Giuseppe Longo

Giuseppe Longo

Œuvres

Characterization and burden of Campania children health migration across Italian regions during years 2006–2010: chance and/or necessity?

Characterization and burden of Campania children health migration across Italian regions during years 2006–2010: chance and/or necessity?

Variations on the theme of invariants: conceptual and mathematical dualities in physics vs biology1 Giulia Frezza* Giuseppe Longo Through the looking glass: could physics and biology reflect each other When one tries to elaborate a mathematical theory apt to explain some aspects of biology though starting from the clue of one and unique materiality anyhow one becomes aware of some peculiarities Especially during the elaboration of theoretical extensions of physics by new observables Bailly and Longo which gives an account in possibly new mathematical terms of living beings singularity some characteristic polarizations have been enlightened and verified A key aspect of this approach is the claim of a duality: a conceptual opposition between some theoretical aspects of the two disciplines Table synthetically shows a representation of some conceptual dualities or could we say borrowing the term from biology a crossing over between physical and biological theories For example in our approach biological time and its irreversibility are viewed as constitutive operators of biological complexity while energy is analyzed as a parameter in contrast to the understanding of time as a parameter and energy as an operator in quantum physics Bailly and Longo As a matter of fact energy appears as a parameter in allometric scaling laws in biology Savage et al Moreover as a conceptual symmetry to entropy we also proposed in the same paper the notion of anti entropy as a measurable local reversal of physical entropy production corresponding to increasing biological Human Evolution vol n pp Dipartimento di Filosofia Università degli Studi di Roma TRE Université Denis Diderot Paris fr CNRS ENS Paris Dépt d Informatique et CREA– Polytechnique fr

Variations on the theme of invariants: conceptual and mathematical dualities in physics vs biology1 Giulia Frezza* Giuseppe Longo Through the looking glass: could physics and biology reflect each other When one tries to elaborate a mathematical theory apt to explain some aspects of biology though starting from the clue of one and unique materiality anyhow one becomes aware of some peculiarities Especially during the elaboration of theoretical extensions of physics by new observables Bailly and Longo which gives an account in possibly new mathematical terms of living beings' singularity some characteristic polarizations have been enlightened and verified A key aspect of this approach is the claim of a duality: a conceptual opposition between some theoretical aspects of the two disciplines Table synthetically shows a representation of some conceptual dualities or could we say borrowing the term from biology a crossing over between physical and biological theories For example in our approach biological time and its irreversibility are viewed as constitutive operators of biological complexity while energy is analyzed as a parameter in contrast to the understanding of time as a parameter and energy as an operator in quantum physics Bailly and Longo As a matter of fact energy appears as a parameter in allometric scaling laws in biology Savage et al Moreover as a conceptual symmetry to entropy we also proposed in the same paper the notion of anti entropy as a measurable local reversal of physical entropy production corresponding to increasing biological Human Evolution vol n pp Dipartimento di Filosofia Università degli Studi di Roma TRE Université Denis Diderot Paris fr CNRS ENS Paris Dépt d'Informatique et CREA– Polytechnique fr

