In 1930, Johann von Neumann, together with Rudolf Carnap and Arend Heyting, participated in a conference held in Königsberg, called “Second Seminar on the Epistemology of Exact Sciences”. The idea behind the reunion of these three researchers was to compose a fairly faithful picture of the three main foundational programs of mathematics at the time: formalism, logicism, and intuitionism. The main objective of this paper is to propose an analysis of the text “The Formalist Foundation of Mathematics” presented by von (...) Neumann in this conference. We show how, from von Neumann's perspective, the problem of the consistency of mathematical theories results from the transformation of a genuinely philosophical question into a genuinely logical-mathematical question and how a proof of consistency could be seen as the solution to the foundational problem. In the final part of the text, we present some considerations about Gerhard Gentzen’s consistency proofs and their importance to contemporary proof theory. (shrink)

Dedekind’s methodology, in his classic booklet on the foundations of arithmetic, has been the topic of some debate. While some authors make it closely analogue to Hilbert’s early axiomatics, others emphasize its idiosyncratic features, most importantly the fact that no axioms are stated and its careful deductive structure apparently rests on definitions alone. In particular, the so-called Dedekind “axioms” of arithmetic are presented by him as “characteristic conditions” in the _definition_ of the complex concept of a _simply infinite_ system. Making (...) sense of Dedekind’s method may be dependent on an analysis of the classical model of deductive science, as presented by authors from the eighteenth and early nineteenth centuries. Studying the modern reconstructions of Euclidean geometry, we show that they did not presuppose deductive independence of the axioms from the definitions. Authors like Wolff elaborated a mathematics _based on definitions_, and the Wolffian model of deductive science shows significant coincidences with Dedekind’s method, despite the great differences in content and approach. Wolff had a conception of definitions as _genetic_, which bears some similarities with Kant’s idea of synthetic definitions: they are understood as positing the content of mathematical concepts and introducing thought objects (_Gedankendinge_) that are the objects of mathematics. The emphasis on the spontaneity of the understanding, which can be found in this philosophical tradition, can also be fruitfully related with Dedekind’s idea of the “free creation” of mathematical objects. (shrink)

Since the application of Postulate I.2 in Euclid’s Elements is not uniform, one could wonder in what way should it be applied in Euclid’s plane geometry. Besides legitimizing questions like this from the perspective of a philosophy of mathematical practice, we sketch a general perspective of conceptual analysis of mathematical texts, which involves an extended notion of mathematical theory as system of authorizations, and an audience-dependent notion of proof.

This paper analyzes the theory of area developed by Euclid in the Elements and its modern reinterpretation in Hilbert’s influential monograph Foundations of Geometry. Particular attention is bestowed upon the role that two specific principles play in these theories, namely the famous common notion 5 and the geometrical proposition known as De Zolt’s postulate. On the one hand, we argue that an adequate elucidation of how these two principles are conceptually related in the theories of Euclid and Hilbert is highly (...) relevant for a better understanding of the respective geometrical practices. On the other hand, we claim that these conceptual relations unveil interesting issues between the two main contemporary approaches to the study of area of plane rectilinear figures, i.e., the geometrical approach consisting in the geometrical theory of equivalence and the metrical approach based on the notion of measure of area. Finally, in an appendix logical relations among equivalence, comparison and addition of magnitudes are examined schematically in an abstract setting. (shrink)

A theory of magnitudes involves criteria for their comparison, equivalence and addition. We examine these aspects from an abstract viewpoint, stressing independence and definability. These considerations are triggered by the so-called De Zolt’s principle in the theory of equivalence of plane polygons.

Theoria, Volume 88, Issue 3, Page 629-652, June 2022.

A theory of magnitudes involves criteria for their equivalence, comparison and addition. In this article we examine these aspects from an abstract viewpoint, by focusing on the so-called De Zolt’s postulate in the theory of equivalence of plane polygons (“If a polygon is divided into polygonal parts in any given way, then the union of all but one of these parts is not equivalent to the given polygon”). We formulate an abstract version of this postulate and derive it from some (...) selected principles for magnitudes. We also formulate and derive an abstract version of Euclid’s Common Notion 5 (“The whole is greater than the part”), and analyze its logical relation to the former proposition. These results prove to be relevant for the clarification of some key conceptual aspects of Hilbert’s proof of De Zolt’s postulate, in his classicalFoundations of Geometry(1899). Furthermore, our abstract treatment of this central proposition provides interesting insights for the development of a well-behaved theory ofcompatiblemagnitudes. (shrink)

