## History of the triangle inequality

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I am currently preparing a talk that revolves around the triangle inequality.

Because this inequality is so well-established, I do not want to (in my talk) belabor too much upon the importance it enjoys. For example, I learned some useful views here. But, these concerns are currently too advanced---for my purposes, I am seeking first some historical background; specifically,

Approximately when, where, and how did the concept of a triangle inequality get formalized, and its importance recognized?

EDIT It seems that the above question is not precise or clear enough. How about the slightly clarified question:

When was it realized (was it Fréchet's 1906 paper cited in the comments?) that the triangle inequality should be a fundamental axiom for defining distances?

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Is this question identical to the same question with "triangle inequality" replaced by "metric space"? – Qiaochu Yuan Aug 8 2011 at 22:53
Hi Qiaochu, could you link to that question? (or do you mean that I should augment / alter the title of this question?) – S. Sra Aug 8 2011 at 23:04
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? – Todd Trimble Aug 8 2011 at 23:07

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Here's the link to Fréchet's original paper: dx.doi.org/10.1007/BF03018603 see also this thread mathoverflow.net/questions/51494/… for some comments. – Theo Buehler Aug 8 2011 at 23:43

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@Qiaochu: I mean $d(a,b) \le d(a,c)+d(b,c)$ for some distance function $d$ (not necessarily $R^n$) – S. Sra Aug 9 2011 at 0:43
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## 3 Answers

There is a discussion of this issue in Dieudonné's History of Functional Analysis, p. 115:

It may seem obvious to us that the results of Hilbert are but one step removed from what we now call the theory of Hilbert space; but if, in fact, the birth of that theory almost immediately followed the publication of Hilbert's papers, it seems to me that it is due to the fact that this publication precisely occurred during the emergence of a new concept in mathematics, the concept of structure.

Until the middle of the XIXth century, mathematicians had been dealing with well determined mathematical "objects": numbers, points, curves, surfaces, volumes, functions, operators. But the fact that algebraic manipulations on different kinds of "objects" had a strikingly similar appearance soon attracted attention (cf. chap.IV, §3), and after 1840 it gradually became clear that the essence of these manipulations did not lie in the nature of the objects, but in the rules to be followed in handling them, which might be the same for very different types of objects. However, a precise formulation of this idea had to wait for the adoption of the set-theoretic concepts and language; and it is only in 1895 that our definition of a group, on an arbitrary underlying set, was formulated by Weber [225]. The trend towards the definition of algebraic structure then gained momentum, and around 1920 all fundamental notions of present-day Algebra had been defined.

In Analysis, no similar development had yet occurred in 1900. The extensions of the ideas of limit and continuity which had been formulated always were relative to special objects such as curves, surfaces or functions. The possibility of defining such notions in an arbitrary set is an idea which undoubtedly was first put forward by Fréchet in 1904 [69], and developed by him in his famous thesis of 1906 [71].

If I may summarize: the idea that one should talk about mathematical objects in terms of the axioms they should satisfy was itself quite new around 1900, and the specific application of this idea to the triangle inequality seems quite likely to have originated with Fréchet for that reason.

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Thanks Qiaochu; The above quotation is spot-on, and the very last sentence of the quote is essentially the attribution that I was searching for. – S. Sra Aug 9 2011 at 16:32

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It was not Frechet, since the statement that "a line is the shortest distance between two points" (which is obviously equivalent to the triangle inequality) is one of Euclid's axioms, and since Euclid is widely viewed as more of a scribe than the discoverer, presumably it goes back further than that.

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Indeed, that is, as you say, obviously equivalent to the triangle inequality for the sense of d(x, y) as meaning "length of straight-line path from x to y". One might also consider the sense of d(x, y) as meaning "length of shortest path from x to y" (or, more pedantically, "infimum of path lengths"). But then the triangle inequality is tautologous! (Though there may still be a notable first explicit remark upon this tautology...) Perhaps we need to figure out which "triangle inequality" exactly is of interest to our question-asker Suvrit. – Sridhar Ramesh Aug 9 2011 at 2:13

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"a line is the shortest distance between two points" ... is one of Euclid's axioms ... it is not! – Gerald Edgar Aug 9 2011 at 12:27

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It is Book I, Proposition 20. – Gerald Edgar Aug 9 2011 at 12:31

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@Gerald. I eat my words. – Igor Rivin Aug 9 2011 at 13:34

Here is a suggestion, to get the idea across in an informal way---it is what I always tell the students when I introduce the triangle inequality: I tell them that its essential content, and the way it gets used, is that "things close to the same thing are close to each other".

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This ought to be a comment, because it is not an answer to the question.  – Todd Trimble Aug 9 2011 at 11:37
Actually Dick, I am already quoting this statement of yours (I saw it in your response to a previous triangle-ineq. related question)!  – S. Sra Aug 9 2011 at 16:28
@suvrit Right,sorry,I missed the indirect reference. – Dick Palais Aug 9 2011 at 20:57
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