[FOM] From theorems of infinity to axioms of infinity

[Timothy Y. Chow]

Arnon Avron wrote:

> On Wed, Mar 20, 2013 at 10:23:27PM -0400, Timothy Y. Chow wrote:
>> The difference between wanting proof and having doubt can be seen
>> even in the context of famous conjectures, e.g., P != NP or the
>> Riemann hypothesis.  Although there is not quite enough consensus
>> about these statements for them to achieve axiomatic status, in
>> practice they are treated much like axioms, in that people feel free
>> to assume them whenever they need to.  There's still an intense
>> desire to find proofs for them, even among people who are totally
>> convinced that the statements are true.
>
> I prefectly agree with the content of the two first paragraphs above.
> However, I think that the two examples given in the third are bad.
> The difference should be clear: the truth of those given in the
> first two examples had never been in doubt before they were adopted
> as axioms. The consensus reached in their case was about the need
> to take them as axioms, not about their truth (and even that need
> has actually been *proved*!). But concerning the two last examples
> the "not quite enough consensus" is about their *truth* -
> and once a proposition has not been self-evident,
> its truth can be established in mathematics only by a *proof*.
> No "consensus" can be a substitute for this!

There's certainly a qualitative difference between famous conjectures and 
axioms.  I wasn't giving P != NP or the Riemann hypothesis as examples of 
potential axioms, but as examples of the distinction between wanting proof 
and having doubt.

Having said that, I'm not sure the distinction you're drawing is quite as 
sharp as you make it out to be.  Consider something like the axiom of 
choice or the consistency of ZF.  For a while, these propositions were not 
self-evident, so by your argument, it would seem that we would never reach 
the point of accepting them as axioms, and would forever insist on finding 
proofs.  Well, as we all know, the trouble is that in these particular 
cases, we have since learned that finding proofs is a hopeless enterprise. 
Given that, mainstream mathematics has in effect granted them axiomatic 
status because of a consensus about their truth.

In the case of P != NP or the Riemann hypothesis, the difference is that 
we have no reason at all to believe that the search for a proof is 
hopeless.  But if in the unlikely event we were to come to believe that a 
proof was in principle unattainable, I could imagine them coming to be 
accepted as axioms.

Tim