Reflections on Concrete Incompleteness

Reflections on Concrete Incompleteness

PENSEE LAÏQUE ET ABSOLUS LES PARADIGMES DES SCIENCES

PENSEE LAÏQUE ET ABSOLUS LES PARADIGMES DES SCIENCES

Randomness and Determination from Physics and Computing towards Biology

Randomness and Determination from Physics and Computing towards Biology

Critique of Computational Reason in the Natural Sciences

Critique of Computational Reason in the Natural Sciences

1PENSIERO LAICO ED ASSOLUTI: I PARADIGMI DELLA SCIENZA

1PENSIERO LAICO ED ASSOLUTI: I PARADIGMI DELLA SCIENZA

Mathematical Infinity “in prospettiva” and Spaces of Possibilities1

Mathematical Infinity “in prospettiva” and Spaces of Possibilities1

Quelques spécifications théorique de l état vivant de la matière Giuseppe Longo CNRS Dépt Informatique ENS et CREA Polytechnique Paris http: www di ens fr users longo Nous résumons ici de façon très schématique et informelle trois propositions plus ou moins mathématisées qui pourraient aider saisir sans aucune prétention de complétude descriptive certains aspects du vivant Le cadre conceptuel pour ces idées se trouve en partie dans Bailly Longo mais les détails techniques sont développés dans les articles cités ci dessous Notre but très trop ambitieux est bien de saisir de façon prétendument scientifique l unité de l organisme vivant en l exprimant dans les termes de sa “singularité physique” La situation critique étendue par laquelle nous caractérisons l état du vivant et les processus qui en sont le siège Bailly Longo 2008c ne se trouve pas dans les théories physiques car pour ces dernières les transitions critiques y sont en général définies par des valeurs précises des paramètres de contrôle elles y sont représentables par un point pour chaque paramètre pertinent Dans notre cas la situation critique que nous considérons demeure aussi longtemps que le vivant en question un organisme par exemple perdure elle serait représentable par un volume non nul dans l espace des paramètres température pression ou toute variable d état pertinent en ce sens qu aucun paramètre ne se réduit un point ce qui intuitivement correspond aux capacités d adaptabilité et la plasticité du vivant Intuitivement toujours elle correspond aussi au fait que tout robuste qu il puisse être relativement des modifications des paramètres dans les limites donc d un intervalle plus o moins grand le vivant est toujours dans une situation critique pris au sens de fragilité cette fois relativement sa propre survie Les propriétés de la criticité dont nous faisons état pour le vivant bénéficient sans s y réduire de certaines propriétés de la criticité physique et notamment de la tendance la divergence des longueurs de corrélation ce qui nous permet ...

Quelques spécifications théorique de l'état vivant de la matière Giuseppe Longo CNRS Dépt Informatique ENS et CREA Polytechnique Paris http: www di ens fr users longo Nous résumons ici de façon très schématique et informelle trois propositions plus ou moins mathématisées qui pourraient aider saisir sans aucune prétention de complétude descriptive certains aspects du vivant Le cadre conceptuel pour ces idées se trouve en partie dans Bailly Longo mais les détails techniques sont développés dans les articles cités ci dessous Notre but très trop ambitieux est bien de saisir de façon prétendument scientifique l'unité de l'organisme vivant en l'exprimant dans les termes de sa “singularité physique” La situation critique étendue par laquelle nous caractérisons l'état du vivant et les processus qui en sont le siège Bailly Longo 2008c ne se trouve pas dans les théories physiques car pour ces dernières les transitions critiques y sont en général définies par des valeurs précises des paramètres de contrôle elles y sont représentables par un point pour chaque paramètre pertinent Dans notre cas la situation critique que nous considérons demeure aussi longtemps que le vivant en question un organisme par exemple perdure elle serait représentable par un volume non nul dans l'espace des paramètres température pression ou toute variable d'état pertinent en ce sens qu'aucun paramètre ne se réduit un point ce qui intuitivement correspond aux capacités d'adaptabilité et la plasticité du vivant Intuitivement toujours elle correspond aussi au fait que tout robuste qu'il puisse être relativement des modifications des paramètres dans les limites donc d'un intervalle plus o moins grand le vivant est toujours dans une situation critique pris au sens de fragilité cette fois relativement sa propre survie Les propriétés de la criticité dont nous faisons état pour le vivant bénéficient sans s'y réduire de certaines propriétés de la criticité physique et notamment de la tendance la divergence des longueurs de corrélation ce qui nous permet ...

