue) and creativity. (articles 1, 2, 3) A czech friend who translated it into german recommended “Edison” by Nezval on “the anxiety that comes with knowledge and the risks and courage required for any creative work” as good illustration of the bipolar mindset.

Posted by: Thomas on November 23, 2009 12:38 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Bipolar in the medical sense is a lot more than creativity. Beware the misuse of technical terms - cf. the Jesuit astronomer and the use of _hell_.

Posted by: jim stasheff on November 23, 2009 1:52 PM | Permalink | Reply to this

### Re: Mathematical Emotion

For synesthesia and much more, I strongly recommend

Musicophilia by oliver Sachs

Posted by: jim stasheff on November 23, 2009 1:50 PM | Permalink | Reply to this

### Re: Mathematical Emotion

I must admit that I only like to do maths for which the appearance of the written symbols on the page is aesthetically pleasing. This is so subjective I cannot even put words to what this constitutes, but there is definitely ‘ugly’ and ‘pretty’ mathematics for me.

Posted by: David Roberts on December 2, 2009 11:29 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Posted by: Thomas on December 17, 2009 3:36 PM | Permalink | Reply to this

### Re: Mathematical Emotion

I completely disagree that “mathematics shapes emotions for the better”. Mathematics puts its emphasis on justifying everything by logical argument. In my experience, this tends to shape mathematicians into people who place excessive value on logical argument in all aspects of life. This is a very dangerous emotional state to be in, in my opinion.

Posted by: Eugenia Cheng on November 28, 2009 4:17 PM | Permalink | Reply to this

### Re: Mathematical Emotion

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!

Posted by: David Corfield on November 28, 2009 4:57 PM | Permalink | Reply to this

### Re: Mathematical Emotion

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.

Posted by: Eugenia Cheng on November 28, 2009 5:28 PM | Permalink | Reply to this

### Re: Mathematical Emotion

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

Posted by: Thomas on November 28, 2009 9:22 PM | Permalink | Reply to this

### Re: Mathematical Emotion

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.

Posted by: bane on November 29, 2009 11:32 PM | Permalink | Reply to this

### Re: Mathematical Emotion

I object to the term “cleavage”. I think it is at best childish and at worst sexist and offensive to use such suggestive terminology in mathematics.

Posted by: anon on November 28, 2009 5:50 PM | Permalink | Reply to this

### Re: Mathematical Emotion

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

Posted by: David Corfield on November 28, 2009 5:56 PM | Permalink | Reply to this

### Re: Mathematical Emotion

I don’t think it is as bad when the term “cleavage” is being used in a field where it literally means “place where something is cleaved”. However, in mathematics terminology is chosen to conjure up an emotional link with something in ordinary life. A cleavage in mathematics is not literally where something is cleaved, but it is supposed to conjure up the notion of something being cleaved. However, if you give people the word “cleavage” and let them conjure up an image in their mind, what happens??

Posted by: anon on November 28, 2009 8:14 PM | Permalink | Reply to this

### Re: Mathematical Emotion

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.

Posted by: anon on November 28, 2009 8:18 PM | Permalink | Reply to this

### Re: Mathematical Emotion

This gives one the strong impression that someone just wanted to get the word “bra” into mathematics somehow.

Can you tell us why this is “wrong”? :-)

(meta: there should be away to put smileys in the text)

Posted by: J-L Delatre on November 29, 2009 12:51 PM | Permalink | Reply to this

### Re: Mathematical Emoticons

J-L wrote:

(meta: there should be away to put smileys in the text)

There are lots of emoticons to choose from here, and I encourage everyone to use them to express their mathematical emotions. If you use the default text filter, this:

$\lt$img src = “http://math.ucr.edu/home/baez/emoticons/tongue.gif” alt = “”/$\gt$

will produce this:

If you think this procedure is too complicated, contact The Boss.

Here are some other smileys for other moods:

There are also other less happy expressions.

Posted by: John Baez on November 30, 2009 7:00 AM | Permalink | Reply to this

### Re: Mathematical Emoticons

I suppose emoticons are better than nothing, but I’d be a lot happier if people tried to write well.

Posted by: David Corfield on December 1, 2009 10:15 AM | Permalink | Reply to this

### Re: Mathematical Emotion

I have my doubts that Dirac was trying to get ‘bra’ into mathematics or physics language as some kind of joke. His book The Principles of Quantum Mechanics, where the term appears, was first published in 1930, which I guess is several years before the abbreviation ‘bra’ for ‘brassiere’ began to catch on (earliest citation in the OED is 1936, and other sources seem to support that estimate). Additionally, Dirac doesn’t strike me as the kind of person who would knowingly insert a winking reference to a woman’s undergarment as a permanent addition to the math/physics lexicon.

On the other hand, he might have preferred three letters to four on grounds of symmetry, or maybe ‘brac’ was not acceptable to him because of the phrase bric-a-brac.

Posted by: Todd Trimble on November 29, 2009 2:07 PM | Permalink | Reply to this

### Re: Mathematical Emotion

My understanding is that the bra-ket notation was only introduced in the second edition of “The Principles of Quantum Mechanics”, which wasn’t published until 1958, well after the abbreviation “bra” had caught on for underwear.

Not knowing Dirac, I can’t say whether or not he thought it would be amusing to get a reference to women’s underwear into mathematics/science. In any case I think it is childish and/or offensive to women that nobody thought to point out to him that it was a bad idea. This terminology prevents me, as a woman, from ever doing research in this field because there is no way I am ever going to say the word “bra” in a talk. Likewise “cleavage”. I feel uncomfortable in every talk in which either of these pieces of terminology is used.

Posted by: anon on November 29, 2009 11:15 PM | Permalink | Reply to this

### Re: Mathematical Emotion

I had my doubts before, but now I'm sure that you're making this up.

Posted by: Toby Bartels on November 29, 2009 11:28 PM | Permalink | Reply to this

### Re: Mathematical Emotion

What is it that you think I’m making up??

Posted by: anon on November 29, 2009 11:47 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Actually Wikipedia claims the bra-ket terminology was first introduced in a paper in 1939. While this is possibly late enough for “bra” to have become widespread for women’s underwear, Wikipedia also informs me that Dirac was a bit autistic, which supports the theory that he simply didn’t notice that he was using inappropriate terminology. Perhaps there is a similar reason behind the awful “cleavage” terminology as well.

In any case this sort of autistic insensitivity is not a whole lot more comforting than childish joke-making.

Posted by: anon on November 29, 2009 11:25 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Yes, I see that in the article on Paul Dirac, he apparently introduced the bra-ket notation in 1939 (see note 6), and OED informs us that “bra” as slang term had indeed been introduced a few years earlier. I’m not sure how widespread or widely known the usage would have been in 1939, but I suppose there’s a good chance that some of his colleagues would have known both it and Dirac’s usage, and if so one could then certainly argue that they should have clued him in right away. But there may be other complicating factors, making it hard to judge.

Terminology, once established, is often hard to supplant. Here on this blog, there was discussion a few years ago on how inapt the phrase “schizophrenic object” is in category theory (it was meant to conjure “split personality”, which is a misconception which does disservice to those who suffer from actual schizophrenia), and there was discussion of possible more appropriate alternatives. For the time being there has been some agreement here to use terms such as “ambimorphic object” instead. But I have no idea if this campaign has resulted in widespread change in the categorical community: terminology, once established, has a lot of inertia.

However, let me say that in observing your being discomfited by “this sort of autistic insensitivity” on Dirac’s part, I’m not seeing much evidence of sympathy or sensitivity to the problems autistics face (if for example you were the parent of an autistic, as I am, then I think you wouldn’t be writing those words). IMO, the problems autistics face in being misunderstood by society are rather greater than the problems autistics may occasionally inflict by using inappropriate words.

Nitpick: the 2nd edition of Dirac’s book came out in 1935. The bra-ket notation appears in the 3rd edition (1947), and the 4th edition appeared in 1958.

Posted by: Todd Trimble on November 30, 2009 2:34 AM | Permalink | Reply to this

### Re: Mathematical Emotion

I too am a parent with a child on the autistic spectrum and found Anon’s comment highly insensitive.

Posted by: David Corfield on November 30, 2009 11:41 AM | Permalink | Reply to this

### Re: Mathematical Emotion

It is telling that I am not picking up the slightest bit of sympathy or understanding about the offensive nature of the “bra” and “cleavage” terminology to women. I have never mentioned this in public because I suspected I would just be either ridiculed or criticised in return. Criticism of me and my use of language may well be justified, but does not address the point in question.

Posted by: anon on November 30, 2009 10:01 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Just as with our campaign to replace ‘schizophrenic object’ by ‘ambimorphic object’, I’m sure most of us here would back calls for replacing other insensitive terminology. I don’t see much point in attributing blame to the originators.

Do you have any suggestions for replacements? I can’t even recall the definition of cleavage, something to do with splitting fibrations?

Posted by: David Corfield on November 30, 2009 11:49 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Just as with our campaign to replace ‘schizophrenic object’ by ‘ambimorphic object”, I’m sure most of us here would back calls for replacing other insensitive terminology.

As some of you may know, I am bisexual. The term ‘bisexual’ is often abbreviated ‘bi’. So in order to be sensitive to me, I hereby request that we all develop new terms to replace the following:

• bi-brane
• bi-pointed
• biactegory
• bialgebra
• bialgebroid
• bias (a terrible insult, this one!)
• bibundle
• bicategory
• biclosed
• bicomplex
• bicrossed product
• bigroupoid
• biholomorphic
• bijection
• bimodel
• bimodule
• bimonoid
• bimorphism
• bilinear
• binary
• bioctonion
• biproduct
• biquaternion
• birational (not an insult, but still inappropriate)
• biring
• bisimplicial
• bisimulation
• bitopological
• bivector

And these are only the ones that happen to have an entry (or a link for an entry) in the nLab!

I grudgingly accept ‘biology’ as from a different root, not to mention a different academic field. But please don't expect me to say it in public!

</sarcasm>

Seriously, there is a big difference between terminology developed to invoke a mistaken (however unknowingly so) idea about people, and terminology developed from a word in ordinary language that also happens to have other meanings referring to people. ‘schizoprhenic object’ is the former; ‘cleavage’ (like ‘bra’ and everything in the list above) is the latter.

