[FOM] 605: Integer and Real Functions

[John Baldwin]

Chow and Steiner have discussed multiplication as repeated addition (or not)

This is a major feature of Chapter 8 of the book I announced yesterday.


http://homepages.math.uic.edu/~jbaldwin/pub/philbook83015.pdf

I argue that the geometric and arithmetic conceptions of multiplication are
the same and that
a major result of Hilbert's grundlagen is their unification over suitable
domains.


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 Mon, Aug 31, 2015 at 9:59 AM, Timothy Y. Chow <tchow at alum.mit.edu> wrote:

> 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
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