FOM: reply to Vaughan Pratt on "7+5=12"
neilt at hums62.cohums.ohio-state.edu
Mon Nov 3 12:01:46 EST 1997
Vaughan Pratt responds:
How about ... no natural numbers at all?
but then he goes on to concede
I don't *think* Hersh would claim that you can't land
on that planet and eventually convert the natives to
your brand of math
by which he means a brand of math that acknowledges natural numbers.
So Vaughan's reply fails to address my point. If the concession is
genuine, then the aliens' conceptual and mathematical scheme can be
augmented by the introduction of numbers, and the NECESSARY truths
governing them, such as "7+5=12", will inexorably flow from such
augmentation. It is absolutely irrelevant if there is a community of
mathematicians, however sophisticated in their alternative math, who
are innocent of the natural numbers. The point is that *any*
conceptual scheme involving predicates that "divide their reference"
(i.e. that are satisfied by identifiable objects) can be extended by
conceptually controlled number-talk.
This is a genuine foundational insight, owed to Frege. I don't care
whether you go fuzzy in your logic, or see everything in terms of
topoi. If you see any THINGS at all, and can re-identify them and
distinguish them from one another by means of concepts, then you can
COUNT them (modulo any concept). Thus arithmetic comes in as a
NECESSARILY POSSIBLE extension of your conceptual scheme. And it
contains NECESSARY truths, whose necessity is actually rooted in
This is another foundational insight of Frege's, although he worked
out the details incorrectly. Where Frege identified #xFx as the class
of all classes in 1-1 correspondence with the class of all Fs, I
simply take #xFx 'synthetically' as the number of Fs. Frege's method
led to contradiction; the alternative one does not. It invites one to
think of the natural numbers as sui generis; indeed, as *logical*
objects that exist necessarily.
What Vaughan has yet to produce is any plausible possible scenario in
"7+5=12" is false, and yet the expressions involved
all have their standard meanings.
Why is the necessity of arithmetic of such foundational significance?
One answer is that it involves the truths of a unique intended
structure. There is a contrast between arithmetic and, say, the theory
of groups or rings. The latter theories are pursued precisely because
of the variety of non-isomorphic structures satisfying the
characteristic group- or ring-axioms. Moreover, the real mathematical
interest in this case resides 'one level up', in the study of the
interrelationships among such structures (homomorphisms, quotient
Arithmetical truth seems, then, to be inherent in the operation of any
conceptual scheme whatsoever, provided that it permits the
re-identification and distinguishing of objects by means of concepts.
Can anything similar be said about, say, geometry? Or topology? Or set
theory? What foundational considerations bear out *their* truths as
necessary? Now *that* is a real foundational question!
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