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ciphergoth comments on Rationality Quotes January 2010 - Less Wrong
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There is a similar story -- whether true or not I don't know -- told at Oxford about Cambridge and at Cambridge about Oxford. Someone wrote a thesis on anti-metric spaces, which are like metric spaces, except that the triangle inequality is the other way round. He proved all sorts of interesting facts about them, but at the viva, the external examiner pointed out that there are only two anti-metric spaces: the empty set and the one-point set.
It is recounted that the student passed, but his supervisor was criticised for not having picked up on this earlier.
I... don't mathematicians usually have more than one interesting example of a mathematical object before they decide to study it?
Generally yes. But not always. Sometimes there's only a single such object. For example, there's a largest sporadic simple group. It is a very interesting object. But there's only one of it.
To use a slightly less silly example, up to isomorphism there's only one ordered complete archimedean field. We call it R and we care a lot about it.
Also, sometimes you lack enough data to know if there are other examples of what you care about. But yes, you should generally try to figure out if a non-trivial example exists before you start studying it.
Not when the question is whether any examples exist!
OK, but it takes two minutes to prove that an anti-metric space with more than one point can't exist. If x != y, then d(x, y) + d(y, x) > d(x, x).
Unless you allow negative distances, in which case an anti-metric space is just a mirror image of a metric space.
Non-Euclidean geometries? IIRC the questions of "what can you still/now prove with this one postulate removed" were studied for centuries before hyperbolic or elliptic geometries were really understood.
Or maybe I'm misremembering. That always did seem odd to me. I guess hyperbolic geometries can't be isometrically embedded in R^3, which makes them hard to intuitively comprehend. But the educated classes have known the Earth was a sphere for millennia; surely somebody noticed that this was an example of an otherwise well-behaved geometry where straight lines always intersect.
The fact that they didn't notice that Earth is an example of a non-Euclidean geometry is especially ironic when you consider the etymology of "geometry".