FOM: determinate truth values, coherent pragmatism

Martin Davis martin at
Tue Sep 5 02:15:35 EDT 2000

At 08:31 PM 9/4/00 -0400, Harvey Friedman wrote:
>Reply to Davis 1:43PM 9/4/00:

>I, personally, am not comfortable with any notion of "determinate" that is
>independent of any specification of how one can, even theoretically,
>determine. That doesn't mean that I reject the concept out of hand. It just
>means that I think the concept needs clarification.

In the natural sciences, the means for determining the truth of specific 
propositions is often available long after the proposition itself arises. 
Your position is reminiscent of the failed positivist program that sought 
to equate meaning of empirical propositions with the availability of means, 
at least in principle, of verifying them. The world we live in and the 
world of mathematics are both vast complex entities in which we with our 
finite minds can make only modest inroads. But to identify what is out 
there with what we are able to ascertain (even in principle) is entirely 

Anyhow, in the cases we are discussing, Goedel already did indicate how one 
could "theoretically" hope to determine. By deducing from new axioms for 
the adoption of which there are compelling reasons. See the work of Harvey 
Friedman. And of course there's the famous example of Projective 
Determinacy and descriptive set theory.

> >I find myself optimistic about the direction of Woodin's
> >recent work on CH. At last an effort is being made to get around the
> >plethora of models of set theory the forcing method made it possible to
> >produce which seemed to make progress on CH impossible. I agree that at
> >this preliminary stage my optimism is hardly warranted by the facts. But so
> >be it.
>I view Woodin's and related programs by the set theorists as serious
>attempts to argue for new axioms for set theorists along the lines of
>convenience and usefulness. In a way, it promises to be somewhat similar to
>the process that I described in my previous e-mail for the adoption of new
>axioms by the mathematical community. Only it is aimed at the set theory
>community. It is not aimed at the general math logic community or the math
>community, neither of which is likely to find such things either convenient
>or useful.


> >>Actually, one should ask: generally accepted by who? I take it that you
> >>mean "the mathematical community." Then I will bet against you very
> >>strongly.
> >
> >Yes that's what I mean. Think how much it has changed over the past twenty
> >years. And your own work will probably do more than anything else to lead
> >them to accept methods going beyond ZFC.
>The principal relevant change in the mathematics community is the
>intensification of the move away from set theoretic problems or problems
>with any nontrivial set theoretic content - that started in earnest in the
>1960's - and towards concrete mathematics.

And intellectual trends never reverse themselves? I didn't claim the change 
was in the direction I was talking about. Only that any static snapshot of 
that kind gives no reliable evidence concerning future developments.

> >I agree: no "self evident" principles. I believe (and again your own work
> >will play a key role) that mathematicians will become comfortable with the
> >view that they must work with principles that are not "self evident".
>Only because they have so much to gain by the acceptance of new axioms for
>what they regard as normal mathematics. They are not going to entertain new
>axioms for the purposes of abnormal mathematics.

As I said, they will only accept them if convinced that they give reliable 
results. And once accepted, they will naturally also accept other 
consequences of these axioms even if they lie in what you call "abnormal" 
mathematics and even if these consequences are remote from their interests.

> >As you know very well, the existence of non constructible sets is implied
> >by large cardinal axioms (specifically the existence of measurable
> >cardinals). Now there is nothing self evident about these axioms. But the
> >way in which they line up in linear order with respect to various criteria
> >suggests very strongly that they are true.
>This way of using "true" in this manner is utterly foreign to the general
>mathematical community. They will take an entirely pragmatic position.

And they may be wrong. Emil Post said that Goedel's work "must inevitably 
result in at least partial reversal of the entire axiomatic trend of the 
late nineteenth and early twentieth centuries, with a return to meaning and 
truth as being of the essence of mathematics."  The question of how to 
regard the large cardinal hierarchy forces the issue. Using words like 
"useful" or "convenient" or "pragmatic" just dodges the issue. What reason 
is there for believing that the combinatorial consequences you draw from 
these axioms are correct except that the axioms are in some sense true.


