FOM: why numbers are objective (contra Feferman)
neilt at mercutio.cohums.ohio-state.edu
Tue Jan 27 12:15:58 EST 1998
>I would say that mathematics is the science of skillful operations
>with concepts and rules invented just for this purpose. The principal
>emphasis is on the invention of concepts.
>This seems to point to the subjective origin of mathematical concepts
Why should the invention of concepts be a *subjective* matter?
Feferman owes us an explanation for assuming that it is so. After
all, Frege gave the first convincing account of the "invention" of the
(second-order) concept NUMBER OF ___S precisely in order to show that
the concept of number was *objective* rather than subjective. He would
have been horrified at any charge of psychologism or subjectivity
concerning the way he pedigrees the numbers as logical objects for
I am going to offer here a neo-Fregean story about the invention of
the *objective* concept of (natural) number. (Cognoscenti please note:
There will be no appeal to "Hume's Principle" in what follows. Nor
will this account provide for the existence of any infinite numbers.)
The concept of natural number is "invented" by extending any language
for discourse about ordinary (concrete) things, so that the extended
language includes a (number term)-forming operator that forms terms
from predicates, and binds a variable in the process. Thus, given one
of the earlier predicates "x is an apple", we can now form the term
"#x(x is an apple)". This is the regimentation, in the newly extended
language, of the English phrase "the number of apples".
Note that in the unextended language, provided only that it has
quantification, the usual connectives, and the identity predicate, it
is possible to express claims of the form "There are exactly n Fs" for
any natural number n. For example, "There are exactly two Fs" can be
(two Fs) (Ex)(Ey)(~x=y & Fx & Fy & (z)(Fz -> (z=x v z=y)).
Note that there is no reference to the *number* 2 in this
sentence. "Adjectival twoness" is instead expressed by a suitable
quantificational complex. So also for 0, 1, (2 already dealt with), 3,
4, ... ad infinitum.
By extending the language so as to include the operator #x(...x...),
we now achieve a new way of expressing the thought that there are
exactly two Fs. (Note: such a thought is Fregean, i.e. objective. It
has objective truth conditions.) One can say "the number of Fs is 2";
(2 Fs) #xFx = s(s(0)).
This last statement has to be logically equivalent to the earlier one;
that is, we should be able to inter-deduce (two Fs) and (2 Fs). It is
precisely this required equivalence between (2 Fs) and (Two Fs) that
confers objective status on the former. That status is inherited from
the latter. But the former "carves up" the conceptual content in a
different way than the latter. For (2 Fs) introduces reference to an
abstract object, namely a number; while there was no such reference to
be discerned in (Two Fs). It is also the purpose of this equivalence,
by the way, to ensure that #xFx means the NUMBER of Fs rather than,
say, the SET of Fs or the MEREOLOGICAL UNION of Fs, or the DEMOCRATIC
TELEPATHIC CONSTITUENCY of Fs, or THE POLYGON DESCRIBED BY THE
BIRTHPLACES of Fs ... .
But I get ahead of myself. Let us backtrack to the point at which we
adopted the new term-forming operator #x(...x...), and were wondering
how to make it mean what we would have it mean.
One has to lay down rules governing the inferential behavior of the
new term-forming operator #x(...x...). How, for example, does one
establish the truth or falsity of identity statements involving the
new kind of terms (those of the form #xFx, with dominant occurrences
of #)? Well, there is a special new constant term one can
introduce---"0"---which satisfies the rule that the number of Fs is 0
just in case there are no Fs. Moreover, there is a new one-place
function symbol---"s( )"---which satisfies the rule that s(t) is the
number of Gs just in case whenever t is the number of Fs there is a
1-1 correspondence of the Fs with all but exactly one of the
Gs. (These meaning constraints on 0 and s can be codified by means of
appropriate rules of natural deduction; I shall spare the reader the
details. They are spelled out in my book Anti-Realism and Logic, OUP,
1987, in the chapter on constructive logicism.)
The new symbols 0 and s allow one also to form arbitrarily long but
finite terms of the form s(s(...(0)...)), which we can call numerals.
The numeral with n occurrences of s is called n*.
A further requirement on any adequate (*objective*) theory of numbers
is that it should entail every instance of the following schema:
(Schema N) #xF(x)=n* iff there are exactly n Fs.
THIS is what makes #xFx MEAN "the (natural) number of Fs".
G.E.Moore got impatient with skeptics about the external world, and
proved that there were at least two things---by holding up his hands.
Let L be the first-order language based on identity and the predicate
"x is a hand of Moore". From the vantage point of the novice user of L
extended by the operator #, Moore's gesture *also* established the
existence of the number 2. For, there are exactly two hands of Moore
iff the number of hands of Moore is 2. But there *are* exactly two
hands of Moore! Hence, the number of hands of Moore is indeed 2. So,
the number 2 exists.
But of course, there would have been an alternative proof of the
existence of the number 2 available to the novice upon suitable
reflection. She could have reasoned thus, in blissful ignorance of
Moore and his hands:
Nothing is not identical to itself. Therefore the number of things not
identical to themselves is 0. There is exactly one thing identical to
0, and for any F the number of Fs is 0 if and only if there are no Fs;
so the number of things identical to 0 is s(0). 0 cannot be identical
to s(0), on pain of any concept F whose number they both are having to
hold both of no things and of exactly one thing, which is
impossible. So there are exactly two things that are either identical
to 0 or identical to s(0). Thus there is exactly one more thing either
identical to 0 or identical to s(0) than there are things identical to
0, the number of these latter being s(0). Hence the number of things
that are identical to 0 or identical to s(0) is s(s(0)). That is to
say, 2 exists (and, it would appear, of necessity). QED
What is "subjective" about any of the foregoing reasoning? Does it not
command rational assent, regardless of how anyone *feels* about
anything? The reasoning actually reveals that natural numbers are even
more objective than any physical things, like sheep and cows (or
little clay replicas of the same), whose need to be totted up by early
Babylonian intellects led to our discovery of numbers. For the
reasoning would hold good in any universe, including one in which
there was just one disembodied Cartesian soul reflecting on the tedium
of non-physical existence. Allow such a soul a sortal predicate or
two, and she could reproduce the reasoning above, to establish the
necessary existence of each and every natural number. It may not
relieve the tedium very much; but for a while at least, the
abstraction could provide her with some distraction.
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