[FOM] From theorems of infinity to axioms of infinity

[Arnon Avron]

On Wed, Mar 20, 2013 at 10:23:27PM -0400, Timothy Y. Chow wrote:
 
> But I think that the desire to provide a proof isn't always
> motivated by doubt, and the axiom of infinity is just an example of
> that.  For another example, consider Euclid's parallel postulate.
> For a long time, many people struggled to prove it from the other
> axioms.  None of them ever doubted that it was true.  They just had
> a strong intuition that it should follow from the other axioms and
> that postulating it separately was redundant and inelegant.
> 
> Similarly, Russell never doubted the axiom of infinity, but just had
> a strong intuition that it was redundant to postulate it separately.
> When this intuition proved to be wrong, it should not be bewildering
> to find him effectively shrugging his shoulders and saying, "Oh
> well, I guess we'll just have to postulate it separately after all."
> 
> 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!

Arnon