Lesbianism, logically
One evening in Carvoeiro last week, I was telling one of the other grad students (an American girl called Emily, who's just finished Part III at Cambridge) about a memorable logic exercise I'd done in my first year as an undergraduate. The story goes that lesbianism was never made illegal in Britain (unlike homosexuality between males), because Queen Victoria refused to believe it happened. "Only men admire only women," she apparently declared, and refused to sign the bill.
But here's the thing: she almost certainly didn't mean that. Let's break it down a bit: the sentence "Only men admire only women" means that the only people who admire only women are men. In other words,For every person x, if x admires only women, then x is a man.
Nowhere does this say that women can't admire women. It only says that women can't admire only women. Interpreted properly, Queen Victoria said nothing about the existence of bisexual women (whom the legislation presumably was also intended to criminalise), only about exclusively lesbian women. More formally yet:For every person x, and every person y, if "x admires y" implies that y is a woman, then x is a man.
What she presumably meant to say was "Only men admire women", or For every person x, and every person y, if x admires y and y is a woman, then x is a man.
My Dad always says that wherever possible, one should present one's results in the form of a table, so let's do that now. Queen Victoria was saying the following:
MenWomen
Admires only men??
Admires both men and women??
Admires only women?Don't exist.
Admires neither men nor women?Don't
exist.
The cells containing question marks are where she remained silent: as far as she's concerned there might be people in those cells, or there might not. Note the last line - the statement "If A then B" is false if and only if A is true and B is false, according to the rules of classical logic (and we spent far, far too long arguing about that in philosophy tutorials, so I really don't want to get into it now), so if some person x doesn't admire anybody, then "x admires y" is false for all y, so "if x admires y, then y is a woman" is true for all y, just as if x were an exclusively lesbian woman or an exclusively straight man.
The upshot of all this is that if you want to prove Queen Victoria wrong, it doesn't matter how many bisexual women you can exhibit: you have to show her a woman who's exclusively attracted to other women (or, less satisfyingly but still correctly, to nobody). The thing is, I sometimes wonder if she wasn't actually right: though I've known a few women who described themselves as lesbians, I think they've all turned out on closer examination to admire women more than men, but not to the point of not admiring men at all. "Lesbian" seems to be more of a political statement (either "I'm out and proud", or "I get less hassle from the gay community if I don't call myself bi") than an accurate description of reality1. Of course, I'm extrapolating from a small and biased sample, so firm conclusions should not be drawn.
This may seem an absurdly picky point, but this is the kind of thing mathematicians really need to be careful about. The sentence fragments "for all" and "there exists" are called quantifiers, and we've just seen an example of the kind of problem that can arise when you get them slightly wrong. Hence most universities teach real analysis in the first couple of terms, as a sink-or-swim approach to teaching proper use of quantifiers. Real analysis is the study of when calculus works and when it doesn't, and it's full of sentences likef is continuous if, for all real numbers x and all ε > 0, there exists some δ > 0 such that |f(x) - f(x + h)| < ε for all real numbers h with |h| < δ.
andf is uniformly continuous if, for all ε > 0, there exists some δ > 0 such that, for all real numbers x, |f(x) - f(x + h)| < ε for all real numbers h with |h| < δ.
Without getting into the details, let's just say that confusing continuity with uniform continuity would be a Bad Thing :-)
hat's one good feature of quantifiers: they make it harder to make that kind of mistake. Their really great feature, however, is the way they can allow you to find proofs more easily. Surprisingly often, it's possible to construct a proof of a statement by expressing it formally using quantifiers (which quickly becomes second nature), then unwinding the quantifiers. Every time you see "For all x satisfying foo and spoffle..." you write down "Take an x satisfying foo and spoffle...", and every time you see "there exists a y such that blah...", think "Given the data I have already, is there an obvious choice of y for which blah is true?" If there is, write it down, and if there isn't, write down "Then ___ is such that blah" and come back to it later when you know what should go in the space. In easy (but possibly complex) cases, this procedure can give you a complete proof; in harder cases, it at least gives you a framework for your proof, and prevents you from wasting energy proving the wrong thing.
