[FOM] The irrelevance of Friedman's polemics and results

John Steel steel at Math.Berkeley.EDU
Sun Feb 5 17:29:10 EST 2006

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?

  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.

       How are we supposed to do that? What this calls to my mind is
calibrating consistency strengths. Is that what FOM'ers should be doing?
Are we talking about certainty that the theory is true, or certainty
that is is consistent, or something in between?

  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"?

  but I think I have
> made it clear that in many (most?) applications one does not need
> absolute certainty.

       Virtually nothing in life is absolutely certain.:) Fortunately, you 
never need it. You need to be right, with enough confidence to act on your 

Which applications do you think require absolute certainty?

> until
> (and if) a formulation of predicative mathematics will be developed
> which is (almost) as convenient  to use as the conventional
> approach. If this happens then indeed it will be a good idea to
> encourage doing things predicatively whenever this is possible.

     This looks highly unlikely to me.

> If this is what you have understood from my messages, then I did a poor
> job.

     I joined the discussion late, missing your earlier messages, so
it's probably my fault.

John Steel

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