[FOM] The "mythological predicativist"

Arnon Avron aa at tau.ac.il
Fri Apr 21 05:53:21 EDT 2006

I have recently refrained from writing to FOM, because I am not
sure anymore whether the people who write to FOM are really interested
in what have always taken to be the real problems concerning the 
foundations of mathematics.  I do want however to make some brief comments 
concerning the recent debate on the limits of predicativity:

1) In principle, a true predicativist will never be able to
accept any given formal system as an adequate representation
of well-founded/absolute/predicativist mathematics. Because
if s/he accepts such a system s/he will be able to go beyond it
(e.g. by accepting as  absolutely proved any  arithmetical
sentence whose truth can be recognized as  equivalent to the
consistency of that system).

  It follows that a non-predicativist A will never be able to convince
a real predicativist B that a certain formal system (no matter which)
exactly represents B's mathematics. Any debate between them
can never reach an agreement (or at least A will never be able to
convince B).

2) I think that practically everyone will agree with my first point.
So how come the debate continues between Weaver the predicativist and some
other non-predicativists? Well, it seems to me that this is partially
because non-predicativists take "the predicativist" as a kind of a 
mythical creature. So they discuss what this mythical person 
can or cannot understand and see. But actual predicativists
are no less and no more mathematicians than non-predicativists. They can see
exactly what non-predicativists can see. The only difference
is that they  accept as absolute only what is really absolute, and 
not based on some metaphysical beliefs. Hence whatever mathematically 
sound claim  a (philosophical) non-predicativist may find concerning absolute 
mathematics - a (philosophical) predicativist will also be  able to 
see the soundness of such of the same insight. Hence Nik Weaver is very right
in denying the possibility of some mathematician understanding
that every theorem of some systems is "predicatively acceptable"
(i.e.: *absolutely* true) without predicativists being able
to understand this too. 

3) Exactly as Platonist mathematics  is open-ended, so is predicative
mathematics (I would prefer to call it "absolute mathematics"). 
In both cases if there are formal systems that capture what 
human mathematicians can know (which I doubt), it is impossible that 
human mathematicians will be able to find such a formal system  and *know* 
that it has this property. For the time being nobody knows even whether
there is any true arithmetical sentence the truth of which 
cannot in principle be established by absolutely reliable means
(i.e.: predicatively acceptable means)! 

4) One last point. It is common nowadays to claim that no foundational
program works. Well, the predicativist program *does* work and is
successful in providing secure foundations to the major and
most important (and interesting) parts of mathematics - and it does
so for a very low cost. As far as I can tell, no such claim 
can be made about any other foundational program.

Arnon Avron

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