[FOM] Infinity and the "Noble Lie"
Andrew Boucher
Helene.Boucher at wanadoo.fr
Wed Dec 14 01:37:26 EST 2005
On 13 Dec 2005, at 7:04 PM, joeshipman at aol.com wrote:
>
> Obviously the
> Prime Number Theorem itself, as usually stated, involves the real
> function log(x), but an equivalent version can be stated that speaks
> only about finite objects, and the Erdos/Selberg proof transforms in
> the same way.
OK thank you. (This is what I meant when I spoke about the Eda
proof, which uses the Harmonic Series instead of log(x).) Still it
is not clear to me that the Eda proof does not use notions of
infinity. Specifically, can the PNT be proven in second-order
arithmetic minus the Successor Axiom? (The Successor Axiom, by
saying that every natural number has a successor, gives the natural
numbers its infiniteness.)
>
> But mathematicians commonly do use "proven" in an absolute sense. For
> every mathematician, there is a set of theorems she is willing to say
> are simply "proven", because they follow from axioms she accepts as
> "true".
My guess is that mathemticians who have doubts about infinity are
saying a theorem is "proven", not when it follows from axioms which
he is willing to accept, but when it follows from axioms which are
generally accepted by the community of mathematicians, i.e. ZFC or
some such. Otherwise it becomes too difficult to communicate with
other mathematicians. Imagine the following dialogue. Doubter:
"Fermat's Last Theorem is still an open problem." Mathematician:
"There's a flaw in Wiles' proof?" D: "Yes, it uses the Axiom of
Infinity." M: "Huh?"
> No one bothers to say, when referring to Euclid's theorem on
> the infinitude of the primes or Gauss's Quadratic Reciprocity Law,
> that
> those theorems have been proven from a particular set of axioms
Well, actually, I would! Euclid's Theorem on the infinitude of
primes, when stating that there are an infinite number of primes, is
not provable without the Successor Axiom. However, there are other
versions (for instance, "for any two numbers n and m, where m equals
(n!) + 1, there exists a prime between n and m") which are provable
in this system. (The Quadratic Reciprocity Law can also be proven in
this system.) But I would agree with you that it's pretty close to
"no one except for a few stragglers."
> If you disbelieve in actual infinities, you shouldn't give a statement
> proven from the Axiom of Infinity a higher epistemological status than
> the Axiom itself.
I would agree with you if the statement is equivalent to the Axiom of
Infinity. On the other hand, one can imagine there exist statements
where the only known proofs use the Axiom of Infinity, but which are
not in fact equivalent. And one could believe these statements for
other reasons besides the proofs, i.e. they are useful in
applications of the real world. Then one might give S a higher
epistemological status than the Axiom of Infinity, which doesn't have
any direct applications (except allowing for proofs of statements
that have applications!).
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