friedman at math.ohio-state.edu
Sat Jan 17 22:02:25 EST 1998
This is a reply to the post of John Steel, 1:23PM, 1/15/98.
> 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.
The way I use the terminology, progress in f.o.m. is always of the kind we
most value, whereas progress in ph.o.m., at least for me, is measured by
its impact on f.o.m. E.g., very careful disucssion of the nature of
mathematical proof did lead to the invention of predicate calculus. The
former was, in my use of terminology, some ph.o.m. which turned into some
This leaves the question: is there intrinsic interest in ph.o.m.
independently of what it does for f.o.m.? Well, people in ph.o.m., or in
philosophy generally, seem to not talk in terms of solutions to their
problems. On the other hand, they generally don't accept the idea that
philosophy is to judged by its impact on f.o.m. or any other foundational
study. I am uncomfortable with this. I am so much more moved by
philosophical discussion that leads into foundational advances - not
necessarily mathematics - than philosophical discussion that doesn't, no
matter how clever it is, no matter how incisive the debate, that for all
practical purposes, I am tempted to take the hard line: Philosophy is to be
measured by the foundational advances thereby generated, be it in
foundations of mathematics, mechanics, relativity, quantum mechanics, law,
computer science, statistics, psychology, finance, economics, political
science, ethics, musical expression, evolutionary biology, etcetera. I
would like to hear from philosophers about this.
>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.
Basically yes. Only I would put it a little differently: replace
"mathematical problems" with "problems in f.o.m." For an interpretation of
f.o.m., see my posting of 11:55PM 1/14/98.
> 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 daily support system for mathematicians, Realism is just fine. Since
such a little bit goes such a long way (in terms of axiomatizations), at
this time in the history of mathematics, for all practical purposes there
is no problem in the day to day activity of mathematicians. Even if there
were to be a shift in this regard - say, because one of my big programs
works to an extreme degree - one will redefine restricted forms of
mathematics where again there are no practical problems with realism.
However, matters get strained when we focus on the exceptions, which,
unfortunately for set theorists, come rather quickly in its (set theory)
history, and rather severely. Fortunately for the mathematicians, the
continuum hypothesis is totally against the spirit of investigation of
mathematics today and virtually since its inception. I don't deny that the
difficulties with such set theoretic problems probably had a role in
shaping this spirit away from such things, of course. But just look at the
different intellectual style and feel that "concrete" mathematics has from
seriously set theoretic mathematics.
>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
I think that you underestimate how intriguing it is. After all, you claim
to be working on the continuum hypothesis. Certainly nobody other than a
small minority of professional set theorists are working on it at all. And
you even have confidence that you are not wasting your time investigating
it from a Godelian naive realist point of view. Wow! And you are a naive
realist all the way through the cumulative hierarchy, right? So, it is not
a matter of choice of humans how far to go up the cumulative hierarchy.
There is merely "all the way" and any proper initial segment that one may
for special reasons restrict attention to, right? I.e., there definitely is
a measurable cardinal or there definitely is not a measurable cardinal; and
in fact, we know that there is a measurable cardinal, right? I think that
you're beliefs are very intriguing.
> 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
OK, you've cut away a lot of conceptual something or another. But you are
left with funny looking positions on the nature of questions such as
continuum hypothesis, Souslin's hypothesis, the existence of countably
additive extensions of Lebesgue measure on all sets of reals, the existence
of measurable cardinals, etcetera.
> 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. ... This is a place where phil. of math. might
>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
I like this. I agree that we need an accompanying analysis of different
notions of solution to the Continuum Problem. But I think that in a way,
you are backing off of Godelian naive realism.
> 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."
The "theory of pictures" or "pictorial set theory" quote is meant to be
another (later) statement of (aspects of) neo relativism, which is looking
more and more attractive to me given the alternatives that I see. But I
need a breakthrough idea here. Where is it?
> Harvey didn't explicitly subscribe to neo-relativism, but he does say
>"...[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
> 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.
Yes, but the standards have to be very high. We can't easily go from the
totally intriguing, thrilling, and focused axiomatic set theories - even
with some large cardinals - to something whose statement involves all sorts
of technical notions. If the technical notions are the unique way to say
something - well then, it can be said without the technical notions.
>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.
I think you underestimate just how bad this is for your point of view. You
may have to eventually compete with things being done in a variety of
subjects inside and outside f.o.m., and inside and outside mathematics,
which are understandable to the layman compared to what you have in mind,
and you may suffer greatly in the inevitable comparisons. It's a serious
problem for your approach.
>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.
Ah, now that's more like it! If you are going to devote your life to this,
don't lower your standards!
>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".
In pictorial set theory, it is hopefully going to be a theorem that all
pictures are compatible in some interesting sense. E.g., if they contain a
base part about N, then the sentences that follow about N in the pictures
are always compatible under inclusion.
>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.
I don't think there is any "picture" that "solves" CH, and that should be a
theorem. However, you may think so. And what I worry about is that there is
a picture of yours that proves that c is real valued measurable, and a
picture of yours that proves MA. They are incompatible. In fact, I can't
tell whether or not, from your point of view, which of these two statements
is true and which are false. Or how you intend to decide them.
Steel quotes me as follows:
>>"There is absolutely no attempt to say that one has
>>completely defined or delineated any of these concepts, or even stated any
I was saying that under pictorial set theory, or neo-relativism, there are
simply pictures, and what principles are inherent in pictures, and what
logical consequences follow from such principles. There is no concept of
true/false. Of course, there is the concept of "true in this picture," but
we don't have the disjunction, "true in this picture" or "false in this
picture." So CH is neither true or false. There is an identification of
"true in this picture" from "follows formally from the principles inherent
in this picture." Practical consequence: from this point of view, one stops
working on CH, and instead worships the theorem that CH is not true in any
picture, in this formal sense. And then extends such results to a massive
number of other set theoretic problems. Another practical consequence: one
can still talk like a realist about massive numbers of things. And, there
is the possibility that mathematicians may be very happy indeed because of
the following. All concrete problems they encounter can be solved in a
picture. Furthermore, all concrete problems in concrete areas of
mathematics that I develop and get them to appreciate and value, also are
solved in a picture. Then the fact that the CH can't be solved by pictures
becomes a mere curiosity to the working mathematician.
>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.
Some people think we already have this, but I think you are dubious. E.g.,
ZFC + large cardinals + V = inner model, versus ZFC + large cardinals +
lots of generic objects exist.
>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.
I don't see why, from your point of view, you say that this would solve the
Continuum Problem. Certainly not in any way Cantor or Godel envisaged.
>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.
That is the feeling that many people on the fom are having. But I thought
you had full confidence that it did express a definite proposition. Are you
now saying that "you don't know?" And you just want to have a positive
attitude and continue your investigations?
>5. It may be that no such universal framework theory exists, as Harvey
>seems to be maintaining.
I am not quite sure what I am maintaining according to this manner of speaking.
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