In this paper is presented and discussed the notion of true finite by opposition to the notion of theoretical finite. Examples from mathematics and physics are given. Fermat’s infinite descent principle is challenged.

ABSTRACTIn this paper, I offer an account of propositional learning: namely, learning that p. I argue for what I call the “Three Transitions Thesis” or “TTT” according to which four states and three transitions between them characterize such learning. I later supplement the TTT to account for learning why p. In making my case, I discuss mathematical propositions such as Fermat's Last Theorem and the ABC Conjecture, and then generalize to other mathematical propositions and to non-mathematical propositions. I also (...) discuss some interesting applications of the TTT, and reply to some noteworthy objections. (shrink)

En partant du problème des partis, conçu comme un dialogue entre partenaires, nous ferons ressortir le tissage de sa structure avec des considérations religieuses et éthiques; puis nous reconsidérerons l’argument dit « du pari » en l’interprétant, mutatis mutandis, comme une préfiguration de la théorie des jeux plutôt que comme la résolution d’un problème de probabilité, ce qu’il est plus évidemment sous la plume de Fermat qui, à la limite, ne résout pas le même problème que Pascal, quoique les (...) deux épistoliers aient pu en avoir l’impression. (shrink)

This paper introduces a little-known episode in the history of physics, in which a mathematical proof by Pierre Fermat vindicated Galileo’s characterization of freefall. The first part of the paper reviews the historical context leading up to Fermat’s proof. The second part illustrates how a physical and a mathematical insight enabled Fermat’s result, and that a simple modification would satisfy any of Fermat’s critics. The result is an illustration of how a purely theoretical argument can (...) settle an apparently empirical debate. (shrink)

The first half of the 17th century was a time of intellectual ferment when wars of natural philosophy were echoes of religious wars, as we illustrate by a case study of an apparently innocuous mathematical technique called adequality pioneered by the honorable judge Pierre de Fermat, its relation to indivisibles, as well as to other hocus-pocus. André Weil noted that simple applications of adequality involving polynomials can be treated purely algebraically but more general problems like the cycloid curve (...) cannot be so treated and involve additional tools–leading the mathematician Fermat potentially into troubled waters. Breger attacks Tannery for tampering with Fermat’s manuscript but it is Breger who tampers with Fermat’s procedure by moving all terms to the left-hand side so as to accord better with Breger’s own interpretation emphasizing the double root idea. We provide modern proxies for Fermat’s procedures in terms of relations of infinite proximity as well as the standard part function. (shrink)

Essay Review of “Les Arithmétiques de Diophante. Lecture historique et mathématique” by Roshdi Rashed and Christian Houzel, and Histoire de l’analyse diophantienne classique : d’Abū Kamil à Fermat by Roshdi Rashed.

Pierre de Fermat is known as the inventor of modern number theory. He invented–improved many methods useful in this discipline. Fermat often claimed to have proved his most difficult theorems thanks to a method of his own invention: the infinite descent. He wrote of numerous applications of this procedure. Unfortunately, he left only one almost complete demonstration and an outline of another demonstration. The outline concerns the theorem that every prime number of the form 4n + 1 (...) is the sum of two squares. In this paper, we analyse a recent proof of this theorem. It is interesting because: it follows all the elements of which Fermat wrote in his outline; it represents a good introduction to all logical nuances and mathematical variants concerning this method of which Fermat spoke. The assertions by Fermat will also be framed inside their historical context. Therefore, the aims of this paper are related to the history of mathematics and to the logic of proof-methods. (shrink)

A truly accessible introduction to the fundamentals of classical philosophy and metaphysics. In a series of three small books, Pierre-Marie Emonet brings the riches of philosophy within reach of us all, not just those with the time and inclination to work through erudite philosophical language.

