FOM: Friedman on Realism/Philosophy
steel@math.berkeley.edu
steel at math.berkeley.edu
Thu Jan 15 16:23:36 EST 1998
This is a reply to a post of Harvey Friedman from 1/12/98.
First, I agree that the philosophy of mathematics which leads to
mathematical progress is by far the most interesting kind. Without this
connection, philosophy of math tends to fall into an endless and fruitless
war of metaphors. In addition to Godel, Hilbert is a good example
of the value of connecting philosophy of math. to definite mathematical
problems. Hilbert made a great contribution by being wrong in a reasonably
definite way. Had he stuck to vague pronouncements on instrumentalism, he
might never have been proven wrong, but he would have contributed much
less. I suspect most people on this list agree with this point of view.
(?)
To my mind, Realism in set theory is simply the doctrine that there are
sets, and that these sets do not depend causally on us (or anything else,
for that matter). Virtually everything mathematicians say professionally
implies there are sets, and none of it is about their causal relations to
anything. (I liked Bill Tait's line from a while back, that Realism is a
defense of mathematical grammar.) As a philosophical framework, Realism is
right but not all that interesting. Both proponents and opponents
sometimes try to present it as something more intriguing than it is, say
by speaking of an "objective world of sets". Such rhetoric adds more heat
than light.
Ironically, it is Realism which makes the important use of Occam's
razor, by cutting away the conceptual clutter that goes with the
assertion that sets in some way depend on us (or idealized
intelligent beings).
( While we're at it, I'd like to make two modest proposals which, if
adopted, might cut short some of the pointless war over metaphor. Namely,
henceforth no one defending or attacking Realism is allowed to bring up
theology, and most importantly, no one is allowed to use the phrase
"out there". Even if you only substitute equivalent phrases or analogies,
at least you'll be off automatic pilot for a minute.)
I think one "shifting boundary of what is understood" at which
the "character of mathematical truth" is "problematic" lies at the
level of Sigma^2_1 sentences, and is marked most famously by the
Continuum problem. ( The quotes are from Martin Davis' characterization
of fom.) This is a place where phil. of math. might contribute to
mathematical progress--a solution to the Continuum Problem will
probably need some accompanying analysis of what it is to be a solution
to the Continuum Problem. Realism, by itself, doesn't get us anywhere
here. I think it does, however, contribute to a willingness to face the
problem.
Concerning what he aptly calls the "pre-foundations" relevant to the
Continuum Problem, Friedman says:
"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. ....Under these circumstances, doesn't moving to a pictorial
interpretation of set theory seem attractive? "
This hearkens back to his post of , in which he describes
"neo-relativism" as a philosophical approach toward what it means to
be a solution to the Continuum Problem:
"3. Neo relativism. There is no uniform objective reality to mathematical
objects. There is only the following spectacular phenomena. One identifies
certain pre-formal concepts which are yet to be explained and worked with.
Then one makes some explanatory remarks connecting these concepts with
other concepts that have been long been discussed and worked with by
people. After these explanatory remarks, one then enunciates a number of
intuitively clear principles about them. These principles are not to be
thought of as evident or true - but rather as explanatory as to how to
work with these concepts. There is absolutely no attempt to say that one
has completely defined or delineated any of these concepts, or even stated
any truths. Rather, these principles attempt to fix aspects of a picture
that these concepts invoke, so that people can work together on a common
ground. Furthermore, these principles try to be strong enough so that all
fundamental aspects of the picture are fixed, in order to facilitate
communication. So that different people will not have different pictures.
One finds that this process is unexpectedly and spectacular successful -
that a little bit goes a long way as discussed above. That the usual
axioms we work with in f.o.m. go a very long way. But not far enough for
CH. Thus this process is only more or less successful. One has relative
success, depending on the context. In arithmetic and real/complex algebra,
etcetera, it is much more successful than in the set theory of sets of
real numbers. The CH is a an instance of failure."
Harvey didn't explicitly subscribe to neo-relativism, but he does say
later:
"...[I expect someday it will be] a THEOREM that many of the set theoretic
statements such as CH cannot be settled through any coherent conceptual
picture."
I am sympathetic to some but not all of this. Here are some comments:
1. One man's technical jargon is another's deep insight. Immediacy is not
necessarily the same thing as being fundamental. We are probably past the
days when new axioms for set theory will be as easily understandable to
the layman as the Zermelo axioms. Moreover, I think metamathematical
considerations will play a role in guiding us to and justifying those
axioms. The evolution of set theory is likely to be much more
self-conscious in the future. Nevertheless, new fundamental principles may
emerge, and be rationally justified.
2. One ambition in foundations is to construct a universal framework
theory in which all mathematical theories can be naturally interpreted.
We want to have ONE picture, so that, as Harvey put it, "people can
work together on a common ground". If truly different "pictures" arise,
the problem of putting them together will become of immediate
importance. In fact, as far as I know this just hasn't happened. One
can get different natural interpretations of the language of set theory
by "restricting" the notion of set, but this is not a case of
incompatible "pictures" emerging.
3. To advocate phi for inclusion in such a framework theory committs one
to the view that phi is true. In this connection, I don't understand the
lines:
"There is absolutely no attempt to say that one has
completely defined or delineated any of these concepts, or even stated any
truths."
To say "snow is white" seriously committs one to ""snow is white" is
true". (Though not perhaps to the assertion that the concept of snow has
been completely delineated.)
4. I think it may be that the meaning of the language of set theory is
not completely determined in some sense. ( But if so, that needs to be
EXPLAINED. The analogy with vague general terms like "bald" is just not
close enough, so far as I can see.) One way this could manifest itself
is in the emergence of equally good universal framework theories
deciding CH in different ways. Such theories would be naturally
interpretable in one another, in virtue of universality. Developing any
one such theory would be the same as developing them all, so in a
practical sense finding any such theory would solve the Continuum Problem.
Nevertheless, one could maintain that in this scenario, the formal
sentence "CH" does not, as the language of set theory is currently used,
express a definite proposition.
5. It may be that no such universal framework theory exists, as Harvey
seems to be maintaining.
John Steel
More information about the FOM
mailing list