LÓGICA DE HACES &

TEORÍA DE CATEGORÍAS

CALI SIMPOSIO

FILOSOFÍA DE LA LÓGICA DE HACES

UNIVERSIDAD DEL VALLE OCTUBRE 19-21, 9.00AM-1.00PM

RALF KRÖMER, JEAN-PIERRE MARQUIS, COLIN MCLARTY

XAVIER CAICEDO, ANDRÉS VILLAVECES, FERNANDO ZALAMEA

PROGRAMA Y ABSTRACTS ANEXOS

BOGOTÁ MESA REDONDA

HISTORIA DE LA TEORÍA DE CATEGORÍAS

UNIVERSIDAD NACIONAL, DEPARTAMENTO DE MATEMÁTICAS, SALÓN 405-202 OCTUBRE 22, 3.00-6.00PM

RALF KRÖMER, JEAN-PIERRE MARQUIS, COLIN MCLARTY

TERCER CONGRESO COLOMBIANO DE FILOSOFÍA UNIVERSIDAD DEL VALLE – CALI, COLOMBIA, OCTOBER 19-21 2010

PHILOSOPHY OF THE LOGIC OF SHEAVES

PURPOSE

Classical Logic and Set Theory influenced deeply the emergence of Analytical Philosophy at the beginning of XXth Century. After a hundred years, it may be time that the Logic of Sheaves and Category Theory help to create alternative forms of practicing Philosophy.

Closer to the concepts of movement, flow, passage, boundary and contamination, Sheaf Logic and Category Theory must reinvigorate some regions of Philosophy abandoned by the Analytical dogmae. “Naturalized” Phenomenology, “Mathematical” Metaphysics, “Semiotical” Hermeneutics, between others, may thus emerge with new instruments at hand.

The Symposium Philosophy of the Logic of Sheaves hopes to establish the state of the question, profiting from a creator of the Logic of Sheaves (Caicedo) and some major specialists in the Philosophy and History of Category Theory (Krömer, Marquis, McLarty). Forty years after Grothendieck’s departure from the IHES, the Symposium will be devoted in part to his legacy.

PROGRAM

Tuesday 19

9.00-10.00 Colin McLarty – Case Western Reserve University Grothendieck's 'incorrigible naivety' in building worlds for Mathematics 10.00-10.30 Discussion

11.00-12.00 Xavier Caicedo – Universidad de los Andes / Universidad Nacional The Model Theory of Sheaves or the Logic of Variable Structures 12.00-12.30 Discussion Wednesday 20

9.00-10.00 Jean-Pierre Marquis – Université de Montréal Sheaves, Spaces and Logic 10.00-10.30 Discussion

11.00-12.00 Andrés Villaveces – Universidad Nacional Bridging a gap between Physics and Logic: the role of Sheaves 12.00-12.30 Discussion Thursday 21

9.00-10.00 Ralf Krömer – Nancy Université / University of Siegen From Cantor to Sheaves. The development of the concepts of direct and

inverse limits – a case study on shifts in mathematical methodology in the prehistory of Category Theory

10.00-10.30 Discussion

11.00-12.00 Fernando Zalamea – Universidad Nacional Sheaf Logic: a mathematical conspectus and a philosophical prospectus 12.00-12.30 Discussion

