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There are many applications of "pairwise", for instance different, disjunct, orthogonal, independent, intersecting, connected, and many more. Some of them like "pairwise intersecting" or "pairwise connected" seem meaningful. But most of them appear to express no more information than with "pairwise" deleted. Who introduced this expression in mathematics in what framework?

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    $\begingroup$ This is basically an English question... $\endgroup$ Feb 4, 2013 at 7:23
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    $\begingroup$ Unoriginally, but frankly, I couldn't give a damn. $\endgroup$
    – Yemon Choi
    Feb 4, 2013 at 7:35
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    $\begingroup$ Judging by a quick search in google scholar and jstor, the phrase "pairwise distinct" may be first due to Leonard Blumenthal. I see LM Blumenthal using this term as early as 1937, so if you are truly curious about its origin, you might start by trying to find something published in 1936 or earlier. Google books had one hit for 1932, but the publication isn't online, so I cannot verify it either way. Mostly, though, I don't think this question belongs here. (And personally I am not invested enough to extend my five minute search.) $\endgroup$ Feb 4, 2013 at 8:49
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    $\begingroup$ "Pairwise disjoint" and "pairwise relatively prime" are ones very commonly used to remove possible ambiguity: do we mean $\bigcap S_i = \emptyset$ (or gcd$(a_1, \ldots, a_n)$ = 1), or something stronger? I would not say "pairwise distinct" (as in your other question); I'd simply say "let $x_1, \ldots, x_n$ be $n$ distinct elements" -- you'd really have to be perverse to misconstrue that. $\endgroup$
    – Todd Trimble
    Feb 4, 2013 at 10:46
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    $\begingroup$ The goal has just been raised. Find an older source than 1903, Hilbert: archive.org/stream/grunddergeovon00hilbrich/… GRUNDLAGEN DER GEOMETRIE I expect that is possible. $\endgroup$
    – Hilbert7
    Feb 4, 2013 at 12:32

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However "distinct" may have the weaker meaning of not all coinciding. So, in case I would therefore use pairwise, for clarity (see e.g. here), like in the other situations you listed.

The fact is that, in lack of a standard agreement on a definition or a notation, people is led to use more specific forms than needed. For instance: some people use $\subset$ for inculsion, some for strict inclusion. Result: some use $\subseteq $ for weak and $\subsetneq $ for strict inclusion, to avoid any doubt. (Or, I once heard somebody -maybe myself, using the expression, for a topology which is (comprable and) not stronger than another, weakly weaker ).

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    $\begingroup$ @Pietro. There is indeed a notational inconsistency in writing < for strictly less, ≤ for less or equal and ⊂ for contained in. However, the (where-the-hell-does-it-come-from?)- terminology concerning partially ordered set is absolutely ridiculous and confusing. While it may do no harm for totally ordered sets, in the non-totally ordered, it leads to terrible expressions like the one you use: from two topologies, none being stronger than the other, would you say that both are weakly weaker than the other? No, no! This should be avoided. $\endgroup$
    – ACL
    Feb 4, 2013 at 10:13
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    $\begingroup$ The discrete topology on a set -- is that called the weakest or the strongest? I can never remember. $\endgroup$ Feb 4, 2013 at 13:38
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    $\begingroup$ @Tom: I believe I've seen both uses of "weak" and "strong", even though they're diametrically opposed. I tend to say that the discrete topology is the finest (and I use "coarser" for the opposite of "finer"). Of course, if one takes literally the definition saying that a topology is the family of open sets (not the closure operator, the family of neighborhood filters, or any of the many other cryptomorphic variants), then one can say that the discrete topology is the largest. $\endgroup$ Feb 4, 2013 at 14:09
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    $\begingroup$ @PietroMajer, I hope that was a typo—nobody uses $\subseteq$ for strict inclusion, right? $\endgroup$
    – LSpice
    Aug 22, 2020 at 13:26
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    $\begingroup$ In the last sentence of your question, are you identifying "not stronger" with "weakly weaker"? What if two topologies are incomparable? Then one is not stronger than the other but also not (even weakly) weaker. $\endgroup$
    – bof
    Aug 22, 2020 at 22:58
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After some pondering about my question (and after finding out that this expression turns up in one of my books) I would like to revise my position a bit: "Pairwise orthogonal" seems redundant, but that may depend on the implicit understanding of quantifiers that have to be added to colloquial speech. "A set of orthogonal vectors" could in principle mean that for every vector there is an orthogonal one. Of course this is not the meaning attached to the word in general in mathematics. But can it be excluded a priori?

Nevertheless my question remains open (until it will be closed): Who invented the word "pairwise" or "paarweise"? I do not believe that Hilbert 1903 was the first, but do not know either and am curious to know it.

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    $\begingroup$ Who invented the word 'paarweise'? It is an adverb used in all kinds of contexts. Goethe used it, it is in Grimms' dictionarry (19 century), referencing Hederich's dict from mid 18th century...for questions on German go to german.stackexchange.com $\endgroup$
    – user9072
    Feb 5, 2013 at 11:35
  • $\begingroup$ Thank you. I should have looked there myself. Perhaps my question, implicitly concerning the use in mathematics, will have no final answer. $\endgroup$
    – Hilbert7
    Feb 5, 2013 at 12:53
  • $\begingroup$ Regarding the use in mathematics there would also be the question for what one asks precisely. If somebody would not say "$a_1, \dots , a_n$ pairwise different" but "$a_1, \dots , a_n $ any two of them different" would this qualify; it does not use pairwise but then the initial motivation seemed to be rather the (perceived) redundancy, which is present, than the particular word. $\endgroup$
    – user9072
    Feb 5, 2013 at 13:36
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    $\begingroup$ The earliest English use of the word 'pairwise' in the OED is by Carlyle in 1831. $\endgroup$
    – HJRW
    Feb 5, 2013 at 14:02
  • $\begingroup$ Then the first use of "pairwise" in mathematical context may be by Gauss: "... oder wenn man sie paarweise so verbindet, dass jede ..." [Carl Friedrich Gauss: "Beiträge zur Theorie der algebraischen Gleichungen", published in 1849 and reprinted in Werke: Band III, Göttingen (1866) p. 84] $\endgroup$
    – Hilbert7
    Feb 13, 2013 at 21:05
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When something is defined as a binary relation, "pairwise" is strictly-speaking required in order to apply it to a set larger than two. That's one advantage. Another is that in normal English "different" is the opposite of "equal".

That said, I think "pairwise different", and many similar things, are unnecessarily pedantic. If the meaning of "different" is so clear that most mathematicians wouldn't even pause to think about it, we don't need "pairwise".

The earliest appearance I found this 1941 paper of von Neumann, but I bet someone will find it in German much earlier. MathSciNet has 168 uses starting in 1949.

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  • $\begingroup$ I also suspect that the first use has been in German, paarweise, but don't know more about it. Could be that someone here just is an expert in that field or has come across that wording by accident. $\endgroup$
    – Hilbert7
    Feb 4, 2013 at 10:12

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