Burgess's comments on anti-foundationalism

[Stephen G Simpson]

I asked John Burgess what the postmodernists have to say about
foundations of mathematics.  He sent me the following response and
asked me to forward it to this group.

  I'm not sure whom you're counting as post-modernists, but I know of
  none of the usual suspects who has anything much to say about
  mathematics, except insofar as it is included in their general
  (negative) orientation towards science (as in the Sokal hoax). Gross
  & Levitt ("Higher Superstition") is probably the best source to
  consult if you're really interested.  It has a section on "feminist
  algebra".  Also there was a review in the JSL a while back of a book
  by someone named Nye called "Words of Power: A Feminist History of
  Logic". I myself reviewed Tiles "Mathematics and the Image of
  Reason", a much more moderate works, but not uninfluenced by Paris
  fashions, in the Math Monthly a few years back.  For the most part,
  "Anti-Foundationalism" in philosophy of mathematics is a home-grown
  phenomenon, not a French import.  It is represented by Tymoczko,
  ed., "New Directions in Philosophy of Mathematics", various works of
  Philip Kitcher, and especially by Davis & Hersh, who are
  professional mathematicians and amateur philosophers.  I reviewed
  their book (negatively) in a recent issue of Philosophia Mathematica
  (this was paired with a positive review by a British
  social-constructivist).  Other public manifestations of
  "anti-foundationalism" among mathematicians can be found in some of
  the responses to the controversial piece on "Theoretical
  Mathematics" in the AMS Bulletin a couple years back.  Since the
  appearance of the notorious Scientific American article on "the
  death of proof", some of these people have backpedaled a bit to say,
  well, they didn't really mean THAT.  Tymoczko and I (representing
  the opposite sides) were to have edited a special issue of
  Philosophia Mathematica on the role of proof in mathematics (whether
  and if so how it matters), but this was prevented by his premature
  death.

I must have overlooked the "theoretical mathematics" article in the
AMS Bulletin, and the response to it.  Ditto for the "death of proof"
article in Scientific American.  John, can you supply volume and page
numbers?

-- Steve