Formal topology aims at developing general topology in intuitionistic and predicative mathematics. Many classical results of general topology have been already brought into the realm of constructive mathematics by using formal topology and also new light on basic topological notions was gained with this approach which allows distinction which are not expressible in classical topology. Here we give a systematic exposition of one of the main tools in formal topology: inductive generation. In fact, many formal topologies can be presented in (...) a predicative way by an inductive generation and thus their properties can be proved inductively. We show however that some natural complete Heyting algebra cannot be inductively defined. (shrink)

We present a manuscript of Paul Lorenzen that provides a proof of consistency for elementary number theory as an application of the construction of the free countably complete pseudocomplemented semilattice over a preordered set. This manuscript rests in the Oskar-Becker-Nachlass at the Philosophisches Archiv of Universität Konstanz, file OB 5-3b-5. It has probably been written between March and May 1944. We also compare this proof to Gentzen's and Novikov's, and provide a translation of the manuscript.

The general framework of this paper is a reformulation of Hilbert’s program using the theory of locales, also known as formal or point-free topology [P.T. Johnstone, Stone Spaces, in: Cambridge Studies in Advanced Mathematics, vol. 3, 1982; Th. Coquand, G. Sambin, J. Smith, S. Valentini, Inductively generated formal topologies, Ann. Pure Appl. Logic 124 71–106; G. Sambin, Intuitionistic formal spaces–a first communication, in: D. Skordev , Mathematical Logic and its Applications, Plenum, New York, 1987, pp. 187–204]. Formal topology presents (...) a topological space, not as a set of points, but as a logical theory which describes the lattice of open sets. The application to Hilbert’s program is then the following. Hilbert’s ideal objects are represented by points of such a formal space. There are general methods to “eliminate” the use of points, close to the notion of forcing and to the “elimination of choice sequences” in intuitionist mathematics, which correspond to Hilbert’s required elimination of ideal objects. This paper illustrates further this general program on the notion of valuations. They were introduced by Dedekind and Weber [R. Dedekind, H. Weber, Theorie des algebraischen Funktionen einer Veränderlichen, J. de Crelle t. XCII 181–290] to give a rigorous presentation of Riemann surfaces. It can be argued that it is one of the first example in mathematics of point-free representation of spaces [N. Bourbaki, Eléments de Mathématique. Algèbre commutative, Hermann, Paris, 1965, Chapitre 7]. It is thus of historical and conceptual interest to be able to represent this notion in formal topology. (shrink)

. The effort in providing constructive and predicative meaning to non-constructive modes of reasoning has almost without exception been applied to theories with full classical logic [4]. In this paper we show how to combine unrestricted countable choice, induction on infinite well-founded trees and restricted classical logic in constructively given models. These models are sheaf models over a $\sigma$ -complete Boolean algebra, whose topologies are generated by finite or countable covering relations. By a judicious choice of the Boolean algebra we (...) can directly extract effective content from $\Pi_2^0$ -statements true in the model. All the arguments of the present paper can be formalised in Martin-Löf's constructive type theory with generalised inductive definitions. (shrink)

Semantical arguments, based on the completeness theorem for first-order logic, give elegant proofs of purely syntactical results. For instance, for proving a conservativity theorem between two theories, one shows instead that any model of one theory can be extended to a model of the other theory. This method of proof, because of its use of the completeness theorem, is a priori not valid constructively. We show here how to give similar arguments, valid constructively, by using Boolean models. These models are (...) a slight variation of ordinary first-order models, where truth values are now regular ideals of a given Boolean algebra. Two examples are presented: a simple conservativity result and Herbrand's theorem. (shrink)

We present a proof of Goodmanʼs Theorem, which is a variation of the proof of Renaldel de Lavalette [9]. This proof uses in an essential way possibly divergent computations for proving a result which mentions systems involving only terminating computations. Our proof is carried out in a constructive metalanguage. This involves implicitly a covering relation over arbitrary posets in formal topology, which occurs in forcing in set theory in a classical framework, but can also be defined constructively.

