Meta-mathematics

[Timothy Y. Chow]

Joe Shipman wrote:

> Thesis: the most simply stated description of the main thrust of work in 
> foundations of mathematics is "consider new axioms".
>
> Of course there's other stuff we do, but if we never make any progress 
> towards deciding, given two axioms whose conjunction is refutable, which 
> to prefer, it's not very satisfying.
>
> I would like to ask something one level up: for what kinds of new axioms 
> is it conceivable that some kind of persuasion could even occur?

I suggest a slightly different spin on this question.

"Persuasion" above might mean "persuaded to add the axiom to our list of 
fundamental axioms of mathematics."  People are supposed to "really 
believe" the axiom.  My view is that the window has closed for this kind 
of persuasion.  To a first approximation, nobody cares any more.  A 
significant fraction of people who do care, are trying to *discard* axioms 
and not add to their number.  For example, maybe they profess not to 
believe in infinite sets.

On the other hand, there's a different possible sense of "persuasion." 
People could be persuaded to use the axiom in their research.  Statements 
such as P != NP or the Generalized Riemann Hypothesis illustrate this 
point.  Nobody calls them axioms, but they kind of function like axioms, 
in the sense that they are frequently used to prove things, and nobody is 
bothered that the statements themselves have no proof.

With this reformulation, mathematicians could be persuaded to "adopt" a 
new axiom (in the practical sense of using it a lot, whether or not they 
profess to "believe" it) if it leads to a wealth of interesting results 
about concrete topics that mathematicians are standardly interested in, 
such as PDEs, number theory, CW complexes, algebraic varieties, planar 
graphs, etc.

> There is no physical experiment which could ever provide persuasive 
> evidence for or against statements of set theory that are not absolute. 
> Mathematicians who firmly disagree about such a statement, but who agree 
> about fundamental physics, will never be able to settle the matter by 
> doing computations or any other physical processes.

I'd suggest that you could strengthen this proposition by changing 
"absolute" to "Pi^0_1."

Tim