

What Exactly is Multiplication? - RiderOfGiraffes
http://www.maa.org/devlin/devlin_01_11.html

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ubasu
Devlin has now written 4 (or 5?) columns trying to figure out what
multiplication means, and fighting with his readers over it, including arguing
at one point that he is not able to get his point across because he is British
and his readers are Americans.

Here's my understanding of what the confusion is. Normally, we define
multiplication at several levels: (i) define multiplication on integers using
e.g. Peano arithmetic, then (ii) define it on rational numbers using the
definition of multiplication on integers, (iii) then on reals numbers, complex
numbers and so on.

The important point to note is that each of these operations or binary
functions are distinct functions because the domain of operations are
different. The confusion arises because the operator symbol and the name have
been overloaded for these different functions. At no point in his discussion
does Devlin seem to make this distinction, that multiplication means different
things for different domains.

The further issue is that certain common properties of these operations have
been abstracted out to form the concept of a group operation, and this is the
perspective that Devlin hangs on to, somehow seeming to think that the
multiplication operation for integers, rationals, reals etc. follows from this
group-theoretic conception, rather than the other way round.

Which I find strange, considering that he is a mathematician. Perhaps I am
misunderstanding his position, but that's not for lack of effort.