We introduce the notion of Boolean measure algebra. It can be described shortly using some standard notations and terminology. If B is any Boolean algebra, let BN denote the algebra of sequences , xn B. Let us write pk BN the sequence such that pk = 1 if i K and Pk = 0 if k < i. If x B, denote by x* BN the constant sequence x* = . We define a Boolean measure algebra to be a Boolean (...) algebra B with an operation μ:BN → B such that μ = 0 and μ = x. Any Boolean measure algebra can be used to model non-principal ultrafilters in a suitable sense. Also, we can build effectively the initial Boolean measure algebra. This construction is related to the closed open Ramsey Theorem 193–198.). (shrink)

We present a new and constructive proof of the Peter-Weyl theorem on the representations of compact groups. We use the Gelfand representation theorem for commutative C*-algebras to give a proof which may be seen as a direct generalization of Burnside's algorithm [3]. This algorithm computes the characters of a finite group. We use this proof as a basis for a constructive proof in the style of Bishop. In fact, the present theory of compact groups may be seen as a natural (...) continuation in the line of Bishop's work on locally compact, but Abelian, groups [2]. (shrink)

This work concerns constructive aspects of measure theory. By considering metric completions of Boolean algebras – an approach first suggested by Kolmogorov – one can give a very simple construction of e.g. the Lebesgue measure on the unit interval. The integration spaces of Bishop and Cheng turn out to give examples of such Boolean algebras. We analyse next the notion of Borel subsets. We show that the algebra of such subsets can be characterised in a pointfree and constructive way by (...) an initiality condition. We then use our work to define in a purely inductive way the measure of Borel subsets. (shrink)

We show that the set of points of an overt closed subspace of a metric completion of a Bishop-locally compact metric space is located. Consequently, if the subspace is, moreover, compact, then its collection of points is Bishop-compact. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

We present a possible computational content of the negative translation of classical analysis with the Axiom of (countable) Choice. Interestingly, this interpretation uses a refinement of the realizability semantics of the absurdity proposition, which is not interpreted as the empty type here. We also show how to compute witnesses from proofs in classical analysis of ∃-statements and how to extract algorithms from proofs of ∀∃-statements. Our interpretation seems computationally more direct than the one based on Godel's Dialectica interpretation.

We associate with any game G another game, which is a variant of it, and which we call . Winning strategies for have a lower recursive degree than winning strategies for G: if a player has a winning strategy of recursive degree 1 over G, then it has a recursive winning strategy over , and vice versa. Through we can express in algorithmic form, as a recursive winning strategy, many common proofs of non-constructive Mathematics, namely exactly the theorems of the (...) sub-classical logic Limit Computable Mathematics [6], Hayashi and Nakata [7]). (shrink)

A proof-theoretical analysis of elementary theories of order relations is effected through the formulation of order axioms as mathematical rules added to contraction-free sequent calculus. Among the results obtained are proof-theoretical formulations of conservativity theorems corresponding to Szpilrajn’s theorem on the extension of a partial order into a linear one. Decidability of the theories of partial and linear order for quantifier-free sequents is shown by giving terminating methods of proof-search.

Hintikka makes a distinction between two kinds of games: truthconstituting games and truth-seeking games. His well-known game-theoretical semantics for first-order classical logic and its independence-friendly extension belongs to the first class of games. In order to ground Hintikka’s claim that truth-constituting games are genuine verification and falsification games that make explicit the language games underlying the use of logical constants, it would be desirable to establish a substantial link between these two kinds of games. Adapting a result from Thierry (...) Coquand, we propose such a link, based on a slight modification of Hintikka’s games, in which we allow backward playing for ∃loïse. In this new setting, it can be proven that sequent rules for first-order logic, including the cut rule, are admissible, in the sense that for each rule, there exists an algorithm which turns winning strategies for the premisses into a winning strategy for the conclusion. Thus, proofs, as results of truth-seeking games, can be seen. (shrink)

