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 t

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.