FOM: Realism/philosophy

[Harvey Friedman]

This is a brief response to John Steel's posting of 19:33 AM 1/9/98. Most
of this touches on methodological matters not specifically in response to
John.

For most people, the realist postion gets more and more difficult to
swallow as one moves up to more and more unrestricted notions of set.
Consider the following sample of restricted notions of set. (I freely use
set theory in order to identify these restrictions).

1. Elements of the power set of the power set of the power set of the power
set of the empty set. (There are 16 of these).
2. Elements of the power set of the power set of the power set of the power
set of the power set of the empty set. (There are 2^16 of these).
3. Elements of the power set of the power set of the power set of the power
set of the power set of the power set of the empty set. (There are 2^(2^16)
of these).
4. Hereditarily finite sets.
5. Sets of hereditarily finite sets.
6. Sets of sets of hereditarily finite sets.
7. Elements of the cumulative hierarchy up to omega + omega.
8. Elements of the cumulative hierarchy up to the first fixed point of the
beth function.
9. Elements of the first power admissible set.
10. Elements of the cumulative hierarchy on ordinals that are accessible
(i.e., below the first strongly inaccessible cardinal if it exists).
11. Elements of the cumulative hierarchy.

Most people's realism gets severely strained somewhere in this list. As one
moves down the list, it is easier and easier to come up with basic
unanswered questions which require more and more (conflicting) technical
expansions of the usual obvious axioms one writes down. Almost everybody I
know of balks at realism for 11, except for some of the professional set
theorists.

As many of you are well aware, I have devoted a considerable chunk of my
life to a program towards establishing an equivalence between context 2 and
11 - at least as far as questions of consistency are concerned, and taking
into acount lengths of proofs - where the equivalence preserves
"intellectual intelligibility." I just got back from the Baltimore ASL
meeting where I talked on an equivalence between 4 and 11 in this sense.
The quality of this equivalence is now incomparably greater than anything I
have previously claimed, and I except further breakthroughs this year.

Exactly what this ultimately says about realism is of course totally
unclear. However one thing is clear: it would change the nature of the
debate forever and must be taken into account in any serious discussion.
That is my business. That's what I do for a living. Seeking those
scientific findings which change the nature of the debate forever. This is
red blooded FOMT (foundations of mathematical thought). Right now, in
replying to you, I am doing a tiny sample of what I call productive
philosophy - or pre foundations.

In fact, I try to do a lot of productive philosophy behind the scenes. I
have to. However, I emphasize that it is very focused in order to be
productive, and bears only some resemblance to the normal activities of
professional philosophers. Nevertheless it is best thought of as
philosophy. Frege did some of this. For me, it is behind the scenes. I
judge its quality totally in terms of what productive developments in
foundations emerge from it. I don't value it highly as an activity in and
of itself. I judge it solely in terms of its use in foundational advances.
I get a sense of what kind of conceptual issue is productive to press, and
what kind of issue is not productive to press. I press the productive ones.
I get very bored with the others.

This is why I value Frege greatly - but not, comparatively, Wittgenstein.
He looks weak compared to Frege and Godel.

I mention Wittgenstein because I was truly startled to find out in
Baltimore that he has become a kind of cult figure for some dear friends of
mine who have become somewhat jaded over the years about FOMT. One of the
main objectives I had for this e-mail list was to raise the excitement
level of FOMT throughout the world. I see a new golden age in FOMT,
comparable only to the 1930's.

Wittgentein - like many "great" philosophers - is a kind of mirror. You see
yourself in his writings. You make of it what you make of yourself. So
arguing about the merits of Wittgenstein with a Wittgenstein lover - as
well as in the case of other "great" philosophers - is like arguing about
the merits of your opponent. And generally the only situation in which A
wins arguments against B about the merits of B is when B is severely
depressed. But that's an unfair contest since, in such a context, B wants A
to win.

As an example, witness the following intense exchange at the Baltimore
meeting with a Philosopher, PHIL. The context was that Juliet Floyd had
just given a provacative historical talk about Wittgenstein and Godel.

HMF: What is this Wittgenstein worship about in connection with Godel?

PHIL: Godel was a great mathematician. But Wittgenstein was by far the
greater philosopher. There is no comparison.

HMF: But how can you say this? Look at the results!

PHIL: Godel didn't know what he was doing in philosophy. He was incompetent
compared to Wittgenstein.

HMF: All right, let me put it another way. Godel made an incomparably more
valuable and lasting contribution to the history of ideas than
Wittgenstein.

PHIL: Not at all. Just the opposite. Wittgenstein is incomparably more
important than Godel. Godel was *just* a mathematician.

HMF: This is just silly. What happened to the old xxxxx? [xxxxx is the name
of PHIL].

PHIL: You don't see the light because you (HMF) are nothing in philosophy.
Never were. Never will be. A philosophical zero. Like Godel. He was nothing
in philosophy. You are nothing in philosophy. I was always a much better
than philosopher than you ever were. You were never anything in philosophy.