Yet our anonymous writer comes to the Café and pretends that using ordinary words in their ordinary meanings is ‘childish’. She (?) tries to display to us all of the worst stereotypes about women in professional fields, that they have delicate sensibilities that cannot survive in the world of men, where one is liable to hear the word ‘bra’ without warning.

In reality, just as I am able to discusses biproducts without thinking about my sexuality, so women are able to discuss cleavages without thinking about their breasts. That is what men do, after all, and women are just as capable as men, despite anon's sexist pretence to the contrary.

Posted by: Toby Bartels on December 4, 2009 2:12 AM | Permalink | Reply to this

### Re: Mathematical Emotion

That’s powerfully argued, I’d say, and it would be interesting to hear back from anon.

I for one am annoyed by the fact that I will never again be able to hear the word “cleavage” (as used in the mathematical sense) without conjuring an image which before simply hadn’t occurred to me. I resent that. It’s like the word ‘gay’ (“don we now our gay apparel”), or ‘queer’ – words which had perfectly acceptable nonsexual meanings, but which are now practically unusable for any except the pretty limited range of meanings they have taken on.

The more I think of it, the more I suspect that Mark Meckes was right: anon heard some stupid boyish joke and now has trouble seeing past it. Just like I will have trouble forgetting ‘cleavage’ now. But is expunging the word entirely really the most rational response? A good roll of the eyes and a withering stare or a few choice words would seem to be more like it.

Posted by: Todd Trimble on December 4, 2009 3:19 AM | Permalink | Reply to this

### Re: Mathematical Emotion

While we’re at it, I might remark that I’m against the common usage of ‘childish’ or ‘boyish,’ because I’m a rather serious follower of the romantic world view.

Posted by: Minhyong Kim on December 4, 2009 10:37 AM | Permalink | Reply to this

### Re: Mathematical Emotion

“the common usage of ‘childish’ or ‘boyish’” - IMO adult’s ideas of ‘childlike mentality’ are in fact often just that of a ‘retarded adult mind’. That makes didactics/advice often so disgusting for the target group.

Posted by: Thomas on December 4, 2009 11:36 AM | Permalink | Reply to this

### Re: Mathematical Emotion

That’s fine. It should be clear from context that I meant ‘childish’, which is to be clearly distinguished from ‘childlike’. (Is English your native language?)

Posted by: Todd Trimble on December 4, 2009 1:02 PM | Permalink | Reply to this

### Re: Mathematical Emotion

I understand, but the common usage of ‘childish’ suggests a dim view of the state of being ‘childlike,’ and can be found at least as objectionable as any of the other terms we’ve been discussing. This may sound like a joke, especially since I make poor attempts at humor rather often on this blog, but I do mean it rather seriously.

Posted by: Minhyong Kim on December 4, 2009 1:26 PM | Permalink | Reply to this

### Re: Mathematical Emotion

I understand, but the common usage of ‘childish’ suggests a dim view of the state of being ‘childlike’

Well, not to me. To me, the common usage of ‘childish’ just refers to aspects of youthful behavior which would be better to have outgrown as an adult. Other aspects of youthfulness (relating to the world with the whole of one’s person, the ability to look at something as if for the first time, playfulness, letting go completely, forgetting one’s self in the midst of absorption, and so forth) are of course incredibly valuable and even to be cultivated throughout one’s life.

As in Matthew 18:3 – “unless you turn and become like children, you will never enter the kingdom of heaven”. The sense of that should be clear, and it’s not the same as saying, “unless you remain as children…”. Big difference!

Posted by: Todd Trimble on December 4, 2009 1:58 PM | Permalink | Reply to this

### Re: Mathematical Emotion

It seems to me that pretty much any of the examples that have come up could be explained away in roughly analogous fashion. (Even though I hope no one challenges me to do it.)

“The sense of that should be clear.”

Could I trouble you to explain that clear sense? Perhaps it will help to explain just a bit my own feelings about the issue, without being too much of a bore. Obviously, when people refer to someone as “childish,” they have certain bad behavior or temperament in mind. Well, it seems to me that people who have such a temperament tend largely to retain it into adulthood, picking up plenty of other shortcomings in the process. Here, I am not entirely serious, but almost so. In short, the problems have little to do with childhood.

Posted by: Minhyong Kim on December 4, 2009 2:27 PM | Permalink | Reply to this

### Re: Mathematical Emotion

As in Matthew 18:3 – “unless you turn and become like children, you will never enter the kingdom of heaven”.

Contrast this with another Bible verse: 1 Corinthians 13:11 – “when I became a man, I put away childish things”.

Unless you're a Christian, of course, there's no reason why you shouldn't expect a contradiction between the words of Jesus and the words of Paul. But I am not a Christian, yet I see no contradiction here. They are simply using different metaphors.

Posted by: Toby Bartels on December 4, 2009 7:26 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Here as well, I’m rather curious about the connotations in Greek of the term translated as ‘childish.’

Posted by: Minhyong Kim on December 5, 2009 12:51 AM | Permalink | Reply to this

### Re: Mathematical Emotion

I second Todd’s distinction between “childish” and “childlike”. To my ears (and I think those of most native English speakers), “childish” is a close synonym for “immature”.

Posted by: Mark Meckes on December 4, 2009 2:32 PM | Permalink | Reply to this

### Re: Mathematical Emotion

I guess I should weaken my previous objection, because the word ‘childish’, offensive to children or not, is definitely clearer to my mind than ‘immature.’ My puzzlement may have to do with limitation in my English, but I suspect more a difference of ethos.

Sorry to sound argumentative, but it is rather odd when an interminable recent discussion concerned the reality of cats and the like, and then roughly the same group of people regard notions like ‘maturity’ or ‘childish vs. child-like’ as self-evident.

Posted by: Minhyong Kim on December 5, 2009 1:12 AM | Permalink | Reply to this

### Re: Mathematical Emotion

roughly the same group of people regard notions like ‘maturity’ or ‘childish vs. child-like’ as self-evident

I certainly don't regard these concepts as self-evident. It's just that English has two words: one for similarity to a child in a bad way (‘childish’) and one for similarity to a child in a good way (‘child-like’). And having both, using one doesn't preclude the applicability of the other to other situations, so if you know that, then neither of them should imply that similarity to a child is good or bad in general.

Posted by: Toby Bartels on December 5, 2009 3:13 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Now I’m thoroughly puzzled by this logic. If you examine a correspondence like

persistent use of ‘childish’ in a pejorative sense $\approx$ suggestion of bad things about a child, inducing ignorance of a child’s nature

persistent use of ‘autistic’ in a pejorative sense $\approx$ suggestion of bad things about a person with autism, inducing ignorance of such a person’s nature

persistent use of ‘X’ in a pejorative sense $\approx$ suggestion of bad things about X, inducing ignorance of X’s nature

am I to understand that the first link is especially weak for some reason? (I strongly sympathize with David and Todd’s sentiments, by the way.)

Posted by: Minhyong Kim on December 5, 2009 3:59 AM | Permalink | Reply to this

### Re: Mathematical Emotion

am I to understand that the first link is especially weak for some reason?

Yes, and the reason is the existence of the word ‘childlike’.

I can understand that it's confusing for someone who hasn't grown up with it, but in English these two words form a pair; it's hard to even think about one (to actually think about what it means and where it comes from, rather than merely to use it casually) without thinking about the other. If I start thinking why something is called ‘childish’ and whether that's fair to children, I naturally begin to think about the word ‘childlike’ and how they have almost opposite meanings.

The word ‘autistic’ means ‘similar to a person with autism’, neither more nor less. The world ‘childish’ does not mean ‘similar to a child’, nor does the word ‘childlike’. Anybody who wrote such a definition into a dictionary would simply be wrong about those words, at least in their contemporary meanings. I don't even know what word I would use if I wanted something neutral; I have to use a phrase such as ‘similar to a child’.

I don't use either word ‘childish’ or ‘childlike’ myself, since they both appeal to stereotypes about children. And I would be suspicious of someone who consistently used only one but not the other; perhaps they have some pro-child or anti-child feelings. But they're not in the same class as ‘autistic’. (For one thing, the people who use those words have all actually been children.)

Posted by: Toby Bartels on December 5, 2009 9:27 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Well perhaps you have a point. But since we’re both mathematicians, let me put your argument in abstract form, just to make sure I understand it.

When using

1. a pejorative word w(A) making comparison to group A;

and

2. a pejorative word w(B) making comparison to group B;

If a word w’(A) exists in the language that suggests positive comparison to group A, and no such word w’(B) exists for B (which itself may be a tough case to make), then case 1 is less offensive than case 2.

Does this capture your position correctly?

Posted by: Minhyong Kim on December 5, 2009 10:07 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Obviously I can’t speak for Toby, but I think the issue is obscured by framing it in terms of “using pejorative words”, as opposed to “pejorative use of words”.

Let me then reframe the distinction. Consider:

1. pejorative use of a word w(A) which refers to some, but not all, characteristics of (possibly some but not all) members of group A;

versus

2. pejorative use of a word w(B) whose primary meaning is “of, like, or pertaining to members of group B”.

Regardless of what alternative words do or do not exist to describe either group, I believe case 2 is more offensive because it carries a stronger implication that the pejorative meaning applies to all members of group B in all circumstances.

Thus pejorative use of “autistic”, which may be defined roughly as “of or pertaining to people with autism”, is more offensive than pejorative use of “childish”, which is understood as referring specifically to certain negative qualities possessed by some children.

Posted by: Mark Meckes on December 5, 2009 4:57 PM | Permalink | Reply to this

### Re: Mathematical Emotion

As I’ve said many times, it’s not that I doubt that a certain perspective can be argued for and against all we want. Regarding the two examples, someone else might have taken up a position exactly opposite to yours (whether we are referring to pejorative words or their use: I’m not sure I understand the significance of your distinction). He/she might have said ‘autistic’ obviously refers to specific qualities pertinent to a specific disability, and carries no intent to put down a person or a class of people as a whole. Meanwhile, words like ‘childish’ or ‘immature’ are so vague that one can only conclude that some ill is thought of the whole group thus implicated.

After a certain amount of dialectical give and take, the temptation would be quite strong for someone with a different ethos to detect only a predisposition to confine critique to words that are already blacklisted in your peergroup.