Abel Lasalle Casanave’s comments on the rhetorical and dialectical aspects of the discussion of proof in chapters 19 to 21. I center most of my response on two issues raised in his concluding questions; the first being the hermeneutic aspect of the development of mathematics and of mathematical knowledge, and the second “the symbolic conception of mathematics” and the “algebraic mode of thought”.As considerações de Abel Lasalle Casanave dizem respeito aos aspectos retóricos e dialéticos da discussão (...) da noção de prova nos capítulos 19 à 21, e minha réplica está centrada em duas questões levantadas em suas observações finais. A primeira refere-se ao aspecto hermenêutico do desenvolvimento da matemática e do conhecimento matemático; e a segunda à “concepção simbólica da matemática” e ao “modo algébrico de pensar”. (shrink)

In §1 I discuss Dedekind and Frege on the logical and structural analysis of natural numbers and present my view that the logical analysis of the notion of number involves a combination of their analyses. In §2 I answer some of the specific questions that Abel raises in connection with Chapter 9 of Logical Forms.

Resumen Este artículo analiza si en la teoría de los cuatro ethe de la modernidad capitalista, desarrollada por Bolívar Echeverría, no está contenida una debilidad de principio. Se trata de saber si para arribar a sus más altas aportaciones, esta teoría no termina pagando un precio que debiera dar que pensar. ¿Acaso renuncia a la crítica de la ideología, por lo menos en la radicalidad realizada por los autores de la Teoría crítica, inspirados en este punto originalmente por Georg Lukács?This (...) article is analyzing if in the theory of the four ethe of capitalist modernity, developed by Bolívar Echeverría, isn’t included a basic weakness. It is about understanding if for to arrive to its highest contributions, this theory is not paying a too high prize. Does it possibly renounce to the critique of ideology, at least in the radicality fulfilled by the authors of Critical theory, inspired in this point originally by Georg Lukács? (shrink)

Since the application of Postulate I.2 in Euclid's Elements is not uniform, one could wonder in what way should it be applied in Euclid's plane geometry. Besides legitimizing questions like this from the perspective of a philosophy of mathematical practice, we sketch a general perspective of conceptual analysis of mathematical texts, which involves an extended notion of mathematical theory as system of authorizations, and an audience-dependent notion of proof.

In his doctoral dissertation, O. Chateaubriand favored Dedekind’s analysis of the notion of number; whereas in Logical Forms, he favors a fregean approach to the topic. My aim in this paper is to examine the kind of logicism he defends. Three aspects will be considered: the concept of analysis; the universality of arithmetical properties and their definability; the irreducibility of arithmetical objects.

En este artículo proponemos que el examen del concepto de demostración de Oswaldo Chateaubriand en los capítulos 19, 20 y 21 de la Parte II de Logical Forms incluye aspectos retóricos y dialécticos .In this paper we argue that Oswaldo Chateaubriand’s conception of proof, in chapters 19, 20, and 21 of Part II of Logical Forms, incorporates rhetorical aspects , and dialectical aspects.

We review Ferreirós and Lassalle Casanave's recently published book "El árbol de los números". The book is a result of the Brazilian-Spanish conference "Sobre la elucidación del concepto de número: cognición, lógica y práctica matemática" hosted in Sevilla in 2013, and collects new papers on History and Philosophy of Mathematics as well as Mathematical Practice. These papers present results of investigations in Cognitive Sciences, Logic and Epistemology of mathematical certainty.

Fue en Inglaterra donde apareció por vez primera un individualismo virtuoso comprometido con la defensa pública de la libertad frente a la amenaza del absolutismo. Allí surgió un discurso político liberal-republicano que defendió que el bien público y el interés privado fueran de la mano. Así, el liberalismo nació como un discurso público y privado de la virtud individual que tenía la vocación de frenar cualquier arrogancia despótica. Pero en la segunda mitad del siglo XX una tendencia neoliberal y libertaria (...) convirtió el mercado en una abstracción dogmática que justificaba un egoísmo descontrolado y sin límites. En Liberales, José María Lassalle expone la necesidad de que el liberalismo del siglo XXI vuelva a los principios virtuosos de sus padres fundadores, John Locke, Adam Smith y Edmund Burke. Los liberales tienen por delante la responsabilidad de enfrentarse a sus propios fantasmas y liderar nuevamente la defensa de una política del deber, y no del beneficio. Una política al servicio de la libertad: preocupada por el c ontrol del poder; que asegure el establecimiento de mecanismos institucionales que impidan la corrupción y las conspiraciones contra el mercado que se urden a las sombras de los gobiernos; que combata el dogmatismo y que defiende la tolerancia como una seña de identidad de nuestra cultura. Ante la mayor crisis de las últimas décadas, urge recuperar la virtud y los valores, una tarea para la que los liberales están mejor capacitados que nadie. (shrink)

