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)

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.

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)

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

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.

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)

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)

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.

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.

This textbook is a flexible and affordable collection of classic and contemporary primary sources in philosophy. The readings cover seven basic topics of Western Philosophy. The selections are long enough to present a self-contained argument but not so lengthy that students lose track of the main point. Each reading has an outline with study questions, questions for reflection and discussion, and an annotated bibliography. The book includes a glossary and an appendix on logic and argumentation.

Nous partageons le problème de savoir comment faire face à autant d'humains si semblables et si différents. A autant d'êtres qui ne peuvent interpréter le fait d'exister sans se comparer les uns aux autres, sans se distinguer les uns des autres, et qui doivent néanmoins cohabiter. Et de savoir comment ces humains peuvent d'autant plus se distinguer qu'ils prennent la place successivement les uns les autres, qu'ils reprennent les mêmes traces et doivent les réinterpréter. La première formulation du problème est (...) plutôt celle de la problématologie, dans la mesure où elle place au centre de la condition humaine le langage ordinaire, l'incessante différence de la question et de la réponse. C'est par elle que les humains mesurent ce qui les éloigne ou les rapproche et qu'ils règlent leurs désaccords. La seconde formulation du problème est plutôt celle de l'herméneutique, dans la mesure où elle place au centre de la condition humaine la condition interprétative, où chaque génération doit réinterpréter le monde où elle se découvre, pour reprendre la conversation rompue par la mort. En exposant ces deux " gestes " élémentaires du questionnement, en construisant l'écart entre deux traditions philosophiques de styles aussi hétérogènes, sans chercher à les subordonner l'un à l'autre, il s'est agi de penser une éthique de l'interrogativité. Ce sens de l'interrogation peut seul orienter nos " éthiques de responsabilité " et nous remettre ensemble dans un monde où nous comparaissons d'autant plus heureux de nous distinguer que nous nous effaçons devant l'autre. Ces deux attitudes font notre civilité. (shrink)

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)

Friedrich Nietzsche has emerged as one of the most important and influential modern philosophers. For several decades, the book series Monographien und Texte zur Nietzsche-Forschung has set the agenda in a rapidly growing and changing field of Nietzsche scholarship. The scope of the series is interdisciplinary and international in orientation reflects the entire spectrum of research on Nietzsche, from philosophy to literary studies and political theory. The series publishes monographs and edited volumes that undergo a strict peer-review process. The book (...) series is led by an international team of editors, whose work represents the full range of current Nietzsche scholarship. (shrink)

This paper concerns the conditions under which realism is an artistic merit in perceptual narratives, and its consequences for the practice of non-traditional casting. Perceptual narratives are narrative representations that perceptually represent at least some of their contents, and include works of film, television, theatre and opera. On certain construals of the conditions under which realism is an artistic merit in such works, non-traditional casting, however morally merited, is often artistically flawed. I defend an alternative view of the conditions under (...) which realism is an artistic merit in perceptual narratives. I identify the two forms of realism at issue in debates about the artistic merits of non-traditional casting, and identify the artistic norms that determine the conditions under which each constitutes an artistic merit. I argue that, independently of the relation between moral merits and artistic merits, non-traditional casting violates these norms less often than is sometimes assumed. Moreover, in certain circumstances, non-traditional casting affects realism in artistically meritorious ways. I conclude by considering the implications of my view for the practice of whitewashing. (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.

This article explores the arguments for architecture having an identifiable essence equivalent to Heidegger's concept of 'dwelling' and place identity, versus those who claim it is no more than a 'hybrid' collection of many different technical, social and cultural practices. The opposing positions in architecture are compared by analogy with the opposing theories of language which underly them, the latter position being associated with the universalist theory of language propagated by Chomsky, while the purpose of architecture in identity formation is (...) compared with Wittgenstein's theory of the interdependence between language and human behaviour as a 'form of life'. As an example, the author points to colonial architecture as a clear case where people retain 'that part of their identity which is their architecture'. The author concludes that, in so far as objects also have meanings, then the physical environment must also be taken into account in the evolution of mind. (shrink)

This textbook is a flexible and affordable collection of classic and contemporary primary sources in philosophy. The readings cover seven basic topics of Western Philosophy. The selections are long enough to present a self-contained argument but not so lengthy that students lose track of the main point. The book includes a glossary and an appendix on logic and argumentation.

