[FOM] 605: Integer and Real Functions

[W.Taylor at math.canterbury.ac.nz]

Quoting Mitchell Spector <spector at alum.mit.edu>:

> 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".
>
> 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.)

These note that (linguistically) a natural number can be either a noun
or an adjective.  I suppose one of the goals of "pure math" is to remove
this distinction.  I know Thomas Forster suspects the key to Paris-Harrington
may be in the conflation of these two aspects.

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

Another trivial remark concerning the above three excerpts.

Multiplying ordinals A and B, in that order, we might think pictorially of

"A of B", meaning A copies of B, end-to-end;    OR

"A by B", meaning B copies of A, end-to-end.

(So       "omega of 2" = "2 by omega" = w,
  whereas  "2 of omega" = "omega by 2" = w2.)

It is the latter word-picture we use to define the actual operation,  AB .
I find this a useful mnemonic, "by", (even though maybe "of" is more natural,
linguistically?)

None of any of this addresses the fact that ordinal ADDITION is also
non-commutative - and I know of no other examples of "addition"
where this is so.   Are there any?

-- Bill Taylor


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