[FOM] predicative foundations

[Nik Weaver]

My preceding message is a response to Harvey Friedman's
assertion that "The claims that predicativity has some special
place in the robust hierarchy of logical strengths ranging from
EFA through j:V into V are unjustified."  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.

One can also question why we should accept the structure of the
natural numbers as primitive --- 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.  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.  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, 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.)

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.  This view
was exhibited, apparently with approval, in a recent post by
Curtis Franks.  So it is important to emphasize that this
early view on predicativism is 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.  As
a result, predicativism is actually in much better accord with
core mathematics than any other major foundational stance.

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.  My preceding message
together with this one, if one took the time to read them, should
make it clear that this is not the case.  This basic misunderstanding
leads Friedman into all sorts of irrelevant directions, which I will
now address.

> On can stop a lot earlier than "predicativity", say, stop at
> ACA0 or RCA0, or one can stop somewhat higher than "predicativity",
> say at one inductive definition, or Pi11-CA0. Or one can stop even
> higher at, say, the theory of a recursively inaccessible, or what
> have you.
>
> All of these stopping places, and many more, have very "nice"
> stories. All of these stories have advantages and disadvantages.
> These advantages and disadvantages make sense and have their
> proponents, both mathematically and philosophically.

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), but the vast majority of predicatively valid normal
mathematics is already available in ACA_0.  This might be significant
if there were a coherent foundational stance corresponding to ACA_0,
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.  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?

> For example, Weaver did not answer the following questions that
> I raised in my last posting responding to him. I will raise them
> again.

Forgive me for not responding immediately.

> 1. Give me one example of a theorem in what you call "normal"
> or "core" mathematics (I make a distinction between the two,
> and I don't know if you do), that is handled in "predicativity"
> but not in ACA0. I asked if there was a significant range of such.

I just did so, and I also indicated why this is only a relevant
question if there is a basic philosophical stance that is
represented by ACA_0.  You claim that such a stance exists.

Incidentally, I have been using the terms "normal" and "core"
interchangeably.  What is the distinction you draw between them?

> 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.

> 3. Clearly 1,2 relate to the issue of how special the stopping
> place of predicativity really is. E.g., why not stop earlier?

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.

> I don't agree that the world of sets is fictional, and I don't
> agree that the word of sets is not fictional. We just don't
> know nearly enough about f.o.m. to make such a determination.
>
> There are obviously equally good arguments on both sides of this.
> Godel took the non fictional view. Is there something that compels
> you to so confidently disagree with Godel? On what basis?

I outlined an answer to this in the preceding message.  For more
extensive discussion, I refer you to all of the messages I've
been posting on this list for the past several months.

I find the underlying attitude (argument from authority --- how
dare I defy Godel) distasteful.

Incidentally, as a young man Godel held anti-realist views.

> Set theory arose out of the (at least then) core mathematical
> activity of working with closed sets of real numbers, in
> connection with trigonometric series. Do you wish to repudiate
> the notion of closed sets of real numbers, or trigonometric
> series in some way?

I've never advocated "repudiating" anything.

> There is no doubt that the exceptions - or exceptions you wish
> to marginalize - will grow as time proceeds.

Nor do I wish to marginalize anything, and your repeated assertions
that I do, despite my denial and your inability to identify
anything I've written that would indicate this, is unhelpful.

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