FOM: listing foundational issues

[Stephen G Simpson]

Robert Tragesser 6 Mar 1998 08:17:02 writes:

 > Could a list of something like all "important" foundational issues
 > be given?

Thanks for this question.  Let me take a stab at answering it.  This
is only preliminary.  Much more can be said.

The answer to this question depends on our understanding of
"foundational" or "f.o.m."  As a working definition of f.o.m. for the
FOM list, I have proposed:

  Foundations of mathematics (f.o.m.) is the systematic study of the
  logical structure and most basic concepts of mathematics, with an
  eye to the unity of human knowledge.

(See also my elaboration at
http://www.math.psu.edu/simpson/hierarchy.html.)

So, one key foundational issue is: what are the most basic concepts of
mathematics?

Currently the orthodox Bourbaki view is that sets are the most basic,
because all other mathematical concepts are to be defined in terms of
them.  However, our idea of what is basic may change over time as a
result of historical developments (e.g. arithmetization of analysis in
the 19th century), and we may also want to consider alternative or
non-orthodox foundational schemes.  I have therefore proposed the
following tentative list of basic mathematical concepts:

    number
    shape
    set
    function
    algorithm
    mathematical definition
    mathematical axiom
    mathematical proof

This list is intended to be philosophically neutral and to accommodate
not only set-theoretic foundations but also a wide range of
alternative foundational ideas.

In these terms it seems pretty clear how to generate a list of
important foundational issues.  Among them would be: What is a number?
What is a shape?  What is a set?  What is a function?  What is an
algorithm?  What is a mathematical definition?  What is a mathematical
axiom?  What is a mathematical proof?  What are the appropriate axioms
for numbers?  What are the appropriate axioms for shapes?  What are
the appropriate axioms for functions?  What are the appropriate axioms
for sets?  Are the axioms consistent?  Are the axioms complete?  Are
additional axioms needed?  etc etc.

Among the historic high points of research on key f.o.m. issues:

  clarification of mathematical proof -- Frege

  the hierarchy of consistency strengths -- G"odel

  clarification of computable functions -- Church, Turing

  intuitionistic logic -- Heyting

  axioms for set theory -- Zermelo, von Neumann 

  set-theoretic independence results -- G"odel, Cohen, Solovay, ...

  finitistic independence results -- Paris/Harrington, Friedman 

etc etc.  Much has been done, but obviously there is much more to
come.  This rich and exciting subject has the potential of becoming
even richer and more exciting.

A warning:

I believe that the picture of f.o.m. that I have painted above is
reasonably mainstream and reasonably neutral as between competing
foundational schemes.  But no matter what you say on the FOM list,
somebody is bound to disagree, and some people are prepared to dispute
almost anything.  For example, McLarty claims to believe that topos
theory and Chow's lemma on projective complex analytic varieties are
foundational.  I think this is because McLarty does not understand the
concept "basic mathematical concept".  See also my earlier postings on
the list 2 fallacy.

Despite the existence of off-the-wall dissenters and envious
detractors, I truly believe that f.o.m. is a wonderful subject which
is full of interesting problems, issues, and programs.

-- Steve