Dear Jean, It was clumsy of me to say "The question is bad". Your question in itself is fine. (Just as the question "What is a fish?", in itself, is fine.) The real problem is how the question was understood in the past, and maybe still is. Good mathematicians - such as Adamek - have produced answers to it, and you also asked a well-posed question - "What were their answers?". Jiri said "faithful and amnestic", and from the discussion in Joy of Cats we see that that includes "set of points" functors from categories of topological spaces to Set. I imagine that many of the answers, maybe most, were proposed on the understanding that they should include those topological examples. Answers like that are the ones that I think we should now begin to view as problematical because they "try to include the whales with the fishes". Specifically, it is a problem with trying to understand topological spaces in the point-set way, with faithful functors to Set. Here's the mathematics. We know - (1) If you try to do point-set topology in a topos then in general the topology is impoverished (no Heine-Borel, no Tychonoff). (2) If you do it point-free then you recover those key theorems, (2a) but important spaces such as the reals are liable to be non-spatial - the "set of points" functor is not faithful. Fourman and Grayson gave counterexamples, if I remember correctly. (3) Joyal and Tierney tell us that internal point-free topology in an elementary topos is equivalent to localic bundles over that topos. (4) That gives access to a fibrewise topology of bundles by doing internal point-free topology - which is surely a good thing. (5) Hence issues with point-set topology in toposes have mathematical significance. They are not just curiosities of logic. Does that help? All the best, Steve.
On 14 Feb 2017, at 17:26, Jean Benabou <jean.benabou@wanadoo.fr> wrote:
Dear Steve,
I can understand you don't like my question. Some of the answers I received prove that other mathematicians I respect approve of it, have thought about, and have given partial answers to it. If you have any objections, take a stand and give MATHEMATICAL reasons for your objections.
All you say is;
I'm saying the same can happen in mathematics. Prove that it DID happen in my question!
By the way, I'm no zoologist but, by fishing through your mails, I found a big gap in your refutation. Neither in your mails nor in Fred's did I find any reference to the octopus. If there is any good reason for this omission, mathematical OR zoological, please make it public and justify it.
All the best,
Jean
Le 14 févr. 17 à 09:48, Steve Vickers a écrit :
Dear Fred,
A good answer, but my point was that it was a bad question.
You see this once you start pressing at the details. Are seals and turtles fish? No, but on your definition it depends on whether flippers count as legs or not. What about sea snakes? Obviously not - they're snakes, that just happen to live in the sea. But then eels do seem a bit more fishy.
A meticulous zoologist would start piling on the subclauses to pin it down more precisely, but we know that that does not actually refine our understanding of zoology. It just amplifies the misconceptions underlying the original question.
All the best,
Steve.
On 11 Feb 2017, at 20:42, Fred E.J. Linton <fejlinton@usa.net> wrote:
Steve, et al.,
If you want
a definition of "fish", but on the understanding that it has to include whales
let me offer: "legless marine vertebrates" :-) .
Cheers, -- tlvp
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