FOM: the necessary truth of "7+5=12"
Vaughan R. Pratt
pratt at cs.Stanford.EDU
Mon Nov 3 10:20:15 EST 1997
>If you find fault with any part of intuitionistic relevant logic then
>you will be refusing to accept inferences that are directly expressive
>of the meanings of the logical operators. If you refuse to accept the
>left-to-right direction of Schema N, then you are refusing to
>participate in natural number discourse altogether, or at least
>refusing to acknowledge that the natural numbers can be used to count
>finite collections. If you refuse to accept the operational definition
>of +, then you are so changing the meaning of that arithmetical sign
>that your claim is devoid of any interest. If you refuse to accept
>the two Peano-Dedekind axioms indicated, then you are so changing the
>meanings of the arithematical signs 0 and s that your claim is once
>again devoid of any interest.
>So which is it to be?
How about the first, no natural numbers at all?
You seem to be assuming that our putative aliens, or ourselves a few
thousand or million years hence, will continue to perceive things as
neatly partioned into countable entities. But suppose for the sake of
argument that the community has adopted, lock, stock and barrel,
Michael Barr's very nice proposal to make fuzzy logic internally
consistent by moving all reasoning into a topos with subobject
classifier the reals. (In other words *all* propositions are real
valued instead of 0-1 valued, including all references to propositions,
even self-references.) In this logic counting ceases to be a
meaningful activity, and *everything* is judged fuzzily, no longer any
neat partitions into "there are two of us in this room". You can put
counting into this system by introducing a natural numbers object, but
it is self-contained without it.
Mathematics may well continue to exist and produce interesting, useful,
and/or deep theorems. But the mathematicians of that era or that
planet may be as unaware of the notion of counting as I am of the
quaint theorems and paradoxes that those alien mathematicians get their
kicks from by living and breathing Barr's system.
My interpretation of Hersh's position is that he is contemplating the
possibility of just such alternative mathematics on other planets, with
systems that, while internally consistent and serving their users well
by satisfactorily reflecting *their* perceptions of the universe, need
bear no relationship to what we regard here and now as inevitable and
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. They could
bring Martin Gardner with them and have him get the ball rolling by
demonstrating how to count dinosaurs or whatever entities come to hand
that he and you and I would have no trouble counting. And if they're
deaf to boot, he could try explaining to them about falling trees
making a sound even when there's no one around.
However the very fact that we're having this argument should indicate
that it might take a while. :)
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