[FOM] Tolerance Principle

[Joseph Vidal-Rosset]

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I want to thanks f.o.m. subscribers for this thread from my previous
email. 

Just some words here in order to make clear the relevance of this
reference to Carnap's principle in the debate on predicativism. 

> Principle of Tolerance: It is not our business to set up
> prohibitions, but to arrive at conventions …In logic there are no
> morals. Everyone is at liberty to build up his own logic, i.e. his
> own language, as he wishes. All that is required of him is that, if
> he wishes to discuss it, he must state his methods clearly, and give
> syntactical rules instead of philosophical arguments. (Logical Syntax
> §17) (quoted from http://www-csli.stanford.edu/~weisberg/Carnap2/ )

And, in his draft Weisberg comments this Principle rightly:

> For example, we can build a language (what Carnap calls a logic) for
> discussing the foundations of mathematics that is either nominalist
> or realist about numbers, a language for science that is positivist
> or realist about electrons, and a language for everyday life that is
> thing-laden or idealist. Since the difference between these is one of
> syntax, or the rules that govern the language, we should attach no
> cognitive value to the choice. There is no fact of the matter. There
> are only pragmatic factors to consider when making the choice. Asking
> us to defend the choice on ``absolute objective grounds'' is
> meaningless. Meaningful questions can only be asked from within
> languages. Hence we have, once again, arrived at the
> internal/external distinction.

So, if we agree with Carnap, it's then very difficult to imagine how a
predicativist can discuss about impredicative systems in order to show
they are "insecure", but without adopting the impredicative language.
(Quine in his famous paper "On what there is" pointed out the same
remark about the difficulty of the nominalist when he tries to refer to
the platonist ontology.) 
Nevertheless, in this debate about predicativism, it seems to me that
there is a burden of proof is on the predicativist which claims as Arnon
Avron: 

On Tue, 7 Feb 2006 00:27:49 +0200
aa at tau.ac.il (Arnon Avron) wrote:

> For *me* (again, I am not entitled to speak in the name of others), 
> "absolutely certain"
> is indeed the "definition" of "predicative". This does not mean
> identifying it with any current brand of predicative mathematics.
> But it means recognizing the deep truth of the main thesis 
> of predicative mathematics (at least in the sense of Poincare, Weyl 
> and Feferman)  concerning where is the main gap (not "borderline") 
> between the absolute and certain and what is not: it is the
> gap between concepts like  a natural number or a formal proof in 
> a given axiomatic system (which are well understood, meaningful
> and safe) and concepts like an  arbitrary set of natural numbers,
> or an arbitrary real number - which have always been problematic
> and are strongly connected with all the 3 foundational crises 
> in the history of mathematics.

The burden of proof to which I think is to show in impredicative
systems like ZFC a least one theorem which is not "absolutely certain"
even in the language of ZFC (because it makes no sense to criticize
the security of ZFC from the point of view of a weaker predicative
system). If such a proof could be done, one can maybe asserts
that impredicative systems are less secure, less meaningful and less
safe than predicative systems. But does it make sense to hope in finding
such a proof? I am afraid that the reply is "no". Am I right?

In the same email, replying to John Steel, Arnon Avron wrote:

> Are you certain that Godel's theorems are absolutely certain? If not -
> how come you have been so certain above that Hilbert's program
> is impossible?

It's true that Gödel noted at the end of his famous paper (1931) that
his demonstration of the incompleteness of PA does not refute
Hiblert formalist point of view presupposing the existence of a
finitist demonstration of consistency, but just because one can conceive
that exist finitist demonstrations which one cannot express in PA
formalism. But again the burden of the proof of the existence of
such demonstrations is on Hilbert's philosophy.


- -- 
Joseph Vidal-Rosset
Université de Nancy 2
Département de philosophie
Bd Albert 1er
F-54000 Nancy

page web: http://jvrosset.free.fr
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