[FOM] Infinity and the "Noble Lie"

[Andrew Boucher]

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!).