[FOM] predicative foundations

[Harvey Friedman]

On 2/15/06 7:12 PM, "Nik Weaver" <nweaver at math.wustl.edu> wrote:

> As I've explained,
> the predativist prohibition on circular definitions is a simple
> consequence of a disbelief in a platonic universe of sets.
> Friedman has a right to be unaware of this justification but it
> is unwise of him to baldly claim that no justification exists.

PROHIBITION! Is that related to banning?

Predicativity is NOT a simple consequence of disbelief in a platonic
universe of sets. It is highly complicated, far more complicated than, say,
Z_2. 

Predicativity is also far from any kind of unique consequence of disbelief
in a platonic universe of sets.

In addition, you have a view of what predicativity is that is at substantial
odds with the standard accepted view. One reason for this impasse is I doubt
that there is any notion of "predicativity" that is sufficiently clear to
decide this dispute that you have with the standard setup.
 
> One can also question why we should accept the structure of the
> natural numbers as primitive ---

This leads to another, equally natural, or perhaps more natural, stopping
place. A huge amount of mathematics can also be done in finitist and even
ultrafinitist frameworks.

>either why we won't accept more
> complicated sets, like the power set of N, as primitive, or why
> we shouldn't stop earlier.

>The answer to the first question is
> easy: you can't even say what P(N) is without resorting to
> platonistic ideas.

It's not hard to make up stories that creep up to more than the standard
setups for predicativity. For instance, you have alreay done so. Under the
old discussion of predicativity, you have already allowed "Platonistic
conceptions" in the door. One can also create stories surrounding the first
recursively inaccessible ordinal, admitting yet more "Platonistic ideas".
There is also the work of Clifford Spector, which has a certain kind of
coherence also, which goes up through Z_2 with heavily non Platonistic
ideas. 

Of course, there is no convincing objection with what you call "platonistic
ideas". Also relevant is that P(N) is hugely weaker than the full cumulative
hierarchy. Conflating the two is an error.

>I can describe N in a metaphysically
> uncontroversial way in terms of making marks on paper.  But
> there is no, or at least no obvious, way to say what P(N) is
> without invoking "sets of numbers" as abstract entities.

Yes, I agree that P(N) is higher up the hierarchy than N. But it is child's
play to create controversy surrounding the idea of indefinitely long series
of marks on paper. Wittgenstein was good at this sort of thing.

So already, there is huge controversy that can be created (and has been)
concerning finite series of marks on paper. If you can't get past that
without substantial controversy, how can you claim to even get to PA, let
alone what you call "predicativity"?

>In
> the opposite direction, in principle I regard finitism as
> coherent and defensible.  (But I must add that for reasons
> explained in my post
> http://www.cs.nyu.edu/pipermail/fom/2005-October/009263.html
> I think true finitists have to use intuitionistic logic,

HAVE TO? What does that mean? Related to banning? Certainly they don't have
to.

>and
> I think that purported finitists who are willing to use
> classical logic in first order arithmetic betray an implicit
> acceptance of N as a well-defined structure.)

BETRAY? This is just totally unconvincing. Certainly there is no betrayal.
> 
> Now one major objection to predicativism is that it renders
> large portions of core mathematics illegitimate and hence
> should be at least viewed with extreme skepticism.  ...
>now known to be false, and that
> in fact, generally speaking, core mathematics (for example:
> anything that might be covered on any of the graduate qualifying
> examinations in my department) can be done predicatively.

Qualifying exams in some Departments probably touch on closed sets of reals
and perfect sets. Also, you still have to modify the mathematics involved to
make it predicative. So the same can be said of much more restrictive
positions than predicativity. You also modify the mathematics there, and
after that, it also matches well.

>As
> a result, predicativism is actually in much better accord with
> core mathematics than any other major foundational stance.

This is misleading. Depends on "accord" and "stance".

Suppose you want to do math the way it is generally written. A very good
choice, incomparably better than predicativity, would be ZC = Zermelo set
theory with choice. You will be missing something very interesting things:

THEOREM (ZFC, but not in ZC). Let S be a set in the plane that is symmetric
about the line y = x. Then S or its complement has a Borel selection.