Note that I’m not accusing you of this at all. It’s just that for your true intent to show through, I feel the case needs to be far better made.

Incidentally, in case someone wants unsolicited advice on discussing with a child entering the teens, general acccusations of ‘immaturity’ do seem to be rather ineffective to a child with any kind of a logical disposition. Far better to refer to the specific nature of misbehavior (rightly or wrongly perceived): selfish, thoughtless, mean, destructive, etc. Among other things, this gives them the clear opportunity to judge your own actions on a fair footing, that is, to check for minimal consistency, an essential ingredient in making a convincing case (assuming a convincing case is there to be made).

Posted by: Minhyong Kim on December 5, 2009 6:58 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Does this capture your position correctly?

It captures what I said, or at least what I started out saying, but actually Mark's analysis is better.

The existence of $w'(A)$ highlights that $w(A)$ does not mean “of, like, or pertaining to members of group $A$”, but what really matters is that $w(A)$ does not mean that.

Incidentally, in case someone wants unsolicited advice on discussing with a child entering the teens, […]

I agree with all of that. As I remarked earlier, I don't really use the word ‘childish’ myself.

Posted by: Toby Bartels on December 5, 2009 8:43 PM | Permalink | Reply to this

### Re: Mathematical Emotion

As far as I can see, Mark is claiming it to be evident that the term ‘childish’ refers to some obvious negative characteristics possessed by some children, a position you , earlier seemed to deny. The situation is further confusing because you then claim to agree with my general assessment that the meaning of ‘childish’ and ‘immature’ are ambiguous and the words are better avoided. (I apologize if it looks like I’m employing a number of tactics just to win an argument. That’s not my intention. At least, I don’t think it is.) Oh well, I accept of course that everyone in this discussion basically means well.

Perhaps one could compare a word like ‘childish’ to something like ‘beastly’, used even by people who appreciate the noble qualities of many a beast. Nevertheless, it is rather plausible that the word arose in association with a negative view of beasts. It might be instructive to compare the word also with one like ‘girl,’ which carries no a priori negative connotation, but which I gather is rude in certain precise contexts. In our present discussion, I am in any case told that the precise limits of ‘childish’ should be so clear to any reasonable user of English that words of caution can safely be dismissed. (OK, that’s probably a small exaggeration of the actual assertions.)

Perhaps I should now admit that I started out the discussion of ‘childish’ in a slightly disingenuous manner with the intention of illustrating the difficulty of laying out normative ethics. In particular, the Confucian side of me doesn’t really have such a rosy view of childhood, even as it is genuinely dismayed that some people who accuse others of immaturity appear to believe that cultivation comes simply with growing old. (And the Taoist side really does sympathize with Wordsworth.)

So maybe I’ll attempt to close out my (lack of) contribution with a personal anecdote, somewhat in the vein of what John was calling for, but not exactly. It might strike you as irrelevant and boring, but a number of people I have told it to orally over the years seemed to find it of at least passing interest.

————————————-

In the years when I taught in New York City, I had considerable contact with Korean-American university students. I myself was somewhat immersed in various kinds of activism at the time, such as the Central American Solidarity Association, or similar groups. I would occasionally recommend these to the young K-A’s. It never seemed to appeal to them, even as they nodded politely. During an evening tea party at my apartment on 113th street, one of the students, with considerable hesitation, started to explain his feelings about the matter. The key point was that he, and many of his friends, felt much more comfortable with *conservative* groups, such as church societies, than with liberal/radical ones. (Only a few of them were actively religious, by the way.) The reasons were somewhat rambling, with different students contributing their opinions, but they could be roughly summarized as follows: They found the means to be accepted or to win respect much more transparent in the conservative groups. It involved something like being good students, socializing in families, gaining admittance to prestigious schools, knowing something about serious books, etc. Meanwhile, they found liberal/radicals groups around NY well-nigh impenetrable to people with a non-local background. One of the students said something to the effect that ‘You had to be cool. But since I came from Korea only when I was in middle school, there was no way to figure out what was cool. And then, if I tried too hard, it would have looked ridiculous anyways.’ I don’t remember in detail the examples of difficulty they brought up, but many skirted around the issue of language, including unknowing use of various taboos. At times it was something as simple as referring to some behaviour as ‘bad,’ (as I still do) when the accepted description might have been ‘inappropriate.’ (One other point I recall is that the Korean students were usually much more into ‘classical’ music than the local average, a pretty big deal given the enormous weight of music in the lives of young people.)

For me, this particular conversation was quite revealing, and did permanently affect my understanding of North American political discourse in ways that would be a bit tiresome to explain in detail here. I did increasingly notice, however, that there was a remarkable degree of segregation in the city when it came to socializing, at least in relation to the high demographic diversity physically present.

—————————————-

The map reveals this blog to be quite far-reaching. If we take into account the multiple identities emphasized by Amartya Sen, the number of groups participating in any given discussion (some silently) must be truly staggering. IMHO (first time I’ve used this charming abbreviation) the only way for us all to move forward with the constructive parts of the discourse is then for each individual to accept a commonsensical degree of asymmetry in our judgement of courteous usage. This of course includes the need to react sensitively and sensibly when our own words run afoul of someone’s sensibility. The asymmetry, however, refers to a need not to expect the ‘same’ of others, who may, for all we know, be doing the best they can within the constraints of their own ethos. In short, be not quick to take offense.

Whew! That was indeed a good deal of rhetoric merely to say something obvious!

Posted by: Minhyong Kim on December 6, 2009 1:46 AM | Permalink | Reply to this

### Re: Mathematical Emotion

As far as I can see, Mark is claiming it to be evident that the term ‘childish’ refers to some obvious negative characteristics possessed by some children, a position you earlier seemed to deny.

I agree with him that it is evident, but (as I wrote) I disagree that it is ‘self-evident’.

I mean that I would expect the strictly negative meaning of ‘childish’ (and the strictly positive meaning of ‘childlike’) to be evident to somebody who grew up speaking English (at least my dialect), but I would not expect it to be evident to somebody who was merely familiar with English, examining how the words are obviously constructed by adding suffixes to ‘child’. I would expect a person learning English to have to be told (or to have to check a dictionary, or at least to have to see many examples before the distinction dawned on them).

I may be wrong.

[…] the Confucian side of me […] the Taoist side […]

I had to check whether your name was ‘Minhyong’ or ‘Yinyang’. (^_^)

They found the means to be accepted or to win respect much more transparent in the conservative groups.

That surprises me. I understand it, once you say it, but it never occurred to me that finding acceptance would be difficult in a campus activist group; they always seem so proselytising. But surface acceptance is different from real respect.

One other point I recall is that the Korean students were usually much more into ‘classical’ music than the local average, a pretty big deal given the enormous weight of music in the lives of young people.

So unlike the radical long-haired hippies, they were into long-hair music? (^_^)

the only way for us all to move forward with the constructive parts of the discourse is then for each individual to accept a commonsensical degree of asymmetry in our judgement of courteous usage. This of course includes the need to react sensitively and sensibly when our own words run afoul of someone’s sensibility. The asymmetry, however, refers to a need not to expect the ‘same’ of others, who may, for all we know, be doing the best they can within the constraints of their own ethos. In short, be not quick to take offense.

Abraham Lincoln apparently said ‘We should be too big to take offense and too noble to give it.’; I've seen some Internet guidelines that urge everyone to try not to offend others and try not to be offended (although I can't find one now; I must have the wording wrong).

I've allowed anon to offend me. Perhaps I should stop writing for a while.

Posted by: Toby Bartels on December 6, 2009 3:33 AM | Permalink | Reply to this

### Re: Mathematical Emotion

“But surface acceptance is different from real respect.”

Yes, that was the point the students made. Of course I urged them not to be too sensitive about such differences, and everyone agreed in principle. But It was apparently difficult to get over them in the long run, and when it came to close friendships.

With regard to liberal groups, I think the critique was phrased roughly as follows: ‘Any actual skills they had in tolerance and diversity were quickly cancelled out by their firm belief that they had it all figured out.’

My own feeling was that the pertinent portions of liberalism do arise from genuinely good intentions, and hence, that such a complaint was a bit too harsh in nature. However, the harshness of the experience seems to have been real enough. I’ve been told many times that my own perspective on life in the US as a foreigner (or as a math nerd, for that matter) is permamently rosier than that of people who survived their teen years there.

Posted by: Minhyong Kim on December 6, 2009 9:51 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Minyong described in some detail how

the young [Korean-American university students] […] felt much more comfortable with *conservative* groups, such as church societies, than with liberal/radical ones.

That’s absolutely fascinating, Minhyong. Unlike Toby, I’m not at all surprised by the basic idea, but I don’t think I’ve ever seen this sort of thing laid out in quite this sort of detail before. Culture is very powerful.

The asymmetry, however, refers to a need not to expect the ‘same’ of others, who may, for all we know, be doing the best they can within the constraints of their own ethos.

Yes; just because people aren’t shouting at you, it doesn’t mean you haven’t offended them; just that they value a civil discourse with you more highly than arguing some moral or cultural point. I try to remember this, and I do feel I should try harder.

Posted by: Tim Silverman on December 6, 2009 11:11 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Tim wrote:

Yes; just because people aren’t shouting at you, it doesn’t mean you haven’t offended them…

Indeed, there are plenty of people who become more and more icily polite the more you offend them. I do this myself in my role as newsgroup or blog moderator — since it sets a really bad example to yell at people.

Unfortunately, there are people who seem unable to detect ‘icy politeness’.

(Don’t worry, I’m not talking about you!)

Posted by: John Baez on December 7, 2009 1:15 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Minyong described in some detail how

the young [Korean-American university students] […] felt much more comfortable with *conservative* groups, such as church societies, than with liberal/radical ones.

That’s absolutely fascinating, Minhyong. Unlike Toby, I’m not at all surprised by the basic idea, but I don’t think I’ve ever seen this sort of thing laid out in quite this sort of detail before. Culture is very powerful.

The asymmetry, however, refers to a need not to expect the ‘same’ of others, who may, for all we know, be doing the best they can within the constraints of their own ethos.

