[FOM] Reply to Putnam 2

[Harvey Friedman]

Hilary Putnam gave the lead plenary talk at OSU on May 14, 2009 at my  
60th birthday celebration.

I replied with FOM posting http://www.cs.nyu.edu/pipermail/fom/2009-June/013780.html

Putnam's reply was presented in the FOM posting http://www.cs.nyu.edu/pipermail/fom/2009-June/013823.html

Here is my second reply.

> Dear Harvey,
>  Thank you for the clarification, which I will save. It is very
> nicely stated. Yes, we agree about the unjustified claims of some
> category theorists. And your position is happily more nuanced than I
> thought. I do think though that the realism-antirealism issue is one
> of intrinsic intellectual interest. But even there we may not
> disagree, because I think it does make a difference to systematic
> knowledge in an indirect way.

> ...
>
>            Be well, and keep in touch,
>                  Hilary

I agree that a number of philosophical issues such as realism- 
antirealism are of intrinsic intellectual interest.

The contentious issue may be just how one ought to productively go  
about tackling issues like realism-antirealism. Or whether a  
particular approach to such issues is productive.

I remain somewhat skeptical about there being deep insights into  
issues like realism-antirealism which do not take into account or say  
something significant about or relate in some significant way to our  
present body of systematic knowledge.

There may be such purported deep insights, which have not yet been  
interwoven with systematic knowledge. Perhaps we would disagree about  
their value. But as soon as its implications for or connections with  
systematic knowledge are clarified, I would immediately recognize  
their value.

Generally speaking, when looking at what philosophers say about such  
issues, I do the following kinds of things:

1. Firstly, see what the discussion looks like if it is specialized to  
realism-antirealism in mathematics. This is the obvious and natural  
move for me, since I have specialized in the foundations of  
mathematics, at least up to now. This specialization of the discussion  
usually makes sense, and various points can be strengthened or  
refuted. Sometimes this leads to a new kind of investigation into the  
foundations of mathematics. Usually this is very remote from the  
points being made in the philosophers' discussion - a clear case of  
where philosophy is being used by me as a method, and not a subject.

2. Secondly, I look to see if the discussion forms the basis - or can  
be molded or modified to form the basis - for systematic knowledge in  
the realm intended, which is often physical theories, linguistics,  
probability/statistics, ethics, mind, etcetera. Here my competence to  
judge this, especially on the fly, is quite limited, but I hope  
nonzero. For most philosophy talks, I just don't see it, but I  
(almost) never leave the room before the talk is finished, and usually  
stay for the question period.

> ... I think
> that, for example,  the realism issue is important for science (I
> argue this, for instance, in the paper “Science and Philosophy”—
> forthcoming in a book of papers of mine edited by Mario De Caro and
> David Macarthur).

I am looking forward to seeing this paper.

> For example, whether one is a realist or an instrumentalist makes a
> difference to the paradigm science of physics itself, and not only
> to what philosophers say about physics. I think that when anti-
> instrumentalism began to defeat logical positivism, and some
> physicists—especially J.S. Bell—tried to understand physics
> realistically, saying “We want to understand quantum mechanics not
> just as a prediction tool, we want a picture of the world, we want
> to make sense of a world in which this crazy tool works”, a great
> many good things happened in physics that would not have happened
> otherwise. Bell was interested in all the foundational approaches—he
> was interested in Bohm’s interpretation, he was interested in GRW’s
> (i.e. Ghirardi, Rimini and Weber’s) spontaneous collapse theory. I
> don’t believe that the so-called “Many-Worlds interpretation” of
> quantum mechanics works, but that  attempt did lead to the discovery
> of the decoherence theorems which certainly are going to be part of
> any explanation of why the macroscopic world we experience is as it
> is, and that interpretation was proposed because its inventors, Hugh
> Everett III, and  Cecil M. DeWitt were willing to take the question
> as to what quantum mechancs actually says about reality seriously.
> And the list goes on and on.