Hintikka makes a distinction between two kinds of games: truthconstituting games and truth-seeking games. His well-known game-theoretical semantics for first-order classical logic and its independence-friendly extension belongs to the first class of games. In order to ground Hintikka’s claim that truth-constituting games are genuine verification and falsification games that make explicit the language games underlying the use of logical constants, it would be desirable to establish a substantial link between these two kinds of games. Adapting a result from Thierry (...) Coquand, we propose such a link, based on a slight modification of Hintikka’s games, in which we allow backward playing for ∃loïse. In this new setting, it can be proven that sequent rules for first-order logic, including the cut rule, are admissible, in the sense that for each rule, there exists an algorithm which turns winning strategies for the premisses into a winning strategy for the conclusion. Thus, proofs, as results of truth-seeking games, can be seen. (shrink)

Valorising the biocultural heritage of common goods could enable peasant farmers to achieve socially and economically inclusive sustainability. Increasingly appreciated by consumers, peasant heritage products offer small farmers promising opportunities for economic, social and territorial development. Identifying the obstacles and levers of this complex, multi-scale and multi-stakeholder objective requires an integrative framework. We applied the panarchy conceptual framework to two cases of participatory research with small quinoa producers: a local fair in Chile and quinoa export production in Bolivia. In both (...) cases, the “commoning” process was crucial both to bring stakeholders together inside their communities and to gain outside recognition for their production and thus achieve social and economic inclusion. Despite the differences in scale, the local fair and the export market shared a similar marketing strategy based on short value chains promoting quality products with high identity value. In these dynamics of biocultural heritage valorisation, the panarchical approach revealed the central place as well as the vulnerability of the community territory. As a place of both anchoring and opening, the community territory is the privileged space where autonomous and consensual control over the governance of common biocultural resources can be exercised. (shrink)

The present paper examines how discursive representations and emotive constructions underpin an argumentative dynamic that emerges from apparently non-argumentative statements, like those found in newspaper headlines. Our data comes from Greek broadsheet newspapers in the polarized context of the Greek crisis. First, we outline an analytic synergy that scrutinizes representational meaning and the semiotization of emotions in headlines. We then move towards the reconstruction of the inferential passage, contained in the headlines, that unites the implicit standpoint with its supporting argument.

L’Introduction à l’analyse des lignes courbes algébriques de Gabriel Cramer, paru en 1750, a tout de suite bénéficié du soutien de D’Alembert qui l’a inclus dans les références bibliographiques de ses articles de mathématiques de l’Encyclopédie portant sur les courbes. Ainsi choisi et légitimé par l’entreprise encyclopédique et ses reprises, l’ouvrage de Cramer devient objet patrimonial au tournant du xixe siècle pour les mathématiciens, amateurs, professionnels ou enseignants qui travaillent sur les courbes algébriques. Le suivi sur le temps long des (...) mentions de l’Introduction dans les encyclopédies, dictionnaires ou histoires des mathématiques en langue française, ainsi que dans quelques textes d’enseignement, nous permettra de comprendre que ce processus de patrimonialisation n’induit pas sa conservation, sa protection, sa valorisation à de seules fins mémorielles, en figeant l’ouvrage dans un passé immuable. Au contraire, il permet une historicisation critique de la part des acteurs qui les met en position d’interroger et de réactualiser les connaissances et les pratiques que l’œuvre contient, au gré des mouvements d’émergence, de réorganisation et des spécialisations des savoirs mathématiques, quitte à en oublier certains aspects pour en mettre de nouveaux en valeur. (shrink)

