Who invented the expression "pairwise different" and what is its advantage over "different" - MathOverflow

Who invented the expression "pairwise different" and what is its advantage over "different"

2013-02-04T07:01:19Z

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?

Answer by Brendan McKay for Who invented the expression "pairwise different" and what is its advantage over "different"

2013-02-04T09:09:41Z

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.

Answer by Pietro Majer for Who invented the expression "pairwise different" and what is its advantage over "different"

2013-02-04T09:11:57Z

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 $ and $\subsetneq $, to avoid any doubt. (Or, I once heard somebody -maybe myself, using the expression, for a topology not stronger than another, weakly weaker ).

Answer by Rhett Butler for Who invented the expression "pairwise different" and what is its advantage over "different"

2013-02-05T08:31:40Z

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.