From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4591 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Re: Bourbaki and Categories Date: Tue, 16 Sep 2008 07:24:43 -0400 (EDT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241020045 13967 80.91.229.2 (29 Apr 2009 15:47:25 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:47:25 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Tue Sep 16 21:20:26 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 16 Sep 2008 21:20:26 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KfkgZ-0004KF-IW for categories-list@mta.ca; Tue, 16 Sep 2008 21:14:23 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 61 Original-Lines: 130 Xref: news.gmane.org gmane.science.mathematics.categories:4591 Archived-At: I don't know what to say about the suggestion that a circle and a line=20 make a sketch of which Euclidean plane geometry is a model. I would thin= k=20 you would need a point too, since intersections are crucial. Maybe=20 complex projective geometry since then two lines intersect in one point=20 (unless they coincide), a line and a circle in two (unless they are=20 tangent or equal) and every pair of circles in four (ditto). Maybe the=20 exceptions could be handled in some sketch. At any rate, it wold e=20 interesting to try to sketch this in detail. At any rate, I never though= t=20 about this before. Michael On Tue, 16 Sep 2008, Andre.Rodin@ens.fr wrote: > > When one defines, say, a group =E0 la Borbaki, i.e. structurally, it us= ually goes > without saying that the defined structure is defined up to isomorphism.= The > notion of isomorphism plays in this case the role similar to that of eq= uality > in the (naive) arithmetic. In most structural constexts the distinction= between > the "same" structure and isomorphic structures is mathematically trivia= l just > like the distinction between the "same" number and equal numbers. It ma= y be not > specially discussed in this case exactly because it is very basic. The = notion > of admissible map, say, that of group homomorphism, on the contrary, re= quires a > definition, which may be non-trivial. > The idea to do mathematics up to isomorphism is not Bourbaki's inventio= n; it > goes back at least to Hilbert's "Grundlagen der Geometrie" of 1899. In = this > sense the modern axiomatic method is structuralist. In his often-quoted= letter > to Frege Hilbert explicitely says that a theory is "merely a framework"= while > domains of their objects are multiple and transform into each other by > "univocal and reversible one-one transformations". Those who trace the = history > of mathematical structuralism back to Hilbert are quite right, in my vi= ew. > I have in mind two issues related to CT, which suggest that CT goes in = a > *different* direction - in spite of the fact that MacLane and many othe= r > workers in CT had (and still have) structuralist motivations. The first= is > Functorial Semantics, which brings a *category* of models, not just one= model > up to isomorphism. From the structuralist viewpoint the presence of > non-isomorphic models (i.e. non-categoricity) is a shortcoming of a giv= en > theory. From the perspective of Functorial Semantics it is a "natural" = feature > of mathematical theories to be dealt with rather than to be remedied. > The second thing I have in mind is Sketch theory. I cannot see that Hil= bert's > basic structuralist intuition applies in this case. In my understanding= things > work in Sketch theory more like in Euclid. Think about circle and strai= ght line > as a sketch of the theory of the first four books of Euclid's "Elements= ". I > would particularly appreciate, Michael, your comment on this point sinc= e I > learnt a lot of Sketch theory from your works. > I have also a comment about the idea to rewrite Bourbaki's "Elements" f= rom a new > categorical viewpoint. Bourbaki took Euclid's "Elements" as a model for= his > work just like did Hilbert writing his "Gundlagen". In my view, this is= this > long-term Euclidean tradition of "working foundations", which is worth = to be > saved and further developed, in particular in a categorical setting. I'= m less > sure that Bourbaki's example should be followed in a more specific sens= e. > Bourbaki tries to cover too much - and doesn't try to distinguish betwe= en what > belongs to foundations and what doesn't. As a result the work is too lo= ng and > not particularly usefull for (early) beginners. I realise that today's > mathematics unlike mathematics of Euclid's time is vast, so it is more > difficult to present its basics in a concentrated form. But consider Hi= lbert's > "Grundlagen". It covers very little - actually near to nothing - of geo= metry of > its time. But at the same time it provided a very powerful model of how= to do > mathematics in a new way, which greatly influenced mathematics educatio= n and > mathematical research in 20th century. In my view, Euclid's "Elements" = and > Hilbert's "Grundlagen" are better examples to be followed. > > best, > andrei > > > le 15/09/08 12:59, Michael Barr =E0 barr@math.mcgill.ca a =E9crit : > >> I don't know about this. I took several courses in the late 1950s tha= t >> seem to have been influenced by the structuralist ideas (certainly >> categories weren't mentioned; I never heard the word until Dave Harris= on >> arrived in 1959) and each of them started by defining an appropriate >> notion of "admissible map". I do not recall any special point being m= ade >> of isomorphism and I think in general it was used for what we now call= a >> bimorphism (1-1 and onto) even in cases, such as topological groups, w= hen >> they were not isomorphisms. >> >> To be sure Bourbaki was not mentioned either, but this structuralist >> influence seemed strong. >> >> Michael >> >