This paper assesses the problem of the possible tension between development and environmental protection, especially for developing countries. Some leaders of these countries like Jacob Zuma claim for example that poor countries should only join the fight against climate change if it does not compromise their economic development, thus suggesting that environmental protection is more often than not an obstacle to economic development. I argue that this argument is if not misleading, at least incomplete because it does not take the (...) diversity of environmental protection seriously and reduces development to economic development. In fact, some aspects of environmental protection may clearly clash with some kinds of economic development. In that line, when countries are still very poor, their economic development should take priority over some ecological-friendly policies. But this is only one side of the truth. The other side of the truth is that some ecologically-friendly policies can also make poor countries richer, more democratic and fairer. (shrink)

Thierry Patrice, 57 ans, médecin, nommé professeur à 32 ans, Lauréat de la Faculté, est internationalement connu pour ses travaux concernant l'action de la lumière sur les tissus vivant en cancérologie. Il a reçu plusieurs Prix pour l'étude du rôle de l'oxygène dans différentes maladies, dont le diabète, mais aussi lors du vieillissement.

Devenir borgne! Difficile de ne pas vivre ce handicap comme un drame. Sans la vision binoculaire et stéréoscopique, le réel semble n'avoir plus d'épaisseur. L'auteur rapporte ce qu'il a éprouvé au cours de sa prise en charge ophtalmologique et interroge en même temps les peintres malvoyants et les philosophes : qu'est-ce que voir? Le handicap visuel permettrait-il de dépasser l'ordinaire de la vision? Pour répondre à ces questions, Monet, Degas, Victor Brauner sont interpellés dans leurs oeuvres ou encore Bruegel et (...) Rembrandt, en même temps que sont convoqués Platon, Descartes, Alain, Merleau-Ponty ou encore James Joyce et William Blake, sans oublier les Évangiles. Les bien-voyants ne se rendent pas compte qu'ils sont condamnés à vivre la tyrannie du réel. Et si les yeux empêchaient de voir en vrai? Le genre humain n'a peut-être pas vocation à voir mais à entrevoir. Dans ces conditions, ne devrait-on pas considérer la malvoyance comme un atout? Ne serait-il pas temps de voir autrement qu'avec les yeux? Peut-être, alors, sentira-t-on la présence de l'Être émerger de la pénombre"--Page 4 of cover. (shrink)

L’article envisage d’abord les conditions et conséquences d’une ressaisie proprement philosophique du concept sociologique d’artification. Trois propositions en découlent. 1˚) Contrairement à l’approche sociologique de l’artification, cette ressaisie relève d’une problématique de la légitimation (mais on parle alors de légitimation spécifiquement théorique et non pas idéologique). 2˚) La question de l’artification suppose une définition spéculative de l’art, laquelle possède nécessairement une vertu normative. 3˚) Seules des pratiques peuvent subir un processus d’artification, et non des objets (à moins de les considérer (...) en tant que fonctions d’une pratique). On passe ensuite à la question de la « transposabilité » historique et culturelle du concept d’artification. Dans une optique sociologique, ce sont les pratiques contemporaines voire actuelles qui, pour l’essentiel, retiennent l’attention. Mais la présence même d’objets anciens et non-occidentaux dans nos musées d’art invite à réfléchir à la manière dont ils sont ou peuvent être qualifiés d’artistiques (ou requalifiés comme tels) ; il s’agit aussi de se pencher sur les processus apparentés à l’artification dans les cultures étrangères à la modernité occidentale. (shrink)

This root-and-branch reevaluation of Darwin's concept of sexual selection tackles the subject from historical, epistemological and theoretical perspectives. Contributions from a wealth of disciplines have been marshaled for this volume, with key figures in behavioural ecology, philosophy, and the history of science adding to its wide-ranging relevance. Updating the reader on the debate currently live in behavioural ecology itself on the centrality of sexual selection, and with coverage of developments in the field of animal aesthetics, the book details the current (...) state of play, while other chapters trace the history of sexual selection from Darwin to today and inquire into the neurobiological bases for partner choices and the comparisons between the hedonic brain in human and non-human animals. Welcome space is given to the social aspects of sexual selection, particularly where Darwin drew distinctions between eager males and coy females and rationalized this as evolutionary strategy. Also explored are the current definition of sexual selection (as opposed to natural selection) and its importance in today's biological research, and the impending critique of the theory from the nascent field of animal aesthetics. As a comprehensive assessment of the current health, or otherwise, of Darwin's theory, 140 years after the publication of his Descent of Man, the book offers a uniquely rounded view that asks whether 'sexual selection' is in itself a progressive or reactionary notion, even as it explores its theoretical relevance in the technical biological study of the twenty-first century. (shrink)