HMF: You know what Godel said about philosophy to me? He said "what
philosophy?" Not a genuine subject yet. [I think Godel could have been
saying this primarily in the context of philosophy of mathematics and
related areas. I don't have precise enough recollection to be useful for
historians].

PHIL: Yeah, and that's precisely why I say that Godel was a philosophical
zero - like you!

So what started off as a discussion of the comparison between Godel and
Wittgenstein quickly shifted to a diatribe by PHIL about the comparison
between PHIL and HMF! In an instant, Wittgenstein became PHIL, and Godel
became HMF! I'm flattered.

I am making another posting shortly called "Attacking Godel." I claim to be
expert in attacking Godel, how to defend him, and how to extend him. Every
intellectual pioneer is attacked, defended, and extended. Not just Godel.
Every breakthrough suggests more possibilities than it realizes, and is
subject to all kinds of attacks. Wright Brothers fly a plane 100 feet?
Useless. Who would want to fly a plane for a few hundred feet? And it is
not cost effective. Besides, its dangerous, and walking is much better for
your health. People are generally satisfied with the current transportation
system.

Now only certain attacks against the Wright Brothers are productive and
lead to the next major steps. This is where the real advances are made.
This is how attacks against the Wright Brothers are judged.

I claim to know the difference between productive attacks and pedestrian
attacks. My attacks on Godel are far more focused and devastating than the
pedestrian attacks by Wittgenstein. I turn mine into FOMT. Wittgenstein
turned his into mirrors.

Hey Wittgenstein lovers. Getting jaded about Godel and classical FOMT? Just
contact HMF and he can help. Or contact simpson at math.psu.edu for a free
subscription to the fom or fom-digest and let's get it on. Let's see how
well you and Wittgenstein do.

Steel writes:

>We do
>use facts about real numbers to build bridges and send men to the moon.

What do you think of the following: we could replace all such uses
systematically with the theory of 64 bit arithmetic, together with
appropriate physical laws, experimentation, simulation, and modelling. The
latter four are necessary even if real numbers with infinite precision are
used. Furthermore this is the trend.

>The Realist has deep suspicions that anything interesting can be made
>of "qualified" existence for mathematical objects, such as existence "only
>in our imagination" ( or only in our minds, or only in our social
>conventions). Different kinds of existence amount to no more than the
>existence of different kinds of things.

But imagine the following development, which I think it is a good bet. The
theory of pictures, as I have outlined in previous postings on the fom,
gets developed as I indicated, with deep results about complete pictures
and the like. No technical jargon. No technical constructions. Just good
old fashioned completeness and unifying examples. Meanwhile, the realist
approach gets bogged down on issues like CH with technical proposals that
don't have the immediate fundamental character one is looking for. E.g., I
don't know why or why not "existence of a nontrivial countably additive
measure on all sets of reals extending Lebesgue measure" is or is not an
axiom that solves CH. Do you? Under these circumstances, doesn't moving to
a pictorial interpretation of set theory seem attractive? You have to agree
that, inevitably, researchers gravitate away from technical morasses and
towards the neat, simple, and clear. It inevitably colors their
philosophical outlook.

>One of the main differences is of course that one cannot attribute
>spacetime locations to sets in any useful way.

I'm not so sure. It would be interesting to try to do this in a useful and
informative way. Of course, an attempt to do so can be considered as
involving an expanded notion of set.

>Similarly, one cannot
>attribute causal relations to pure sets and physical objects (such as
>ourselves) in any useful way.

I'm not so sure. It would be interesting to try to do this in a useful and
informative way. Of course, an attempt to do so can be considered as
involving an expanded notion of set.

>The interesting things that CAN be said
>about sets are said in set theory, and the sciences which apply it.

What do you make of the currently grossly limited, trivial use of sets in
the sciences?

> There
>are lots of really useful things to be said in this domain--that's why
>society supports mathematicians.

Are you saying that society supports mathematicians because of set theory?

>Virtually everything said in this domain
>logically implies that there are sets. None of it is about how
>these sets are related to our imaginations or social conventions.

The theory of pictures has yet to be developed. Would you be excited about
working on it?

>   If one shifts to the question of how we know about sets, then perhaps
>imagination, in some sense, plays a role which it does not play in other
>areas. The Realist objects, however, to confusing this question with the
>question of whether there are sets, as seems to be done when one says that
>they "exist only in our imagination".

But this shift may be necessary to make further substantial progress in the
foundations of set theory - or, rather, the foundations of set theoretic
thought. Pictorial set theory is a branch of set theoretic thought, if not
a branch of set theory. And with regard to applications, it may well turn
out that any genuine application of set theory can be done equally well as
a corresponding application of pictorial set theory. It's just that
pictorial set theory may perhaps be developed in a much further and more
convincing way than ordinary set theory.