Subject: categories: ST vs CT

From: Giuseppe Longo <[email-masked]>
Date: Mon, 5 Feb 2001 14:12:31 +0100

The key issues raised by Bill Lawvere may be further specified if we  
try to make explicit the different "philosophical projects" which  
underlie the Set Theory vs Category Theory debate.

ST is "newtonian": it proposes a fixed-absolute universe (ZF or  
alike) where all of present (and future!) mathematics should be  
embedded (described).  Its perspective is (basically) "laplacian":  
given the (current) axioms of set-theory (or arithmetic) we can  
prove/decide/predict all future states of affairs (theorems of  
mathematics).

CT originates from a rather different perspective.  Riemann first  
proposed to found mathematics (geometry) on key regularities of the  
world to be turned into conceptual invariants, as "hypothesis" about  
the structure of physical space (connectivity, continuity, isotropy  
... ; H. Weyl added symmetries to this).  Then, one had to single out  
the "invariant preserving transformations" (Kleine, Clifford,  
Poincare' ...).
The objects, as structural invariants, and the morphisms of CT, as
transformations (including isomorphisms ...; functors; natural
transformations ...), came out directly from this tradition; that  
is, CT followed the riemannian project who had replaced the absolute
axioms/universe of euclidean geometry by the "relativizable" notions  
of invariant and transformation (as founding manifolds).
In CT, the unity of mathematics, not as an absolute pre-fixed  
universe, but as an open world of ongoing new concepts and  
structures, is constantly re-constructed, by (interpretation)  
functors, which relate new categories to old ones (moreover, natural  
transformations (adjunctions ...) help to correlate these functors).

Clearly, Newton and Laplace were immense scientific personalities,  
but ...,  since then, many things have happened.  In Physics, of  
course, but also within ... ST or in the proof-theory of Arithmetic,  
such as the independence theorems or the remarquable results on the  
"unremovability of machinery", which, even if within ST, force us  
outside the "rational mechanics" of PA ....

More on this may be found in a two pages note of mine on "Laplace (vs
Hilbert)" (or, reflections on rational mechanics and  
incompleteness), an item inserted in the Dictionary at the end of JY  
Girard's paper "Locus Solum" (MSCS, n. 11.3, to appear) and in a  
paper on "The reasonable effectiveness of Mathematics ..." (both  
downloadable from my web page).

Giuseppe Longo
Lab. et Dept. d'Informatique
CNRS et Ecole Normale Superieure
(Postal addr.:  LIENS
45, Rue D'Ulm
75005  Paris   (France) )
http://www.dmi.ens.fr/users/longo
  et :
CENtre d'Etude des systemes Complexes et de la Cognition
de l'ENS (CENECC)
http://www.cenecc.ens.fr/

e-mail: [email-masked]
(tel. ++33-1-4432-3328; fax  -2151)