Symétries: Pour une épistémologie aux

Symétries: Pour une épistémologie aux

Incompletezza1 Giuseppe Longo http: www di ens fr users longo Laboratoire et Département d Informatique CNRS et Ecole Normale Supérieure Paris et CREA Ecole Polytechnique Introduzione Il teorema di incompletezza di Gödel del non solamente un grande risultato di Logica Matematica ma puo anche divenire il punto di partenza di una riflessione che va oltre la Matematica e la questione dei suoi fondamenti e le correla a problemi e metodi in altre discipline Riferendoci ad esso faremo qui una “storia critica delle idee” ovvero una rilettura esplicitamente a posteriori di alcuni passaggi del pensiero scientifico moderno in cui l audacia di proposte di conoscenza si scontrata con problemi dimostrabilmente insolubili e risultati limitativi o negativi i quali pero a loro volta hanno aperto nuovi orizzonti del sapere Rifletteremo cioè ad alcuni grandi paradigmi scientifici per coglierne un aspetto comune l incompletezza appunto nei rispettivi ambiti e nei suoi diversi significati vedremo i modi in cui essa stata dimostrata e in alcuni casi superata Un analisi puntuale benché informale del teorema di Gödel e di una riflessione di Turing sarà dunque solo un elemento di questo testo In esso pur evitando si spera abusi e contaminazioni improprie si estenderà il tipo di lettura proposto alle analisi scientifiche ed epistemologiche di Laplace ed al loro limite nel grande teorema negativo di Poincaré cosi chiamato dal suo autore quindi alle tesi di Einstein sulla non completezza della Meccanica Quantistica termine usato e tema analizzato in un celeberrimo articolo in collaborazione con Podolski e Rosen Si parlerà infine della presunta completezza delle descrizioni molecolari in Biologia ovvero del DNA inteso come luogo della informazione ereditaria e programma completo dell ontogenesi Da Laplace a Poincaré L unità del metodo e dell Universo secondo Laplace va trovata nell identità delle leggi della Fisica alla scala della nostra percezione e di quelle che governano le particelle microscopiche Tutti i fenomeni osservabili ...

Incompletezza1 Giuseppe Longo http: www di ens fr users longo Laboratoire et Département d'Informatique CNRS et Ecole Normale Supérieure Paris et CREA Ecole Polytechnique Introduzione Il teorema di incompletezza di Gödel del non solamente un grande risultato di Logica Matematica ma puo' anche divenire il punto di partenza di una riflessione che va oltre la Matematica e la questione dei suoi fondamenti e le correla a problemi e metodi in altre discipline Riferendoci ad esso faremo qui una “storia critica delle idee” ovvero una rilettura esplicitamente a posteriori di alcuni passaggi del pensiero scientifico moderno in cui l'audacia di proposte di conoscenza si scontrata con problemi dimostrabilmente insolubili e risultati limitativi o negativi i quali pero' a loro volta hanno aperto nuovi orizzonti del sapere Rifletteremo cioè ad alcuni grandi paradigmi scientifici per coglierne un aspetto comune l'incompletezza appunto nei rispettivi ambiti e nei suoi diversi significati vedremo i modi in cui essa stata dimostrata e in alcuni casi superata Un'analisi puntuale benché informale del teorema di Gödel e di una riflessione di Turing sarà dunque solo un elemento di questo testo In esso pur evitando si spera abusi e contaminazioni improprie si estenderà il tipo di lettura proposto alle analisi scientifiche ed epistemologiche di Laplace ed al loro limite nel grande teorema negativo di Poincaré cosi' chiamato dal suo autore quindi alle tesi di Einstein sulla non completezza della Meccanica Quantistica termine usato e tema analizzato in un celeberrimo articolo in collaborazione con Podolski e Rosen Si parlerà infine della presunta completezza delle descrizioni molecolari in Biologia ovvero del DNA inteso come luogo della informazione ereditaria e programma completo dell'ontogenesi Da Laplace a Poincaré L'unità del metodo e dell'Universo secondo Laplace va trovata nell'identità delle leggi della Fisica alla scala della nostra percezione e di quelle che governano le particelle microscopiche Tutti i fenomeni osservabili ...

Causality and programs in Molecular Biology

Causality and programs in Molecular Biology

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 ...

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 ...

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 ...

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 ...

Some Topologies for Computations

Some Topologies for Computations

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

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

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 ...