Within foundations/philosophy of mathematics, there are some similar  
stories. Cantor's development of set theory represented a bold realist  
approach to mathematics, where, e.g., there is a supposedly clear  
notion of function independently of expressions or of modes of  
definition. On the other hand, things obviously got out of hand with  
Frege's overly general comprehension axiom, resulting in Russell's  
Paradox. But then we have Russell (simplified) type theory, followed  
by Zermelo and Zermelo/Frankel set theory. Such axiomatic set theories  
are reasonably trumpeted as victories for the realist position. After  
all, where can those axioms come from other than from realist truth?

As you know, I am a logical whore, so I am agnostic on such issues as  
realist-antirealist. But I focused on the fact that when teaching set  
theory, it is common to motivate the axioms of set theory by saying  
that they are all obvious in the finite except for Infinity.

So this could form the basis of an antirealist explanation for ZF -  
provided one can show that ZF can be axiomatized by taking "all  
'axioms' true in the finite" and merely adding Infinity.

THEOREM (see http://www.cs.nyu.edu/pipermail/fom/2009-January/013343.html) 
. The following are logically equivalent.
i. All comprehension (function) schemes true in the hereditarily  
finite sets (demonstrably true in a very weak set theory) whose  
defining property uses at most one quantifier (i.e., A or E).
ii. All comprehension (relation) schemes true in the hereditarily  
finite sets (demonstrably true in a very weak set theory) whose  
defining property is EA.
iii. All comprehension (relation-function) schemes true in the  
hereditarily finite sets (demonstrably true in a very weak set theory)  
whose defining property is E...EA.
iv. Extensionality, pairing, union, power set, separation,  
replacement, and no epsilon cycles. I.e., all axioms of ZF without  
Infinity, with a weakened form of Foundation.

So ZF (with weakened foundation) can be axiomatized by i, ii, or iii,  
together with Infinity.

This clearly gives the antirealist a boost vis a vis the realist, in  
foundations of mathematics. So I am a good whore.

The realist here has some attacks, including the obvious: what about  
the axiom of choice and full foundation - they are not covered! The  
antirealist here has some defenses, including the obvious: prove a  
sharper form that treats the axiom of choice and full foundation.

There is a lot more to say about this situation, but I just wanted to  
give a recent example of what I have in mind.

> Thus, if the question of realism and anti-realism is a metaphysical
> question – and at least since Hume and Berkeley it has been a
> metaphysical question (we did not have the modern kind of anti-
> realism in the Greek time, but for sure that question has been with
> us for three hundred years—there’s a straight line from Hume to
> Mach, and  it entered physics itself with a vengeance)—then this
> metaphysical question is one that cannot simply be dismissed as a
> philosopher’s “confusions”, “misuse of language”, or whatever. And
> if it isn’t a metaphysical question?—but I don’t know any other name
> for that sort of question.

So why don't I give such examples of my own for physical theories?

I started my career with the intention of doing this sort of thing in  
all areas of systematic knowledge. I rightly thought that it was best  
for me to start with mathematics - and then move on.

Unfortunately (or fortunately, we don't know which yet), I never got  
past mathematics (and related excursions into computer science). But  
more recently, it seems like I am having some sort of late music  
career, and we will see if that gets integrated with my mathematical  
work and foundational perspective.

At least I have the good intention of dealing seriously with physical  
theories during 2010-2020.

> ...And I think that Wittgenstein himself was deeply in
> the grip of a metaphysical picture—for example, when he claimed, as
> he does, on my reading at least, that the only genuine kind of
> necessity is linguistic necessity.

 From what little that I know of W's position on necessity, I think it  
likely that a careful analysis will yield important new areas of  
systematic knowledge. This would include a new kind of succinct  
foundation for actual computer systems, as well as various delicate  
investigations into circularities in mathematical logic; e.g., using  
and to define and, etcetera. In fact, when I first met Hilary in Fall,  
1964, as a first semester freshman at MIT, I was focused on the  
circularities in logic, and asked, innocently, "how does logic start?"  
Hilary said that he didn't have anything like a complete answer to  
this question, but rightly gave me references to a couple of books  
that do the basic setup with much more care than normal.

