FOM: g.i.i.; f.o.m.; genetic defects; ignorance; the boxing match

[Stephen G Simpson]

General intellectual interest:

  Colin McLarty writes:
   > Those on fom who insist "fom is of more general interest than other
   > math" have still not even distinguished the many things this could
   > mean, let alone picked one it does mean. Does it mean:
   > 
   > 1) More people actually want to hear about fom than other math.
   > 
   > 2) More people would want to hear about fom than other math, if
   >         they knew more about it.
   > 
   > 3) People would benefit more from learning fom than other parts of math
  
  All of statements (1)-(3) are true.  And it is also true that
  
    (4) F.o.m. (foundations of mathematics) is of much greater general
    intellectual interest than pure mathematics.
  
  But (1)-(3) are not the essence of (4).  They are only consequences of
  (4).  And to understand (4), we need to first understand very clearly
  what f.o.m. is.  After that, the reasons for (4) will be obvious.


Definition of "foundations of mathematics":

  What is f.o.m.?  So far as I am aware, nobody on the FOM list,
  except me, has proposed a definition of f.o.m.  I proposed such a
  definition way back in September and October, when the FOM list was
  just getting started.  I have said repeatedly that I would welcome
  discussion of alternative concepts of what f.o.m. is, but nobody
  here has offered one.
  
  To repeat my definition:
  
     F.o.m. is the systematic study of the most basic mathematical
     concepts and the logical structure of mathematics, with an eye to
     the unity of human knowledge.
  
  A couple of explanatory notes:
  
  1. The word "basic" refers to the hierarchy of concepts.  All of
  human knowledge is organized in a hierarchy of concepts, which is a
  partial ordering.  This is elaborated in my essay
  
     http://www.math.psu.edu/simpson/Hierarchy.html
  
  The "most basic mathematical concepts" are those mathematical
  concepts that are closest to the base of the conceptual hierarchy.
  As such, the most basic mathematical concepts are necessarily
  involved in all connections between mathematics and the rest of
  human knowledge.  This is why they are essential for the unity of
  human knowledge.
  
  2. As a tentative list of the most basic mathematical concepts, I
  offered the following:
  
     list 1:
  
     number
     shape
     set
     function
     algorithm
     mathematical axiom
     mathematical proof
     mathematical definition
  
  Naturally this list is open to revision, but I think the intent is
  clear.
  
  With this much explanation, I think it's clear why f.o.m. is of such
  enormous general intellectual interest.  The reason is that some
  knowledge of concepts such as those on list 1, qua basic
  mathematical concepts, is essential to an understanding of the unity
  of human knowledge.  Nobody can claim to be educated without some of
  this insight.


"Genetic defects":

  Some FOMers (Pillay, Marker, McLarty, ...) have raised a radical
  objection to list 1.  They have asserted that list 1 is radically
  incomplete and needs to be supplemented as follows:
  
     list 2:
  
     cohomology
     projective analytic variety
     Riemannian manifold
     ...
  
     (List 2 seems to be open ended.)
  
  It's clear to me that this proposal has no merit.  I think it's
  pretty obvious why the concepts on list 1 are much more basic than
  the concepts on list 2.  One way to appreciate this is to note that
  the usual textbook definitions of the concepts on list 2 are
  formulated in terms of chains of concepts that begin with concepts
  on list 1.  It's also clear that there is a need for a field of
  study devoted to basic mathematical concepts a la list 1.  That
  field of study is f.o.m.  List 2 is of a very different character.
  
  I'm at a loss to understand what might be called the "list 2
  mind-set".  I don't know how anyone can claim that, for example, the
  concept "Riemannian manifold" is just as basic as the concept "real
  number".  My best analysis of the situation is that these people
  "just don't get it", i.e. they don't understand the concept "basic
  concept".  Perhaps they don't appreciate the idea of a hierarchy of
  concepts.  Are they under the spell of Tennant's "Quinean holism"?
  I don't know.
  
  In making this point about the list 2 mind-set, I once engaged in
  some hyperbole or figurative language.  I said that some people are
  "congenitally incapable" of grasping the distinction between basic
  and non-basic concepts.  Here of course I wasn't referring to any
  genetic defect, but rather to a certain mind-set, the list 2
  mind-set, which appears to prevent them from understanding a crucial
  distinction.  This mental condition may even be correctible.
  Perhaps all that is needed is for these people to admit (to
  themselves at least) that f.o.m. is a legitimate field of study,
  distinct from pure mathematics.


Ignorance:

  I'm not the only one who has used rough language.  Both Dave Marker
  and Lou van den Dries have accused their opponents of being
  "ignorant" of certain mathematical developments.  Dave later
  apologized.  Lou's latest accusation is as follows:
  
   > My *experience* in the course of 30 years has indeed made me
   > suspicious of certain wide spread instincts that Harvey and Steve
   > may have in mind here, and which I share: these instincts,
   > covered by a thin veneer of questionable philosophy, are often
   > used to justify ignorance of major developments in mathematical
   > thought of the last 200 years that are outside of the *relatively
   > minor* and *exceedingly familiar* FOM-line: Cantor, Frege,
   > Goedel, ...
  
  I don't know how one could "justify ignorance" of major mathematical
  developments.  Obviously students of f.o.m. need to be aware of
  mathematical developments insofar as they have an impact on
  foundational issues.  On the other hand, if the developments in
  question are of interest only for particular branches of pure
  mathematics, then non-mathematicians need not be concerned.

  I also don't know what "wide spread instincts" Lou is referring to,
  or in what sense Lou "suspects" and/or "shares" those instincts.
  Lou, I think you need to explain yourself.


The boxing match:
   
  I'm completely on Harvey's side in the boxing match with Lou.  I
  can't understand how anyone can deny that f.o.m. questions such as
  P=NP and the role of large cardinal axioms in finite combinatorics
  are of incomparably greater general intellectual interest than
  Fermat's theorem, etc.


Sincerely,  
-- Steve

Name: Stephen G. Simpson
Position: Professor of Mathematics
Institution: Penn State University
Research interest: foundations of mathematics
More information: www.math.psu.edu/simpson/