[FOM] Tolerance Principle

[Arnon Avron]

On Sat, Feb 04, 2006 at 10:30:45AM +0100, Joseph Vidal-Rosset wrote:
 
> A Tolerance Principle ? la Carnap could maybe solve the problem from a
> strict scientific point of view: "truth" in mathematic or mathematical
> logic depends on the axiomatic system (there is no Moral in Logic). 
> 
> My question is both to  Arnon Avron and Harvey Friedman: is Carnap's
> Tolerance Principle wrong (and then intolerable) ? 

I am not sure that Carnap has intended to apply this principle
to the whole of mathematics,  but my knowledge here is too limitted 
to tell.  Regardless of this, axiomaic systems for the the certain,
unquestionable  parts of mathematics (like the natural numbers)
cannot be chosen arbitrarily. They should be true, and there can
be no tolerance about that. Also the question whether a certain
object is  a proof in a given axiomatic system A, or whether
a certain proposition is or is not a theorem of A,
should have an absolute answer, not depending on the question
what is the axiomatic system B which we use to answer the questions
about A. Now to define the notion of a proof one should understand
finitary inductive definitions, and so already understand (and 
rely on the ceretainty) of a significant part of predicative 
mathematics. I dont see how a principle of tolerance can apply 
to the necessary background which makes its very formulation possible. 

Arnon Avron
of mathematics