ABSTRACTS

Xavier Caicedo. The Model Theory of Sheaves or the Logic of Variable Structures. As envisioned by Lawvere, and established in firm ground by Reyes, Freyd and Joyal, sheaves over topological spaces or, more generally, sheaves of a Grothendieck topos yields a natural model theory of variable or extended structures which provides a geometric foundation for intuitionistic logic based in locality of truth and trascending classical constructivist foundations. New logical operations arise naturally in this context that we attempt to classify, particularly in the case of logical connectives which reflect local properties of the domain of variation and go beyond Grothendieck topologies. We have introduced a notion of generic filter on the domain of variation of a sheaf of structures with respect to any fragment of first order infinitary logic. The transit of a sheaf to a static limit via one of these filters and the corresponding generic model theorem, a common generalization of the Forcing Theorem in set theory and Los Theorem on ultraproducts, yields and unifies both set theoretic forcing constructions and the basic model existence theorems of classical finitary or infinitary logic: completeness, compactness, omitting types, joint consistency, model theoretic forcing, Fraïssé limits, the use of consistency properties, etc. It illuminates also the need of countable conditions in some cases. We have justly called this transit the fundamental theorem of (classical) model theory. It provides also a semantical explanation of Gödel's double negation interpretation of classical logic into intuitionistic logic. However, fundamental aspects of the model theory of sheaves proper, the full knowledge of its intrinsic intuitionistic logical operations, the construction and classification of sheaves qua interpretations of genuine intuitionistic theories, and other possible applications, await deeper exploration. Ralf Krömer. From Cantor to Sheaves. The development of the concepts of direct and inverse limits – a case study on shifts in mathematical methodology in the prehistory of Category Theory. This talk is on the role of the notions of direct and inverse limits in the early development of Category theory. In the first part, these constructions are traced back to two seminal ideas of Cantor central to his contributions to the theory of Fourier series (giving rise to his work on set theory), namely his definition of the reals and his nowhere dense perfect set. After a short discussion of the historical methodology, it is shown how throughout the first third of the twentieth century, these two constructions have been imitated and transformed in order to be applied in various fields including algebraic number theory, the theory of topological groups, Galois theory and homology theory. Special attention is given to the epistemological assumptions underlying these applications. The second part concerns the claim that the two types of limit constructions have been crucial at three turning points in the history of Category theory, namely in Eilenberg and Mac Lane's 1942 introduction of functors and natural transformations, in the introduction and development of the sheaf concept in the work of Leray, Cartan and Grothendieck, and in Kan's 1958 introduction of adjoint functors. This claim is made more precise, and supplied with evidence. Jean-Pierre Marquis. Sheaves, Spaces and Logic. The introduction of sheaves in logic seems to be a simple generalization of standard set-theoretical semantics. However, it opened the door to new concepts, for instance the generic model and the classifying topos and new relations between logical theories, spaces (as conceived by Grothendieck) and various areas of mathematics. It thus brought about a new unification of mathematics and logic that is still going a vigorous development. Colin McLarty. Grothendieck's 'incorrigible naivety' in building worlds for Mathematics. We look at three sources for categorical logic in Alexander Grothendieck's algebraic geometry of the 1960s. In two cases Grothendieck suggested new ideas for formal logic to organize reasoning in geometry. The third case is the most conceptual. It is topos logic which first became explicit a few years later in William Lawvere's view of a topos as a 'universe of variable sets.' But it is implicit in Grothendieck's own strategy for understanding the vast new geometric and arithmetic 'worlds' he was creating. Andrés Villaveces. Bridging a gap between Physics and Logic: the role of Sheaves. The interaction between Physics and Logic (the search for a meaningful Logic for Physics, the construction of a true Model Theory of Physics) has always been laden with issues, impasses, apparent paradoxes. However, recent developments (Zilber, Connes, etc.) and new connections between Non-Commutative Geometry, Model Theory and Mathematical Physics have revived the possibility of a better, deeper conformation of just such a dream: a genuine Model Theoretical Analysis of Physics. I will describe (in an external, not so technical, way) some of those recent developments, and will attempt to show the philosophical role that sheaves play in bridging such a gap. Fernando Zalamea. Sheaf Logic: a mathematical conspectus and a philosophical prospectus. (A). Sheaf theory emerged in the 1940’s, through Leray and Cartan, propelling afterwards (1950-60’s) a reconstruction of algebraic geometry by Serre and Grothendieck. Lawvere’s (1971) elementary version of Grothendieck’s topoi launched an investigation of the logic of topoi, and, particularly, (pre)sheaves’ topoi. After Freyd’s (1972) construction of connectives and quantifiers in topoi, a reformulation by Joyal (early 1970’s) of Kripke’s modal and intuitionistic semantics (1965) became the main tool for formalizing the many intermediate, internal, logics of topoi. A more flexible version of Kripke-Joyal semantics was afterwards proposed by Caicedo (1994), who, in his model theory of sheaves, has constructed a generic model for sheaves of first order structures. (B). From a philosophical standpoint, we contend that the logic of sheaves combined with category theory may stand, for new forms of philosophy in the XXIst century, as classical first order logic combined with set theory stood for the emergence of analytic philosophy in the XXth century. In our talk, we will present briefly (conspectus) the main conceptual ideas behind the mathematical developments described in (A), and discuss briefly (prospectus) some of the philosophical import (B) of the logic of sheaves, around three main issues: transitory ontology and sheafification epistemology; naturality of sheaf logics (“real” neighborhood semantics, real fibers), against artificiality of classical logic (“ideal” pointwise semantics, ideal limit); elucidation of a Grothendieck’s transformation in the evolving analytical/ synthetical understanding of mathematical concepts. Further information and contact: fernandozalamea@gmail.com