Ce livre cherche à montrer comment l'eau, à la fois réalité et symbole, nourrit la pensée de Platon. Quelle est l'origine de cette thématique et quelles en sont les finalités? A travers ces interrogations, il s'agit de mettre en lumière la fonction de cette matière particulière dont les aspects sont déclinés de manière originale dans l'ensemble de son oeuvre. En rassemblant la multiplicité des images de l'eau, l'auteur interroge la nature et la valeur des images en s'efforçant de montrer comment (...) Platon, écrivain et poète, met en oeuvre avec inventivité métaphores, allégories et mythes. De quoi alimenter un questionnement sur le réel, auquel l'eau donne un fil directeur. On y voit aussi se défaire une certaine idée du platonisme. Critique à l'égard de la rhétorique et des arts, Platon vise dans le miroir les effets illusoires de l'image et de l'imitation, mais Socrate utilise lui-même les images et les reflets tout au long des dialogues, et on en découvre avec étonnement toutes les variations. La métaphore hydrique conditionne l'univers de la parole et de la science, reflète le mouvement de la pensée et ses dérives. Dans les images métaphysiques et morales, ce sont les flux, les fleuves et le monde marin qui reflètent la nature et la qualité des âmes ; mythes et allégories déplient les symboles de la corruption, de l'oubli et de l'incontinence, où se jouent une idée de la matière et la valeur de l'existence. Sur le plan ontologique, l'eau dénoue les figures de l'instable qui mettent en oeuvre la réfutation des «liquides» - théoriciens du flux - et l'intelligibilité du devenir. Enfin, dans l'ordre politique, la critique de la thalassocratie, les mythes (l'Atlantide, la mer de la dissemblance) et les images nautiques suivent les lignes fluctuantes de l'histoire athénienne, mais tracent aussi les contours de la Cité juste et le visage du vrai gouvernant. Dans le scintillement des eaux, à la croisée du sensible et de l'intelligible, se construit l'iconologie platonicienne. L'eau figurée, transfigurée dans le fil liquide de la poésie, forme les multiples reflets de ce qu'il faut vraiment voir pour se libérer du spectacle des ombres et penser le chemin d'une sagesse, qui ne renonce jamais au monde. (shrink)

à travers l’analyse d’une œuvre-projet de Francis Alÿs, cette étude vise à mettre en lumière la complexité dialectique inhérente au concept d’authenticité. Lorsqu’elle s’applique à l’art, celle-ci peut renvoyer, selon les cas à des valeurs ou à des principes d’évaluation divers. Elle peut, par exemple, se dire soit de l’objet soit de l’artiste. Or, dans ses usages courants, ces multiples significations tendent à s’amalgamer et à subir des retournements dialectiques pas toujours contrôlés. Théoriciens et artistes peuvent dès lors vouloir dissoudre (...) certains de ces amalgames afin de mettre en évidence le fonctionnement de cet opérateur axiologique complexe et d’attirer l’attention sur certains de ses aspects particuliers qui n’apparaissent pas lorsqu’ils sont confondus avec d’autres. C’est ainsi qu’à travers l’exposition de dérivations populaires d’un tableau de Henner représentant sainte Fabiola, l’artiste Francis Alÿs a entrepris de mettre en lumière l’idée d’authenticité iconique pure, distincte de l’authenticité artistique, qui s’est retrouvée fondue à l’intérieur de celle-ci au cours des Temps Modernes. De manière paradoxale, c’est précisément le caractère dérivé, lié à l’expression d’une piété visant une sorte d’icône moderne, qui constitue le vecteur de l’authenticité particulière qui s’attache à l’objet iconique et à l’acte de représentation de son prototype. (shrink)