Yes; just because people aren’t shouting at you, it doesn’t mean you haven’t offended them; just that they value a civil discourse with you more highly than arguing some moral or cultural point. I try to remember this, and I do feel I should try harder.

Posted by: Tim Silverman on December 6, 2009 11:28 PM | Permalink | Reply to this

### Re: Mathematical Emotion

(One line was deleted from the previous post, for come reason.)

We do appear to live in an age when psychological categories trump all others in their ontological status!

Posted by: Minhyong Kim on December 5, 2009 1:16 AM | Permalink | Reply to this

### Re: Mathematical Emotion

By the way, no, English is not my native language.

Posted by: Minhyong Kim on December 4, 2009 1:34 PM | Permalink | Reply to this

### Re: Mathematical Emotion

“there is a big difference between terminology developed to invoke a mistaken (however unknowingly so) idea about people, and terminology developed from a word in ordinary language that also happens to have other meanings referring to people. ‘schizophrenic object’ is the former”

Is this really true? Part of what puzzled me a bit about this situation is that the common mistake appears more faithful to the etymology. Of course there’s no need to be stubborn about etymology, but it goes somewhat against your claim. Of course, I’m no expert in Greek.

Posted by: Minhyong Kim on December 4, 2009 10:27 AM | Permalink | Reply to this

### Re: Mathematical Emotion

If you have schizophrenia, then there is a split between your mind and reality, not a split within your mind.

Posted by: Toby Bartels on December 4, 2009 6:56 PM | Permalink | Reply to this

### Re: Mathematical Emotion

I didn’t doubt there was an explanation that vindicates the medical usage. The question was whether the common use is indeed ‘developed to invoke a ‘mistaken idea about people.’ But I admit, I didn’t follow the earlier discussion on this blog.

Posted by: Minhyong Kim on December 5, 2009 12:02 AM | Permalink | Reply to this

### Re: Mathematical Emotion

By the way, perhaps this would then get you labeled a schizophrenic.

(I myself stand in grave danger of appearing imbalanced with this rapid string of posts.)

Posted by: Minhyong Kim on December 5, 2009 1:32 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Toby:

The term “bisexual” is often abbreviated ‘bi’. So in order to be sensitive to me, I hereby request that we all develop new terms to replace the following:

<long list … >

I’d add that, as an Englishman, and therefore heir to a particular tradition of humour, I know that it’s possible to find a double entendre pretty much anywhere if you’re looking for one. However, as a general rule, people don’t, unless they are either a) 15 years old or b) listening to/watching an English comedy program.

As to being 15 years old: I can still vividly recall the French class where our teacher, discussing the Palace of Versailles, carelessly described the Hall of Mirrors as a place where Louis XIV “held his balls”—a remark which resulted in literally about five minutes of uncontrollable laughter from the class. Since we were 15, there was nothing the teacher could do but stand there smiling and looking slightly exasperated until we’d finished. However, I do not draw the conclusion that the word “ball” should be withdrawn from usage, let alone the many other ordinary words with far more obscene slang meanings

As to English comedy: I remember some time ago hearing on some radio program hearing some BBC comedy guidelines from, I’d guess, the 40’s or 50’s. It was attempting to clamp down on doubles entendres, and gave as an example “Winter draws on”, which to a non-rhotic speaker is homophonous with “Winter drawers on”—“drawers” being a reference to underwear (ha ha). Again, I would not advocate eliminating the word “draw” (or “drawer”). That would be terribly inconvenient, and outside the context of a comedy show, alternative meanings are not likely to be salient.

It’s one thing if an obscene or embarrassing meaning becomes so dominant that it naturally springs to the mind of the vast majority of people who use it: then I think we can legitimately correct the usage of people who don’t notice it. It’s quite another if noticing some potential double entendre is only the idiosyncratic response of a handful of people. In my youth, when I had not left my fifteenth year too far behind, I would regularly spot potentially obscene meanings behind some innocent expression that someone had used. But, unless the context was one where it was appropriate to make a dirty joke, my natural feeling was one of mild embarrassment at my own dirty-mindedness, not anger at the “insensitivity” of the innocent speaker.

If a large number of people assure me they’re offended by something I say, then, naturally, politeness requires (in the absence of some overriding moral imperative) that I change to accommodate them (in fact, I hope I’d be sensitive enough to notice without being told.) But this doesn’t give random strangers a license to order me to disrupt my life for their slightest personal convenience—let alone to issue such orders to an entire community. After all, any number of people cause me mild inconvenience or offence all the time, but there has to be a certain amount of live and let live if we are to get along with each other.

Posted by: Tim Silverman on December 4, 2009 12:35 PM | Permalink | Reply to this

### Re: Mathematical Emotion

To be fair, the terminology anon finds objectionable is not like your double entendres, Tim. Bra and cleavage have only one meaning in everyday language, and they both refer to women’s breasts.

Posted by: Tom Leinster on December 4, 2009 12:57 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Bra and cleavage have only one meaning in everyday language, and they both refer to women’s breasts.

Maybe this is another American/British difference, but that's not true of ‘cleavage’ in the language that I know. I can imagine someone who, although a native English speaker, is uneducated and ill read, who does not really know the meaning of hard words like ‘schizophrenia’ and ‘cleavage’; for such a person, ‘schizophrenic object’ and ‘cleavage’ in the mathematical sense will both seem appropriate metaphors that insensitively refer to people. But I expect educated English speakers to know these words, or at least to look them up if they don't.

Posted by: Toby Bartels on December 4, 2009 7:41 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Tom riposted:

the terminology anon finds objectionable is not like your double entendres

“Bra” I’ll grant—it did initially strike me as a bit of a weird piece of terminology (though, on the other hand, I was about 15 when I first encountered it …).

“Cleavage”, on the other hand, when I encountered it crystallography and geology, seemed so obviously related to the verb “cleave” that the “everyday” meaning simply didn’t spring to mind at all. In fact, it wasn’t until I came across this thread that the two meanings had ever been linked in my mind.

It’s not that I can’t see why someone mightn’t be taken aback on first hearing the word in an unfamiliar sense. It’s the “You must all change your usage or I will be forced to give up my research!” that seems a bit OTT. Also, “bra” and “cleavage” don’t strike me as particularly rude words in their everyday senses, if you’re grown up. I guess they might be annoying if you’re constantly surrounded by snickering adolescents.

Posted by: Tim Silverman on December 4, 2009 9:05 PM | Permalink | Reply to this

### Re: Mathematical Emotion

They might be annoying if you’re constantly surrounded by snickering adolescents.

Careful: “snicker” is dangerously close to “knicker”, and anon might take offense.

Posted by: John Armstrong on December 4, 2009 10:25 PM | Permalink | Reply to this

### Re: Mathematical Emotion

I think most women (and men) would not immediately associate “snicker” with “knicker” because the candy bar brand neutralizes the word. Also, it’s not clear that many people know what the word means.

The word “bra,” on the other hand, has only one commonly known definition. To use it in a professional context not relevant to underwear can be uncomfortable to most women.

Posted by: st on December 4, 2009 11:02 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Said st:

The word “bra,” on the other hand, has only one commonly known definition. To use it in a professional context not relevant to underwear can be uncomfortable to most women.

It can be—I mean, I can imagine that happening—but is it? Anybody can say their personal prejudices are shared by more than half the human race (or, more modestly, half the local community), but getting the votes to prove it is another matter entirely. Since I’m not inclined to take the Bolsheviks or the Moral Majority at their own valuation; and I don’t like other people appointing themselves as my representatives without consulting me; so I need something more than “can be” before I go covering up the legs of pianos to avoid alarming nice young ladies of delicate sensibility.

Posted by: Tim Silverman on December 4, 2009 11:33 PM | Permalink | Reply to this

### Re: Mathematical Emotion

I’d never associated the mathematical sense of ‘cleavage’ with women’s breasts until this discussion. The reason is that I first heard about the mathematical term as part of a big discussion about split exact sequences, splittings, cloven fibrations, and so on — so if you’d asked me what ‘cleavage’ meant in math, I would have explained all that stuff and maybe also the use of this term in crystallography, where you can split or cleave a crystal… never thinking about its other meaning.

But now of course I can’t help it.

But that will go away in a while.

Posted by: John Baez on December 4, 2009 9:33 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Tom Leinster wrote in part:

Bra and cleavage have only one meaning in everyday language, and they both refer to women’s breasts.

I know that I already responded to this comment, but …

One reason that I would like to keep the word ‘cleavage’ in its mathematical usage is my surprise that a native English speaker would write what Tom wrote above about its meaning in ordinary language. I don't want ‘cleavage’ to lose what I regard as its ordinary meaning in favour of a highly specialised colloquial anatomical usage. If it's the right word for the job, then we should use it.

(Incidentally, I have no opinion as to whether ‘cleavage’ is the right word for the job related to fibred categories; I have no feel for that concept yet. Maybe it's a lousy term —no disrespect meant to lice.)

Posted by: Toby Bartels on December 6, 2009 1:41 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Tim’s example underlines the possibilities of unintentional double entendres between US English and British. I had at least 2 encounters with this when I first went to Oxford as a grad student.

Posted by: jim stasheff on December 4, 2009 1:31 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Jim wrote:

Tim’s example underlines the possibilities of unintentional double entendres between US English and British. I had at least 2 encounters with this when I first went to Oxford as a grad student.

Okay… if nobody else will, I’ll do it:

What were they?

Posted by: John Baez on December 5, 2009 4:14 AM | Permalink | Reply to this

### Re: Mathematical Emotion

I’m willing to bet “pants” was among them.

Incidentally, that also throws “infinite pair of pants” into the reject bin.

Posted by: John Armstrong on December 5, 2009 5:42 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Right - pants was one I said in mixed company
conversely a fellow student (female) in suggesting she’d stop by to get me for an outing the next day said she would stop by and knock me up !
but the worst was: i had picked up the math expression to bugger a construction’
while at Princeton and used it in a lecture in Oxford with JHCW in the audience. He felt complelled to explain to me in Latin! the British meaning

Posted by: jim stasheff on December 5, 2009 1:58 PM | Permalink | Reply to this

### Re: Mathematical Emotion

LOL!!!

Reminds me of Max Kelly, when I was giving a seminar talk in Sydney and Max was attempting to find problems with it, made some sort of objection and concluded, “now you’re buggered.” Followed by some hasty backpedaling that he didn’t mean it literally!

