FOM: natural numbers
neilt at hums62.cohums.ohio-state.edu
Fri Oct 24 12:09:38 EDT 1997
A propos Harvey's interesting question about the nature of number, I'd like
to suggest a fundamental conceptual constraint on the notion of
natural number. First, some preliminaries about notation.
One needs a language with at least the following expressions, or ones
equivalent to them:
0 for the number zero;
s( ) for the successor function;
#x(..x..) variable-binding term-forming operator on formulas;
plus the usual logical symbols, including the identity predicate.
`s(...s(0)...)', with n occurrences of s, will be the numeral for the number n. We shall call this numeral n*. The numeral, remember, is a piece of notation,
not the natural number itself.
Suppose one is given a one-place predicate F. Everyone knows how to
find formulae involving just the quantifiers, connectives, identity
predicate and the predicate F that express the claims
There are no F's
There is exactly one F
There are exactly two F's
There are exactly n F's
Note that in these formulae there is no occurrence of any numeral n*;
nor need the quantifiers be ranging over numbers. To say
There are exactly two apples
I simply write down
ExEy(-x=y & Apple(x) & Apple(y) & (z)(Apple(z) -> (z=x v z=y))).
I shall abbreviate this to
So in general (E_n x)F(x) says there are exactly n F's.
This statement need not (in general) involve any mention of, or
quantification over, numbers themselves.
Of course, we *might* be talking about numbers, saying something like
"There are exactly two numbers with the property F", but the conceptual
point is clear. The mention of "two" in my example here is not *itself*
a reference to numbers. "Two" here is adjectival, not substantival.
ADEQUACY CONDITION ON ANY THEORY OF NATURAL NUMBER
(compare with Tarski's Adequacy Condition on any theory of truth)
Any adequate theory of number must yield as a theorem each instance
of the following schema:
Schema N: (E_n x)F(x) if and only if #xF(x)=n*
Only with such a theory does one pin down the essential conceptual role
of natural numbers: they are used to count finite collections.
By the way, perhaps we shouldn't even distinguish natural numbers as
ordinals from natural numbers as cardinals. Maybe that distinction only
makes sense for the transfinite. If you claim grasp of the `ordinal'
sense of natural numbers---claiming to be able, say, to tell me who the
n-th person in a queue is---it follows that you are also committed to
claiming grasp of the `cardinal' notion. You can't locate the n-th person
without being able to tell me that there are n persons ahead of or identical
to him, hence that the number of these persons is n*. Conversely, if you know
that the number of persons ahead of or identical to him is n*, then you know
that he is the n-th person in the queue.
For those interested, the adequacy condition on number was stated in my
book `Anti-Realism and Logic', OUP, 1987.
It also appears in a recent paper `The Necessary Existence of Numbers',
Nous 31, 1997, pp. 301-336. A .dvi version of this paper is available from
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