FOM: Does Mathematics Need New Axioms?

[Stephen G Simpson]

This is a reply to John Steel's posting of today.

Harvey Friedman envisions a future scenario where large cardinal
axioms would be pervasive throughout core mathematics.  Steel asks:

 > From the tone of your letter, I take it that you find the latter
 > scenario implausible. Do you?

I find Harvey's scenario somewhat implausible, for a number of
reasons, some of which I touched on in my message.  But I have to
admit that Harvey has single-handedly made some very interesting
progress toward his scenario.  And I would also say that the set
theory community has not been sufficiently appreciative of those
efforts.

Concerning whether set theory is part of mathematics proper, or
whether it is legitimate to distinguish set theory from core math,
Steel says:

 > 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.)

These things don't happen by accident.  A while back, there was a good
discussion on FOM of how and why the MSC committee decided to abolish
Set Theory and merge it with Logic and Foundations.  Andreas Blass,
who was on the MSC committee, contributed to the FOM discussion.  You
can look it up in the FOM archives.

The current MSC scheme includes several other subjects which, like
Logic and Foundations, are also not part of mathematics proper:

 01-XX   History and biography [See also the classification number -03
         in the other sections]
 03-XX   Mathematical logic and foundations
 04-XX   This section has been deleted {For set theory see 03Exx}
 62-XX   Statistics
 68-XX   Computer science {For papers involving machine computations and
         programs in a specific mathematical area, see Section -04 in
         that area}
 70-XX   Mechanics of particles and systems {For relativistic mechanics,
         see 83A05 and 83C10; for statistical mechanics, see 82-XX}
 73-XX   This section has been deleted {For mechanics of solids, see 74-XX}
 74-XX   Mechanics of deformable solids
 76-XX   Fluid mechanics {For general continuum mechanics, see 74Axx, or
         other parts of 74-XX}
 78-XX   Optics, electromagnetic theory {For quantum optics, see 81V80}
 80-XX   Classical thermodynamics, heat transfer {For thermodynamics of
         solids, see 74A15}
 81-XX   Quantum theory
 82-XX   Statistical mechanics, structure of matter
 83-XX   Relativity and gravitational theory
 85-XX   Astronomy and astrophysics {For celestial mechanics, see 70F15}
 86-XX   Geophysics [See also 76U05, 76V05]
 90-XX   Operations research, mathematical programming
 91-XX   Game theory, economics, social and behavioral sciences
 92-XX   Biology and other natural sciences
 93-XX   Systems theory; control {For optimal control, see 49-XX}
 94-XX   Information and communication, circuits
 97-XX   Mathematics education

Steel:

 > > What are the appropriate axioms for *all* of mathematics?
 > 
 > Does this sound like a question which belongs in a "small and
 > obscure corner"?

>From the standpoint of general intellectual interest, the question is
of tremendous importance.  But from the standpoint of the core or
mainstream mathematician, the question belongs to a small and obscure
corner.  That corner is known as 03-XX, Logic and Foundations.

 > I don't follow your "core" mathematician's reasoning.

We need to follow this reasoning (not necessarily agree with it), in
order to understand the goals and standards which have guided and will
continue to guide the development of core mathematics, for the
forseeable future.

 > Does answering the age-old foundational question have no special
 > importance?

It has very great special (and general!) importance.  But it will not
be of much interest to core mathematicians qua core mathematicians,
unless and until it seriously impacts their core mathematical
concerns.  Call them narrow-minded if you will, but recognize that
their outlook is decisive with respect to the question ``Does
mathematics need new axioms?''.

I said:

 > > The core mathematician is perfectly justified in raising this
 > > question.
   [whether large cardinals will ever have any impact on core math]

Steel responded:

 > Of course. My short answer would be that the theory of projective
 > sets one gets from large cardinal hypotheses demonstrates their
 > power and scope. 

This ``short answer'' cannot sway the core mathematician, because
projective sets are not close enough to core math (geometry, number
theory, differential equations, etc).  Would the long answer be more
convincing?  I have my doubts.

Steel:

 > As for more concrete applications, time will tell.

``Time will tell'' both understates and overstates what we know right
now.  Without waiting 500 years, we know right now that core math is
thriving with no need for input from the ``new axioms'' (Logic and
Foundations) community.  We also know that there are some big
obstacles to finding such applications, one of the obstacles being the
Shoenfield Absoluteness Theorem.  We also know that, despite the
obstacles, Harvey has made some progress.

Steel:

 > The process of codifying mathematics into ZFC was largely complete
 > before metamathematics, as we understand it now, arrived on the
 > scene. ...

How could codifying mathematics into ZFC take place before ZFC arrived
on the scene?  

But, OK, we seem to be arguing over terminology.  Steel is apparently
taking ``metamathematics'' to mean the study of formal systems such as
ZFC, PA, subsystems of Z_2, intuitionistic systems, etc.  I was taking
it more broadly to mean essentially the same thing as f.o.m.

 > metamathematics is part of mathematics too, so none of this serves
 > Steve's larger point.

Not really.  F.o.m. is not really part of mathematics proper.  It is
better viewed as a *mathematical science*, like statistics or computer
science.  F.o.m. researchers acknowledge this when they distinguish
metatheorems from theorems.  The entire outlook (perspective, goals,
values, terminology, way of thinking, ...) of f.o.m. is very different
from that of core math.  The distinction is vital.

Steel:

 > How can anything important hinge on a distinction between
 > mathematics and "mathematics proper"?

I think I have answered this, but let me repeat myself.  

F.o.m. is *not* part of mathematics proper.  The attitudes and goals
are very different.  If we as f.o.m. researchers want to understand
the attitudes of core mathematicians, it is vital that we fully
assimilate the distinction.  On the other hand, if we as
f.o.m. researchers decide to bury our heads in the sand and insist
that our favorite topics in logic and f.o.m. (proof theory, model
theory, set theory, recursion theory) *are* part and parcel of
mathematics, and that the core mathematicians are simply too
narrow-minded or short-sighted to realize this, then we are very
seriously deluding ourselves.

Steel:

 > Do you attach value to your own work, Steve?

Yes.  A lot of my work in recent years has been in the area of reverse
mathematics, and I posted several times on FOM about the value of this
kind of investigation.  In those postings I emphasized that f.o.m. is
very different from mainstream math (the ``two cultures'' point) and
is not to be evaluated by the same standards.  See for instance

    Subject: FOM: what is the value of reverse mathematics? 
    From: Stephen G Simpson <simpson at math.psu.edu> 
    Date: Thu, 5 Aug 1999 20:39:49 -0400 (EDT) 

which was in the context of the brawl with Soare last year.

Steel:

 > Do you consider it concrete or computational?

Not really.  Reverse math is in a sense *close to* core or mainstream
math, because it deals with mainstream (e.g., first-year graduate
curriculum) mathematical structures and theorems.  But reverse math is
not *part of* mainstream math.  What sets it apart is its
f.o.m. perspective, goals, results, methods, ....

Steel:

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

No wonder Steel does not want to discuss these issues.

-- Steve