Thanks, Max.

Also reminds me of a crude attempt in some Japanese newspaper to translate an Australian politician’s side remark of someone, “he’s a funny bugger, isn’t he?”, which when translated back involved the words “comical homosexual”.

Posted by: Todd Trimble on December 5, 2009 3:15 PM | Permalink | Reply to this

### Re: Mathematical Emotion

The contributor creating the $n$Lab entry with the most offensive math terminology wins.

I have created associative operad. Can you top this?

Posted by: Urs Schreiber on December 4, 2009 8:11 PM | Permalink | Reply to this

### Re: Mathematical Emotion

The contributor creating the nLab entry with the most offensive math terminology wins.

I have created associative operad. Can you top this?

Or how about the various other terms for building?

(Now, that’s an interesting case in the light of some of the comments here…)

Posted by: Urs Schreiber on December 4, 2009 8:27 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Urs said:

most offensive math terminology wins

and

building

OK, now you’re making anon’s point for her. I don’t want to play this game.

Posted by: Tim Silverman on December 4, 2009 9:09 PM | Permalink | Reply to this

### Re: Mathematical Emotion

OK, now you’re making anon’s point for her.

And here I was thinking that I was listing examples for the absurdity of that point.

I suppose my attempt to make you at least collect some useful technical information on the terms that you are discussing failed, too.

Posted by: Urs Schreiber on December 5, 2009 2:26 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Urs said,

And here I was thinking that I was listing examples for the absurdity of that point.

Sorry Urs, didn’t mean to pick a fight with you. I think I was starting to feel guilty about piling on so hard, and projected that onto you, for which I apologise.

If someone had merely admitted feeling a bit embarrassed about the word “bra”, I think (I hope) I’d have been a bit more sympathetic. As it was, the complaint about the (more innocuous) word “cleavage”, the melodrama (“now I can never work in that research area” … my life is ruined! Ruined!) and the somewhat insulting reference to sufferers from autism made me a bit more reckless.

I suppose my attempt to make you at least collect some useful technical information on the terms that you are discussing failed, too.

One day …. Thanks for trying.

Posted by: Tim Silverman on December 5, 2009 5:49 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Actually, in this case I do wonder who first came up with the abbreviation for the associative algebra operad, and whether it was offered up as some sort of (puerile, juvenile – you supply the word) joke. How are you supposed to pronounce it? It’s $Ass$ staring you straight in the face. Couldn’t he (I’ll bet it’s a he) have written $Assoc$ (and pronounce it uh-SŌSH)? Or even just $As$?

Since I’m not British, I’m overwhelmingly reminded not of a donkey but of the vulgar term, and I simply can’t bring myself either to say it or write it under anything like professional circumstances, and I find it somewhat irritating every time I see it in mathematical text. Fie on the individual who introduced it.

In the case of the famous mathematician Jacques Tits, it’s of course completely different. It’s a French surname, not something made up.

Posted by: Todd Trimble on December 5, 2009 4:39 PM | Permalink | Reply to this

### Re: Mathematical Emotion

I’ve definitely seen the associative operad written as $As$.

In the case of the famous mathematician Jacques Tits, it’s of course completely different. It’s a French surname, not something made up.

I suppose the same applies to the Adams Spectral Sequence.

Posted by: Mike Shulman on December 5, 2009 5:07 PM | Permalink | PGP Sig | Reply to this

### Re: Mathematical Emotion

Do people abbreviate Adams Spectral Sequence in the way I think you’re suggesting? Seriously?

Posted by: Todd Trimble on December 5, 2009 5:40 PM | Permalink | Reply to this

### Re: Mathematical Emotion

To me, $Ass$ just seems like the obviously correct abbreviation. I'm not sure that I could formalise the rules for such things (although including both of a double consonant seems to be one, since $Comm$ also seems obviously correct), but $Assoc$ seems too long (although reasonable) and $As$ simply wrong (I want to edit the page and correct it).

I can look at that abbreviation and know that it has another meaning (well, two other meanings), but that meaning is not relevant, is it? We're obviously not trying to invoke that metaphor, so I ignore it.

Posted by: Toby Bartels on December 5, 2009 8:38 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Some abbreviations, like this, are probably not really worth the trouble.

I understand the purely linguistic points you’re making, Toby, and in some ways I envy your ability to totally tune out the other associations. But it’s not a pure world we live in, and IMO this particular abbreviation is so bald and in-your-face, it’s asking for trouble, far more trouble than it’s worth in supposed time or space savings, and therefore I think it’s just a dumb or at least extremely ill-advised idea.

(As a thought experiment, suppose the most reasonable abbreviation for some term, from a purely linguistic point of view, were $Boob$. Does that mean we should just go ahead and use it?)

I urge my fellow Lab technicians not to shut their eyes to these aspects. I myself plan on using $Assoc$, for the time being.

Posted by: Todd Trimble on December 5, 2009 11:26 PM | Permalink | Reply to this

### Re: Mathematical Emotion

As a thought experiment, suppose the most reasonable abbreviation for some term, from a purely linguistic point of view, were $Boob$. Does that mean we should just go ahead and use it?

It's hard to be sure without seeing the original; I think that I would rather abbreviate a term that begins with the letters ‘boob’ as $Boo$. But then that might lead to expressions like $Boo B$ or $Boo_b$ (which are both spelt and pronounced like a slang term for breasts); I could certainly see myself using those. Even $Boob$, if it really looks right (as $Ass$ in fact does to me).

Although most mathematics is written in English these days, and abbreviations like $Ass$ are derived from English (although that one could be derived from many other languages), it still feels rather provincial to me to reject notation (as opposed to terminology) becomes it spells out a word in one language. Notation should be universally applicable, and I don't want to worry about what it spells out in English, French, or (for diagrams) Chinese, if none of that is relevant.

How many mathematicians, while capable of reading and writing mathematics in English, wouldn't know about words like ‘ass’ and ‘boob’? I have no good idea about that, but even if they look up $Boo$ when they introduce it, I can hardly expect them to check again when they write $Boo_b$. But maybe there are not very many people in this position.

I myself plan on using $Assoc$, for the time being.

As I wrote, I find that perfectly reasonable. But I would object to editing other people's writings, even this if that's all that you did, just as I refrain from fixing everybody's choices of quotation mark style, spelling, and notation throughout the Lab.

Posted by: Toby Bartels on December 6, 2009 1:22 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Fine as those points are, I’ll just reiterate that insofar as we don’t live in a world of pure logic, $Ass$ IMO seems more trouble than it’s really worth. Same with ‘bra’ actually; it’s also slightly silly just from a linguistic standpoint, I feel. ‘Cleavage’ I’m less exercised about.

But I would object to editing other people’s writings, even this if that’s all that you did, just as I refrain from fixing everybody’s choices of quotation mark style, spelling, and notation throughout the Lab.

I agree, and if I’m not mistaken, I’ve also honored that principle pretty consistently.

Posted by: Todd Trimble on December 6, 2009 2:35 AM | Permalink | Reply to this

### Re: Mathematical Emotion

I’ve also honored that principle pretty consistently.

Yes, and I don't mean to imply that you haven't.

Posted by: Toby Bartels on December 6, 2009 3:03 AM | Permalink | Reply to this

### Re: Mathematical Emotion

I see we have it as an undefined term at small fibration on nLab.

Posted by: David Corfield on November 30, 2009 11:52 AM | Permalink | Reply to this

### Re: Mathematical Emotion

I don’t know if this makes me sound absurdly innocent or insensitive or both, but I can honestly say that I’ve never (before this discussion) made any mental connection between “cleavage” in the mathematical sense here and a certain visual effect associated with plunging necklines. The way the word is used mathematically is from the beginning (cloven fibrations, etc.) strongly associated with splittings and that’s the sense it’s meant to convey. I wonder if I’m alone in never before making the connection with what you (anon) find offensive.

The only occasion where I’ve made any mental connection between the two senses of ‘bra’ is when I was constructing a crossword, and entered as my clue for ‘ket’: Accessory for Dirac’s bra? It was meant in a lightly humorous vein and certainly not meant to be against women.

All that said, it’s perhaps good that you bring all this out. I probably wouldn’t campaign myself against ‘bra’ since it’s not a piece of terminology I’m in the habit of using, although ‘brac’ would probably be a reasonable substitute. As for ‘cleavage’, I think ‘cleaving’ might work instead, unless that’s taken for some other technical notion. (‘Splitting’ wouldn’t work since that’s already taken, as a very strict form of ‘cleavage’.) Any other suggestions?

Posted by: Todd Trimble on November 30, 2009 1:49 PM | Permalink | Reply to this

### Re: Mathematical Emotion

I have always assumed that both the mathematical sense of “cleavage” Todd cites above, and the sense associated with plunging necklines, were independently borrowed from the presumably much older geological term.

I would speculate, based on anon’s comments, that he or she has heard someone somewhere make a joke about the vertical line between a bra and a ket being called cleavage. I’ve never heard such a joke myself, and it appears that most of the others here haven’t either.

Posted by: Mark Meckes on November 30, 2009 4:22 PM | Permalink | Reply to this

### Re: Mathematical Emotion

I would speculate…

That would make sense. I wondered why an obscure term from the obscure theory of fibrations was causing such offence.

If you’re correct, then the offensive term is not being used in a technical sense, and is presumably mentioned to bolster the flagging attention of the male physics undergraduate. I had an English teacher at secondary school who used a similar technique to interest us in the use of the semicolon. All thoroughly reprehensible.

The next step for our clean up mathematical language campaign is to take on moral graph with its conservative view of unmarried parents as immoral.

Posted by: David Corfield on December 1, 2009 9:15 AM | Permalink | Reply to this

### Re: Mathematical Emotion

This is taking us away from the main point of the thread, but I can’t resist asking: Surely, some serious philosopher has thought through the questions surrounding these sensibilities?