> >This views them like a
> >physicist.
>In order to win the Nobel Prize in physics, it is said that there must at
>least be experimental confirmation. The analog in math here is nonexistent
>or at the most extremely weak compared to what goes on in physics.
> >Set theorists explore this territory as utterly remote from
> >everyday human experience as quarks. They must look for theoretical
> >cohesiveness. As Goedel suggested years ago, such axioms can justify
> >themselves on the basis of their consequences - and here again it is you
> >who are doing so much to bring this about.
>But the justification for the general math community is not along any idea
>of truth. It is rather along the lines of coherent pragmatism. Because the
>possibility of experimental confirmation is nonexistent - or at least
>inconceivable at this point - truth plays no significant role in the
>equation. Only coherent usefulness.
> >Yes. But they will want more. They will hardly regard them as acceptable,
> >however convenient, unless they have reason to believe that their
> >arithmetic consequences are true.
>The only way they have of testing truth of arithmetic consequences is that
>there is no contradiction. So this just amounts to faith in consistency.
>This they will gradually accept after there is enough well worked out
>theory and enough smart people are using them to gain confidence there are
>no problems and to get a vested interest in their use. Notions of truth
>seem utterly irrelevant for this process in the general mathematical
>There is only one other way I can conceive of that some very concrete
>statements may be confirmed, and that is probabilistically. I.e., one may
>derive from a large cardinal an estimate on the fraction of elements of a
>specific finite set that have a computer testable property. And then trials
>can actually be done to confirm the estimate. And if that estimate can be
>rigorously proved using large cardinals but not without - either in an
>appropriate sense necessarily involving lengths of proofs, or at least with
>no known way to remove the large cardinal - then that would amount to
>something dramatic in favor of the large cardinals. However, it would still
>be incomparably weaker than what routinely happens in physics, partly
>because alternatives to large cardinals can also deliver the same
>consequence. E.g., one can always use "continuum is real valued measurable"
>instead of "there is a measurable cardinal" for any such purpose.
>The idea that one can confirm arithmetic propositions via physical
>observation is much too far fetched to be seriously contemplated for a
>variety of reasons.
> >>However, it seems extremely far fetched to think that this process is going
> >>to apply to CH or to any statement that settles CH  - since such statements
> >>appear to be in principle divorced from what the mathematicians view as
> >>normal mathematics.
> >
> >Here you take an inappropriately narrow view. These things change, and we
> >know that even the remotest large cardinal axioms have arithmetic
> >consequences.
>Let me put it more coherently. For every natural mathematical statement -
>at least thus far - there is a large cardinal axiom (including possibly 1 =
>0) which has exactly the same arithmetic consequences (and somewhat higher)
>over ZFC. Because of this, one can without loss of generality just look at
>large cardinal axioms from the point of view of normal mathematics. There
>is not even the remotest hint of a counterexample to this around.
> >I take it that when "Argle" speaks of arbitrary collections, he does not
> >intend them to be restricted to the cumulative hierarchy.
>What is it about the cumulative hierarchy that makes the notion of
>arbitrary collection there any clearer to you  than the notion of arbitrary
>collection anywhere else? Are you willing to say that there is no
>difference to you in the level of determinateness that arbitrary statements
>about subsets of V(kappa + 2) have compared to Pi-0-1 sentences, where
>kappa is the first measurable cardinal?
> >Nevertheless, I'm
> >glad you asked. Without the availability of long well-orderings, the
> >cumulative hierarchy stops short. Methodologically, what the large cardinal
> >axioms supply is precisely the well-orderings needed to extend the
> >hierarchy. We know (Russell Paradox) that there is no end to the process.
>But what does this have to do with truth? Granted, I am showing that
>postulating extensions of the cumulative hierarchy in this way is
>coherently useful and convenient for normal mathematics, with a general
>feeling of consistency.
> >But there is nothing to stop us from going on and on. And that is the great
> >service that the set theorists do.
>Cardinals far beyond our real understanding were already introduced in 1911
>by Mahlo. A real challenge is to make sense of them and to use them for
>normal mathematics. I'm doing the latter, but the former would really be a
>great service, even for tiny ones. The process of making sense of them
>probably needs to be done in a new way already starting at aleph-one. After
>all, that really is a very large ordinal.
>In summary, let me make some remarks:
>1. The analogy with physics is severely strained. There are serious
>differences that make it unconvincing.
>2. The general mathematical community is going to adopt new axioms only on
>the basis of coherent usefulness for normal mathematics, together with some
>feeling of consistency. Truth will not enter into the equation for them. No
>one knows how to get it to enter in any convincing way anyways.
>3. Large cardinals will be the center of attention by the mathematics
>community, since consistency and relations with concrete statements are the
>only issues for them in connection with adoption of new axioms.
>4. I also see the possibility that the general math community may be more
>comfortable with "there is an atomless probability measure on all subsets
>of the unit interval" than with large cardinal axioms, in that this
>involves objects that are far less abnormal than a large cardinal. Of
>course, this particular one and a measurable cardinal are known to be
>equivalent for normal mathematical purposes. (Actually this equvalence
>passes through some sophisticated set theory that is not "convenient" for
>mathematicians, and so there is a need to put things in a form that
>mathematicians will find easy to use). Of course, a byproduct of this
>probability measure is that CH is very badly false. But, when viewed as an
>approach to CH, this is seriously at odds with the approach currently taken
>by the set theorists. After all, if they like it, then the CH would have
>been viewed as being settled long long ago by them. They reject this
>probability measure as an axiom candidate. However, from the coherent
>pragmatism that I see in the math community, they may well accept this when
>shown how to use this in an easy and coherent and effective way for a
>sufficient body of attractive normal mathematics. This would be where the
>set theorists' approach  via some notion of truth and the mathematicians
>approach via coherent pragmatism would clash.

                           Martin Davis
                    Visiting Scholar UC Berkeley
                      Professor Emeritus, NYU
                          martin at
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