Rapports de stage : les meilleurs exemples - Page 22

Symmetries and symmetry breakings: the fabric of physical interactions and the flow of time1 Giuseppe Longo Dépt d Informatique CNRS Ecole Normale Supérieure et CREA Polytechnique Paris http: www di ens fr users longo Summary This short note develops some ideas along the lines of the stimulating paper by Heylighen Found Sci 2010a It summarizes a theme in several writings with Francis Bailly downloadable from this author s web page The “geometrization” of time and causality is the common ground of the analysis hinted here and in Heylighen s paper Heylighen adds a logical notion consistency in order to understand a possible origin of the selective process that may have originated this organization of natural phenomena We will join our perspectives by hinting to some gnoseological complexes common to mathematics and physics which may shed light on the issues raised by Heylighen Note: Francis Bailly passed away recently: his immense experience in physics has been leading our joint work for many years Historically it is with relativist physics that there occurs a “change of perspective”: we pass from “causal laws” to the structural organization of space and time or even from causal laws to the “legality normativity of geometric structures” This understanding of causal laws by the identification of structural organizations stems essentially from the intrinsic duality existing between the characterization of the geometry of the universe and that of energy momentum within that universe By this duality and the putting into effect of the principle of invariance under the differentiable transformations of space time the “forces” are relativized to the nature of this geometry: they will even appear or disappear according to the geometric nature of the universe chosen a priori to describe physical behaviors Now it is similar for quantum physics in gauge theories Here gauge groups operate upon internal variables such as in the case of relativity where the choice of local gauges and their changes enable to define or ... - Giuseppe Longo
Symmetries and symmetry breakings: the fabric of physical interactions and the flow of time1 Giuseppe Longo Dépt d'Informatique CNRS Ecole Normale Supérieure et CREA Polytechnique Paris http: www di ens fr users longo Summary This short note develops some ideas along the lines of the stimulating paper by Heylighen Found Sci 2010a It summarizes a theme in several writings with Francis Bailly downloadable from this author's web page The “geometrization” of time and causality is the common ground of the analysis hinted here and in Heylighen's paper Heylighen adds a logical notion consistency in order to understand a possible origin of the selective process that may have originated this organization of natural phenomena We will join our perspectives by hinting to some gnoseological complexes common to mathematics and physics which may shed light on the issues raised by Heylighen Note: Francis Bailly passed away recently: his immense experience in physics has been leading our joint work for many years Historically it is with relativist physics that there occurs a “change of perspective”: we pass from “causal laws” to the structural organization of space and time or even from causal laws to the “legality normativity of geometric structures” This understanding of causal laws by the identification of structural organizations stems essentially from the intrinsic duality existing between the characterization of the geometry of the universe and that of energy momentum within that universe By this duality and the putting into effect of the principle of invariance under the differentiable transformations of space time the “forces” are relativized to the nature of this geometry: they will even appear or disappear according to the geometric nature of the universe chosen a priori to describe physical behaviors Now it is similar for quantum physics in gauge theories Here gauge groups operate upon internal variables such as in the case of relativity where the choice of local gauges and their changes enable to define or ...
Giuseppe Longo
Mathematical intuition and the cognitive roots of mathematical concepts1 Giuseppe Longo Arnaud Viarouge CNRS et Ecole Normale Superieure Psychology and Human Development Dpt et CREA Ecole Polytechnique Paris Fr Peabody College Vanderbilt University http: www di ens fr users longo Nashville TN USA Abstract The foundation of Mathematics is both a logico formal issue and an epistemological one By the first we mean the explicitation and analysis of formal proof principles which largely a posteriori ground proof on general deduction rules and schemata By the second we mean the investigation of the constitutive genesis of concepts and structures the aim of this paper This genealogy of concepts so dear to Riemann Poincaré and Enriques among others is necessary both in order to enrich the foundational analysis by this too often disregarded aspect the cognitive and historical constitution of mathematical structures and because of the provable incompleteness of proof principles also in the analysis of deduction For the purposes of our investigation we will hint here to the philosophical frame as well as to the some recent advances in Cognition that support our claim the cognitive origin and the constitutive role of mathematical intuition From Logic to Cognition Over the course of the XXth century the relationships between Philosophy and Mathematics have been dominated by Mathematical Logic A most interesting area of Mathematics which from onwards year of one of the major mathematical results of the century Gödelian Incompleteness enjoyed the double status of a discipline that is both technically profound and philosophically fundamental From the foundational point of view Proof Theory constituted its main aspect also on account of other remarkable results Ordinal Analysis after Gentzen Type Theory in the manner of Church Gödel Girard various forms of incompleteness independence in Set Theory and Arithmetics and produced spin offs which are in the course of changing the world: the functions for the computation of proofs ... - Giuseppe Longo
Mathematical intuition and the cognitive roots of mathematical concepts1 Giuseppe Longo Arnaud Viarouge CNRS et Ecole Normale Superieure Psychology and Human Development Dpt et CREA Ecole Polytechnique Paris Fr Peabody College Vanderbilt University http: www di ens fr users longo Nashville TN USA Abstract The foundation of Mathematics is both a logico formal issue and an epistemological one By the first we mean the explicitation and analysis of formal proof principles which largely a posteriori ground proof on general deduction rules and schemata By the second we mean the investigation of the constitutive genesis of concepts and structures the aim of this paper This genealogy of concepts so dear to Riemann Poincaré and Enriques among others is necessary both in order to enrich the foundational analysis by this too often disregarded aspect the cognitive and historical constitution of mathematical structures and because of the provable incompleteness of proof principles also in the analysis of deduction For the purposes of our investigation we will hint here to the philosophical frame as well as to the some recent advances in Cognition that support our claim the cognitive origin and the constitutive role of mathematical intuition From Logic to Cognition Over the course of the XXth century the relationships between Philosophy and Mathematics have been dominated by Mathematical Logic A most interesting area of Mathematics which from onwards year of one of the major mathematical results of the century Gödelian Incompleteness enjoyed the double status of a discipline that is both technically profound and philosophically fundamental From the foundational point of view Proof Theory constituted its main aspect also on account of other remarkable results Ordinal Analysis after Gentzen Type Theory in the manner of Church Gödel Girard various forms of incompleteness independence in Set Theory and Arithmetics and produced spin offs which are in the course of changing the world: the functions for the computation of proofs ...
Giuseppe Longo
  • Univers Univers
  • Ebooks Ebooks
  • Livres audio Livres audio
  • Presse Presse
  • Podcasts Podcasts
  • BD BD
  • Documents Documents