Regarding the suitably sensitive use of language, it’s hard not to receive the impression that many people simply fit into the conventions of their peer groups through some subconscious process of trial and error. But then, the occasional arbitrariness there causes some others to refer sarcastically to ‘political correctness.’ Just to give a concrete example, in the liberal North American environment where I spent many years, pejorative comments touching on race were generally frowned upon, but it seemed perfectly respectable to speak of ‘rednecks.’ Even regarding race, it became obvious after a while that there was some kind of a heierachy in the depth of the frown, depending on the race referred to. Specifically in relation to this discussion, it’s never been clear to me why the mistaken conception of ‘schizophrenia’ tends to cause such an outcry, that is, among all the instances where scientific usage is at variance with the vulgar. (I won’t give a specific example X here since I’d rather not spend energy on a discussion of X vs. schizophrenia.)

Perhaps the analysis of such issues goes beyond philosophy, but I wondered where one might find some sensible literature. Martha Nussbaum, for example, is probably a prototypical philosopher who concerns herself with such matters, but I’ve found what I’ve read rather unsatisfactory.

Posted by: Minhyong Kim on December 1, 2009 10:02 AM | Permalink | Reply to this

### Re: Mathematical Emotion

There’s plenty of work by feminist philosophers – perhaps here is a good place to start.

Posted by: David Corfield on December 1, 2009 10:31 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Sorry if this is off topic (and the true story is a bit more complicated, but allow me to simplify):
German did not have a gender neutral singular word for “parent”, only the plural (“Eltern”). This lead people to talk about father and child (“Vater und Kind”) when they talked about nodes in a graph, instead of parent and child. Since the legislator noticed this problem before the mathematicians did, they invented a word that was intended to be used analog to “parent” (“Elternteil”, don’t get confused: the noun is male, the meaning is neuter).
But since this was an expression that peoply only knew from texts of law, some mathematicians invented their own by deleting the ‘n’ in “Eltern” (appending a ‘n’ after a consonant is a pattern to get plural from singular, so this seemed only logical) and talk about “Elter und Kind”.
The word Elter did not make it to everyday speech (yet), so the only ones who know about this are probably people who took a course on graph theory.

Posted by: Tim vB on December 1, 2009 1:32 PM | Permalink | Reply to this

### Re: Mathematical Emotion

That page has some interesting remarks, but I’m afraid it is generally of the sort that’s become somewhat predictable at this stage. I suppose I was wondering about assessments that attempt to be more comprehensive and detached. Something that could have a grand title like:

‘Foundations of normative ethics for social discourse’

but with a rather down-to-earth discussion. Oh well.

Posted by: Minhyong Kim on December 2, 2009 9:25 AM | Permalink | Reply to this

### Re: Mathematical Emotion

I’ll just point out that it’s a bit more complicated than your text suggests. Part of the issue with the colloquial usage of the term schizophrenia (and to some extent “bipolar”) is that, whilst it’s different from the medical meaning of the term, most people think it’s got the same (possibly simplified) meaning as the medical term. So a big part of the issue is a desire to avoid boosting an incorrect colloquial meaning by attaching it to more technical terms. Its a bit like the way most people actually use “infinite” to mean “a really, really, really large amount”: would you be happy with a renaming of a “googol” to “first iconic infinite number” by someone drawing from colloquial usage of “infinite”? (It’s not a great analogy because if you question people about their usage of infinite, eg, statements like “there’s an infinite amounts of solar energy available” they will admit that they do know they’re using the term wrongly.)

This is slightly different from using terms that could be seen as pejorative or off-putting to certain groups, and the issue of whether people are being oversensitive “on behalf of” others.

Posted by: bane on December 1, 2009 1:41 PM | Permalink | Reply to this

### Re: Mathematical Emotion

As editor (formal and informal),I’ve been waging a battle against the use of so-called’ since it can be used in a slightly pejorative sense.

Posted by: jim stasheff on December 1, 2009 2:06 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Well, this is where a difference of perspectives might lie. If I were to go around correcting usages of ‘infinite’ too often (around friends of my wife, for example), I would be regarded as something of a bore. I’d also rather not discuss the simplistic stereotypes that come up in media-mathematics unless specifically asked about them.

So when people do put in a lot of energy towards correcting other’s usage, I presume it’s motivated by some sense of its social importance. But then, the question of a reasonable balance obviously comes up, since we would like to think that what is important has some meaning that transcends power relations between different sectors. It would be nice to understand the foundations of such meaning.

Posted by: Minhyong Kim on December 1, 2009 2:10 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Minhyong wrote:

Regarding the suitably sensitive use of language, it’s hard not to receive the impression that many people simply fit into the conventions of their peer groups through some subconscious process of trial and error. But then, the occasional arbitrariness there causes some others to refer sarcastically to ‘political correctness.’ Just to give a concrete example, in the liberal North American environment where I spent many years, pejorative comments touching on race were generally frowned upon, but it seemed perfectly respectable to speak of ‘rednecks.’ Even regarding race, it became obvious after a while that there was some kind of a hierachy in the depth of the frown, depending on the race referred to.

[…]

Perhaps the analysis of such issues goes beyond philosophy, but I wondered where one might find some sensible literature.

Since apparently we can’t just pick up a copy of Foundations of Normative Ethics for Social Discourse, maybe someone could mention some examples of these phenomena from cultures rather different than the liberal provinces of North America (which I know all too well, having spent most of my life there).

I think it’s useful to see examples from cultures far removed from ones own, since we’re all too emotionally caught up in the battles of our own culture to examine them dispassionately.

Indeed, maybe that’s one reason your imagined tome hasn’t been written yet!

Posted by: John Baez on December 3, 2009 5:32 PM | Permalink | Reply to this

### Re: Mathematical Emotion

If you’re correct, then the offensive term is not being used in a technical sense, and is presumably mentioned to bolster the flagging attention of the male physics undergraduate. I had an English teacher at secondary school who used a similar technique to interest us in the use of the semicolon. All thoroughly reprehensible.

I’d caution you not to discount the pedagogical value inherent in this sort of trick.

When I took my first calculus class, of course we learned the quotient rule for taking derivatives, where – unlike the product rule – the order of the terms matters. It must be “VEE DEE you minus YOU DEE vee” in the numerator, not the other way around.

As a sort of mnemonic, our instructor made a show of cautiously walking over to the door and checking to make sure nobody was listening in the hall. He turned and in an exaggerated stage whisper advised us: “Venereal Disease is an Ugly Disease”. V-D-U-D.

Prurient, maybe. But none of us forgot the quotient rule. To this day I can’t use it without remembering the scene.

Posted by: John Armstrong on December 1, 2009 4:45 PM | Permalink | Reply to this

### Re: Mathematical Emotion

A cleavage could naturally be called a “pseudo-splitting,” since there is a (colax-idempotent) 2-monad whose strict algebras are split fibrations and whose pseudo-algebras are cloven fibrations.

Posted by: Mike Shulman on November 30, 2009 7:36 PM | Permalink | PGP Sig | Reply to this

### Re: Mathematical Emotion

That’s actually a good idea, Mike. I’d like to hear from other Lab technicians as well.

Posted by: Todd Trimble on November 30, 2009 8:21 PM | Permalink | Reply to this

### Re: Mathematical Emotion

cloven? not cleft?

Posted by: jim stasheff on December 1, 2009 2:08 PM | Permalink | Reply to this

### Re: Mathematical Emotion

I believe that’s the standard terminology, yes. The word puts me in mind of the dietary prescriptions found in Leviticus, but hey, I’m just the messenger here.

Posted by: Todd Trimble on December 1, 2009 3:35 PM | Permalink | Reply to this

### Cleave Unto Duality; Re: Mathematical Emotion

My problem with “cleavage” is that “cleave” is self-dual.

cleave
Pronunciation: \ˈklēv\
Function: intransitive verb
Inflected Form(s): cleaved \ˈklēvd\ or clove \ˈklōv\ also clave \ˈklāv\; cleaved; cleav·ing
Etymology: Middle English clevien, from Old English clifian; akin to Old High German kleben to stick
Date: before 12th century
: to adhere firmly and closely or loyally and unwaveringly
Merriam-Webster Online Dictionary. 2009.

So “to cleave” is for two things to stick together. Yet one can “cleave” one thing apart into two objects, the gap being the “cleavage.”

Wikipedia:Cleavage in general refers to a division or separation of form. Its usage is heavily dependent on cultural context.
* Partial exposure of part of the body:
o Cleavage (breasts), partial exposure of the separation between a woman’s breasts.
+ Cleavage enhancement, methods of making a person’s breast cleavage look more substantial than it really is.
o Buttock cleavage
o Toe cleavage
* Cleavage (crystal), in mineralogy and materials science, is a process of splitting a single crystal.
* Cleavage (geology), Foliation perpendicular to stress as a result of ductile deformation. e.g. shales (shists)
* Cleavage (embryo), in embryology, is the division of cells in the early embryo.
* Cleavage (fiber), in optical fibers.
* Teeth cleavage or tooth cleavage, slang for Diastema (dentistry), the gap between a person’s two front teeth.
* In cell biology, the Cleavage furrow is the indentation that begins the process of cleavage, by which animal cells undergo cytokinesis.
* Bond cleavage in chemistry and biochemistry
* Cleavage (politics) The divisions of society that cause people to vote differently

Posted by: Jonathan Vos Post on December 2, 2009 11:42 PM | Permalink | Reply to this

### Re: Cleave Unto Duality; Re: Mathematical Emotion

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

Posted by: John Baez on December 5, 2009 4:18 AM | Permalink | Reply to this

### Re: Cleave Unto Duality; Re: Mathematical Emotion

And I’m going to think about politics. Now that’s grossly unfair!

Posted by: Mark Meckes on December 5, 2009 5:01 PM | Permalink | Reply to this

### Re: Mathematical Emotion

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

Maybe you think anon was trolling. Maybe you think anon is a man pretending to be a woman. If so, the correct response is none: don’t feed the troll.

But suppose that anon is a genuine person writing sincerely. Then the situation is this. A woman leaves a comment here saying that certain mathematical terminology makes her uncomfortable, for reasons to do with sex and gender. She writes anonymously because she suspects her feelings will be mocked. What happens? Male mathematicians line up to heap ridicule on her.

What makes me even more depressed is that it confirms part of anon’s point: that mathematicians can be socially insensitive. Telling a woman who feels uncomfortable with the word “cleavage” that she shouldn’t because it has some technical meaning in mineralogy is unlikely to make her feel any less uncomfortable.

