FOM: Hersh's fingers

[Charles Parsons]

At 4:57 PM -0800 2/27/98, 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.
>

Sopmewhat to my own surprise, I find myself agreeing at least to some
extent with Reuben Hersh.

First of all Steel assimilates Frege's point to a framing of mathematics in
set theory that was alien to him, in saying that 'five' "expresses a
property of a certain SET of fingers". Frege put the point by saying that
having a certain number is a property of a _concept_; that's a peculiarly
Fregean notion of course, but what the underlying observations come to is
that a statement of number has a logical form involving an operator on a
_predicate_, which Frege expresses as 'the Number belonging to the concept
F'.

Hersh claims, correctly in my view, that there's a simple empirical fact
that there are five fingers on his left hand. This can be expressed as the
result of applying a quantifier-like expression 'there are five' to the
predicate 'finger on [Hersh's] left hand'. It seems to me reasonable to
assume that someone can learn to use such a construction, and to verify
statements of that kind by counting, without having any conception of set,
or associating any entity at all with a predicate such as 'finger on X's
left hand'.

Sets, or other such entities as Frege's 'concepts', come in later in two
possible ways: in setting up a semantical _theory_ about the use of such
constructions (and many others), or in giving a general account of
cardinality within mathematics. But that the latter goes beyond what the
child, or even the ordinary person, learns in acquiring the concept of
number and the ability to make statements of cardinal number is indicated
by the fact that such a theory was only given satisfactorily by Cantor and
his contemporaries in the late 19th century.

There are serious arguments for the claim that something like perception of
a set is involved in verifying by perception such a thing as that there are
five fingers on one's left hand. This is argued by Husserl in his early
work _Philosophie der Arithmetik_ (1891).

Such a view faces two difficulties: (1) Why should it be favored over the
ontologically more noncommittal view sketched above? (2) Granted that one
does perceive or intuit some bearer of number in such a situation, given
the different interpretations available for what is intuited (set,
plurality, concept, possibly mereological sum), how does one know that it's
a _set_ that someone innocent of these distinctions intuits?

A claim with some kinship with Husserl's has been advanced by Penelope
Maddy; see ch. 2 of her _Realism in Mathematics_ (Oxford 1990). Her answer
to (2) is that since set theory is the optimal framework for mathematics as
a whole, it's sets that are perceived. (It's not obvious that that follows;
she gives an argument, which I won't rehearse.)

Some of what I think about these matters is in my paper "Intuition and
number", in Alexander George (ed.), _Mathematics and Mind_ (Oxford 1994),
but the discussion there is combined with addressing issues about intuition
that are irrelevant to the present context.

Maddy's views have changed in many respects since the 1990 book; I'd be
interested in what she now thinks about these issues.

Where does this leave Hersh's claim that 'five' has two meanings, one
empirical and one belonging to "abstract theory"? I think that's much too
crude, but there is something on the right track about it. The paper of
mine mentioned above approaches the concept of number from a genetic point
of view, modeled to some extent on W. V. Quine's _The Roots of Reference_
(1974). That would allow for stages in the development of the concept of
number, which one might distinguish as different meanings (though more than
two), although for reasons that would take us too far afield I don't think
that's the best way to put it. But even at the early stage, 'five' is not
an empirical concept in the usual sense; I represent it as part of a
quantifier, but of course that's idealized.

Where I think Hersh is on the right track is that to make sense of
numerical language as used by someone innocent of theoretical mathematics,
we don't necessarily have to bring in the concepts that are needed in
mathematics as an "abstract theory", and it can often be misleading about
the simpler situation to do so. That's totally independent of Hersh's
social constructionism about theory.

Although I accuse Steel of assimilating Frege to a modern framework alien
to him, I think there's a point where he has Frege right, where Frege too
would disagree with what I'm saying here: Roughly, Frege thought that the
analysis of the concept of number as deployed in the simplest statements of
number had to be one that is adequate for the use of the concept in
theoretical mathematics.

Charles Parsons