This chapter contains sections titled: Introduction Desiderata for an Account of Genre Existing Accounts of Genre An Account of Genre Conclusion Notes References.

To take into account the growing importance of this extraterritoriality paradigm reassessing the idea of territory in literature, culture and languages, this book offers an interdisciplinary and plurilingual journey through four centuries, four continents and a dozen languages.

While scientific inquiry crucially relies on the extraction of patterns from data, we still have a far from perfect understanding of the metaphysics of patterns—and, in particular, of what makes a pattern real. In this paper we derive a criterion of real-patternhood from the notion of conditional Kolmogorov complexity. The resulting account belongs to the philosophical tradition, initiated by Dennett :27–51, 1991), that links real-patternhood to data compressibility, but is simpler and formally more perspicuous than other proposals previously defended in (...) the literature. It also successfully enforces a non-redundancy principle, suggested by Ladyman and Ross, that aims to exclude from real-patternhood those patterns that can be ignored without loss of information about the target dataset, and which their own account fails to enforce. (shrink)

The paper introduces a special issue of the journal Philosophical Problems in Science (ZFN) on the topic of the nature and norms of scientific explanation.

La théorie de la physique chez les physiciens contemporains / par Abel Rey,... Date de l'édition originale: 1907 Le présent ouvrage s'inscrit dans une politique de conservation patrimoniale des ouvrages de la littérature Française mise en place avec la BNF. HACHETTE LIVRE et la BNF proposent ainsi un catalogue de titres indisponibles, la BNF ayant numérisé ces oeuvres et HACHETTE LIVRE les imprimant à la demande. Certains de ces ouvrages reflètent des courants de pensée caractéristiques de leur époque, mais (...) qui seraient aujourd'hui jugés condamnables. Ils n'en appartiennent pas moins à l'histoire des idées en France et sont susceptibles de présenter un intérêt scientifique ou historique. Le sens de notre démarche éditoriale consiste ainsi à permettre l'accès à ces oeuvres sans pour autant que nous en cautionnions en aucune façon le contenu. Pour plus d'informations, rendez-vous sur www.hachettebnf.fr. (shrink)

It has been argued that the approximate number system (ANS) constitutes a problem for the grounded approach to cognition because it implies that some conceptual tasks are performed by non-perceptual systems. The ANS is considered non-perceptual mainly because it processes stimuli from different modalities. Jones (2015) has recently argued that this system has many features (such as being modular) which are characteristic of sensory systems. Additionally, he affirms that traditional sensory systems also process inputs from different modalities. This suggests that (...) the ANS is a perceptual system and therefore it is not problematic for the grounded view. In this paper, I defend the amodal approach to the ANS against these two arguments. In the first place, perceptual systems do not possess the properties attributed to the ANS and therefore these properties do not imply that the ANS is perceptual. In the second place, I will propose that a sensory system only needs to be dedicated to process modality-specific information, which is consistent with responding to inputs from different modalities. I argue that the cross-modal responses exhibited by traditional sensory systems are consistent with modality-specific information whereas some responses exhibited by the ANS are not. (shrink)

In this incisive study of the biological and cultural origins of the human self, the author challenges readers to re-think ideas about the self and consciousness as being exclusive to humans. In their place, he expounds a metatheoretical approach to the self as a purposeful system of extended cognition common to animal life: the invisible medium maintaining mind, body and environment as an integrated 'field of being'. Supported by recent research in evolutionary and developmental studies together with related discoveries in (...) animal behavior and the neurosciences, the author examines the factors that have shaped the evolution of the animal self across widely different species and times, through to the modern, technologically enmeshed human self; the differences between which, he contends, are relations of degree rather than absolute differences. We are, he concludes, instinctive and 'fuzzy' individuals clinging to fragile identities in an artificial and volatile world of humanity's own making, but which we now struggle to control. However, the reasons for humanity’s failure to respond effectively to the climate emergency, he argues, lie not only with the powerful sectors of modern society which continue to exploit fossil fuels, but also the conservative nature of the human self and its deeply rooted resistance to changing familiar modes of life. (shrink)