Early in his mathematical career Leibniz discovered some important methods and results but had to recognize that his findings had been anticipated by other mathematicians such as Pierre de Fermat, James Gregory, Isaac Newton, François Regnauld, John Wallis, etc. This paper investigates the cases of Isaac Barrow and Pietro Mengoli who, earlier than Leibniz, had been familiar with the characteristic triangle, transmutations methods, the inverse connection between determining tangents and areas of curves or the sums of the reciprocal (...) figurate numbers, and the harmonic triangle. To what extent was Leibniz aware of the results and publications of his predecessors? How did he assess their methods and results? Why did Leibniz never acknowledge any influence of these two mathematicians on his own studies? After publication of Leibniz’s manuscripts concerning the prehistory and early history of the calculus in the Academy Edition these questions can be investigated on the solid foundation of original texts. (shrink)

Cet ouvrage collectif sur Artémis et Diane en Grèce ancienne et en Italie, dirigé par Giovanni Casadio et Patricia A. Johnston, regroupe la plupart des interventions du Vergilian Society’s Symposium Cumanum, tenu à la Villa Vergiliana, près de l’ancienne Cumes, en juin 2012. Après une introduction générale des éditeurs sur Artémis puis sur Diane et la présentation des contributions, il est organisé en quatre parties. Le cadre de la première paraîtra très vaste, « Artémis à Mycènes et en Grèce...

Claude Calame nous donne ici une édition révisée de son livre désormais classique, Les Chœurs de jeunes filles en Grèce archaïque, publié en 1977 aux éditions de l’Ateneo. Le livre était la version imprimée de sa thèse présentée pour soutenance à l’Université de Lausanne, rédigée en grande partie quand il enseignait à l’Université d’Urbino, grand centre alors de recherches sur la poétique grecque et la sémiotique. Il se présentait en deux volumes, correspondant à deux parties. Le deuxième vol...

L’étrange histoire du jeune Damon de Chéronée, le « dernier des Péripoltides », et de son masque de suie, que Plutarque a placée en introduction à ses Vies de Cimon et de Lucullus, a suscité des interprétations fort diverses : dernier avatar du « Chasseur noir », témoignage sur les luttes féroces entre factions pro-romaines et pro-pontiques aux temps de la première guerre de Mithridate… On cherche ici surtout à montrer ce que Plutarque a voulu faire en écrivant cette biographie (...) miniature en écho à celle de Lucullus, le sauveur de sa patrie. Entre des figures de fondateur ou de refondateur qui se substituent les unes aux autres, raconter le passage et l’intégration d’une cité grecque dans l’empire romain, faire un travail de mémoire qui conjure les fantômes du passé et contribue à l’homonoia dans la petite cité. Avec une question centrale : Rome cité-tyran ?Plutarch and Damon of Chaironeia: a story, a myth, a text, or something more? The strange story of young Damon of Chaironeia, the ‘last of the Peripoltids’, with his soot-smeared face, which Plutarch placed as an introduction to his Lives of Cimon and Lucullus, has given rise to various interpretations: from a last avatar of the Black Hunter to evidence for the fierce struggle between pro-Roman and pro-Pontic factions at the time of the First Mithridatic War… In this paper we have tried to show what Plutarch’s intention may have been when he wrote this miniature biography, echoing that of Lucullus, the saviour of his home city. Between figures of founders or re-founders substituting for each other, the task was to tell of the passing and integration of a Greek city into the Roman Empire, to make a work of memory, liable to ward off ghosts from the past and to contribute to homonoia in the little city, with a central question : was Rome a tyrant-city? (shrink)

Carnap's ideal of explication has become a key concept in analytic philosophy and the basis of a method of analysis which may be considered as an alternative to various forms of naturalism, including Quine's conception of a naturalized epistemology. More recently, new light has been shed on this aspect of the classical Carnap-Quine debate by contemporary philosophers. Whereas Michael Friedman articulated a notion of relativized a priori which owes much to Carnap's internal/external distinction, André Carus attempted to restate Carnap's ideal (...) of explication in a way that bridges the gap between conceptual engineering and naturalism. On the other hand, Mark Wilson argued that concepts develop in unpredictable ways and cannot be planned in the way classical analytic philosophy has assumed from Frege and Russell onwards. This book consists of a series of defences as well as critiques of Carnap's programme, setting it in its historical context, discussing specific cases of explications, and enriching the on-going debate on conceptual engineering and naturalism in analytic philosophy.Carnap's ideal of explication has become a key concept in analytic philosophy and the basis of a method of analysis which may be considered as an alternative to various forms of naturalism, including Quine's conception of a naturalized epistemology. More recently, new light has been shed on this aspect of the classical Carnap-Quine debate by contemporary philosophers. Whereas Michael Friedman articulated a notion of relativized a priori which owes much to Carnap's internal/external distinction, André Carus attempted to restate Carnap's ideal of explication in a way that bridges the gap between conceptual engineering and naturalism. On the other hand, Mark Wilson argued that concepts develop in unpredictable ways and cannot be planned in the way classical analytic philosophy has assumed from Frege and Russell onwards. This book consists of a series of defences as well as critiques of Carnap's programme, setting it in its historical context, discussing specific cases of explications, and enriching the on-going debate on conceptual engineering and naturalism in analytic philosophy. (shrink)

