bourbaki

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Date: Wed, 21 Apr 93 16:41:02 -0400
From: jt@linus.mitre.org
Message-id: <9304212041.AA05383@malabar.mitre.org>
To: qed@mcs.anl.gov
Subject: bourbaki
Cc: jt@linus.mitre.org
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My own feeling is that the influence of the Bourbaki project -- in the
sense that is being discussed here, that is the elaboration of a
compendium of mathematics from a small number of first principles --
has been relatively insignificant. The "seminaire Bourbaki" on the
other hand is another matter. 

I don't think most of the Bourbaki volumes are as widely read as one
would expect. Here are some examples:

  It was commonly remarked by analysts that Bourbaki's treatment of
  integration on arbitrary locally compact spaces (though quite nice in
  many respects) was done "twice." (Basically, treatment number one is
  too general, requiring treatment number two.) For this reason, analysts 
  usually preferred Rudin's treatment or even the treatment given in Monroe's 
  ancient text. This was particularly hard fact for francophiles like myself 
  to accept.

  Much of the stuff that graduate students have to learn isn't in
  Bourbaki. (The whole Hormander theory of linear partial differential 
  equations is not mentioned, group representations; In fact, you name
  it, it probably isn't there.)  

  Some of the abstractions widely used by Bourbaki are not that
  useful. For example, uniform spaces mentioned previously by John Harrison 
  is one such abstraction in my opinion. (Bourbaki's "Notes historiques" 
  attributes this concept to A. Weil.) Uniform spaces are a general framework 
  for talking about uniform continuity, uniform convergence, completeness etc. 
  However, most structures which are uniform spaces are usually something else 
  as well, and which is easier to deal with (e.g., a metric space, a topological 
  group etc.) Typically analysts prefer to treat structures in these other more 
  direct ways rather than treat them as instances of another abstraction. Some 
  abstractions are helpful, others aren't.

  There are some nice Bourbaki chapters. The treatment of Lie algebras
  is elegant; Chapter 9 of "Topologie generale", "Les applications des
  nombres reels a la topologie" is another one which is very elegant.

I for one do not have a clear idea of the mathematics QED should
attempt to cover. "All of it" I think is utterly unrealistic. A
codification of the Bourbaki volumes is (a) much too restrictive and
(b) almost guaranteed not to be used by anybody.  The mathematics QED
should attempt to cover, is I think, the fundamental issue we should
be discussing now. From a marketing point, this I think makes a lot of
sense.  Using an automobile manufacturer's analogy, let's design and
produce a car for the market, not find a market for a car we designed
and produced without consideration of the market.

The choice of logic is an important issue, but should be decided by
the kind of mathematics which is going to be the main focus of QED.

Javier Thayer