Maybe you think that because anon used “autistic” in an objectionable way (and didn’t apologize), the gloves are off. In that case I’m disappointed. Usually the conversation here is respectful, even in the face of provocation.

Personally I don’t care one way or the other about “cleavage”. But I do care about how people are treated.

Posted by: Tom Leinster on December 5, 2009 9:40 PM | Permalink | Reply to this

### Re: Mathematical Emotion

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.

Posted by: Tim Silverman on December 5, 2009 10:06 PM | Permalink | Reply to this

### Re: Mathematical Emotion

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.

Posted by: Toby Bartels on December 5, 2009 10:51 PM | Permalink | Reply to this

### Re: Mathematical Emotion

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.

Posted by: Mark Meckes on December 5, 2009 11:56 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Yes, that may have been an overly harsh way to put it.

Posted by: Tom Leinster on December 6, 2009 12:55 AM | Permalink | Reply to this

### Re: Mathematical Emotion

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.

Posted by: Todd Trimble on December 6, 2009 12:01 AM | Permalink | Reply to this

### Re: Mathematical Emotion

And ‘brac’ now strikes me as better or at least less needlessly provocative in hindsight, whatever Dirac’s reasons might have been.

Am I the only person who’s always been bothered by ‘bra-ket’ simply because the ‘c’ has disappeared for no apparent reason?

Posted by: Mark Meckes on December 6, 2009 12:52 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Evidently not! That was part of one of anon’s comments.

There are lots of terminological discussions at the nLab, and we might look at this further some time.

Posted by: Todd Trimble on December 6, 2009 2:16 AM | Permalink | Reply to this

### Re: Mathematical Emotion

it’s quite likely that anon is a woman

Anon has claimed to be a woman, in the post that I found simply unbelievable.

Posted by: Toby Bartels on December 6, 2009 1:18 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Ah yes, you’re right; thanks. And obviously there is no reason to call such a claim into question.

Posted by: Todd Trimble on December 6, 2009 2:19 AM | Permalink | Reply to this

### Re: Mathematical Emotion

obviously there is no reason to call such a claim into question.

You bet there is!

When I see an anonymous person on the Internet claiming to be a member of a minority group (which women are, in academic mathematics) and exhibiting exaggerated negative stereotypes of that group (in this case, a delicacy so great that one is unable to enter the field of quantum mechanics for fear of hearing a bad word), then I question their sincerity.

It was wrong of me to say that I know anything for a fact, but it would surprise me to learn that anon really is a woman who was serious in her claims. Do you know any woman (or any person at all) as delicate as that? I don't.

Posted by: Toby Bartels on December 6, 2009 3:20 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Do you know any woman (or any person at all) as delicate as that? I don’t.

Interesting, but yes, I believe I do know people whose behavior or reactions align with what I gather so far from anon.

I don’t know I’d use the word ‘delicate’. It’s possible that anon has overstated her reactions. Another possibility is that sexist talk fills her with such rage that she doesn’t know quite how to deal with it, at least not outwardly in a public setting like a conference or seminar, and instead of exploding she shuts down and fumes in silence. These are just two of many possibilities, which I’d prefer not speculating on much further, as the data we have is pretty thin, and she seems to have withdrawn herself from the discussion.

In any event, what she has said doesn’t seem out of the realm of possibility to me, so my own instinct is to take her at her word and respond accordingly.

Posted by: Todd Trimble on December 6, 2009 1:16 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Todd said:

‘Pseudo-splitting’ seems to me to be worth considering for use in the nLab.

Having publicly lambasted the unfortunate anon for this1, I now feel honour-bound for the sake of equity to say out loud that I am very sceptical about the value of fiddling with terminology in this way. In fact, I think it’s a waste of time and effort—even while sympathising at a purely personal level with Todd’s and anon’s embarrassment.

On the other hand, I don’t really feel I have any right to tell people how to spend their time on their own projects; particularly since, not too long ago, I kind of publicly counted myself out of the nLab for the time being, annoying several people here.

And also, insofar as one of the purposes of the nLab is to rebuild mathematics on an $\infty$-categorical basis, it makes sense to rebuild the terminology at the same time, and while doing that, one might as well remove a few gratuitous personal annoyances as well.

1Although, while agonising over my behaviour on this thread last night, it did become apparent to me that anon had managed to offend me in a really astonishingly large number of separate ways, which Todd (and the other participants) certainly haven’t. However, that is now behind me.

Posted by: Tim Silverman on December 6, 2009 10:59 PM | Permalink | Reply to this

### Re: Mathematical Emotion

I actually consider myself one of the more conservative voices when it comes to fiddling with terminology, and we do have a certain amount of experience at the nLab in handling this sort of thing. Rest assured that whatever happens, the matter will be in good hands. :-)

BTW: you can always change your mind about the nLab. All past annoyances will be forgotten! :-)

Posted by: Todd Trimble on December 6, 2009 11:31 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Todd came back with:

I actually consider myself one of the more conservative voices when it comes to fiddling with terminology, and we do have a certain amount of experience at the nLab in handling this sort of thing. Rest assured that whatever happens, the matter will be in good hands. :-)

Well, even without your assurance, I’m not sitting up worrying that you’ll (collectively) produce something actively bad. For the most part, changing terminology is at worst an annoying inconvenience, and it’s not something I’d normally mention, only … the circumstances seemed to demand it.

you can always change your mind about the nLab

The matter is, as they say, under constant review.

Posted by: Tim Silverman on December 7, 2009 1:23 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Todd Trimble wrote:

‘Pseudo-splitting’ seems to me to be worth considering for use in the nLab.

What about just a ‘cleaving’? I’m not in general a fan of changes of terminology — it’s tough enough already to keep track of the terminology of weak higher — but this seems a rare case where there’s an almost-synonym which (a) is immediately grokkable to anyone who knows the original, (b) isn’t already in use for something else, (c) is in analogy with other usage in the area (split $\rightarrow$ splitting, cleave $\rightarrow$ cleaving), and (d) completely avoids the bad connotations of the original.

(I don’t personally have a big problem with the original — I rolled my eyes when I first heard it, but it doesn’t bother me now — but I can understand why others might.)

Posted by: Peter LeFanu Lumsdaine on December 6, 2009 11:42 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Right, I had suggested the same thing back here (not that I expect everyone to have kept up with this enormous and rambling conversation!). It’s another very reasonable possibility.

Posted by: Todd Trimble on December 7, 2009 12:23 AM | Permalink | Reply to this

### Re: Mathematical Emotion

If we do want to make a change, then I’m pretty indifferent between “cleaving” and “pseudo-splitting.” The former has the advantage of being closer to the existing terminology “cleavage,” but the latter has the advantage that its meaning is (or might be) more obvious to a newcomer.

I’m also pretty indifferent about whether to make a change. I think it’s a bit oversensitive to be offended by “cleavage,” but as I have my own points of oversensitivity I can try to respect other people’s, and I’m in favor of not offending people unnecessarily. But mostly I don’t think the notion is important enough to expend a lot of thought on. The whole idea that a “cloven” fibration is any different in practice from a “non-cloven” one stems, I think, from a misconception about the role of the axiom of choice in category theory, and disappears entirely if you adhere consistently to the use of anafunctors. In other words, every fibration is automatically and canonically “ana-cloven”.

Posted by: Mike Shulman on December 7, 2009 1:44 AM | Permalink | PGP Sig | Reply to this

### Re: Mathematical Emotion

The whole idea that a “cloven” fibration is any different in practice from a “non-cloven” one stems, I think, from a misconception about the role of the axiom of choice in category theory, and disappears entirely if you adhere consistently to the use of anafunctors. In other words, every fibration is automatically and canonically “ana-cloven”.

Ah, no wonder I couldn't tell what the point was!

Any chance that the really correct term is just ‘splitting’, and that we ought to call the usual notion of splitting ‘strict’?

Posted by: Toby Bartels on December 7, 2009 9:03 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Any chance that the really correct term is just ‘splitting’, and that we ought to call the usual notion of splitting ‘strict’?

I think that since “split” is always used to mean the strict notion, using it for the pseudo notion would create more confusion than it would be worth.

Posted by: Mike Shulman on December 7, 2009 9:26 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Like with 2-limit and functor?

Posted by: Toby Bartels on December 7, 2009 9:34 PM | Permalink | Reply to this

### Re: Mathematical Emotion

I think the difference there is that functors and limits for 2-categories are really very important things, so it’s worth a little bit of confusion to get a good terminology for the important notions (e.g. not having to carry around “weak” or “bi-“). I don’t view “cleavings” or “splittings” as of comparable fundamental importance.

Posted by: Mike Shulman on December 8, 2009 12:00 AM | Permalink | Reply to this

### Re: Mathematical Emotion

All right, I think that answers my question.

Posted by: Toby Bartels on December 8, 2009 1:46 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Wow, this is incredible. Anon says they have problems with certain language that appears in mathematics and physics, and 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, finally culminating in the ridiculous complaint by someone that they had actually been insulted by anon. 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.

Posted by: Richard on December 6, 2009 6:09 AM | Permalink | Reply to this

### Re: Mathematical Emotion

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.

Posted by: John Armstrong on December 6, 2009 7:36 AM | Permalink | Reply to this

### Re: Mathematical Emotion

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.’

Posted by: Minhyong Kim on December 6, 2009 11:48 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Allow me to rephrase that last sentence:

Your presumably noble intentions become even harder to sympathize with if you lay down a series of rather aggressive sentences that conclude with the oxymoronic combination

“Get over it…respect others.”

Posted by: Minhyong Kim on December 6, 2009 12:29 PM | Permalink | Reply to this

### Re: Mathematical Emotion

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?

Posted by: Todd Trimble on December 6, 2009 2:20 PM | Permalink | Reply to this

### Re: Mathematical Emotion

As a white male living in Europe I was never part of a discriminated minority - which makes it very hard for me to understand the sensitivity that people have who do make this experience.
A few years ago I became friends with someone from Tansania (native African, i.e. black), who went to Germany to study agriculture.
As some of you may know, racism is despised in Germany, but that does not mean it does not exist! But it is expressed in very subtle ways.
Example: After he finished a lecture during which he had the room dimmed, his professor told him “Well, first thing: turn on the lights, we can’t see you”.
(Implying that you could very well discern the white people in the room).
He was very upset about this, plus he could not talk about it to anyone (well, he talked to me, but - as I said before - it took me quite some time to understand him).

