FOM: Does Mathematics Need New Axioms?

[John Steel]

Reply to Simpson:[

On Wed, 17 May 2000, Stephen G Simpson wrote:

  If large cardinal
> axioms are needed in only a small and obscure corner of mathematics,
> then that is a very different scenario from what Harvey envisions,
> where the need for large cardinal axioms would be pervasive throughout
> core mathematics (number theory, geometry, differential equations,
> etc).
>
    From the tone of your letter, I take it that you find the latter
scenario implausible. Do you?

 
  set theory is better viewed as being part of f.o.m. rather than
> part of mathematics proper.  Recall that the recent reorganization of
> the Mathematics Subject Classification scheme abolished set theory as
> a separate classification (04XX) and merged it into logic and
> foundations (03XX).  


     I don't see how the fact that set theory has been put one place or
another in the Mathematics (!!) Subject Classification scheme is
evidence that it is not part of mathematics. (Or "mathematics proper",
whatever that is.)


>   The goal of this work [set theory]
> is to answer the age-old foundational question:
> 
>    What are the appropriate axioms for *all* of mathematics?
> 

    Does this sound like a question which belongs in a "small and
obscure corner"?
    By the way, I would say the goal of some (not all)  work in set
theory is to make progress on this age-old question. I don't think it is
a question we will ever be finished answering.


> Therefore, it is perfectly reasonable for core mathematicians to think
> that, if new axioms are needed for set theory, then that doesn't
> necessarily have any consequences for particular core math branches.
>
    Therefore? I don't follow your "core" mathematician's reasoning.
 
> Steel spins this a little differently:
> 
   >  > Yes, set theory has special importance because of its foundational
>  > role.
> 

      I object to having my pearls of wisdom characterized as spin. Also,
I can't reconcile your demurral here with your statement immediately
above. Does answering the age-old foundational question have no special
importance? 
    

> Steel:
> 
>  > One should expect the need for new axioms, and the new axioms
>  > needed, to show up here [in set theory] first.
> 
> Why ``first''?  Why not ``only''?  The core mathematician is perfectly
> justified in raising this question.
>  
     
     Of course. My short answer would be that the theory of projective
sets one gets from large cardinal hypotheses demonstrates their power
and scope. As for more concrete applications, time will tell.


>  In the
> 20th century, set-theoretic foundations achieved a certain level of
> acceptance among core mathematicians, but that was a matter of
> convenience rather than deep conviction.  There are alternative
> foundational schemes that many people find attractive.
> 
    Which schemes are you talking about? The various constructivist
schemes are fragments of ZFC. They are not incompatible with ZFC unless
one takes them as outlawing the mathematics which they cannot support.
Is this what you mean by an alternative scheme--the proposal to abolish
some part of ZFC or its large cardinal extensions?



> Here is another exchange.  I said:
> 
>  > > 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.
> 
> Steel replied:
> 
>  > ZFC and its large cardinal/determinacy extensions codify
>  > mathematics, not metamathematics.
> 
> I concede this statement.  But Steel ought to concede that the
> *process of codifying* mathematics within ZFC is metamathematics, not
> mathematics.  And, the *discovery* of the ZFC axiom system and its
> extensions and their uses is metamathematics, not mathematics.  And
> the discovery of significant metatheorems concerning such formal
> systems is metamathematics, not mathematics.  
 
     The process of codifying mathematics into ZFC was largely complete
before metamathematics, as we understand it now, arrived on the scene.
The reduction of all mathematical concepts to sets and membership was
done by Descartes, Cauchy, Dedekind, Frege, and others (this list is not
meant to be definitive). It seems a stretch to say that Cauchy's work
putting Calculus on a firm foundation was metamathematics.
     At any rate, metamathematics is part of mathematics too, so none of
this serves Steve's larger point.


 (I take
> ``metamathematics'' to be synonymous with ``f.o.m.''

    I think this is an idiosyncratic subject classification scheme.
Usually, "metamathematics" refers to the mathematical study of formal
mathematical theories. It is a relative term: you "go to the metalevel"
when you stop working within the theory, developing it, and begin to
study it as an object.


 
>  > Steve seems to be saying that the axioms for all of mathematics do
>  > not themselves belong to mathematics.
> 
> Yes, I am saying something like that..  Let me put it this way: The
> discovery and study of such axiom systems and their use in the
> formalization of mathematics is part of f.o.m.  It is not part of
> mathematics proper.
> 
     How can anything important hinge on a distinction between
mathematics and "mathematics proper"?


>  > If so, at what point in e.g. the proof that [0,1] is compact do we
>  > enter mathematics?
> 
> ``The proof'' that [0,1] is compact?  There are many proofs of this
> well known theorem.

    Take any proof of this fact which starts from the axioms of ZFC.
Which is the first line of the proof that constitutes a statement of
mathematics?



> Another exchange between me and Steel.  I said
> 
>  > > 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.
> 
> and Steel replied
> 
>  > Permanent? How far ahead can you see?
> 
> It is important to try to see as far as possible.

     It is also important not to overestimate how far ahead you can see.

  The question that
> we are discussing is, what is the long term future of abstract,
> non-computational math as contrasted with concrete, computational
> math?  This question is debatable, and I speak only for myself.
> 
      I myself don't want to discuss this question. I doubt that
much would come from doing so. The dividing line beween the two is
vague, to say the least. It is obvious that both sorts have value,
isn't it? (Do you attach value to your own work, Steve? Do you
consider it concrete or computational?) Why should we try to predict
what the balance will be a few hundred years from now?
   

> Philosophically, the idea of mathematics as the study of an abstract
> mathematical realm remote from the real world of entities and
> processes seems to be a dead end.

       To my mind, both "abstract mathematical realm" and "real world
entities and processes" are 95% hot air.


John Steel