FOM: Does Mathematics Need New Axioms?

[Penelope Maddy]

Dear Harvey,

I'm wondering about your attitude towards extrinsic justifications for 
axioms.  In your latest outline, you include:

>View D. We should give up trying to obtain intrinsic justifications for
>large cardinals.
>
>I am optimistic about the program of finding convincing intrinsic
>justifications of large cardinals via some non set theoretic notion such as
>predication. There are some suggestive developments that make me optimistic
>about this. Work on transfer principles shows how large cardinals can be
>obtained by transferring statements about functions on the natural numbers.
>Work on multiple worlds provides new kinds of axiomatizations of large
>cardinals. In both cases, the large cardinals involve are big enough to
>refute the axiom of constructibility.

This by itself might suggest that you think intrinsic justifications are to 
be preferred to extrinsic justifications.  But your reply to Avron includes 
the following exchange:

 >>  Let me add here that personally I cannot be persuaded by Friedman's
 >>arguments
 >>(and impressive achievements) because of my (old-fashioned?) criterion of
 >>accepting a statement as an AXIOM only if it is obviously, self-evidently
 >>TRUE.

 >You are not talking about axioms, but rather "intrinsic axioms," which >is
 >to be distinguished from "extrinsic axioms." I think that the difference is
 >one of terminology. Perhaps you would rather use terminology like >"commonly
 >accepted principles."

 >>And I dont see what can make me suddenly realize that a certain large
 >>cardinal axiom is in fact self-evident (how could I have missed such a
 >>self-evident fact before??

 >I was not talking about self-evidence. This goes on throughout science -
 >that principles become universally accepted without being self evident. >The
 >analogies with physical science are certainly not perfect, but some
 >analogies with physical science work well.

Here I suppose the analogy with physical science is meant to call up the 
idea of justification in terms of consequences, which is what extrinsic 
justifications are.

Putting these two together, my guess is that you think intrinsic 
justifications are of considerable interest, so the search for them should 
not be abandoned, but that extrinsic justifications are legitimate, too, 
and might even stand alone in some cases if they are strong 
enough.  Right?  (If so, we are in complete agreement here.)

I suppose I ought also to comment on this passage:


>View E. Naturalism strongly suggests that mathematics needs large cardinal
>axioms.
>
>I acknowledge that naturalism strongly suggests that set theorists need
>large cardinal axioms. But at the moment, naturalism does not suggest that
>mathematics - in any broader sense - needs large cardinal axioms. However,
>I anticipate that to change radically through the development of Boolean
>relation theory and its extensions. The naturalist point of view needs to
>expand to incorporate normal mathematics. This requires confronting and
>making sense of the proud anti-foundational anti-philosophical perspective
>of the normal mathematician.

The naturalist sees (at least one) justification for large cardinals as 
arising from the foundational goal of set theory (touched on briefly toward 
the end of my comments for the panel discussion).  This idea is put 
beautifully in this passage from John's outline:

  >       What does it mean to adopt large cardinal axioms--just what is
 >the behavior being advocated? One ambition of set theory is to be a >useful
 >universal framework in which all our mathematical theories can be
 >interpreted.  (Of course, one can always amalgamate theories by
 >interpreting them as speaking of disjoint universes; this is the >paradigm
 >for a useless framework.)  To believe that there are measurable >cardinals
 >is to believe "there are measurable cardinals" should be added to this
 >framework; that is, to seek to naturally interpret all theories of sets,
 >to the extent that they have a natural interpretation, in extensions of
 >ZFC + "there is a measurable cardinal".

Harvey's point is that mathematicians in other areas of mathematics do not 
now have identifiable goals that require them to adopt large cardinal 
axioms.  Quite right, but I agree with John that this fact doesn't 
undermine the set theorist's justification for adding them to his theory.

While I'm at it, I'd also like to record my support for John's second 
argument against V=L:

>2. Even if there were a wonderful, useful DST of projective sets based
>on V=L (as there is in fact a wonderful infinitary combinatorics),
>adopting PD in no way eliminates it. The language of set theory as
>used by the believer of V=L can be interpreted in the language of
>set theory as used by the PD believer: one translates phi as (phi)^L.
>The V=L believer will agree that this preserves the meaning of the
>language as he is using it. In this way, all his mathematics becomes a
>chapter in the PD-believer's mathematics. There is, however, no
>meaning-preserving translation in the other direction.
>
>    In this light, we can see that adopting V=L in fact lets us settle NO
>questions which weren't already settled by ZFC! Settling phi on the basis
>of ZFC + V=L is precisely the same as settling (phi)^L on the basis of
>ZFC. Adopting V=L just prevents us from asking as many questions!
>
>    Of course, it might have been reasonable to avoid asking about the
>world outside L ( compare the restriction V=WF given by the foundation
>axiom). But in fact, we have found all sorts of deep mathematical
>structure outside L. Adopting V=L amounts to prohibiting the investigation
>of that structure. Why would we want to do that?

This is the argument I try to spell out at the end of my book (where John's 
influence is much in evidence).  In fact, it seems to me that his outline 
contains an improvement of the crude idea of 'fair interpretation' proposed 
there:

 >In practice, the way a theory S is interpreted in an extension T of the
 >form ZFC + large cardinal hypo. H is: S is the theory of a certain inner
 >model of a certain generic extension of an arbitrary model of T.

Yours,

Pen