#ecole-polytechnique

169 Books
d c t i v i t y - INRIA, Project-Team TREC
D c t i v i t y
INRIA, Project-Team TREC
23 Pages
d c t i v i ty - INRIA, Project-Team TREC
D c t i v i ty
INRIA, Project-Team TREC
28 Pages
On the Relevance of Negative Results1 Giuseppe Longo CNRS et Dépt d
On the Relevance of Negative Results1 Giuseppe Longo CNRS et Dépt d'Informatique École Normale Supérieure Paris et CREA École Polytechnique http: www di ens fr users longo Abstract The access to scientific knowledge is a construction of objectivity which needs the critical insight of “negative results” These consist in the explicit construction of internal limits to current theories and methods We shall hint to the role of some results which in Logic in Physics or Computing opened up new areas for knowledge by saying “No we cannot compute this we cannot decide that The idea is that both the sciences of life and of cognition in particular in connection to Mathematics and Computing need similar results in order to set limits to the passive transfer of physico mathematical methods into their autonomous construction of knowledge and open the way to new tools and perspectives We will compare this perspective with the requirement both at the national and European levels to finalize most all research activities into foreseeable industrial applications Scientific knowledge and critical insight The analysis of concepts conducted on a comparative level if possible as well as the tentative explanation of the philosophical project should always accompany scientific work In fact critical reflections regarding existing theories are at the core of positive scientific constructions because science is often constructed against the supposed tyranny and autonomy of “facts” which in reality are nothing but “small scale theories” Science is also often constructed by means of an audacious interpretation of “new” and old facts it progresses against the obvious and against common sense le “bon sens” it struggles against the illusions of immediate knowledge and must be capable of escaping from already established theoretical frameworks For example the very high level of mathematical technicity in the geometry of Ptolemaic epicycles constructed from clearly observable facts strongly perplexed numerous Renaissance thinkers such as ...
GIUSEPPE LONGO
9 Pages
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
18 Pages
THEOREMS AS CONSTRUCTIVE VISIONS1 Giuseppe Longo CNRS Ecole Normale Supérieure et CREA Ecole Polytechnique Rue D
THEOREMS AS CONSTRUCTIVE VISIONS1 Giuseppe Longo CNRS Ecole Normale Supérieure et CREA Ecole Polytechnique Rue D'Ulm Paris France http: www di ens fr users longo Abstract This paper briefly reviews some epistemological perspectives on the foundation of mathematical concepts and proofs It provides examples of axioms and proofs from Euclid to recent “concrete incompleteness” theorems In reference to basic cognitive phenomena the paper focuses on order and symmetries as core “construction principles” for mathematical knowledge A distinction is then made between these principles and the “proof principles” of modern Mathemaical Logic The role of the blend of these different forms of founding principles will be stressed both for the purposes of proving and of understanding and communicating the proof THE CONSTRUCTIVE CONTENT OF EUCLID'S AXIOMS From the time of Euclid to the age of super computers Western mathematicians have continually tried to develop and refine the foundations of proof and proving Many of these attempts have been based on analyses logically and historically linked to the prevailing philosophical notions of the day However they have all exhibited more or less explcitly some basic cognitive principles for example the notions of symmetry and order Here I trace some of the major steps in the evolution of notion of proof linking them to these cognitive basics For this purpose let's take as a starting point Euclid's Aithemata Requests the minimal constructions required to do geometry: Invited lecture ICMI conference on Proof and Proving Taipei Taiwan May Hanna de Villiers eds Springer
GIUSEPPE LONGO
19 Pages
EPE Newsletter October - Brigitte
3 Pages
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