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 ...

Critica della Ragion Informatica in Scienze della Natura Giuseppe Longo Dépt d Informatique CNRs et École Normale Supérieure Paris http: www di ens fr users longo Sunto Cercheremo in questo testo di mettere brevemente in evidenza alcuni principi costitutivi di quella particolare forma di conoscenza che ci data dalla macchina digitale il moderno computer nel suo rapporto alla matematica da cui origina ed alle scienze della natura fisica e biologia La tesi di fondo che la ricchezza storica e concettuale della teoria che ha permesso la realizzazione pratica di questa straordinaria macchina lungi dall esser “neutra” o “trasparente” rispetto al reale In particolare in relazione alle strutture causali ed alle rotture di simmetrie che le generano strutture centrali dell intelligibilità della Natura la macchina digitale ne propone di proprie Questo permetterà di accennare ad una distinzione fra “imitazione” e “modellizzazione” nell attività di simulazione o formalizzazione e di mettere in evidenza i limiti e le potenzialità della simulazione digitale Dall alfabeto alla macchine La novita straordinaria cui siamo confrontati oggi una macchina frutto di un percorso storico evolutivo molto articolato Questa macchina non c era “prima” nello stesso modo in cui milioni di anni fa non c erano i mammiferi sulla faccia della terra Nella dinamica sempre costitutiva di novita del sistema evolutivo emergono i mammiferi nulla di miracoloso solo una vicenda molto complessa che mescola invarianza e variabilita continuita e cambiamento in parte aleatori in parte ancora non ben classificabili nelle attuali categorie fisiche di determinazione In modo analogo se non piu complesso si sviluppa la storia umana e al suo interno con una continuità discontinuità che ricca di pratiche comuni del linguaggio della cultura simbolica si arriva a questa macchina il calcolatore digitale che sta cambiando il mondo Essa il punto attualmente massimo di un percorso tutto particolare che inizia certo con il linguaggio ma risente soprattutto della ...

Critica della Ragion Informatica in Scienze della Natura Giuseppe Longo Dépt d'Informatique CNRs et École Normale Supérieure Paris http: www di ens fr users longo Sunto Cercheremo in questo testo di mettere brevemente in evidenza alcuni principi costitutivi di quella particolare forma di conoscenza che ci data dalla macchina digitale il moderno computer nel suo rapporto alla matematica da cui origina ed alle scienze della natura fisica e biologia La tesi di fondo che la ricchezza storica e concettuale della teoria che ha permesso la realizzazione pratica di questa straordinaria macchina lungi dall'esser “neutra” o “trasparente” rispetto al reale In particolare in relazione alle strutture causali ed alle rotture di simmetrie che le generano strutture centrali dell'intelligibilità della Natura la macchina digitale ne propone di proprie Questo permetterà di accennare ad una distinzione fra “imitazione” e “modellizzazione” nell'attività di simulazione o formalizzazione e di mettere in evidenza i limiti e le potenzialità della simulazione digitale Dall'alfabeto alla macchine La novita' straordinaria cui siamo confrontati oggi una macchina frutto di un percorso storico evolutivo molto articolato Questa macchina non c'era “prima” nello stesso modo in cui milioni di anni fa non c'erano i mammiferi sulla faccia della terra Nella dinamica sempre costitutiva di novita' del sistema evolutivo emergono i mammiferi nulla di miracoloso solo una vicenda molto complessa che mescola invarianza e variabilita' continuita' e cambiamento in parte aleatori in parte ancora non ben classificabili nelle attuali categorie fisiche di determinazione In modo analogo se non piu' complesso si sviluppa la storia umana e al suo interno con una continuità discontinuità che ricca di pratiche comuni del linguaggio della cultura simbolica si arriva a questa macchina il calcolatore digitale che sta cambiando il mondo Essa il punto attualmente massimo di un percorso tutto particolare che inizia certo con il linguaggio ma risente soprattutto della ...

