[FOM] 605: Integer and Real Functions

[John Baldwin]

I have been asked to expand on my comments regarding Hilbert, addition and
multiplication.
Here is a capsule of my view.

here are two fundamental intuitions about multiplication

i) (arithmetic) multiplication is repeated addition

ii) (geometric) multiplication is similarity (scaling)

The two are quite different: the second is a group since inverses are
evident in the intuition.

Hilbert proved that in any Euclidean field one can define a
multiplication by similarity of triangle that distributes over
addition. Thus he defines a field of line segments.

Identifying a segment by an end point, the segments become numbers
and the arithmetic interpretation is fully encompassed by the
geometric.


I have taken down the link to my book because of concerns by publishers
about `prior publication'.

 The issues above are expanded in the slides at
http://homepages.math.uic.edu/~jbaldwin/multnoaddnd.pdf






John T. Baldwin
Professor Emeritus
Department of Mathematics, Statistics,
and Computer Science M/C 249
jbaldwin at uic.edu
851 S. Morgan
Chicago IL
60607

On Wed, Sep 2, 2015 at 12:00 AM, <W.Taylor at math.canterbury.ac.nz> wrote:

> 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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