FOM: book by Leo Corry

[Stephen G Simpson]

McLarty cited a book ``Modern Algebra and the Rise of Mathematical
Structures'' by Leo Corry, 1996.  I got it out of the library.  From
the preface it seems that Corry has at least partially accepted the
dubious premise of ``categorical foundations''.  He wants category
theory to assume a place alongside subjects that have serious
f.o.m. content, such as proof theory and set theory.  He doesn't use
the phrase ``categorical foundations'', but he does say:

  Although category theory is not usually included under the label
  ``metamathematics'', it is obviously a metamathematical theory in
  the etymological sense of the word. ....  Many disciplines usually
  included under the heading of ``metamathematics'' have been the
  object of considerable historical and philosophical research;
  category theory until now has received too little attention.

So Corry set out to write a historical/philosophical study of category
theory.  But he got sidetracked, and Part One of the book is a history
of ideal theory (Dedekind, Hilbert, Noether, ...).  I have read only a
little of that discussion.  Part Two traces the history of some
general notions of mathematical structure (Ore, Bourbaki, ...) and of
category theory (Eilenberg, MacLane, ...).  Model theory is dismissed
as irrelevant, because of its reliance on mathematical logic.  The
historical/philosophical/mathematical discussion seems superficial at
best.  There are some interesting extended quotes from unpublished
internal documents of Bourbaki.

As McLarty has noted, Corry speculates on page 332 that Bourbaki
stayed away from categories because it's difficult to reconcile them
with structures.  In actual fact, such a reconciliation is difficult
only if you arbitrarily decide that, while structures must be sets,
categories may be proper classes.  There is no serious mathematical
reason forcing anyone to make such a decision, so Corry's speculation
seems incorrect.  

The obvious reason why Bourbaki stayed away from categories is
mentioned by Dieudonne in a quote on pages 382-383.  According to
Dieudonne, categories were not useful enough in the branches of
mathematics that Bourbaki proposed to treat.

McLarty 21 Apr 1999 12:42:13

 > It was primarily homological algebra which Bourbaki members
 > routinely used but could into get into their system.

If Bourbaki couldn't get homological algebra into their system, then
what is Bourbaki's book ``Homological Algebra'' (Paris, Hermann, 1980)
about?  I haven't actually seen this book, but Corry cites it.

-- Steve