In this article I am not going to try and define introspection. I am going to try to state as precisely as possible how the practice of introspection can be improved, starting from the principle that there exists a disjunction between the logic of action and of conceptualization and the practice of introspection does not require that one should already be in possession of an exhaustive scientific knowledge bearing upon it. . To make matters worse, innumerable commentators upon what passes (...) for introspection do not seem to have practised it and have certainly never contributed anything to its development. My aim is therefore to bring to light a procedure for progressive improvement in the practice of introspection when it is employed in a programme of research. (shrink)

This volumes aim is to provide an introduction to Carnaps book from a historical and philosophical perspective, each chapter focusing on one specific issue. The book will be of interest not only to Carnap scholars but to all those interested in the history of analytical philosophy.

Infants' own activities create and actively select their learning experiences. Here we review recent models of embodied information seeking and curiosity-driven learning and show that these mechanisms have deep implications for development and evolution. We discuss how these mechanisms yield self-organized epigenesis with emergent ordered behavioral and cognitive developmental stages. We describe a robotic experiment that explored the hypothesis that progress in learning, in and for itself, generates intrinsic rewards: The robot learners probabilistically selected experiences according to their potential for (...) reducing uncertainty. In these experiments, curiosity-driven learning led the robot learner to successively discover object affordances and vocal interaction with its peers. We explain how a learning curriculum adapted to the current constraints of the learning system automatically formed, constraining learning and shaping the developmental trajectory. The observed trajectories in the robot experiment share many properties with those in infant development, including a mixture of regularities and diversities in the developmental patterns. Finally, we argue that such emergent developmental structures can guide and constrain evolution, in particular with regard to the origins of language. (shrink)

Through a detailed study of Spinoza's concept of 'experience', Moreau shows how Spinoza extends the power of reason to capture the singularity of individuals: their lives, languages, passions and societies.

This first English-language translation captures the excitement of the original text-already a contemporary classic, and will likely become a standard reference in the history of eighteenth-century thought, politics, and society, and in the ...

What, if anything, is wrong with high executive compensation? Is the common “lay reaction” of indignation and moral outrage justified? In this paper, my main goal is to articulate in a more systematic and philosophical manner the egalitarian responses to these questions. In order to do so, I suggest that we take some insights from recent debates on two versions of egalitarianism: a distributive one, according to which no one should be worse off than others because of unfair distributions of (...) goods and resources, especially ones based on matters of luck or arbitrary factors, and a relational one, which maintains that egalitarian justice requires members of a society to relate to one another as equals. Drawing on recent attempts to highlight the tricky nature of managerial authority, I argue that high inequalities in pay are not simply a distributional matter but should also be analyzed through a relational lens. I also attempt to show that relational egalitarians are well-equipped to question the now dominant “incentives” view of CEO compensation. (shrink)

In mass democracies, voting—in elections or referendums—is the main way in which most citizens can publicly express their political preferences. And yet this means of expression is sometimes perceived by them as highly frustrating, partly because it does not allow for much expression. Dominant voting methods lead to a reduction of options, pressure citizens to vote tactically at the cost of expressing their genuine preferences, and fail to convey what they really think about different candidates, parties, or options. Yet citizens (...) do not merely have a right to vote; they have a more fundamental right to political expression from which one can derive a right to a voting method that offers the most opportunities for expression among those satisfying other important requirements. The aim of this article is therefore to add this consideration about the importance of political expression to debates about voting methods that have mainly been conducted from the perspective of social choice theory. To illustrate what is at stake, it introduces evaluative voting methods (allowing voters to grade all of the candidates, parties, or options) as a promising way of honoring the right to political expression without jeopardizing other important properties of voting methods. (shrink)

