FOM: Does Mathematics Need New Axioms?

[John Steel]

Reply to Simpson:

On Wed, 10 May 2000, Stephen G Simpson wrote:

> This is a quick reply to John Steel's posting of earlier today.
> 
> Steel said:
> 
>  > If any part of mathematics needs new axioms, then mathematics needs
>  > new axioms. Of course, the importance of the need relates to the
>  > importance of the areas in need.
> 
> and
> 
>  > If set theory needs new axioms, then mathematics needs new axioms.
> 
> Steel's argument seems to be that math *obviously* needs new axioms,
> because set theory does, and set theory is (an important?)  part of
> math.
> 
     Yes, although it is you who are emphasizing the importance of field
divisions within mathematics. I took the question to be whether the
current axioms provide an adequate foundation for all of mathematics.
I agree that it's also interesting to look at what's needed in various
parts.


> I would comment that there is a distinction between set theory qua
> branch of math versus set theory qua f.o.m.  As a branch of
> mathematics, set theory is relatively small and unimportant compared
> to other branches such as geometry, number theory, and differential
> equations.  Think of the history of these branches.  The main
> importance of set theory derives from its contemporary foundational
> role, as a proposed foundation for *all* of mathematics.

    Yes, set theory has special importance because of its foundational
role. ( "Proposed" understates the truth here.) One should expect the
need for new axioms, and the new axioms needed, to show up here first.

  In this
> sense, set theory, like the rest of logic and f.o.m., is not really
> *part of* mathematics.  It is a mathematical science, but not
> mathematics.  The term ``metamathematics'' comes to mind.

     This is way off. ZFC and its large cardinal/determinacy extensions
codify mathematics, not metamathematics. The objects of study are
natural numbers, real numbers, sets of real numbers,..., not formulae,
theories, proofs, etc. Of course, set theorists often study the
metamathematics of set theory, and it is conventional to call that
study set theory. But this part of set theory needs no new axioms at
all, and in fact can be carried out in Peano Arithmetic!     
     Steve seems to be saying that the axioms for all of mathematics
do not themselves belong to mathematics. If so, at what point in e.g.
the proof that [0,1] is compact do we enter mathematics? 

 
> It is a mistake to interpret Harvey's view as a call for market
> research or a shallow public relations campaign.  As Steel later
> acknowledges, there are serious intellectual issues here.  It is wrong
> to ignore these issues or try to sweep them under the rug.
>
    I think the serious question is how important the new axioms will
be in the long run. I don't think that has much to do with what people
who know little about them think of them now. I don't think it has
much to do with how well they are advertised now. We can say a few
intelligent things about the long-run importance of large cardinals, but
we should acknowledge the uncertainty in such predictions. Fifty or a
hundred years from now would be a much better time to make such
judgements. Five hundred would be better still.

 
>  > How can you know views which "began in earnest in the 1960's" are
>  > fixed and fundamental? 
> 
> There are sound philosophical and historical reasons for thinking that
> the shift back to concrete and computational math, which began in the
> the 1960's, is permanent and long overdue. 

     Permanent? How far ahead can you see?

 Of course this is
> debatable, but it seems to me that the burden of proof is on the
> proponents of abstract, non-computational math.
> 
     The main burden on the proponents of abstract, non-computational math
is to produce good mathematics of that sort.



John Steel