The n-Category Café

This reminded me of a conversation I once had with a chemist who asked me why so much effort was spent on teaching undergraduates useless mathematics, such as calculus and probability theory, instead of teaching the very useful stuff – ie, group theory and algebraic topology!

I was amazed how much mathematics was being consumed by chemists - especially graph theory and algebraic topology.

So where does an algebraic topologist look to get hired into the chemistry biz?

This whole “pure mathematics is useless” thing is largely a pose

I don’t think so. You don’t get people saying “I went into nursing because there’s a small but non-zero probability that at some point, decades or centuries in the future, somebody practising a profession somewhat related to mine might do something of benefit to mankind.” I don’t think that’s how vocations work. Maybe some people feel a vocation to study algebraic topology because they want to be of help to chemists, but not many, surely…. And that’s all that’s needed for Collingwood to make his point.

the true reason why societies tolerate and even pay for mathematicians is that in utilitarian terms the payoffs have been big

Is that the reason the Ptolemies supported the library of Alexandria? Surely societies support all manner of activities that are useless in utilitarian terms. Sometimes they even invent fake reasons why these activities must secretly be useful. (“These jewels are so beautiful that they must have magic powers!” “If you can predict the positions of the planets, you must be able to foretell the fate of kings!” “This economic model is so elegant that it must hold the secret of wealth, health and happiness!”)

I don’t think everybody’s values are purely utilitarian. I’m not sure if that is even philosophically coherent (it makes it seem as though everything derives its value from being useful for something else, but nothing is of value for itself…). Well, I don’t think you meant that exactly—but I do feel that the claim that everything that people support is useful seems like just as much of a pose as that people sometimes do (or support) things because they’re useless.

Indeed. I stumbled over that phrase and had to try reading it again. This does not have the feel of a ‘generic’ “he” but rather seems like an overt statement that pure mathematics, as a useless venture, is only taken up by males.

AK wrote: “Sheesh. Even in 1939 Collingwood should have had enough examples of female mathematicians.”

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I saw a list of famous women mathematicians and there were six, ending with Emmy Noether who died 1935; only three, and in the last two hundred years, were accomplished pure mathematicians. This may be a prejudice but people think of pure mathematicians as exceptional rather than mediocre and that’s the bias I read into Collingwood. When he wrote this piece there was no living famous or highly competent woman pure mathematician.

Likely this had something to do with the prejudice of male dominated institutions of higher learning, but is that the sole cause? Women think differently than men and have different physical brain structure. I don’t think Collingwood meant an average man or group of male mathematicians elected to pursue pure mathematics, so then he wouldn’t mean an average woman or a group of women pure mathematicians. By far, there were many more accomplished male pure mathematicians at the time while women in the same class were rare, and one doesn’t generalize a rule from the exceptional female examples. So perhaps he shared in the common academic male opinion of the potential of women for mathematics which was held in the time he wrote this, but based on the evidence available to him, what he wrote is true if one accepts the premise that he was describing a group that had demonstrated abstract accomplishments (nearly all men) rather than a rare individual female, but certainly not a group of women who had demonstrated accomplishments in the abstract area of pure mathematics.

Collingwood wrote:”to a man or group of men who will undertake it.”

That portion of his remark is found in the sentence which talks about ‘pure mathematicians’. Your criticism uses the
phrase “should have had enough examples of female mathematicians.”

Female mathematicians are not necessarily pure mathematicians. So “enough examples”? There was not one living example of a gifted female pure mathematician at the time Collingwood expressed his opinon. And only 3 such in the two hundred years preceding his statement. That he means he could have made such a statement using “he” based only on the evidence and without any prejudice of political correctness at all. Perhaps your hunch is right, blatant political incorrectness, but that is to be inferred from his era and not from what he wrote. Was he supposed to claim that potentially women should be included without substantial evidence, but on egalitarian grounds?! You are going uphill and against the evidence to assert that the discrepancy of male genius pure mathematicians, there are more than women, is due solely to environmental factors.
http://www.nybooks.com/articles/2176
“Women have proportionately smaller brains than do men, but apparently have the same general intelligence test scores. Thus, I have proposed that the sex difference in brain size relates to those intellectual abilities at which men excel. Women excel in verbal ability, perceptual speed, and motor coordination within personal space: men do better on various spatial tests and on tests of mathematical reasoning. It may require more brain tissue to process spatial information.”

Oddly, I think Collingwood could be read as saying there is a light in which the movement towards an appreciation of ‘social impact’ could be seen as positive, hard though it may be to do so for a directive emanating from the Prince of Darkness.

Can you say of the citizens of Sheffield that they are

…grateful to the pure mathematician for doing his job, and proud of him for being so clever as to be able to do it; not that every one else should rush in to share his life, but that even if his neighbours feel no call to share it they should honour him for living as he does. The fact that they do so honour him is a proof that they want a life of that kind to be lived among them, and feel its achievements as a benefit to themselves.

A conspiracy to queer the data seems to be a delightfully mischievous endeavor. Those of us in the colonies who want to help our colleagues across the pond can simply throw in a few random citations. We could even include a sentence in the introduction, “As is standard within this area, the authors point out that this work has nothing whatsoever to do with the \cite{FavBrit1,FavBrit2}.”

I really enjoyed this paper; Gowers seems so grounded including his choice not to introduce the word “fractal” which might have proved to be a distraction. Thanks!

I did, and was disappointed. They took as their starting point the story of Ramanujan and Hardy which had heaps of dramatic potential, and who were already fascinating characters in their own right, and danced most post-modernly and brazenly about several timeframes trying to dramatise the nature of inquiry itself, with characters it proved hard to care a jot about. Too much spectacle, and way too little of the real meat of drama - exegesis of character and motive.