foundations of mathematics

[Stephen G Simpson]

Thanks to John Burgess, I think we are finally finished with the
digression into postmodernism.  Let's return to the main topic:
foundations of mathematics.

Anand seems to be hinting that the
Frege-Russell-Hilbert-G"odel-... line of research in foundations of
mathematics may be out of fashion.  Obviously I disagree.  I present
two broad considerations.

  (1) Atiyah and Horgan notwithstanding, the logical structure of
  axiom-definition-theorem-proof remains an essential, indispensable
  component of mathematical practice.  There is a timeless, objective
  need for a discipline devoted to elucidating this logical structure
  and to discovering the nature and role of the most basic
  mathematical concepts.  Fads and fashions aside, if the F-R-H-G line
  were not already in existence, we would have to invent it.
  
  (2) In actual fact, the F-R-H-G line of has been and continues to be
  very fruitful.  For examples I would point to the Friedman volume,
  "Harvey Friedman's Work in the Foundations of Mathematics,"
  North-Holland, 1985.

Let me also comment on another of Anand's claims: that certain
relatively recent mathematical developments (penetration of
cohomology, Weil conjectures, Lang conjectures, Langlands program,
possibly others) can be considered "foundational" in the same sense as
Cartesian geometry.  To me this claim seems wildly exaggerated.  So
far as I know, cohomology et al are interesting and important
mathematical developments, but that's all they are.  I don't see how
they can be called "foundational."  First, they are very far removed
from the most basic mathematical concepts.  Second, I don't see how
they could ever have anywhere near the same general intellectual
impact as Cartesian geometry or even G"odel's second incompleteness
theorem.  Am I missing something?  I'd like Anand or somebody in his
circle (Angus? Charlie? Lou?) to explain how Anand's claim can be
taken seriously.

I recognize of course that a lot of this depends on your notion of
"foundational."  If you were to define "foundational" very broadly to
refer to any interesting mathematical idea whatsoever, then obviously
cohomology et al would be "foundational" in that sense.  But this
would be a pointless word game, and nobody here has proposed such a
definition.  So far as I know, there is only one serious concept of
"foundational" on the table, namely mine presented in

    www.math.psu.edu/simpson/Hierarchy.html

or, to put it in a nutshell: "foundations of mathematics" is the
systematic study of the most basic mathematical concepts.  I would
argue that this definition of "foundational" is the correct one.  But
of course, if somebody else wants to present an alternative notion of
"foundational," I'd be glad to hear it.

-- Steve