[FOM] predicative foundations

Nik Weaver nweaver at math.wustl.edu
Mon Feb 13 03:29:39 EST 2006

Harvey Friedman wrote:

> In particular, there are claims, particularly by Avron, and
> implicitly by you, that predicativity has some special place
> in the robust hierarchy of logical strengths ranging from EFA
> through j:V into V.

I've made this claim explicitly in numerous posts, for example

You specifically derided the last message in the above list in
your "Role of Polemics" post.  Did you read it?

> Proponents of predicativity generally speak of an obvious
> absolute objectivity of the natural number system, and all
> first order sentences in the semiring of natural languages.
> I don't see anything particularly compelling about that view
> either  - it is simply one particularly stopping place in the
> robust hierarchy of commitments one may wish to make.

It is a stopping place based on accepting the natural numbers,
which have been considered obviously absolutely objective for
thousands of years, while not accepting the infrastructure of
Cantorian set theory which (1) is based on highly dubious
metaphysical beliefs about a fictional world of sets, (2) is
inconsistent in its naive form, and (3) is essentially irrelevant
to core mathematics.

You may not find this view compelling, but you should at least
recognize that it is of basic philosophical significance, and
not merely one out of countlessly many possible stopping places.

> > ... very confused ideas about predicativism, such as that
> > we are out to "ban" impredicative mathematics, or to classify
> > various kinds of mathematics as "good" or "bad", etc.
> Who is "we"? Have you any comments about Pollard's recent
> http://www.cs.nyu.edu/pipermail/fom/2006-February/009698.html ?

"We" is, in the first place, the person to whom you actually
attributed these views, namely, me.  I have never advocated
banning impredicative mathematics nor have I ever said it is
bad mathematics.

What I have said is that core mathematics is 99% predicative, and
I have made the observation that most core mathematicians generally
regard genuinely impredicative mathematics as pathological.  That's
an observation, not a value judgement.  I've also argued that one
has a clear philosophical basis for believing in the correctness
of predicative mathematics and one does not have such a basis for
impredicative mathematics.  That's not the same as saying it should
be banned!

> > However, we now know that virtually all ordinary mathematics
> > is predicatively justified --- say, 95% of all theorems that
> > appear in the Annals of Mathematics, or 99% if we exclude the
> > rare set theory/logic paper.
> The same can be said of systems substantially weaker than
> predicativity - although the specific numbers you cite could
> use some investigation.
> E.g., what about a system corresponding to ACA or ACA0? Also 95%?
> I think the percentage is also very large for RCA and RCA0.

Yes, surely 95% of the theorems that appear in the Annals could be
recovered in ACA_0.  I wonder what your point is here.

The relevance of your comment about RCA and RCA_0 is lost on me.

> > Conversely, 99% of what gets lost in the world of predicativism
> > is mathematics that ordinary working mathematicians would regard
> > as set-theoretically pathological.
> Of course, the exceptions here are what is important. E.g., Kruskal's
> theorem and the graph minor theorem.

As I've pointed out to you before, I have shown that Kruskal's
theorem is predicatively provable in my paper "Predicativity
beyond Gamma_0".  Whether the graph minor theorem is predicatively
provable is open.

I claim to have decisively refuted the Feferman-Schutte analysis
of predicativism.  You continue to refer to it as if it were
established fact.  Are you willing to directly assert that I
am wrong?

> I have no doubt that the situation will look entirely different
> by the end of the century. At some point, there will be a special
> kind of breakthrough that shows how to use higher principles in a
> huge variety of contexts to gain sharper information of a valued
> kind.

I don't believe this, but it would be very important if it did
happen and I sincerely wish you luck.

I do wonder whether that sharper information will be true information.
I specifically wonder what grounds there are for believing that
arithmetical consequences of large cardinal axioms are actually
true, if one is not a platonist.

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

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