[FOM] 605: Integer and Real Functions

[Timothy Y. Chow]

Mark Steiner wrote:

> 3.  To say that multiplication is grounded in areas seems to be to get
>    the matter backwards: areas of what?  We do not want to ground
>    arithmetic in one of its empirical applications, nor do we want to
>    assume that space is Euclidean.  From Greek mathematics till
>    Descartes, the numbers were not closed under multiplication, the
>    product of two linear magnitudes was, indeed, a two dimensional
>    magnitude.
[...]
> Using the set theoretical approach it is clear that multiplication can 
> be thought of as a function on the Cartesian product of two sets, and 
> thus is clearly commutative.

This discussion of multiplication reminds me of a series of articles that 
Keith Devlin wrote some years ago about what multiplication is.  The most 
relevant for the current discussion is, I believe, the final article in 
the series:

https://www.maa.org/external_archive/devlin/devlin_01_11.html

This article links to previous articles in the series (which focus on 
arguing that multiplication, whatever it is, is *not* "repeated 
addition"), but the links are broken.  The correct links, for those who 
want them, are:

https://www.maa.org/external_archive/devlin/devlin_06_08.html
http://www.maa.org/external_archive/devlin/devlin_0708_08.html
http://www.maa.org/external_archive/devlin/devlin_09_08.html

Devlin's bottom line is that multiplication is a multi-faceted concept, 
but for him, the dominant concept is that of *scaling*.  In particular, he 
does not necessarily regard multiplication *of natural numbers* as the 
purest and most basic manifestation of multiplication.

Devlin does not claim to have found the unique correct concept of 
multiplication but I think he makes some astute observations and his 
article is worth reading.

I have one additional comment of my own to make.  Although at first glance 
it might seem that it is desirable to define multiplication in a way that 
make its commutativity obvious, there is a possible argument for the 
opposite point of view.  Namely, there are many operations that 
mathematicians are inclined to call "multiplication" that are *not* 
commutative.  If multiplication is defined in a way that is too strongly 
tied to commutativity then it may be hard to explain the temptation to use 
the same word for these other noncommutative operations.

Tim