Simone Weil (1909-1943) foi filósofa, escritora, ativista política e humanista. Nasceu em Paris, no seio de família judaica. Formou-se em filosofia pela Université de Sorbonne e se tornou a primeira mulher catedrática da França. Militou fervorosamente pela causa dos trabalhadores fabris e, posteriormente, lutou na Guerra Civil Espanhola. Faleceu aos trinta e quatro anos por motivos de saúde.[1] [1] Para mais informações, vide texto disponível em:. Acesso em: 25 de junho de 2019.

En Occident, à partir de 1520, l’Ecclesia una se fragmente en confessions chrétiennes rivales. Cette réalité religieuse nouvelle ne peut être pensée sans mots; mais les néologismes lexicaux nécessaires sont lents à s’imposer. La situation paroxystique de ceux qui changent effectivement d’Église dans les années 1520-1650 est pourtant favorable à de telles créations. Elle permet de trouver les mots pour dire, et donc pour admettre, l’existence de l’Autre Église.

Toute nation participant à la division internationale du travail retire nécessairement un gain économique supérieur à une situation d’autarcie. C’est le message fondamental, normatif, de toutes les écoles théoriques ayant œuvré à la compréhension des fondements de l’échange entre les pays. Le protectionnisme est par conséquent très souvent vilipendé par les économistes qui y voient un frein à la circulation internationale des biens et des services, bloquant ainsi la croissance, l’emploi et l’élévation des niveaux de vie. Le questionnement actuel portant (...) sur les gains que les nations et les populations ont réellement retiré de la mondialisation, sur les inégalités que cette mondialisation aurait occasionnées, suggèrent de rouvrir l’œuvre de Johann Gottlieb Fichte, qui, pour avoir été présent dans certaines écoles de pensée économique, est aujourd’hui totalement absent. Dans L’État commercial fermé, il adresse en effet une critique virulente aux économistes qui, soit prônent le libre-échange, soit le protectionnisme. Fichte estimait que les avancées de la métaphysique devaient être appliquées à la vie réelle. C’est pourquoi son ouvrage contient l’une des premières doctrines du socialisme d’État. La situation présente de la mondialisation peut redonner du lustre à une approche aujourd’hui oubliée des économistes. (shrink)

The essential junkiness of our culture and biology.

Jesuits in China adopted key Confucian terms to express Christian notions; for example, Tian 天 or Shangdi 上帝 was considered an equivalent for God, and guishen 鬼神 for angels. A Terms controversy started among the Jesuits and other missionaries and developed into the famous Rites Controversy. However, all the missionaries agreed in rejecting the Neo-Confucian concept of Taiji 太極, which was believed to be materialistic, pantheistic, or atheistic. The Flemish Jesuit François Noël, after a careful study of Neo-Confucian texts, interpreted (...) Taiji with the Western concept of nature, and even claimed that it held theistic meaning. We shall analyze and evaluate here this early Western attempt in giving a positive meaning to Taiji. (shrink)

Second Scholasticism greatly developed the medieval theory of continuous quantity as the Aristotelian notion for thematizing spatial extension, paving the way for the idea of space as extension in early modern natural philosophy. The article analyzes the section related to the category of continuous quantity in the Coimbra commentary on the Dialectics (1606), showing that it is indebted to the novel theory of Francisco Suárez on quantity as bestowing extension to a body in a particular sense, something which had been (...) overlooked by previous research. The scholarly debate on quantity was brought to China, and here the Chinese translation is examined of the section on quantity in the fourth volume of the Mingli Tan, published in China in 1636-­1639. (shrink)