Issue 77 Contents
Proving that women can prove things
You remember Larry Summers, former US Treasury Secretary, latterly president of
Harvard? About three ago, at a conference on diversifying science and
engineering, he made some silly remarks about how women might be disadvantaged
in the top-scientist job market. Silly, not because he said they might be
disadvantaged – of course, in some ways they were and still are – but because he
said the disadvantage might be ‘innate’.
Two responses suggest themselves immediately to this kind of nonsense, one logical and the other sociological. As a mathematician I am more comfortable with the former so that is what I really want to talk about; but I will say what I mean by ‘sociological’ because unfortunately it undermines logic.
The sociological response is to ask why somebody who is obviously so well-versed in so many of the arts: getting your voice heard, getting your own way, getting your own office, getting your own back…, why somebody like that should say something so crass. The question “is A innately better-suited than B (that is hard-wired more suitably) for science?” is not even a meaningful question, because nobody can agree on what ‘suitable’ means, nor even on what ‘science’ means. It is strictly a question for the chattering classes after their second bottle of plonk, not for Harvard presidents or Secretaries of State, or World Bank Chief Economists (yes, he was that too).
So why did Larry? My guess is, because the prejudice that science and mathematics are a boy-thing is in the very air we all breathe: it’s out there waiting to creep into the pontificatings of anyone who has not suffered the ill-effects of the prejudice or thought long and hard about what can be done about it. It is a miasma, waiting to infect your half-baked ideas and cause them to cough up nasty, petty-minded, pronouncements. If you look for this miasma, you will see it everywhere. An example: try and buy a calendar about mathematics at the website calendarclub. co.uk. You will find it, “The Mathematics Calendar 2008”, right below “Great Big Diggers”. It is under the category “For Dads”.
Logic is powerless against the sociological fact of mass mathematical misogyny. But, for what it is worth, here is the mathematician’s refutation of Summers’ Conjecture. A conjecture, in mathematical parlance, is what somebody thinks is true but cannot prove irrefutably. It may survive unrefuted for many years (even hundreds of years, like Fermat’s Last Theorem, of which more in a minute) but it remains a conjecture unless and until some clever person produces a logical proof. And if instead that somebody produces so much as a single counterexample, an example for which the conjecture is false, the conjecture dies.
Summers’ Conjecture says, essentially, ‘female wiring makes top-flight science hard for women’. Put like that, logical refutation becomes a stroll: produce somebody who is wired as woman who finds top-flight science easy. She is a counterexample. To be on the safe side I will produce two.
In 1776, a little over a hundred years after the French mathematical genius Pierre de Fermat died, Sophie Germain was born in Paris. You can read about her life among a collection of biographies of women mathematicians compiled by a very different Larry (type ‘Larry Riddle’ into Google and you will findhim and his collection). By the early eighteen hundreds she was manifesting a prodigious talent for mathematics. This must have been hard-wired: any nurture involved was entirely self-nurture; she was strenuously, cruelly even, discouraged by her parents. Only by pretending to be Monsieur LeBlanc could she gain the attention of the world’s pre-eminent mathematicians, Legendre and Gauss. Through them she announced the first real breakthrough in over a hundred years in the problem that Fermat posed with his notorious ‘Last Theorem’. What is now known as ‘Germain’s Theorem’ involves a tremendous piece of mathematical lateral thinking: don’t think of how to prove the theorem; try to rule out one way of showing it is false. She invented what are still known as ‘Germain prime numbers’ and showed that they split counterexamples to Fermat’s Last Theorem into Case I and Case II. She ruled out Case I for a huge range of numbers.
Germain died of breast cancer in 1831 at the age of fifty-five, just too soon to receive an honorary degree from Göttingen, Gauss’s university. Her theorem is, as far as I can tell, the first major mathematical result by a woman that is recorded by history. Larry Summers could fight a rearguard action, as those do whose conjectures are demolished: he could assert that “women were hard-wired unsuitably for top-flight science until 1776.” Whatever.
My other counterexample is from a very different time and place. Cheryl Praeger was born in a little Queensland town called Toowoomba in 1948. She became one of the world’s top mathematicians in a rapidly moving and crowded field, the theory of permutation groups. In 1983 she proved, with three collaborators, a pivotal result about permutation groups, known at the time (and, by a quirk of fate, still known) as Sims’ Conjecture. The result is very deep and its proof was all the more spectacular because it used something called the Classification of the Finite Simple Groups, which was absolutely at the leading edge of what mathematicians were doing at the time; only a handful of mathematicians would have been up to the technical challenge.
Apart from mathematically, Praeger seems to me to be very different from Germain. She is a mother of two (Germain never married) and is enormously active in the sociological cause of mathematics and women in science. You would speculate that her hard-wiring differs in many ways from Germain’s. But mathematically she exhibits exactly the hardwiring of a top-flight mathematician. The collaboration that proved Sims’ Conjecture was, I am certain (I know Cheryl and one of the other members of that team well enough to feel certain) just the blue-print of a collaboration among top mathematicians. In fact, the rest of the team was male. It is just irrelevant.
Irrelevant except for one thing. By an amazing coincidence, one of the other mathematicians of the team also came from Toowoomba. Praeger and Peter Cameron never met until they were both researchers at Oxford University but, as far as nurture is concerned, they had identical backgrounds in very material ways. As with Germain, we can rule out any special privileges of education or opportunity that might have allowed Praeger to surmount the innate disadvantage of Summers’ Conjecture. She is my second counterexample: a top-flight scientist who happens to be a woman.
I only needed one counterexample to kill Summers (excuse me, mathematicians often drop the ‘conjecture’ for brevity). You only need one Dorothy Hodgkin in chemistry, or one Joan Robinson in economics, or one Judit Polgár in chess. Actually when a mathematical conjecture is crushed overwhelmingly, resourceful mathematicians begin to wonder: “well, perhaps it is essentially never true”. Perhaps, essentially all women are potentially, innately, good at science.
Issue 77 Contents