Timeline for Extensional theorems mostly used intensionally
Current License: CC BY-SA 2.5
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 23, 2015 at 3:02 | answer | added | goblin GONE | timeline score: 1 | |
| Jun 28, 2010 at 13:31 | answer | added | Nick | timeline score: 2 | |
| Jun 28, 2010 at 12:09 | answer | added | Russell O'Connor | timeline score: 3 | |
| Jun 10, 2010 at 18:50 | answer | added | T.. | timeline score: 2 | |
| Apr 14, 2010 at 23:59 | answer | added | Jacques Carette | timeline score: 4 | |
| Mar 21, 2010 at 1:02 | answer | added | Terry Tao | timeline score: 17 | |
| Mar 20, 2010 at 18:30 | comment | added | Reid Barton | OK, good! I did not (entirely) mean to be contrary, but neither did I want to "silently assent" to the position in your question, since I assume the actual beliefs of mathematicians are among the things of interest under the category "math-philosophy". | |
| Mar 20, 2010 at 18:20 | comment | added | Jacques Carette | @Reid: That this viewpoint is not unusual amongst mathematicians is a large part of my motivation for posting this question. At least I am in good company in worrying, as many mathematicians, some of great reknown, have written extensively on this point. I could add Church, and Kripke, to that list, and more recently P. Aczel and W. Lawvere. | |
| Mar 20, 2010 at 18:03 | comment | added | Reid Barton | I don't believe in the distinction you draw in your example, and I would be surprised if my viewpoint were an unusual one among mathematicians. The other points discussed seem to center around the fact that mathematical notation is a human and not totally formal activity, which certainly does cause problems for computer algebra software that tries to mimic it. | |
| Mar 20, 2010 at 16:58 | comment | added | Jacques Carette | @Gerald: Your second comment captures exactly what I meant. Mathematicians do this (correctly) instinctively, but when you try to mechanize mathematics, these issues become extremely important, and when misunderstood lead to bad bugs in software. This is the source of many bugs which are unlikely to ever be fixed in either Mathematica or Maple. | |
| Mar 20, 2010 at 16:56 | comment | added | Jacques Carette | @Gerald: good point about polynomials, I have added 'over fields of characteristic 0'. And yes, in algebra, the point that $x^2$ as a polynomial in $(\mathbb{Z}/2)[x]$ is not the same polynomial as $x$ is taught. Though the emphasis (when I was taught this) is usually put on the fact that $\forall x:Z/2. x^2\equiv x (\mod 2)$, which I had to 'unlearn'. | |
| Mar 20, 2010 at 16:53 | answer | added | Noam Zeilberger | timeline score: 5 | |
| Mar 20, 2010 at 16:51 | history | edited | Jacques Carette | CC BY-SA 2.5 |
clarify
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| Mar 20, 2010 at 16:51 | comment | added | Gerald Edgar | How about... It makes sense to say "$\sum_{n=1}^\infty 2^{-n}$ converges", and we say $\sum_{n=1}^\infty 2^{-n}=1$, but we don't say "1 converges". Is this what you mean? – Gerald Edgar 0 secs ago | |
| Mar 20, 2010 at 16:44 | comment | added | Gerald Edgar | Polynomials... interesting. Map $x \mapsto x^2$ on field $Z/2$ is the identity function, but polynomial $x^2 \in (Z/2)[x]$ isn't considered to coincide with polynomial $x$. This, at least, is covered in basic algebra courses. | |
| Mar 20, 2010 at 15:51 | history | asked | Jacques Carette | CC BY-SA 2.5 |