[FOM] The unreasonable soundness of mathematics

Daniel Mehkeri dmehkeri at gmail.com
Sun May 15 14:56:58 EDT 2016

2016-05-14 21:17 UTC, Timothy Y. Chow <tchow at alum.mit.edu>:
> I do not want to get into debates about *what Nelson believed*, because
> then we can get entangled in debates about whether what Nelson believed
> was self-consistent, or sound, or whether the act of asserting that what
> Nelson believed might not have been sound is dishonoring to his memory.
> However, I'm happy to debate with *you*, if you wish to don an
> ultrafinitist hat.

Right, fair. On the other hand, I'm not all that keen on arguing for
(or against) hypothetical positions that nobody ever seriously
advocated. Well, anyway, my hat is on.

>  But ultrafinitists do not generally believe in an
> infinitude of integers.  Do you, Mehkeri-qua-ultrafinitist, believe in an
> infinitude of integers?  If so, why do you still call yourself an
> ultrafinitist?  If not, then please spell out carefully what "sqrt(2) is
> irrational" means in terms that clearly do not implicitly assume an
> infinitude of integers.
> My contention is that you can, for any feasible N, spell out in
> ultrafinitist terms what it means for there to not exist positive A and B
> less than N such that A^2=2*B^2, but that you cannot make the leap to
> asserting this for *all N* in the sense that non-ultrafinitists mean it.
> You have to take one feasible N at a time.  And it is therefore an
> unexplained mystery why, as you keep trying larger and larger values of N,
> the equation A^2=2*B^2 remains insoluble.

Okay, my hat says:

 (1) There are integers.
 (2) They can be added, multiplied, and compared.
 (3) sqrt(2) is irrational, which means A*A =/= B*B*(1+1) or something
equivalent to that. It does make a statement about every rational
number, or every pair of positive integers. The statement is true. I
can prove it.
 (4) The totality of exponentiation has not been proven, in
particular, nothing needed for (3) implies it.
 (5) Although it hasn't been disproven either yet, I tend to think it will.

I say: I guess my hat counts as believer in an infinitude of integers,
though you may have meant something else by that.

Anyway, assume we can infer that belief from (1)-(3). Then (4)-(5) is
a reason to still call him ultrafinitist. Ultrafinitism is about
feasibility, addition and multiplication are feasible operations,
while exponentiation isn't.

Assume we can't infer that belief. Then (3) answers your question as
to what the statement means. He says it does indeed apply to *all the
positive integers*, and he says he can prove it, and I think he can
too. But by assumption, we can not infer belief in an infinitude of
integers from that.


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