That’s why I took anon’s comments seriously.

Posted by: Tim vB on December 8, 2009 2:01 AM | Permalink | Reply to this

### Re: Mathematical Emotion

I can’t seem to keep my mouth shut on this topic even after several attempts, so I guess I do consider it rather important. Speaking as a minority of sorts, the Confucian in me wished to register another comment for the youngsters out there, even if it seems not quite fitting to this forum.

I’ve lived now 24 years away from my home country. During that period, I’ve lived years in the US and UK, and spent at least several months each in France, Japan, Germany, and India. I can say with complete honesty that I’ve never once had a personal encounter with anything that could have been construed as racism. Perhaps this is one reason I get somewhat involved in discussions of these topics. The only racism I’ve definitely witnessed in my life was in fact in the Korea of my youth, when people had very little contact with foreigners. It was unfortunately usual then to hear crude jokes about the appearance and mannerisms of passing Caucasians.

As a result, I didn’t have much occasion to think about these issues deeply until I had children. Then they were somewhat forced upon me, if only because the kids had to discuss such matters at school, both in the US and in the UK. (As far as I know, they’ve likewise never encountered any racism directed against them.)

On the whole, I’ve discouraged them from thinking about such things, except as required by the surrounding institutions. (‘Discouraged’ is maybe too strong a word. It’s just a non-issue in my household.) That is, general categories like ‘kindness’ and ‘meanness’ seem to me much more useful to them for the time being than the various causes and ‘isms.’

Perhaps others find the comparison absurd, but I did mean it rather seriously when reminded Toby of our uncertainty even with physical cateogries like cats or mathematical ones like groups. Even though I joked around about it, of course I understand that the meanings of those terms need sometimes to be examined with detachment. How then can I get worked up about the reality of ‘racism’?

Posted by: Minhyong Kim on December 8, 2009 12:33 PM | Permalink | Reply to this

### Re: Mathematical Emotion

An interesting way to get a more precise idea of one’s detachment-levels.

Posted by: Thomas on December 8, 2009 2:54 PM | Permalink | Reply to this

### Re: Mathematical Emotion

detachment-levels

Now that is interesting! I want to say that everybody should go take those tests, but in fact it is rather depressing to do so.

Posted by: Toby Bartels on December 9, 2009 9:39 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Toby commented:

I want to say that everybody should go take those tests

I tried to take one of them, but none of the questions made any sense to me, so I gave up. They seem to be saturated with profoundly false presuppositions.

Posted by: Tim Silverman on December 9, 2009 2:11 PM | Permalink | Reply to this

### Re: Mathematical Emotion

none of the questions made any sense to me

Each of their tests come in three sections (in random order, it seems). One asks demographic information, and another asks for your concious opinions about things; in both cases, some of them are laughably simplisitic, and I'm pretty sure that you can just leave them blank. The important part is the automatic sorting task.

If you just do one of those, it shouldn't make you depressed, because you can find many explanations for the outcome, including sheer randomness. It is the consistency of results over many tests that I found depressing. The only explanation that I can find in the end is that garbage has seeped into my brain.

Posted by: Toby Bartels on December 9, 2009 8:10 PM | Permalink | Reply to this

### Re: Mathematical Emotion

I tried taking one of those tests, too (an automatic sorting task), but I made too many errors to produce a result. So instead of being depressed by my latent prejudices I guess I should be depressed about my hand-eye coordination.

Posted by: Mark Meckes on December 9, 2009 3:03 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Example: After he finished a lecture during which he had the room dimmed, his professor told him “Well, first thing: turn on the lights, we can’t see you”.

(Implying that you could very well discern the white people in the room).

Perhaps there is more to this story than you reported, but I don't follow the implication.

What should the professor say if it is too dark to see the lecturer well?

Posted by: Toby Bartels on December 9, 2009 3:21 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Sorry, I’m not very articulate, at least in English.
The point is this: If the lecturer has finished and the discussion with the audience is supposed to start, it’s a harmless remark to say “please turn on the lights”. It’s harmless to me, it makes sense, I would never suspect any other implication than what the words say if you understand them verbatim.

It was not the same to my friend, he instantly suspected some subtle racism, but I don’t know if that’s correct. Why the difference? My explanation is that he had developed some sensitiviy towards remarks like that. Being “non caucasian white” in Germany, you will experience situations that are far more explicit than the one described above.

The analogy to the “bra” story is this: I would dismiss any remarks to undergarment during a QM class as an infantile joke. But to anon both this remark and my dismissal might be very upsetting if she has bad experiences with sexism.

Posted by: Tim vB on December 9, 2009 10:53 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Sorry, I’m not very articulate, at least in English. […]

OK, I understand now, thanks.

Posted by: Toby Bartels on December 9, 2009 12:30 PM | Permalink | Reply to this

### Re: Mathematical Emotion

If there’s anyone else out there still interested in the topic of the original post, I had really hoped to read more responses David’s challenge:

Now who has the best example of a mathematical term for which we can construct a similarly intricate account of its emotional expressiveness?

While I haven’t been ambitious enough to try to match the intricacy of Collingwood’s account for ‘atomic proposition’, I’ve been idly trying to come up with the most evocative names I can think of for ‘well-behaved’ objects. I think two of the best I thought of are ‘amenable group’ (agreeable, cooperative, it won’t cause you trouble) and ‘unconditional basis’ (which to me has echoes of ‘unconditional love’, although the original coinage, via ‘unconditional convergence’, was probably conceived more along the lines of an unconditional warranty on an appliance).

Maybe a better example, for German speakers anyway, is ‘schlicht function’. I’ve seen schlicht translated as any of ‘plain’, ‘simple’, ‘artless’, ‘unostentatious’. What do any of the native German speakers here think about this one?

Posted by: Mark Meckes on December 6, 2009 3:47 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Good idea. I’ve been regretting the way the proposed topic has been skittled by other discussions.

There are a number of resonances underlying the word ‘compact’ (the description under Idea of compact space seems almost tactile), and perhaps even more underlying words like closed, closure, and complete. (These nLab pages really only scratch the surface.) I think all these terms are ripe for analysis.

The word ‘perfect’ is sometimes used in curious ways (perfect space, perfect group, perfect number).

Posted by: Todd Trimble on December 6, 2009 5:21 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Right, this is exactly the terrain that interests me. There’s a strong feeling attached to a word like completion. It taps into all sorts of psychological currents, such as the drive of the collector to complete their set.

As you say, we’ve only scratched the surface. Where to go next then? Is it possible that there could be a unified account of completion?

Posted by: David Corfield on December 7, 2009 10:18 AM | Permalink | Reply to this

### Re: Mathematical Emotion

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

Posted by: Toby Bartels on December 6, 2009 8:26 PM | Permalink | Reply to this

### Re: Mathematical Emotion

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.)

Posted by: Minhyong Kim on December 6, 2009 8:52 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Minhyong,

Apology accepted. I should also apologize for sounding a little heavy handed, but I’ve spent so much writing time on necessarily emotionally constrained mathematics in the last few years, it feels good to let loose sometimes! Anyway, I think David must be quite surprised, amused, and maybe even frightened, about all the emotion he stirred up simply by posting about “Mathematical Emotion.” He started it all; maybe he should summarize all of this for us.

Posted by: Richard on December 7, 2009 4:03 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Summarizing ‘all this’ sounds to me a rather hard task. I certainly never imagined that threads would lead off where they did. Even when the topic shifted out of mathematics, I think some useful lines of argument were sketched, and as you mentioned, plenty of emotion expressed.

Minhyong asked for a ‘Foundations of normative ethics for social discourse’. When I asked around my colleagues for suitable material, all I heard suggested was that one listen to Lenny Bruce “the infamous 1950’s NYC stand-up comic who satirized all those prejudicial terms and expressions”. If I hear of any young philosopher wanting to work on the topic, I’ll steer them to this thread.

As for the original thesis, that even with a technical language, to the extent that it is used properly as a language, it will express emotion, perhaps we might have made a little more progress.

Posted by: David Corfield on December 7, 2009 10:09 AM | Permalink | Reply to this

### Re: Mathematical Emotion

In addition to satire of prejudicial terms, we’ve seen plenty of normative attempts being satirized as well, which can be equally amusing or irritating, depending on where you stand. I suppose a sufficiently clever commentator might manage to put together an interesting way to recursively satirize the satire itself, and so on. Something I find rather boring about certain kinds of comedy one finds on TV, say of the Daily Show variety, is how they stop pretty much at level zero. There is the added question of the prejudices that go into the selection of prejudices to satirize, and ad infinitum. (I hear a million voices chiding me: what do you expect from TV?)

In short, I somehow doubt there is more than very limited insight to be gleaned from a study of Lenny Bruce :=)(=:

Posted by: Minhyong Kim on December 7, 2009 10:47 AM | Permalink | Reply to this

### Re: Mathematical Emotion

David writes:

If I hear of any young philosopher wanting to work on the topic, I’ll steer them to this thread.

And don’t forget: the $n$-Category Café is under careful observation by a team of social scientists! I wonder what they think about the reaction to anon’s comments.

Posted by: John Baez on December 7, 2009 9:45 PM | Permalink | Reply to this

### Re: Mathematical Emotion

And don’t forget: the n-Category Café is under careful observation by a team of social scientists!

But for accurate observation, maybe we should forget?

Posted by: Toby Bartels on December 7, 2009 10:08 PM | Permalink | Reply to this

### Re: Mathematical Emotion

Just look into the camera and act natural!

Posted by: John Baez on December 8, 2009 7:19 AM | Permalink | Reply to this

### Re: Mathematical Emotion

Would be interesting if they shared their insights with us, if they think they can take the risk to disturb the object of study.
I’ve already been part of a field study by a student of sociology, she did her master thesis on my collegues and me. (I had thought you had to travel into the wild to do a field study, but as a sociologist you can obviously do that while sitting in an office sipping coffee).

Posted by: Tim vB on December 8, 2009 8:22 AM | Permalink | Reply to this

### Re: Mathematical Emotion

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.

Posted by: Tim vB on December 6, 2009 8:52 PM | Permalink | Reply to this

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