> I am afraid the great majority of
> Wittgenstein’s unpublished remarks on the foundations of mathematics
> are, frankly, junk,. (Not, however, the famous remark on Gödel
> theorems – that’s been widely misunderstood.[i]) What finally led me
> to this harsh verdict was studying his remarks about Dedekind cuts,
> his remarks about Cantor’s proof of the non-denumerability of the
> real numbers, and his remarks about what it means to say there are
> infinitely many integers. When Wittgenstein says “I want to deprive
> set theory of its charm”, one naturally thinks that what he wants to
> give up is just Zermelo Fraenkel set theory. (Not that I would
> agree, even if that were all he meant.)  In fact it turns out that
> what he includes under “set theory” includes Dedekind cuts (hence
> the intermediate value theorem of the calculus), includes the
> standard treatment of the theory of real variables, includes the
> heart of classical mathematics.
>
> How could a great philosopher, one who urged us constantly to be
> sensitive to different “forms of life”, devote perhaps fifty percent
> of his unpublished writing to mathematics, without ever seeking to
> learn anything about what the mathematical form of life is? For
> Wittgenstein Cantor’s Continuum Hypothesis is “metaphysics” in the
> pejoritive sense! —But it seems to me that the metaphysical
> questions; “What is going on in mathematics?”, “Is it really just
> ‘’grammar?”, “Are we merely following certain linguistic rules and
> engaging in certain linguistic practices?”, or “Is there an
> objective truth in mathematics that goes outside of that?” (which is
> my position), are important and rationally discussable. In my view,
> whenever somebody sets out to be consistently “anti-metaphysical” he
> ends up doing bad metaphysics. I believe this is true even of
> Wittgenstein. This seems to be a very profound piece of evidence
> that some metaphysical questions are inescapable.

What you are complaining here about W, is just the kind of thing that  
I complain about with philosophers, generally - although the W case  
here is more blatant than what I often see in mainstream philosophy.  
That is, not informing the discussion sufficiently with the great  
systematic subjects.

Nevertheless, even here, with this W "junk", there is probably  
something interesting from my perspective - if one is willing to dig  
very deeply, and not be concerned about the fact that it is junk on  
reasonable readings. It may suggest alternative readings that bring it  
to life - even if those alternative readings were not even imaginable  
to W.

I find this sort of thing even in classical piano music. I, like  
Vladimir Horowitz, tend to consider classical piano music as an  
opinion of the composer concerning "the musical truth". (I am putting  
some words in Horowitz's mouth - he did say "there are a lot of  
dynamic markings of the composers - [even Beethoven!] - that I don't  
agree with").

> To be sure, the way we cut up cultural space into separate fields
> changes with time. It is well known that questions that were one
> time considered to be philosophical questions later became
> scientific questions. That doesn’t mean that all the questions we
> presently call “philosophical” will eventually be swallowed up by
> some special science. At least at present, that seems to me a
> utopian fantasy. But the fact that it is no longer be tenable that
> there exists a special field of metaphysics, doesn’t mean that
> questions that were traditionally regarded as metaphysical don’t
> continue to interest us. They interest us even when the
> metaphysicians are wrong. For example, consider the premise of
> Kant’s philosophy, the idea that the laws of geometry are a priori
> and unrevisable and yet they refer to objective space, the space in
> which we live and move and have our being, and not just to an “ideal
> space”. I think he identified a real problem, but the fate of that
> problem turned out to be very different then he anticipated.
> Nevertheless, he asked the right questions. When I say there are
> insights in traditional metaphysics, I mean precisely this.

I think of issues like "realism-antirealism" as a gift that keeps on  
giving. It will always be there, but have a steady incessant stream of  
spinoffs in the form of new (kinds of) systematic knowledge.

> [i] See Juliet Floyd & Hilary Putnam (2000). ‘A Note on
> Wittgenstein's "Notorious Paragraph" About the Gödel Theorem’.
> Journal of Philosophy 97 (11), pp. 624-632.

I would like to see some FOM postings analyzing this paper.

Harvey Friedman