[FOM] predicative foundations

Nik Weaver nweaver at math.wustl.edu
Tue Feb 7 08:37:52 EST 2006

It seems that this debate on predicativism has largely run
its course, but I have one or two points to add.  Much of
the previous discussion seems to me not very substantial.
I think it largely stemmed from 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.

(I would have liked to join in the discussion earlier, if only
to express my thorough approval for Arnon Avron's position,
but I was on a trip and didn't have the opportunity.)

Curtis Frank wrote

> ... since from the restrictive standpoint of Predicativism lots
> of ordinary mathematics appears unjustified and even paradoxical,
> whatever is getting a foundation from the Predicativist standpoint
> isn't the same thing as that body of knowledge with a thousands-of-
> years-old reputation for absolute certainty.

This was a description of Hilbert's position and I think it explains
very well the reason that predicativism initially attracted so little
support.  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.  Conversely, 99% of what gets lost in the world of
predicativism is mathematics that ordinary working mathematicians
would regard as set-theoretically pathological.  That's why after
decades of eclipse predicativism is reemerging as a major foundational

Harvey Friedman wrote (referring to his own work)

> ... is there STILL some way of siphoning off these new results
> as not good mathematics, that can be distinguished from real
> mathematics? ... I can imagine experiments like this that just
> might convince a very wide range of people that they can't tell
> the difference in terms of "naturalness", "beauty", etc.

I am not interested in classifying anything as "not good" or
"not real mathematics", but I do think there is a reasonably
well-defined notion of core mathematics.  And probably a better
criterion than "beauty" or "naturalness" for membership in this
class is "having essential connections with other core mathematics"
--- at least, this is a criterion that core mathematicians are
more likely to accept.  On that count the results in question are
unsuccessful at the moment, although it is obviously premature to
draw any definite conclusions about this.

Another basic classification is "can it be concretely visualized
or does it depend on vague metaphysical speculation involving a
ghostly `universe of sets'?"  It is a remarkable fact that this
distinction corresponds so exactly to the preceding core/not core
distinction.  The correspondence is so good that Hilbert's point
about predicativism not fitting ordinary mathematical practice
can now be turned into a point in predicativism's favor: it is
by a wide margin the foundational stance which *best* fits
ordinary mathematical practice.  I'll post a specific example
of this in a separate message.

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

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