[FOM] A Defence of Set Theory as Foundations

[Nik Weaver]

Roger Bishop Jones wrote:

> Weaver seems to think that the iterative conception
> can be construed in only two ways, platonistically
> or constructively. (or does he to think it
> inconsistent because it can only be construed in
> both ways?)

I can't even parse the sentence.


Robert Lindauer wrote:

> The notion of a "constructed" infinite group is, I think, incoherent.
> The operational nature of constructing things leaves us a limited
> amount of time in which to do it, leaving us a limited number of things
> constructed.

I suppose so, if you take "constructible" to mean "physically
constructible in our universe".  No doubt the question of what
is actually physically possible is of the highest scientific
interest, but I don't see it as particularly relevant to the
fundamental nature of mathematics.  I would say that in math
we study the realm of logical possibility, not physical
possibility.

Or do you mean that a construction of length omega is not
even logically possible?  (You use the word "incoherent".)
My position is that we can form a definite picture of an
imaginary world in which constructions of length omega can
be carried out, and this is sufficient to justify their
status as logical possibilities.

I think most mathematicians would agree with that --- it
is on this kind of basis that I believe most mathematicians
accept that all arithmetical statements have a definite
truth-value.  We believe the twin primes conjecture is
definitely true or false because we can imagine running
through the natural numbers and checking it.

This is the conceptualist view.

> We are left, as Mr. Jones rightly puts it, with the option of
> 'semanticizing' our mathematics - making it a kind of word-play.  This
> means giving a new kind of meaning to "true" where the objects in
> question don't exist and don't "really" have the properties they are
> asserted to have.  We therefore have to separate the concept "true in
> reality" from "true in the story" or "true in the system".  We are
> left, however, with something rather unsatisfactory for most
> mathematicians - a true that means "not really true" and perhaps "not
> possibly true since originally fictional".

I have a hard time with this.  Does the story have to be
consistent?  In other words, if we regard ZFC as simply a
"story" we tell about fictional objects, do we care whether
the story is internally coherent?  The comment "perhaps not
possibly true" suggests not, which surprises me.  If not,
this can hardly be regarded as a justification of mathematics.
One could justify anything this way.

I would have thought that people who adopt this kind of
fictionalist approach would want to require that the story
be consistent --- and I would think that the way you determine
that a story is consistent is by imagining a fictional world
in which the story takes place.  That would make you a
conceptualist, because your basic criterion for legitimacy
would be whether you can conceive an imaginary world in which
the axioms being justified hold.

If one takes this point of view seriously, one has to think
hard about what set-theoretic axioms can actually be verified
in a concretely imagined world.  I've analyzed this question
in some detail in papers which are posted on my home page
(a url was given in messages I posted in September).  The
basic conclusion was that ZFC is not justifiable on these
terms, yet the much weaker systems which are conceptualistically
justifiable do in fact suffice for normal mathematical practice.

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