FOM: "The Complete Theory of Everything"

[Stephen G Simpson]

There is a vast gulf between the foundational point of view and the
typical pure mathematician's point of view.  I'd like to illustrate
this with an extreme example, namely the divergent views of Harvey's
paper

 "The Complete Theory of Everything: Validity in the Universal Domain"

which is available at

  http://www.math.ohio-state.edu/foundations/manuscripts.html

Harvey's motivation was (of course) foundational, going all the way
back to Frege's original purpose in inventing first order logic, in
the late 19th century.  Frege's intention was also (of course)
foundational: to give a precise analysis of the philosophical notion
of logical validity, by setting up formal axioms and rules to capture
this.  Much later Frege's analysis was nonconstructively vindicated by
the G"odel completeness theorem.  Harvey revisits and critically
reexamines Frege's work from a fresh perspective, which Frege himself
might have found congenial.

Harvey's new twist on Frege is that, while first order logic captures
validity in structures whose domain is a set, it arguably does not
capture validity in the universal domain, which consists of absolutely
everything that exists.  This is because the universal domain arguably
has certain indiscernibility properties which, for instance, prevent
it from being linearly ordered.  Harvey writes down appropriate and
elegant axioms for the universal domain and then proves completeness
of the resulting axiom system.  It's a clever proof, using familiar
techniques of model theory in a novel way.  And it's a striking
result, which says something of basic philosophical importance about
logical validity.  As might be expected, philosophers interested in
mathematical logic have been captivated.  Also as might be expected,
pure mathematicians have been largely indifferent, although some of
them also get the point.

Recently Harvey visited Anand's university and spoke about "The
Complete Theory of Everything" as well as other results in f.o.m.
Here is Anand's somewhat patronizing description of the encounter.
Read this carefully:

   Harvey - you told me some results when you were in Urbana. What I
   remember is something about the first order validities in a
   structure in which both a linear ordering and pairing function are
   defined. I recall it being an interesting result. The abstract you
   sent around (list of your recent talks) is too vague. In any case,
   if I remember, this looked like a nice result firmly in the sphere
   of modern model theory (which by the way doesn't need to be
   marketed as the "theory of everything" ). Anyway I'd like to know
   more details.

What I see here is that Anand, being a pure mathematician who uses
model-theoretic methods in his own pure mathematical research, is
readily able to grasp and remember the technical details of Harvey's
proof (indiscernibles, linear orderings, pairing functions, etc.)  And
Anand appreciates this technical, mathematical, model-theoretic
aspect; he finds it "nice".  But Anand is totally turned off by and
left completely cold by the foundational aspect, i.e. the analysis of
validity in the universal domain.  To Anand's mind, the foundational
aspect of Harvey's result is comprehensible only as a marketing ploy.
In short, Anand just doesn't get it.

It's also interesting to observe the applied model theorists' reaction
to other foundational programs and results.  Anand's expressed reaction
to "The Complete Theory of Everything" is similar to his expressed
reaction to Reverse Mathematics:

   Rather paradoxically (and provocatively), from what I understand of
   results in reverse mathematics, I tend to view the results as quite
   interesting from the mathematical point of view, but just a
   curiosity from the foundational point of view. But I'd be happy to
   revise my judgement after looking at Steve's book.

In other words, Anand finds the purely mathematical, technical details
of Reverse Mathematics amusing at some level.  But the foundational
aspects of Reverse Mathematics (including a rich and precise analysis
of various classical f.o.m. programs such as finitistic reductionism a
la Hilbert, predicative reductionism a la Weyl, etc., and the
limitations of these programs with respect to mathematical practice)
is totally alien to Anand.  Once again, Anand probably regards the
invocation of Hilbert and Weyl as nothing but a marketing ploy.  Once
again, Anand just doesn't get it.

I view these remarks by Anand as excellent examples of the typical
pure mathematician's mindset with respect to foundational issues.  Or
maybe they are representative of a similar but more combative mindset,
such as might be expressed by a pure mathematician's adoring little
brother.

I hope that nobody here is offended by what Anand and I have said.  In
my opinion, these revealing exchanges are one of the fascinating
aspects of what has been going on here in the FOM list.  I want this
to continue and grow.  Hurrah for the FOM list!

-- Steve