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15Espace, temps et cognitionRevue de Synthèse 124 (1): 61-118. 2003.La cognition humaine paraît étroitement liée à la structure de l'espace et du temps relativement auxquels le corps, le geste, l'intelligibilité semblent devoir se déterminer. Pourtant, ce qui, après les approches physico-mathématiques de Galilée et de Newton, fut caractérisé par Kant comme formes de l'intuition sensible, n'a cessé au cours des siècles qui suivirent de se trouver remis en cause dans leur saisie première par les développements théoriques. En mathématiques d'abord, avec les géométr…Read more
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21Letter to TuringTheory, Culture and Society 36 (6): 73-94. 2019.This personal, yet scientific, letter to Alan Turing, reflects on Turing's personality in order to better understand his scientific quest. It then focuses on the impact of his work today. By joining human attitude and particular scientific method, Turing is able to “immerse himself” into the phenomena on which he works. This peculiar blend justifies the epistolary style. Turing makes himself a “human computer”, he lives the dramatic quest for an undetectable imitation of a man, a woman, a machin…Read more
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1The ongoing MOOC revolution is bound to change the academic world on an unprecedented scale. It is in fact very likely that in the coming decades universities all over the world will shrink in size and number, while professors will assume more and more the role of specialized tutors looking after lectures delivered by a small number of world known academic “superstars”. In what follows we shall analyze some aspect of the phenomenon, focusing on the reasons why, like it or not, the academic world…Read more
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34How Future Depends on Past and Rare Events in Systems of LifeFoundations of Science 23 (3): 443-474. 2018.The dependence on history of both present and future dynamics of life is a common intuition in biology and in humanities. Historicity will be understood in terms of changes of the space of possibilities as well as by the role of diversity in life’s structural stability and of rare events in history formation. We hint to a rigorous analysis of “path dependence” in terms of invariants and invariance preserving transformations, as it may be found also in physics, while departing from the physico-ma…Read more
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79The differential method and the causal incompleteness of programming theory in molecular biologyFoundations of Science 12 (4): 337-366. 2007.The “DNA is a program” metaphor is still widely used in Molecular Biology and its popularization. There are good historical reasons for the use of such a metaphor or theoretical model. Yet we argue that both the metaphor and the model are essentially inadequate also from the point of view of Physics and Computer Science. Relevant work has already been done, in Biology, criticizing the programming paradigm. We will refer to empirical evidence and theoretical writings in Biology, although our argu…Read more
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18On church's formal theory of functions and functionalsAnnals of Pure and Applied Logic 40 (2): 93-133. 1988.
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4Review: C. P. J. Koymans, Models of the Lambda Calculus (review)Journal of Symbolic Logic 52 (1): 284-285. 1987.
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Matematica e scienze della natura, a partire da EnriquesNuova Civiltà Delle Macchine 21 (4): 19-38. 2003.
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6The mathematical continuum, from intuition to logicIn Jean Petitot, Francisco J. Varela, Bernard Pachoud & Jean-Michel Roy (eds.), Naturalizing Phenomenology: Issues in Contemporary Phenomenology and Cognitive Science, Stanford University Press. 1999.
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27On church's formal theory of functions and functionals: The λ-calculus: connections to higher type recursion theory, proof theory, category theoryAnnals of Pure and Applied Logic 40 (2): 93-133. 1988.
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20Fondamenti cognitivi della matematica e analisi matematiche del viventeRivista di Estetica 37 113-124. 2008.Nel presente testo cercherò principalmente di esplicitare la correlazione tra le mie ricerche, così come ho cercato di riassumerle nel testo su La ragionevole efficacia della matematica, e l’impostazione vareliana nei confronti delle scienze cognitive. Nel tentativo di far questo, intendo prendere spunto da un aneddoto che, a dire il vero, pare sia effettivamente accaduto. Si racconta che, intorno ai sette o otto anni, il piccolo Gauss si trovò di fronte un maestro severissimo che, come puniz...
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111Reflections on Concrete IncompletenessPhilosophia Mathematica 19 (3): 255-280. 2011.How do we prove true but unprovable propositions? Gödel produced a statement whose undecidability derives from its ad hoc construction. Concrete or mathematical incompleteness results are interesting unprovable statements of formal arithmetic. We point out where exactly the unprovability lies in the ordinary ‘mathematical’ proofs of two interesting formally unprovable propositions, the Kruskal-Friedman theorem on trees and Girard's normalization theorem in type theory. Their validity is based on…Read more
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208Mathematical intuition and the cognitive roots of mathematical conceptsTopoi 29 (1): 15-27. 2010.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 f…Read more
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Antinomie e polarità, determinazione e aleatorietà nel processo vivente della materiaDiscipline Filosofiche 19 (1). 2009.
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34The mathematics of computing between logic and physicsIn S. B. Cooper & Andrea Sorbi (eds.), Computability in Context: Computation and Logic in the Real World, World Scientific. 2011.
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16Mathematics and the Biological PhenomenaAquinas 43 (2): 331-354. 2000.The first part of this paper highlights some key aspects of the differences in the use of mathematical tools in physics and in biology. Scientific knowledge is viewed as a network of interactions, some than as a hierarchically organized structure where mathematics would display the essence of phenomena. The concept of "unity" in the biological phenomenon is then discussed. In the second part, a foundational issue in mathematics is revisited, following recent perspective in the physiology of acti…Read more
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34Symmetries and Symmetry-Breakings: The Fabric of Physical Interactions and the Flow of Time (review)Foundations of Science 16 (4): 331-333. 2011.This short note develops some ideas along the lines of the stimulating paper by Heylighen (Found Sci 15 4(3):345–356, 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 organiza…Read more
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38Mathematical intelligence, infinity and machines: beyond GodelitisJournal of Consciousness Studies 6 (11-12): 11-12. 1999.We informally discuss some recent results on the incompleteness of formal systems. These theorems, which are of great importance to contemporary mathematical epistemology, are proved using a variety of conceptual tools provably stronger than those of finitary axiomatisations. Those tools require no mathematical ontology, but rather constitute particularly concrete human constructions and acts of comprehending infinity and space rooted in different forms of knowledge. We shall also discuss, albei…Read more
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Andata e ritorno dalla finanza alle matematiche passando per la biologiaNuova Civiltà Delle Macchine 29 (4). 2011.
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35Prototype Proofs in Type TheoryMathematical Logic Quarterly 46 (2): 257-266. 2000.The proofs of universally quantified statements, in mathematics, are given as “schemata” or as “prototypes” which may be applied to each specific instance of the quantified variable. Type Theory allows to turn into a rigorous notion this informal intuition described by many, including Herbrand. In this constructive approach where propositions are types, proofs are viewed as terms of λ-calculus and act as “proof-schemata”, as for universally quantified types. We examine here the critical case of …Read more
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Mathematical Concepts and Physical ObjectsIn Luciano Boi, Pierre Kerszberg & Frédéric Patras (eds.), Rediscovering Phenomenology. Phenomenological Essays on Mathematical Beings, Physical Reality, Perception and Consciousness, Hal Ccsd. pp. 195-228. 2007.
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36What is Turing's Comparison between Mechanism and Writing Worth?In S. Barry Cooper (ed.), How the World Computes, . pp. 450--461. 2012.
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