A study of what Descartes calls émotions in his Passions of the Soul suggests that, rather than just a theory of passions—as Descartes himself explicitly claims to be proposing—he was in practice putting forward a more comprehensive theory of passions-émotions, a unified theory which would be closer to what today should properly be called Descartes’ theory of emotions. I try here to make explicit the grounds of this unity by showing that émotions both fit within the functional account Descartes attributes (...) to what he calls passions; and complement the intentionality of passions by expanding it to new objects. In order to show this I offer also a tentative distinction, functionally and intentionally, between passions and émotions in general, on one hand, and, on the other, between the two apparent types of émotions Descartes refers to in the treatise—interior or internal [intérieure] and intellectual [intellectuelle] émotions. (shrink)

In this article, we discuss recent evidence from experimental economics on the impact of social preferences on workplace behavior. We focus on situations in which a single employer interacts with multiple employees. Traditionally, equity and efficiency have been seen as opposing aims in such work environments: individual pay-for-performance wage schemes maximize of efficiency but might lead to inequitable outcomes. We present findings from laboratory experiments that show under which circumstances partially incomplete contracts can create equitable work environments while at the (...) same time reaching surprisingly efficient outcomes. (shrink)

We study the Iterated Bilateral Reciprocity game in which the need for help arises randomly. Players are heterogeneous with respect to ‘neediness’ i.e. probability of needing help. We find bounds on the amount of heterogeneity which can be tolerated for cooperation (all players help when asked to help) to be sustainable in a collectivity. We introduce the notion of Generalised Reciprocity. Individuals make a costly first move to benefit another under the reasonable expectation that either the other or somebody else (...) will reciprocate. We hope that these tentative attempts at extending Axelrod’s seminal work on cooperation will inspire future efforts in the field of organisational culture and social theory more generally. (shrink)

Abstract:I defend that the distinctive object of our aesthetic evaluation of films is the full emotional experience, taken as a unified whole, that we go through as we watch a film and that I call the viewer's film emotional life. The aesthetic value itself—the positive quality we perceive in the experience of having had a certain film emotional life—is in the significance we experience in that film emotional life insofar as it contributes to the discovery and the exploration of the (...) viewer's human and individual emotional nature. I finally suggest that the explanation of the positive felt quality of such experience is in the progress it represents regarding our resourcefulness as emotional subjects toward the realization of our individual ideal of life. (shrink)

Excerpt from L'E nerge tique Et le Me canisme: Au Point de Vue des Conditions de la Connaissance 1. - Les physiciens contemporains s'accordent tous a reconnaitre la valeur objective complete des resultats purement experimentaux. Ceux-ci constituent veritablement la science de la physique, et, sauf dans des cas exceptionnels, momentanes, dans la science en train de se faire, il n'y a, a leur sujet, aucune divergence. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find (...) more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.". (shrink)

This book argues that the primary function of human thinking in language is to make judgments, which are logical-normative connections of concepts. Robert Abele points out that this presupposes cognitive conditions that cannot be accounted for by empirical-linguistic analyses of language content or social conditions alone. Judgments rather assume both reason and a unified subject, and this requires recognition of a Kantian-type of transcendental dimension to them. Judgments are related to perception in that both are syntheses, defined as the unity (...) of representations according to a rule/form. Perceptual syntheses are simultaneously pre-linguistic and proto-rational, and the understanding makes these syntheses conceptually and thus self-consciously explicit. Abele concludes with a transcendental critique of postmodernism and what its deflationary view of ontological categories—such as the unified and reasoning subject—has done to political thinking. He presents an alternative that calls for a return to normativity and a recognition of reason, objectivity, and the universality of principles. (shrink)

Frontmatter -- Vorwort zur 2. Auflage -- Vorwort -- Inhaltsverzeichnis -- Erster Teil. Die Willen-zur-Macht-Prozesse -- Zweiter Teil. Der Gedanke der ewigen Wiederkehr des Gleichen -- Dritter Teil. Wiederkunftslehre und neuzeitliches Denken -- Literaturverzeichnis -- Personenregister -- Sachregister.