This sounds like a funny “blooper” story, but could just as well be an entirely normal history of the solution to an important problem. Many important theorems are proved by contradiction, and for all we know, the question of the existence of partially uniform k-quandles could have been a difficult unsolved problem.
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.
Likewise there’s the story about the Princeton student defending his thesis on the set of real functions that satisfy the Lipschitz condition for every positive constant C, and being asked by an examiner to compute the derivative of such a function...
My point having been, of course, that the k-quandle story is not (necessarily) of this type.
I don’t think you need to do anything as sophisticated as computing the derivative to prove that the only such functions are constant functions. Consider any distinct x_1, x_2. d(x_1, x_2) is nonzero by the definition of metric spaces. If d(f(x_1), f(x_2)) were nonzero, there would be a K small enough for the condition to be violated; therefore it must be zero for all x_1, x_2.
The humor of asking the student to compute the derivative is that one imagines the student confidently starting to answer the question, until a dawning horror rises on the student’s face as the implications of the answer become evident.
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.
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”.
This sounds like a funny “blooper” story, but could just as well be an entirely normal history of the solution to an important problem. Many important theorems are proved by contradiction, and for all we know, the question of the existence of partially uniform k-quandles could have been a difficult unsolved problem.
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.
Likewise there’s the story about the Princeton student defending his thesis on the set of real functions that satisfy the Lipschitz condition for every positive constant C, and being asked by an examiner to compute the derivative of such a function...
My point having been, of course, that the k-quandle story is not (necessarily) of this type.
I don’t think you need to do anything as sophisticated as computing the derivative to prove that the only such functions are constant functions. Consider any distinct x_1, x_2. d(x_1, x_2) is nonzero by the definition of metric spaces. If d(f(x_1), f(x_2)) were nonzero, there would be a K small enough for the condition to be violated; therefore it must be zero for all x_1, x_2.
The humor of asking the student to compute the derivative is that one imagines the student confidently starting to answer the question, until a dawning horror rises on the student’s face as the implications of the answer become evident.
I… don’t mathematicians usually have more than one interesting example of a mathematical object before they decide to study 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.
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.
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”.