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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 Sender: qed-owner Precedence: bulk

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