[FOM] Predicativity and Certainty

[Arnon Avron]

On Sun, Feb 05, 2006 at 02:29:10PM -0800, John Steel wrote:
> 
> 
> On Sun, 5 Feb 2006, Arnon Avron wrote:
> 
> > Also Hilbert did not intend in his
> > program to convince "core mathematicians" to use only finitary
> > methods. On the contrary: his goal was to allow them to safely use
> > their infinitistic methods with full justification.
> 
>        It's a mystery to me why this is your lead in to "the main task
> of research in FOM". Hilbert's program was shown completely impossible
> by Godel. Why keep trying to do the impossible?


Originally I  gave this only as an example that the fact that someone
classifies certain methods as less safe does not mean that s/he
forbids using them. But since you raise the question: Maybe
Hilbert's program is impossible, as Godel's work indeed indicates
(though it does not prove it - but this is another issue). But
it is still possible to reconstructs the most important parts 
of mathematics on absolutely safe FOUNDATIONS. This is not
just a speculation - it has already been demonstrated by the
work of Feferman and others (starting from Weyl).

> 
>   This is
> > how I see the main task  of the research in FOM. It is *our*
> > task to find out what is the degree of certainty of various
> > pieces of mathematics.
> 

The trouble with short notes of the sort we  post to FOM
is that what one says might be understood in a wrong way.
When I talked about "degrees of certainty" I did not mean
measuring by some numbers or even ordinals. I meant
a  "degree" as a sort of equivalence class of propositions,
and a partial order between them (I am not convinced like
Friedman that the order should be linear. I am convinced that
it has a top element: the absolutely certain propositions).

> Are we talking about certainty that the theory is true, or certainty
> that is is consistent, or something in between?

*I* am talking about Truth (in an interpretation).
Consistency is very important, but it is only a necessary condition.

> 
>   It is true that I identify "predicative mathematics"
> > with "absolutely certain mathematics",
> 
> 
> Can you say exactly what predicative mathematics is? If not, how do 
> you know it is "absolutely certain"? If so, aren't you 
> absolutely certain  of Con(Predicative mathematics)?
> Perhaps predicative mathematics is not recursively axiomatizable?
> Is the identification of "absolutely certain" with "predicative"
> meant to be a definition of "predicative"?

Good questions.

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.

Now you are absolutely right in pointing out that 
predicative mathematics in my sense is not recursively axiomatizable
(or at least that if it is, *we* shall not be able to show this 
fact by recognizing some axiomatic system as codifying 
all predicative mathematics - because once we do that we shall be 
able to add its consistency as a new certain truth). But there
is nothing special here about predicative mathematics. The same is
true, e.g.,  also for Platonist mathematics.
  
>  Virtually nothing in life is absolutely certain.:) 

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?


> Fortunately, you  never need it. You need to be right, 
> with enough confidence to act on your  belief.

This is exactly what most  people (from politicians to  terrorists) 
do: they believe they are right, and have 
enough confidence to act on their  beliefs. 
There is more to *Mathematics* than that.

Arnon Avron