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Monday, July 10, 2006

Discrimination against -oids

From Not Even Wrong, I see that the Institut Henri Poincare in Paris is holding a 3 month program on Groupoids and Stacks in Physics and Geometry. They have included an interesting overview of the subject to motivate the program. I have a particular soft spot for groupoids having studied the case for their admission into the paradise of mathematics in chapter 9 of my book. Groupoids had a strangely difficult childhood, finding acceptance surprisingly late, the oddest explanation for which is Connes' claim that: is fashionable among mathematicians to despise groupoids and to consider that only groups have authentic mathematical status, probably because of the pejorative suffix 'oid'. (Noncommutative Geometry, 6-7)
Rather than this persecution of suffixes, a more common sentiment is that they are really just dressed up groups. In an old e-mail I have from Saunders Mac Lane he adopts just this line, perhaps surprisingly for the co-inventor of category theory, groupoids being categories with invertible morphisms, and very present in Mac Lane's home of algebraic topology.

Someone who had read my book made the good suggestion that I subject matroids to a similar treatment. Now I hear (penultimate comment by Srandby) that Mac Lane didn't like these either.
Once, Mac Lane came to give a talk. During the talk, in front of a packed audience, he stated that matroid theory wasn’t good or important mathematics, pissing off several faculty who worked in matroid theory. I found this comment to be very bizarre. Here was an advocate of a vast generalization of dubious importance dismissing a generalization of vector spaces that has tremendous importance.
Is there something to Connes' anti-oid theory?

Anyone who wants to take up the challenge of assessing matroids should take a look at Coxeter Theory: The Cognitive Aspects, an article by Alexandre Borovik. In section 13 - Combinatorics as non-parametric mathematics - Borovik claims:
The work of three generations of mathematicians confirmed that matroids, indeed, capture the essence of linear dependence. Since linear dependence is a ubiquitous and really basic concept of mathematics, it is not surprising that the concept of matroid has proven to be one of the most pervasive and versatile in modern combinatorics. (p. 23)
This book with Gelfand should no doubt be consulted too: Coxeter Matroids, Birkhauser, xiv+264 pp., ISBN 0-8176-3764-8 (with I. M. Gelfand and N. White), 2003.

I believe that Borovik will include the Coxeter Theory article in a book to appear with Springer. I read a draft of this book a couple of years back and found it wonderfully rich.


Simon said...

On the "perjorative suffix `oid'": the following is contained in the OED entry for `meretricious'.

1992 N.Y. Times Bk. Rev. 26 Apr. 1/1 Lately..words that end in ‘oid’ have become synonyms for the meretricious: sleazoid, Marxoid, tabloid.

July 11, 2006 2:33 PM  
John Baez said...

Someday the philosophy of real mathematics will include a branch called subaltern studies. As you probably know, Antonio Gramsci used the term "subaltern" to denote any group of people of inferior rank or station, and proposed an approach to history that focuses on these people rather than the powerful elites.

Within mathematics, groups are now part of the elite, while groupoids, monoids, semigroups, quasigroups and loops are subaltern.

Similarly, rings are elite, while rngs, rigs, semirings and near-rings are subaltern. And so on!

Get the pattern? Clustered around each elite concept is a cloud of subaltern ones - "defective" versions of the elite one, each with some axioms removed. James Dolan speaks of centipede mathematics: you take a nice concept and see how many legs you can pull off and have it still walk.

The prefixes "near-" , "semi-", "pseudo-", "quasi-", or the suffix "-oid", are warning signs that you're dealing with one of these subaltern concepts. They're not beautiful enough to love - they just limp along sadly. But they're undeniably part of mathematics, so one must know a bit about them now and then....

Except sometimes one of these subaltern concepts is really something with a beauty all its own: it only looks ugly if you mistake it for a defective version of something else. An ugly duckling that's really a swan! A Cinderella waiting for a prince to bring her diamond slippers! I think monoids and groupoids fall in this category: they're not defective groups; they're specially simple and charming categories!

So, if you discover a concept that seems to be really important, give it a name that reflects its essence. Give it a name that says what it really is! Don't call it a something-else-oid.

July 17, 2006 9:37 AM  
John Baez said...

Matroids may be subaltern now, but J. A. Nieto is proposing them as the key to M-theory! That would sure turn things around. I sort of doubt this view will catch on - but at least it would explain the letter "M".

July 17, 2006 9:49 AM  
david said...

I see Sandra Kingan has a webpage devoted to matroids, which links to James Oxley's interesting What is a matroid?. We learn there that Rota wasn't happy with the name matroid, and so "mounted a campaign to try to change the name to 'geometry', an abbreviation of 'combinatorial geometry'." (Could matroids be useful in our categorified geometry?)

It's interesting to see matroids, like groupoids, prompting such explicit advocacy. Exponents clearly perceive them to be misranked subalterns in need of promotion. I wonder what the best example of pressure for demotion might be.

July 17, 2006 10:39 AM  

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