Timeline for History of the triangle inequality
Current License: CC BY-SA 3.0
22 events
| when toggle format | what | by | license | comment | |
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Aug 9, 2011 at 16:33 | vote | accept | Suvrit | ||
| Aug 9, 2011 at 12:24 | comment | added | Tom Leinster | @Suvrit: my main point was that he apparently felt fine about using the term "distance function" in the absence of the triangle inequality, which I'm not sure anyone would do now. The values of his distance functions really are to be thought of as distances, not squares of distances (e.g. when he comes to consider embeddability into Euclidean space). @Todd: possibly, though I don't think he mentions it. | |
| Aug 9, 2011 at 7:57 | answer | added | Dick Palais | timeline score: 2 | |
| Aug 9, 2011 at 4:24 | answer | added | Qiaochu Yuan | timeline score: 7 | |
| Aug 9, 2011 at 1:14 | answer | added | Igor Rivin | timeline score: 3 | |
| Aug 9, 2011 at 0:53 | comment | added | Todd Trimble | @Suvrit: I now see your response to Qiaochu, which appeared while I was composing my last comment. | |
| Aug 9, 2011 at 0:51 | history | edited | Suvrit | CC BY-SA 3.0 |
some clarification...
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| Aug 9, 2011 at 0:51 | comment | added | Todd Trimble | @Tom: could Schoenberg have been contemplating Lorentzian distances as well? @Suvrit: unfortunately, I'm not crystal clear yet on the question either. The algebraic formula for Euclidean distance was, I suppose, written down by Descartes (I'm looking now at an edition of La Geometrie, published by Hermann: books.google.com/…, page 2), and the triangle inequality for Euclidean distance is given as Proposition 20 in Euclid's Elements, which surely Descartes knew. John Stillwell should be asked... | |
| Aug 9, 2011 at 0:48 | comment | added | Suvrit | Maybe it helps, if someone could just tell me: when was it realized that to formalize the notion of a distance, we need to have the triangle inequality as an axiom? | |
| Aug 9, 2011 at 0:44 | comment | added | Suvrit | @Tom: Wasn't Schoenberg actually talking about cpd kernels, or essentially, squares of distance functions? | |
| Aug 9, 2011 at 0:43 | comment | added | Suvrit | @Qiaochu: I mean $d(a,b) \le d(a,c)+d(b,c)$ for some distance function $d$ (not necessarily $R^n$) | |
| Aug 9, 2011 at 0:30 | comment | added | Tom Leinster | Earlier today (mathoverflow.net/questions/72356) I cited a 1938 paper of Schoenberg that centres on spaces equipped with a "distance function" $d$ that is symmetric and satisfies $d(x, x) = 0$, but doesn't necessarily satisfy the triangle inequality. (Nor does it satisfy the separation axiom that $d(x, y) = 0$ implies $x = y$, but that's much less important.) Schoenberg knew Fréchet's definition of metric space, and cites it, but it's instructive to note that Schoenberg was content to talk about "distances" that don't satisfy the triangle inequality. | |
| Aug 9, 2011 at 0:27 | comment | added | Qiaochu Yuan | @Suvrit: I confess I still don't understand your question. When you say "the algebraic inequality," do you mean, for example, the triangle inequality in $\mathbb{R}^n$ in terms of the distance formula? | |
| Aug 9, 2011 at 0:16 | comment | added | Suvrit | @Theo: thanks for your comment and the links. | |
| Aug 9, 2011 at 0:13 | comment | added | Suvrit | @Todd and Qiaochu: I would love to know of the oldest source of the algebraic inequality being written down. Also, given your comments, I guess what will be useful to narrate to an audience will be the first time when somebody laid down the axioms for concepts of distance. The only catch here is that the triangle inequality itself is part of the axioms, which suggests that in some sense, the "fundamentalness" of the triangle inequality was recognized after people had been happily using distances informally for centuries.... | |
| Aug 8, 2011 at 23:43 | comment | added | Theo Buehler | Here's the link to Fréchet's original paper: dx.doi.org/10.1007/BF03018603 see also this thread mathoverflow.net/questions/51494/why-the-name-separable-space/… for some comments. | |
| Aug 8, 2011 at 23:09 | comment | added | Qiaochu Yuan | @Suvrit: what I mean is, if I replaced "triangle inequality" with "metric space" in your question, would that still be faithful to your intended meaning? (Like Todd, I'm not sure what it means to formalize the triangle inequality unless you mean writing down the axioms of a metric space. This was done by Frechet in 1906.) | |
| Aug 8, 2011 at 23:07 | comment | added | Todd Trimble | I'm not really clear on what "gets formalized" means. I suppose the concept itself was known to the ancient Greeks, and the algebraic inequality for Euclidean distance has been written down for centuries. For "gets formalized" to count, do you mean that the concept of real number should have been made rigorous first? And do you mean introduced as a formal axiom for concepts of distance? | |
| Aug 8, 2011 at 23:04 | comment | added | Suvrit | Hi Qiaochu, could you link to that question? (or do you mean that I should augment / alter the title of this question?) | |
| Aug 8, 2011 at 22:53 | comment | added | Qiaochu Yuan | Is this question identical to the same question with "triangle inequality" replaced by "metric space"? | |
| Aug 8, 2011 at 22:49 | history | asked | Suvrit | CC BY-SA 3.0 |