categories: Re: slogans

[Giuseppe Longo <Giuseppe.Longo@ens.fr>]

... about adding slogans:
"there is no mathematics without structures"
and
"there are no structures without transformations (between structures)"

In stating this, I would go back to (Gauss and) Reimann (and Klein).  
 This was a turning point of last century mathematics.  Geometry is  
the analysis of (possibly) curved space and the unity of geometries  
is found on the notion of transformation (over manifolds, say:  
continuous, differentiable ...).
Mathematics is no more found (only) on "quantities", since ratios of  
length and of angles, at the heart of Euclidean geometry, are not  
preserved in non-euclidean frames (their group of automorphisms are  
not closed under omotheties).

Category Theory is the theory which inherited this fantastic  
broadening of perspective.


--Giuseppe Longo
Lab. "Jacques Herbrand"
CNRS et Ecole Normale Superieure
(Postal addr.:  LIENS
45, Rue D'Ulm
75005  Paris   (France) )

http://www.dmi.ens.fr/users/longo
e-mail: longo@di.ens.fr
(tel. ++33-1-4432-3328, FAX 4432-2080)


Upon kind permission of the M.I.T. Press, the book below is
currently downloadable from Longo's web page above (its n-th
edition is out of print...):

Andrea Asperti and Giuseppe Longo. Categories, Types and
Structures: an introduction to Category Theory for the working
computer scientist. M.I.T.- Press, 1991. (pp. 1--300).