Suppose that knowledge and ignorance are complements in the sense of being mutually exclusive: for person S and fact p, either S knows that p or is ignorant that p. Understood in this way, ignorance amounts to a lack or absence of knowledge: S is ignorant that p if and only if it is not the case that S knows that p. Let us call the thesis that knowledge and ignorance are opposites the “Complement Thesis”. In this article, I discuss (...) its deployment in an ingenious new argument advanced by Alvin Goldman and Erik Olsson (2009) which, if sound, establishes that there is a kind of knowledge that amounts to nothing more than true belief. I rebut their argument and in doing so delineate some important epistemological lessons brought to light by the contrast between ignorance and knowledge. (shrink)

Frederick Griffith was an English bacteriologist at the Pathological Laboratory of the Ministry of Health in London who believed that progress in the epidemiology and control of infectious diseases would come only with more precise knowledge of the identity of the causative microorganisms. Over the years, Griffith developed and expanded a serological technique for identifying pathogenic microorganisms, which allowed the tracing of the sources of infectious disease outbreaks: slide agglutination. Yet Griffith is not remembered for his contributions to the biology (...) and epidemiology of infectious diseases so much as for discovering the phenomenon known as ‘transformation’. Griffith’s discovery, for many, was a pure case of serendipity whose biological relevance had also largely escaped him. In this paper, I argue that the key to understanding the significance of bacterial transformation – and the scientific legacy of Fred Griffith – rests not only on it initiating a cascade of events leading to molecular genetics but also on its implications for epidemiology based on the biology of host–parasite interactions. Looking at Griffith’s entire career, instead of focusing only on the transformation study, we can better appreciate the place of the latter within Griffith’s overall contributions. Presented in this way, Griffith’s experiment on bacterial transformation also ceases to appear as an anomaly, which in turn leads us to rethink some of the most prevalent historical conceptions about his work. (shrink)

Dans le traité De la divination dans le sommeil, Aristote entend réfuter la thèse d’une origine divine des rêves qui semblent annoncer les événements futurs. Pour cela, il ne se contente pas de soutenir une thèse différente. Il dénonce également la faiblesse épistémologique de la pratique divinatoire, en proposant un modèle scientifique alternatif : une méthode de classification des rêves et de division par épuisement, qui exclut les rêves inspirés. Cette méthode conduit à souligner le caractère irréductible des faits de (...) hasard et à admettre le caractère inexplicable d’un grand nombre de rêves « véridiques ». La notion de coïncidence ( sumptôma ) joue sur ce point un rôle essentiel. Ce rôle est à la fois descriptif et critique : il permet de définir la classe des rêves dont la dimension véridique est absolument fortuite, et il dénonce l’illusion commune qui les relie à des événements réels. De ce point de vue, le chapitre 1 du De divinatione apporte une contribution, modeste mais spécifique, à la théorie des modalités physiques. (shrink)

Résumé La localisation des implantations des grands éléments d’infrastructure collective pose des problèmes de légitimation juridique spécifique. Ils délimitent en effet plusieurs territoires distincts en fonction de la nature des impacts qu’ils provoquent, lesquels, chacun, sont soumis à leurs propres législations et rentrent dans les compétences d’autorités différentes, y compris celle dont dépend la décision sur l’infrastructure envisagée. De telles situations normatives complexes empêchent la sémantisation légitimante que le droit est censé assurer dans les décisions de l’appareil étatique: elle sera (...) toujours partielle, voire lacunaire. Les compensations ne peuvent être que procédurales, et celles-ci sont insuffisantes à atteindre l’idéal postulé par le principe de légalité des activités publiques. Mais il ne peut en aller autrement pour ces infrastructures. (shrink)

