* ST vs CT
@ 2001-02-05 13:12 Giuseppe Longo
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From: Giuseppe Longo @ 2001-02-05 13:12 UTC (permalink / raw)
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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: longo@dmi.ens.fr
(tel. ++33-1-4432-3328; fax -2151)
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