And some other results in Borel selection theory.

Nevertheless, right now, you don't lose too much by taking ZC instead of
ZFC.

> This last point seems to be the only one that has made an
> impression on Harvey Friedman, and he apparently takes it to be
> my central justification for predicativism.

Other "justifications" are merely reiterating that you think you have found
a nice stopping place among myriads of stopping places that are, in fact,
much simpler and clearer to point to. So other "justifications" are really
not justifications at all for any special status.

> Well, that is an odd claim.  There are a few things that one can do
> predicatively but not in ACA_0 (Kruskal's theorem, the consistency
> of ATR_0), 

Kruskal's theorem cannot be done predicatively. This is an old result of
mine. Con(ATR0) is not normal mathematics.

>but the vast majority of predicatively valid normal
> mathematics is already available in ACA_0.

Hence one is better off stopping at ACA_0 from the point of view of
mathematical practice. It's much cleaner, clearer, attractive, etcetera.

>This might be significant
> if there were a coherent foundational stance corresponding to ACA_0,

Child's play to come up with one that looks as attractive as tortuous
involved controversial ones for predicativity. You must have some special
meaning for "foundational stance". What is a "stance"? Does this involve
banning issues? We now agree that any talk of banning is silly.

> because then you would have an alternative that is at least competitive
> with predicativism in terms of accord with normal mathematical practice.
> 
> I have to admit I've never heard of anyone advocating ACA_0 as
> a basic philosophical stance.

You can make up myriads of philosophical "stances", at many levels of the
hierarchy. I would recommend that you abandon "philosophical stances" as
counterproductive, leading nowhere but argumentation.

We do not know anything near enough about f.o.m. to suggest that we can
focus on a handful of "philosophical stances" with any kind of confidence.
One is merely reduced to making pseudo claims that something or another is
OK and something else is not OK. This is simply pointless.

DISCLAIMER: At some point in the future, we MAY know enough about f.o.m. for
it to make sense to seriously talk about "correct philosophical stances",
but that time is not in sight.

>Yet you tell me there is a "nice
> story" in its favor, which "has proponents both mathematically
> and philosophically".  Can you say who some of those proponents
> are, and what that nice story is, in the case of ACA_0?

Any story you can make about some precise stopping place for predicativity,
and I can make a much better story about stopping at ACA_0.
> 
> Incidentally, I have been using the terms "normal" and "core"
> interchangeably.  What is the distinction you draw between them?

The mathematics community has a definition, reflected somewhat well in their
qualifying exam topics. Kruskal's theorem and the graph minor theorem are
part of NORMAL mathematics, but not part of CORE mathematics. This is well
in accord with math community usage.
> 
>> 2. Give me one example of a theorem aobut finite objects, in what
>> you call "normal" or "core" mathematics (I make a distinction
>> between the two), that is not handled in PA, or even EFA.
> 
> I know that you know plenty of examples of such theorems.  Are you
> quizzing me?  Anyway, the question is irrelevant.

There aren't any at the moment. I believe that EFA is enuf for all finite
core mathematics, today.

> This comment only makes sense if you think that I'm suggesting
> predicativism should be accepted *because* it accords well with
> normal mathematical practice.  Of course I never said this or
> anything remotely approximating it, and have never even heard
> of anyone who advocated such a position.

But accordance with mathematical practice is all you have. There is nothing
compelling about the stopping place intrinsically. By looking at the
controversy about where to stop, between what you write and the classical
accepted systems, you can see this.
> 
> I find the underlying attitude (argument from authority --- how
> dare I defy Godel) distasteful.

Explaining why you disagree so condidentally with Kurt Godel is a good and
fair question that I now reiterate. Where and why did Godel go wrong?

I certainly don't find a lot of what Godel said compelling, but I certainly
don't reject any of it. I think you do reject it.
> 
> Incidentally, as a young man Godel held anti-realist views.

Do you think that  he ever subscribed to one of the controversial stopping
places for predicativity?

Harvey Friedman