[FOM] Tolerance Principle

[Roger Bishop Jones]

On Monday 06 February 2006 20:50, Arnon Avron wrote:
> On Sat, Feb 04, 2006 at 10:30:45AM +0100, Joseph Vidal-Rosset 
wrote:
> >
> > 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.

Carnap's "principle of tolerance" is simply the recognition that 
there are many different languages in which one can reasonably 
talk about the world.  Its prime consequence in Carnap's 
philosophy is that he worked on phenomenalistic and 
physicalistic accounts of physics rather than (as a positivist 
might) taking a dogmatically phenomenalist position.
His tolerance position, and all his work on languages, is 
pluralistic, and in the context of mathematics recognises that 
there are many different languages in which one can do 
mathematics.  This principle, it seems to me, cannot fail to 
apply anywhere. Languages inevitably involve many arbitrary 
choices, some of them trifling (e.g. which symbols to use) 
others not so trifling (and perhaps less "arbitrary").
In the end, for Carnap, it comes down to pragmatics.

Roger Jones