[FOM] Predicativism and natural numbers

[Nik Weaver]

Giovanni Lagnese asked

> How do most predicativists justify their acceptance of an
> impredicative definition of the set of natural numbers?
> Is there a philosophical (not practical) argument that
> justifies this exception?

As an actual predicativist I can give my answer to this question,
although obviously I make no claim about the extent to which others
share(d) my views.

I think the question is important because predicativists have tended
to present their prohibition against circular definitions as _the_
central tenet of their philosophy.  But this only makes sense against
some background of what is initially accepted, so casual observers can
be left with the impression that there is no particular reason to accept
predicativism "given the natural numbers" as opposed to predicativism
relative to weaker or stronger initial assumptions.

In my approach one starts with more basic philosophical principles
and the prohibition against circularity comes out as a consequence.
This is one of the main reasons I use a different name for my stance
("mathematical conceptualism") even though it is in effect predicativism
given the natural numbers.

I start with the idea that words like "set" and "collection" have no
literal referents.  Actually, it surprises me that this is not taken
for granted by modern philosophers.  Ordinary language assertions
involving such words can always be rephrased in equivalent ways which
do not use such words.  So to assume that a phrase like "flock of birds"
actually refers to some non-physical atemporal entity is to make a kind
of mistake that analytic philosophers showed us how to avoid a century
ago.  That one must not be seduced by the grammatical form of a sentence
into inferring the shadowy existence of metaphysical entities is indeed
possibly the most important lesson of twentieth-century philosophy.

Since there is no canonical universe of sets "out there", doing set
theory requires that we explicitly identify a structure which is to be
taken as playing the role of a universe of sets.  I guess this makes me
a structuralist, as I understand the term.  What this actually means is
that one has to precisely identify some (arbitrary) objects which are to
be taken as playing the role of sets and then say for which pairs of such
objects the membership relation is taken to obtain.  Now it is obvious
that at no stage in the process of identifying the desired structure may
we assume it has already been identified.  That comment is what makes me
a predicativist, but you see that it is not my central principle but
merely a trivial consequence of the basic set-up which may even hardly
seem worth mentioning.

What makes my stance "conceptualist" is that I do not require that
one actually carry out the process by which the desired universe
of sets is identified.  The issue of what we, biological beings who
inhabit a particular physical universe, can actually do is quite
beside the point.  The appropriate requirement is rather that the
proposed process should be _logically possible_, and I take the
correct criterion for judging its logical possibility to be whether
we can imagine it taking place in _some_ coherent world.

So the short answer to the question why I accept the natural
numbers is that I can imagine being able to identify a structure
which plays that role.  If you like, I can imagine writing a
mark on an unending strip of paper, making another mark next to
that, and so on.  There is no vicious circle in that conception.
Now one can debate whether it really is possible to even imagine
carrying out a process that requires more than omega steps, but
that is a separate question and I won't argue the point here.

You could also argue that my construction implicitly requires that
I already have a conception of omega and is circular in that way.
But that argument confuses "omega", a particular object that resides
in a set-theoretic universe which is indeed not yet available to me,
with the concept "of length omega", which need not refer to any
special object and which I think is initially available.  Again,
that is a point that I am prepared to argue separately.

In contrast, I can imagine no process which would enable me to
identify a structure which could play the role of the power set
of omega and which could not be further enriched by a longer
process.  In fact I gave an argument in my paper "Mathematical
conceptualism" (available on my website, which can be accessed at
http://math.wustl.edu/~nweaver --- see the bottom of page 10)
which I think more or less conclusively shows that no such
conception is possible.

Solomon Feferman makes the point this way:

"What distinguishes this conception [of omega] ... is that we
have a complete and clear mental survey of all the objects being
considered, together with the basic interrelationships between
them.  In particular we understand what is meant by saying that
all natural numbers have a certain property in a way that is more
special than what we understand when we speak of all functions
(or all sets, or all structures of some kind) having a certain
property." ("A more perspicuous formal system for predicativity",
p. 70.)

My formulation of this would be that we can imagine omega ---
the key phrase being "a complete and clear mental survey" --- in
a way that we cannot imagine the power set of omega.

Nik Weaver
Math Dept.
Washington University
St. Louis, MO 63130 USA
nweaver at math.wustl.edu