[FOM] 605: Integer and Real Functions

[Mitchell Spector]

Timothy Y. Chow wrote:
> ...
> 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.


I would go farther than this and just say that multiplication is not inherently commutative.  Even 
in the simplest case, "m copies of n" is very different from "n copies of m".

Along those lines, the standard algorithm for multiplying two natural numbers treats the 
multiplicand and the multiplier very differently.  In contrast, the standard algorithm for adding 
two natural numbers treats both addends in the same way.

Linguistically, it's common usage in English to apply the same word, "addend", to both numbers being 
added, unlike the distinction that is made between "multiplicand" and "multiplier".  (There is a 
word "augend", but it has fallen into disuse, for good reason.)

The fact that some particular multiplication operations in fact turn out to be commutative is 
arguably a deep property of those specific examples.


It may be instructive to look at the difference between multiplication of cardinals, which is 
commutative, and multiplication of ordinals, which is not.


Mitchell