Timeline for 0 dimensional Dedekind domain?
Current License: CC BY-SA 2.5
14 events
| when toggle format | what | by | license | comment | |
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| Oct 4, 2011 at 12:21 | vote | accept | ashpool | ||
| Jun 20, 2010 at 19:24 | comment | added | Tom Goodwillie | I would not want to encourage indignant questions about why people can't agree on definitions, but I have no objection to getting into the occasional discussion related to extreme cases of a definition. Personally I get both pleasure and insight from looking into those corners of the landscape. | |
| Jun 20, 2010 at 17:29 | comment | added | Pete L. Clark | @Tom: I agree with everything you have said, so you may reasonably conclude that I am not too keen on encouraging such questions. | |
| Jun 20, 2010 at 13:25 | answer | added | Charles Matthews | timeline score: 5 | |
| Jun 20, 2010 at 11:21 | comment | added | Tom Goodwillie | @Pete: Thank you for the information about format. To your other point, I acknowledge that creating a tag for the empty set bordered on frivolity, but I do think that if one wanted a tag for the general topic of "taking trivial cases of definitions seriously" then empty-set is as good a short phrase as any. | |
| Jun 20, 2010 at 7:59 | comment | added | Pete L. Clark | @Prof. Goodwillie: after I removed the new tag "es.empty-set" I looked back and saw that you had added it, not the OP. Let me provide some justification: first, the format for tags with two letters followed by a period followed by a hyphenated explanation is reserved for tags which duplicate the arXiv classification (it is also recommended that every MO question get at least one arXiv tag). Also, if we need a tag for the empty set, I don't think this is the question that shows it, since neither the question nor a good answer need involve $\varnothing$. | |
| Jun 20, 2010 at 7:39 | comment | added | Pete L. Clark | ...Some people (especially Bourbaki) like to phrase definitions so that they do not require ad hoc modifications in trivial cases (for instance, Bourbaki does not explicitly state that the empty set and the whole set are open subsets of a topological space but rather remarks that this follows from their conventions on empty unions and intersections). A lot of other people regard such practices as overly fastidious. So there is never going to be universal agreement on "trivial cases" of definitions. I don't see why this should be bewildering -- in any event, it is a basic fact of life. | |
| Jun 20, 2010 at 7:39 | comment | added | Pete L. Clark | There are a lot of elementary definitions in mathematics for which there is no consensus. This especially happens when the differences in the definition are so small as to make no real difference in the resulting theory: it then becomes mostly a matter of convention and aesthetic... | |
| Jun 20, 2010 at 7:32 | history | edited | Pete L. Clark | CC BY-SA 2.5 |
added 4 characters in body; edited tags
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| Jun 20, 2010 at 5:20 | comment | added | S. Carnahan♦ | Are you looking for a mathematical or a sociological explanation? | |
| Jun 20, 2010 at 4:50 | comment | added | Tom Goodwillie | If you think of a Dedekind domain as a regular (or normal) one-dimensional Noetherian domain, then you know that fields are not allowed. If you think of a Dedekind domain as a domain for which every fractional ideal is invertible, or as a domain in which the monoid of nonzero ideals has a basis, then you know that fields are allowed. Esthetically, both points of view make sense to me. If you adopt the second one then you will have to state trivial exceptions to some theorems. | |
| Jun 20, 2010 at 4:12 | history | edited | Tom Goodwillie |
edited tags
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| Jun 20, 2010 at 3:53 | comment | added | Boyarsky | Perhaps because for any interesting fact about Dedekind domains the potential for confusion is never noticed since whatever is being said is obvious in the field case (or at least very well-known to anyone first learning about Dedekind domains). I doubt at the post-textbook stage of math that anyone would regard fields as Dedekind domains. (I am counting Bourbaki as a textbook, by the way.) A good analogy is whether a differential geometry book implicitly assumes manifolds have positive dimension or not: probably goes either way, depending on the source. | |
| Jun 20, 2010 at 3:30 | history | asked | ashpool | CC BY-SA 2.5 |