The Cognitive Foundations of Mathematics: human gestures in

The Cognitive Foundations of Mathematics: human gestures in

Mathematical Concepts and Physical Objects

Mathematical Concepts and Physical Objects

1The Constructed Objectivity of Mathematics and the Cognitive Subject1

1The Constructed Objectivity of Mathematics and the Cognitive Subject1

Intellectica pp

Intellectica pp

The difference between Clocks and Turing Machines1 Giuseppe Longo

The difference between Clocks and Turing Machines1 Giuseppe Longo

Protention and retention in biological systems

Protention and retention in biological systems

Disponible vers le juin euros

Disponible vers le juin euros

1CERCLES VICIEUX MATH MATIQUES ET

1CERCLES VICIEUX MATH MATIQUES ET

L infini mathématique in prospettiva et les espaces des possibles1

L'infini mathématique in prospettiva et les espaces des possibles1

Giuseppe Longo LIENS CNRS et D pt de Math matiques et Informatique

Giuseppe Longo LIENS CNRS et D pt de Math matiques et Informatique

La rationalité mathématique et les formes de la connaissance esquisse d un projet entre mathématiques et cognition1 Giuseppe Longo CNRS Ecole Normale Supérieure et CREA Ecole Polytechnique Rue D Ulm Paris France http: www di ens fr users longo Résumé Dans ce texte on essayera de mettre en évidence quelques uns des nouveaux défis que les sciences de la cognition posent aux mathématiques Au delà des succès de méthodes mathématiques bien établies largement utilisées dans ces disciplines on soulignera l importance de la recherche de méthodes nouvelles que les nouveaux enjeux demandent Ma thèse est que les mathématiques sont extrêmement plastiques presque autant que notre cerveau et donc qu elles peuvent et sauront se constituer autour des nouveaux problèmes posés La rationalité des mathématiques est dynamique dans l histoire quoiqu elle soit loin d être arbitraire en fait ses racines cognitives la placent au coeur de notre rapport actif de compréhension structuration du monde De plus le rôle paradigmatique des mathématiques parmi nos formes de connaissance permet l analyse de certains aspects de la cognition humaine par analogie avec les analyses fondationelles en mathématiques Ceci est bien le point principal que l on développera une analyse de ce que l imaginerie cérébrale ou la connaissance des activités neuronales ponctuelles peut nous dire analyse conduite en parallèle une réflexion sur le rôle du signe formel dans les fondements des mathématiques Per il volume “Geometria intuizione esperienza Centro Enriques Plus Edizioni Livorno

La rationalité mathématique et les formes de la connaissance esquisse d'un projet entre mathématiques et cognition1 Giuseppe Longo CNRS Ecole Normale Supérieure et CREA Ecole Polytechnique Rue D'Ulm Paris France http: www di ens fr users longo Résumé Dans ce texte on essayera de mettre en évidence quelques uns des nouveaux défis que les sciences de la cognition posent aux mathématiques Au delà des succès de méthodes mathématiques bien établies largement utilisées dans ces disciplines on soulignera l'importance de la recherche de méthodes nouvelles que les nouveaux enjeux demandent Ma thèse est que les mathématiques sont extrêmement plastiques presque autant que notre cerveau et donc qu'elles peuvent et sauront se constituer autour des nouveaux problèmes posés La rationalité des mathématiques est dynamique dans l'histoire quoiqu'elle soit loin d'être arbitraire en fait ses racines cognitives la placent au coeur de notre rapport actif de compréhension structuration du monde De plus le rôle paradigmatique des mathématiques parmi nos formes de connaissance permet l'analyse de certains aspects de la cognition humaine par analogie avec les analyses fondationelles en mathématiques Ceci est bien le point principal que l'on développera une analyse de ce que l'imaginerie cérébrale ou la connaissance des activités neuronales ponctuelles peut nous dire analyse conduite en parallèle une réflexion sur le rôle du signe formel dans les fondements des mathématiques Per il volume “Geometria intuizione esperienza Centro Enriques Plus Edizioni Livorno

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