Political judgments usually combine a normative principle or intuition with an appreciation of empirical facts regarding the achievability of different options and their potential consequences. The interesting question dividing partisans of political idealism and realism is whether these kinds of considerations should be integrated into the normative principles themselves or considered apart. At first sight, if a theorist is concerned with guiding political judgments, non-ideal or realist theorizing can seem more attractive. In this article, however, I argue that ideal theorizing (...) might be considered valuable even by theorists moved by a pragmatic concern because it is less exposed to conservatism. I nonetheless contend that the aim to guide action in the world as it is should not be abandoned. Therefore, I outline a four-step method for proceeding from abstract moral principles to concrete political judgments and apply it to a test case. (shrink)

Decision under conditions of uncertainty is an unavoidable fact of life. The available evidence rarely suffices to establish a claim with complete confidence, and as a result a good deal of our reasoning about the world must employ criteria of probable judgment. Such criteria specify the conditions under which rational agents are justified in accepting or acting upon propositions whose truth cannot be ascertained with certainty. Since the seventeenth century philosophers and mathematicians have been accustomed to consider belief under uncertainty (...) from the standpoint of the mathematical theory of probability. In 1654, Blaise Pascal entered into correspondence with Pierre de Fermat on two problems in the theory of probability that had been posed by the Chevalier De Méré – the first involved the just division of the stakes in a game of chance that has been interrupted, the second is the likelihood of throwing a given number in a fixed number of throws using fair dice. This correspondence resulted in fundamental results that are now regarded as the foundation of the mathematical approach to probability, and historical studies of probabilistic reasoning almost invariably begin with the Pascal-Fermat correspondence. Franklin has no interest in denying the significance of the mathematical treatment of probability – he is, after all, a professional mathematician – but the principal theme in his book is the gradual “coming to consciousness” of canons of inference governing uncertain cases. (shrink)

Ce livre a pour objet d’évaluer l’apport de la physique quantique à l’explication du phénomène de la conscience. Après un état des lieux d’ordre sémantique, philosophique et neurobiologique de la question de la relation entre cerveau et conscience, les principaux modèles « classiques » actuels de la conscience sont exposés. Nous montrons que ces modèles laissent en suspens deux questions importantes : a) celle d’expliquer la synchronisation quasi-instantanée de régions éloignées du cerveau qui semble nécessaire à la constitution de percepts (...) conscients; b) celle du fossé explicatif qui existe entre l’expérience subjective, relevant de ce que le sujet est capable de ressentir de façon privée, et la description de ses corrélats neurophysiologiques dans le langage de la science, à la troisième personne – comment ces derniers « donnent-ils lieu » à l’expérience subjective?Les approches quantiques de la conscience plus récentes sont alors exposées de façon détaillée. Nous examinons dans quelle mesure elles peuvent contribuer à répondre à ces questions et proposer une représentation unificatrice intégrant les deux aspects psychique et physiologique du phénomène de conscience. (shrink)

Depuis la revolution scientifique du XVIIe siecle, la science est tourmentee par le sens a donner a sa propre entreprise. Plusieurs tentatives pour fonder l'intelligibilite de la nature sur les structures pretendument immuables de la raison ont echoue. Mais l'echec philosophique est a la mesure du succes aveuglant des connaissances scientifiques, qui n'ont plus que faire du scrupule de principe. Kant n'est-il pas un des avocats les plus eminents de ce scrupule tombe en desuetude? Face au divorce consomme entre physique (...) et metaphysique, il restait a se demander, avec Kant, ce que la raison a a gagner si chacune se force malgre tout a parler le langage de l'autre. Effort douloureux mais riche d'enseignement: si la physique est poussee dans cet horizon de la metaphysique qui lui est etranger, elle retrouve le moyen de s'aventurer dans une reflexion sur son propre point de depart. Reflexion qui hante encore la science aujourd'hui, meme si c'est a son insu. Dans ce premier geste il doit y avoir quelque chose de definitif: seule la philosophie kantienne de la nature nous met vraiment aux prises avec lui. Pierre Kerszberg est professeur de philosophie a l'universite d'Etat de Pennsylvannie. (shrink)