[FOM] Tolerance Principle
Roger Bishop Jones
rbj01 at rbjones.com
Tue Feb 7 03:40:07 EST 2006
On Monday 06 February 2006 20:50, Arnon Avron wrote:
> On Sat, Feb 04, 2006 at 10:30:45AM +0100, Joseph Vidal-Rosset
> > 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.
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