FOM: reply to Vaughan Pratt on "7+5=12"
Vaughan R. Pratt
pratt at cs.Stanford.EDU
Mon Nov 3 13:25:54 EST 1997
>What Vaughan has yet to produce is any plausible possible scenario in
>which:
>
> "7+5=12" is false, and yet the expressions involved
> all have their standard meanings.
How did "7+5 no longer makes 12" (my wording) turn into "7+5=12 is
false"?
I understood Hersh to be talking about a scenario in which these were
not the same thing, due for example to the extraterrestrials not having
these concepts. In such a scenario, it would amazing not to have
7+5=12 yet to have 8+6=14. This is what I had in mind when I said that
the link
When 7+5 no longer makes 12, 8+6 will no longer make 14.
seemed more necessary to my mind than either of its constituents.
Your point seems to be that the natural numbers structure comes as a
rigid whole and cannot be adjusted locally. I have no trouble with
that, its partial rigidity seems clear enough, in the sense that adding
any new equation will cause some collapse. (But not necessarily total
collapse, if you add the equation 0=2 you may still be able to
distinguish 0 from 1.)
What I don't see in your argument is how you are able to pass from the
partial (or even total) rigidity of such a structure to the absolute
necessity of its individual truths. It seems to me that its truths
cannot be any more absolute than the structure as a whole. Mathematics
is informed by many structures, and I see no a priori reason why any
particular one of them needs to be an absolutely mandatory component of
the mathematics of every civilization.
>Why is the necessity of arithmetic of such foundational significance?
>One answer is that it involves the truths of a unique intended
>structure.
Intended by whom? By what definition of mathematical universe are you
able to prove that it must contain the wherewithal to count?
>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!
I was surprised to see set theory on this list. I thought foundations
made sets more fundamental than numbers. Are you making the case for
reversing this? If so then you might like my definition of sets as
ordinals, which is based on the complete arithmetic of ordinals,
including lexicographic sum and product, exponentiation, division, and
remainder, all of which play an integral and indispensable role in the
definition. For this you need to buy my explanation of the place of
non-ordinal sets such as {6} and the prime numbers, namely that they
are typed objects defined by predicates on their type.
Vaughan
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