Anyway, I had to mentally re-run this conversation to check I hadn't said anything too stupid when Emily's rugby-blue girlfriend showed up the next night :-)
1 taimatsu has possibly the best answer to this conundrum. When I asked her about it, she waved her hand dismissively, and said "Pffft. Labels."
But here's the thing: she almost certainly didn't mean that. Let's break it down a bit: the sentence "Only men admire only women" means that the only people who admire only women are men. In other words,For every person x, if x admires only women, then x is a man.
Nowhere does this say that women can't admire women. It only says that women can't admire only women. Interpreted properly, Queen Victoria said nothing about the existence of bisexual women (whom the legislation presumably was also intended to criminalise), only about exclusively lesbian women. More formally yet:For every person x, and every person y, if "x admires y" implies that y is a woman, then x is a man.
What she presumably meant to say was "Only men admire women", or For every person x, and every person y, if x admires y and y is a woman, then x is a man.
My Dad always says that wherever possible, one should present one's results in the form of a table, so let's do that now. Queen Victoria was saying the following:
MenWomen
Admires only men??
Admires both men and women??
Admires only women?Don't exist.
Admires neither men nor women?Don't
exist.
The cells containing question marks are where she remained silent: as far as she's concerned there might be people in those cells, or there might not. Note the last line - the statement "If A then B" is false if and only if A is true and B is false, according to the rules of classical logic (and we spent far, far too long arguing about that in philosophy tutorials, so I really don't want to get into it now), so if some person x doesn't admire anybody, then "x admires y" is false for all y, so "if x admires y, then y is a woman" is true for all y, just as if x were an exclusively lesbian woman or an exclusively straight man.
The upshot of all this is that if you want to prove Queen Victoria wrong, it doesn't matter how many bisexual women you can exhibit: you have to show her a woman who's exclusively attracted to other women (or, less satisfyingly but still correctly, to nobody). The thing is, I sometimes wonder if she wasn't actually right: though I've known a few women who described themselves as lesbians, I think they've all turned out on closer examination to admire women more than men, but not to the point of not admiring men at all. "Lesbian" seems to be more of a political statement (either "I'm out and proud", or "I get less hassle from the gay community if I don't call myself bi") than an accurate description of reality1. Of course, I'm extrapolating from a small and biased sample, so firm conclusions should not be drawn.
This may seem an absurdly picky point, but this is the kind of thing mathematicians really need to be careful about. The sentence fragments "for all" and "there exists" are called quantifiers, and we've just seen an example of the kind of problem that can arise when you get them slightly wrong. Hence most universities teach real analysis in the first couple of terms, as a sink-or-swim approach to teaching proper use of quantifiers. Real analysis is the study of when calculus works and when it doesn't, and it's full of sentences likef is continuous if, for all real numbers x and all ε > 0, there exists some δ > 0 such that |f(x) - f(x + h)| < ε for all real numbers h with |h| < δ.
andf is uniformly continuous if, for all ε > 0, there exists some δ > 0 such that, for all real numbers x, |f(x) - f(x + h)| < ε for all real numbers h with |h| < δ.
Without getting into the details, let's just say that confusing continuity with uniform continuity would be a Bad Thing :-)
hat's one good feature of quantifiers: they make it harder to make that kind of mistake. Their really great feature, however, is the way they can allow you to find proofs more easily. Surprisingly often, it's possible to construct a proof of a statement by expressing it formally using quantifiers (which quickly becomes second nature), then unwinding the quantifiers. Every time you see "For all x satisfying foo and spoffle..." you write down "Take an x satisfying foo and spoffle...", and every time you see "there exists a y such that blah...", think "Given the data I have already, is there an obvious choice of y for which blah is true?" If there is, write it down, and if there isn't, write down "Then ___ is such that blah" and come back to it later when you know what should go in the space. In easy (but possibly complex) cases, this procedure can give you a complete proof; in harder cases, it at least gives you a framework for your proof, and prevents you from wasting energy proving the wrong thing.
Anyway, I had to mentally re-run this conversation to check I hadn't said anything too stupid when Emily's rugby-blue girlfriend showed up the next night :-)
1 taimatsu has possibly the best answer to this conundrum. When I asked her about it, she waved her hand dismissively, and said "Pffft. Labels."