[FOM] predicativism and functional analysis

[Harvey Friedman]

On 2/21/06 2:18 AM, "Nik Weaver" <nweaver at math.wustl.edu> wrote:

> This is a follow-up to a previous post of mine,
> http://www.cs.nyu.edu/pipermail/fom/2006-February/009778.html
> in which I argued that Cantorian set theory fits normal
> mathematical practice rather poorly, because core mathematics
> takes place only at the very bottom of the cumulative hierarchy.

Finitist: predicativity fits normal mathematical practice rather poorly,
because core mathematics takes place only involving finite objects. Any
exceptions are easily dispensed with, or are of limited significance. Real
numbers can be handled by approximations by rationals. Finitely presented
groups can be dealt with in terms of their entirely finitary elements. And
who cares about general finitely generated groups?

Set theorist: predicativity fits normal mathematical practice rather poorly,
because core mathematics IS NOW BEING NAUTRLLY CONDUCTED using substantial
set theoretic notions, including topoi and Grothendieck universes, as well
as Zornifications even in countable structures (Fields medalist and
friends), and minimal bad sequences even in infinite sequences of finite
trees and finite graphs. Some of this is known to be unremoveable. Also
large cardinals or substantial uncountable cardinals in the theory of Borel
selection, Boolean relation theory, and even in finite graph theory.

My own experience has been as follows. Mathematicians who don't use
Gorthendieck universes and such, want to know what are the minimal axioms
needed to support the usual natural development of mathematics the way it is
generally developed and taught, WITHOUT resorting to coding and other means
of forcing the formalization to be tamer than the uncritical development.

MY ANSWER: Without a doubt, the right vehicle is ZC = Zermelo set theory
with the axiom of choice. You do lose some very interesting things, but
there is no coding involved. The mathematical community is generally very
satisfied with ZC, and to handle exceptions that arise, there is more than
enough confidence in Replacement to have chosen ZFC as the GOLD STANDARD.

> Here I want to illustrate the point that there is a remarkably
> exact fit between predicativism and core mathematics.  I would
> even say a *stunningly* exact fit!

It may look like there is, only if you are trying to force your view of
mathematics to conform to a strong philosophical bias.
 
> (I hasten to add that this is only weak evidence for the actual
> correctness of predicativism, although I do think it is some
> evidence in favor of correctness.)

Set theorist: the fact that uncountable cardinalities and large cardinals
give such good results is some evidence for their existence.

Finitist: the fact that you get so many good results using the full series
of natural numbers, even about big initial segments of it, is some evidence
for the existence of the full series of natural numbers.
 
> At least in my interpretation of predicativism, which I call
> "conceptualism", it is possible to treat things like the real
> line and the power set of N essentially as proper classes.

ACA0 people:  it is possible to treat things like infinite sets of integers
essentially as proper classes.

Set theorists: it is possible to treat things like {x: phi(x)} essentially
as proper classes.

> This problem is easily resolved if E is separable.

Set theorist (ZC level): You are doing coding, and mathematicians find this
unnecessary and ugly and pointless and distasteful. In ZC, no coding is
involved.

>In that case
> there is a countable dense subset and every continuous linear
> functional is determined by its values on that subset, so we
> can effectively restrict these functionals to the subset; this
> renders them sets of ordered pairs and we can then form the class
> of all such functions.

Set theorist (ZC level): Coding in action. Not satisfactory for core
mathematicians.
> 
> Second duals are trickier.
> 
> What I finally realized is that standard second dual techniques
> don't actually make use of the second dual as a Banach space.

Set theorist (ZC level): More coding. Not satisfactory for core
mathematicians because they want their spaces. You can't tell them sometimes
they can have their actual spaces, and other times they cannot. The story
will not be tolerated well by core mathematicians. It will look like an
attempt to conform mathematics to someone's philosophical biases.

> They only use individual elements of the second dual, which is
> predicatively okay!

See above.

> We have a name for the separable sequence space c_0.  We
> have a name for its (separable) dual l^1.  We have a name for
> l^infinity, the (nonseparable) dual of l^1.  We have no name
> for the dual of l^infinity.

Set theorist: NAMES? The only names core mathematicians want to hear about
are the names of the great theorems and the great mathematicians who prove
them and of the next prize winners and of the next hires.
> 
> We have a name ... We have a name ...  We have a name ...

Set theorist: Name, shame.

> We have no name for the dual of B(H).  K(H) and TC(H) are
> separable; B(H) is not.
> 
Set theorist: WHAT? No name now?

> We have a name ... We have a name ...

Set theorist: More names.

>We have no name ...

Set theorist: No name now? Why?
> 
> It goes on and on.

Set theorist: yes, it does!!!

>If you pick a standard Banach space and
> start taking duals, invariably the first one that doesn't have
> a standard name is the first one that's impredicative.

Set theorist: Gee, I wonder if that could be because you are doing the
naming and you are trying to avoid (ban) impredicativity?

>This
> is a *stunningly exact fit* between predicativism and core
> mathematics.
> 
Set theorist: You know, I found this pair of shoes in my closet, and it has
a "stunningly exact fit" to my feet. How could this be? It must be evidence
that those shoes are correct.

Harvey Friedman