[FOM] impredicative definitions/paper announcement

[Arnon Avron]

On Sun, Nov 21, 2010 at 12:52:21PM -0800, Monroe Eskew wrote:
> Aside from being one possible response to Russell's paradox, what is
> the motivation for predicativism?
 
We have discussed this in the past, so let me repeat the main points.
I should also note that I can safely speak only for myself.

So the goal of the predicativist program
is simply to solve the basic problem of FOM in general. This 
problem was formulated by Shapiro in "Foundations without 
Foundationalism" as follows:

 "How to reconstruct mathematics on a secure basis, one maximally
 immune to rational doubts"

Now Shapiro's formulates this problem in order to attack it. He 
acknowledged that this was the historic problem of FOM, but according
to him, its time has passed. As far as I was able to see he gave no reason
why he thinks that this problem is not important anymore - except that it is
not fashionable nowadays to work on it.  However, predicativists (I believe 
concerning this point I can speak in the name of all of them) are not 
interested in fashions. The above was, is, and will remain (until solved
satisfactorily) the main big problem of FOM. Those FOM subscribers
who think like Shapiro that it is not important anymore, or is not 
interesting, or obsolete, need not have any interest in predicativism.
(It is amazing to think that there can be such people in a list
called FOM - but sadly, it seems that this is actually the situation). 

Now what is predicativism?

 Well, a  predicativist is first of all a mathematician who finds as vague
and possibly meaningless the notion of an ``arbitrary subset of S"
in case $S$ is an infinite collection. The use of this notion
means commitment to the "existence" in a mysterious way of objects
we (as humane beings) cannot even describe, refer to, or comprehend. 
Thus "arbitrary" subsets of N (say) are mystic objects that only an infinite
mind (whatever this means) can identify, and belief in their existence
is exactly this: a sort of religious belief. Now beliefs of this sort
may be true of course. God may exist. Angles may exist. Spirits
(good and bad) may exist. One may conduct one's life according to
such beliefs. But pure mathematics, to the extent it is expected
to provide *certain* knowledge, immune to rational doubts, cannot
be based on beliefs of this sort. (Note that I am not claiming
that *all*  of useful/interersting  mathematics should be 100% certain. 
Thus I  see no reason why the mathematics which is used in some scientific
theory should be more certain than other parts of that theory).

 On the other hand a predicativist finds as crystal clear finite
mathematical objects that are  (at least potentially) fully describable,
have concrete finite representations,
and can be identified, distinguished from one another, and manipulated 
in precise ways by any finite mind given enough (finite) time 
and (finite) space.  Examples are the natural numbers, finite
strings of symbols from some finite alphabet, hereditarily finite sets, etc. 
Moreover: predicativists take as meaningful and self-evident 
the use of classical first-order logic at least when the intended 
domain of discourse is a constructively defined collection of such 
objects, as long as it is finite or *potentially* infinite.
This means that the objects of the collection can be obtained 
one by one by an effective process that can be continued indefinitely.
Both Poincare and Weyl (and I can only humbly join them) took this to be
an absolute minimum needed not only for mathematical thinking,
but for thinking in general, including thinking *about* mathematics: 
our notions of a "(formal) proposition" and a "(formal)
proof" are completely based on this minimum!

  Now the main difficulty of the predicativists is that the "real"
numbers, so central to modern mathematics and science, are not 
objects of the crystal clear type described above. Accordingly,
the main problem of the predicativists is how to reconstruct
the theory of real numbers (or at least the most significant parts
of it) and (the most significant parts of) modern analysis 
"on a secure basis, immune to rational doubts". This has  indeed been
the one big problem of FOM in all the foundational crises
since the time of the Greeks - and it still is. (With all the
respect I have to Kruksal Theorem or GMT, they are by far of secondary
importance comparing to the *real* problem: the reals).

The border between "predicativists" and "platonists" 
passed therefore somewhere between N and P(N), between the
rational(s)  and the (arbitrarily) irrational(s), between the
potential infinite and the actual infinite. (On the other hand the
main difference between "classical" predicativists and constructivists 
is that predicativists acknowledge the fact that there are, or at least
can be, meaningful propositions with determined truth-value, without us
knowing  this value or being capable of finding it. Classical 
predicativists also do not pretend that they do not understand the ordinary 
meaning of the word "not").

Now there are of course infinite higher order constructs,
like sets of natural numbers, or set of functions, which are 
*not* arbitrary, and *can* safely be used.  These constructs  
are acceptable only when introduced through definitions that
fully determine them, without assuming a god-given universe
of all mathematical objects. Such definitions cannot of course 
be circular, and one can only refer in them to constructs which  
were introduced by previous definitions.

I have no illusion that this short answer will convince you
about the great importance of the predicativist program. But
I hope at least that you will learn from it its eternal motivation.

Arnon