# FOM: Hersh's fingers

Reuben Hersh rhersh at math.unm.edu
Sat Feb 28 14:46:17 EST 1998

```On Fri, 27 Feb 1998 steel at math.berkeley.edu wrote:

>
> Reuben Hersh writes:
>
>   So the natural numbers as describing physical objects
> are not the same as the natural numbers in pure mathematics.
>
> The fact that I have five fingers on my left hand is an empirical
> observation.  "Five" in that usage is an adjective.  There is no
> conceptual difficulty there, any more than in saying my fingers are long
> or short.
>
>
>
>        "Five" is indeed an adjective here, but it does not express a
> property of fingers, as does "short". It expresses a property of a certain
> SET of fingers . Frege discusses this very clearly in "Grundlagen
> der Arithmetik". Perhaps one reason logicians and foundations specialists
> are shocked by Hersh's views (as I confess I am) is that they are so well-
> publicized despite the fact that they are shot through with such
> elementary blunders.
>
>
> John Steel
>
Dear Professor Steel,

Thank you for responding to my posting.
I wrote, and you quote me correctly, that there is no conceptual
difficulty in saying the fingers on my left hand (or the toes on
your left foot, if you prefer) are five.  This is just an empirical
observation, no more obscure than saying my fingers or your
toes are long or short.  You are shocked at this elementary blunder.

There is no blunder here, elementary or advanced, unless you really
maintain that saying we have five fingers or toes
involves some conceptual difficulty.  By saying that this simple remark
is just as simple as saying your toes or my fingers are short, I
was of course by no means saying that "five" applies separately and
individually to each toe or finger!  I think this bit of everyday
logic was well understood centuries before Frege wrote it up.

Why don't you respond to the serious point I was making?
Some people say that five is an empirical, physical entity, because
it applies to *the set of your toes*  and *the set of my fingers.*  Some
say, with equal or greater force, that five is an abstraction, part of an
abstract theory.  I claim to resolve this contradiction by showing that
"five" has two distinct but related meanings, one empirical, and
one in the so-called "abstract theory" (which is  real and
resides in our collective thinking.)

Korner advanced this distinction in his *Philosophy of Math,* maybe
you would find it more digestible there.

As far as my being publicized, I apologize for that.  I
didn't mean to irritate anyone.

Reuben Hersh

```