En este artículo se investigan los presupuestos ontológicos de aquellos nombres divinos que pueden predicarse de Dios y de las criaturas. En un primer momento, se destaca la anterioridad metafísica del ente como atributo de Dios. Después, al descubrir que el ente es un concepto análogo, se examinan las distintas modalidades de analogía confrontando dos textos: Summa contra Gentiles, I, cap. 34 y De Veritate, q.2, a.11. Finalmente, se muestra que la analogía de proporcionalidad es la que debe preferirse para (...) predicar algo acerca de Dios, y se pone de manifiesto en qué medida se funda sobre la cuarta vía «ex gradibus qui in rebus inveniuntur». (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)

The debate between the amodal and the grounded views of cognition seems to be stuck. Their only substantial disagreement is about the vehicle or format of concepts. Amodal theorists reject the grounded claim that concepts are couched in the same modality-specific format as representations in sensory systems. The problem is that there is no clear characterization of format or its neural correlate. In order to make the disagreement empirically meaningful and move forward in the discussion we need a neurocognitive criterion (...) for representational format. I argue that efficient coding models in computational neuroscience can be used to characterize modal codes: These are codes which satisfy special informational demands imposed by sensory tasks. Additionally, I examine recent studies on neural coding and argue that although they do not provide conclusive evidence for either the grounded or the amodal views, they can be used to determine what predictions these approaches can make and what experimental and theoretical developments would be required to settle the debate. (shrink)

Lewis Carroll: Logic Charles L. Dodgson, 1832-1898, was a British mathematician, logician, and the author of the ‘Alice’ books, Alice’s Adventures in Wonderland and Through the Looking Glass and What Alice Found There. His fame derives principally from his literary works, but in the twentieth century some of his mathematical … Continue reading Lewis Carroll: Logic →.

Lewis Carroll: Logic Charles L. Dodgson, 1832-1898, was a British mathematician, logician, and the author of the ‘Alice’ books, Alice’s Adventures in Wonderland and Through the Looking Glass and What Alice Found There. His fame derives principally from his literary works, but in the twentieth century some of his mathematical … Continue reading Lewis Carroll: Logic →.

En este libro, que se titula «Sobre los nombres divinos [De divinis nominibus]», siguiendo la costumbre de aquellos que han transmitido la ciencia magistralmente [artificiose], Dionisio empieza, en primer lugar, presentando algunas consideraciones necesarias para todo el estudio sucesivo, y, en segundo lugar, prosigue exponiendo el objeto principal [de su tratado] en el capítulo tercero.

En contraposición a la predominancia cultural de lo urbano y en medio de un contexto de desenfrenado capitalismo globalizador, el VIII Congreso sobre Pensamiento Filosófico Contemporáneo se planteó el desafío de recoger algunas voces y discursos a menudo marginados por el canon del pensamiento occidental metrocentrista. A lo largo de esos tres días de octubre, en el congreso se analizaron pormenorizadamente las situaciones de sujetos, comunidades y geografías rurales, situaciones que tienden a ser activamente desescuchadas en un presente espacial que (...) encuentra la globalización, la híper-densificación de las principales zonas urbanas-metropolitanas, y el vaciamiento forzoso de las geografías rurales, como uno de sus marcos de conflicto. Estos análisis profundizaron en las necesidades específicas de dichas comunidades, arrojando luz sobre sus desafíos y ofreciendo una perspectiva crítica a propósito de la complejidad de la interacción entre lo rural y lo urbano. Desde estas inquietudes, el VIII Congreso se articuló como una plataforma a través de la que celebrar la diversidad de discursos filosóficos emergentes producidos en los marcos de lo comúnmente entendido como «rural», reconociendo su importancia crucial en la construcción de un diálogo integral sobre los desafíos que definen nuestro presente; y este monográfico es resultado de ese intercambio de saberes. (shrink)

Este artículo examina las ruralidades queer como campo de estudios en el Estado español. Se abordarán detalladamente algunos de los conceptos clave que fundamentan este campo a nivel internacional, y se proporcionará un análisis del breve, pero a la vez prolífico, desarrollo que este campo de estudios ha experimentado en el Estado español durante la última década. Finalmente, se presentarán algunas de las propuestas culturales, políticas y artísticas que están enriqueciendo las ruralidades queer y ampliando sus horizontes políticos e interpretativos.

Causal approaches to explanation often assume that a model explains by describing features that make a difference regarding the phenomenon. Chirimuuta claims that this idea can be also used to understand non-causal explanation in computational neuroscience. She argues that mathematical principles that figure in efficient coding explanations are non-causal difference-makers. Although these principles cannot be causally altered, efficient coding models can be used to show how would the phenomenon change if the principles were modified in counterpossible situations. The problem is (...) that efficient coding models also involve difference-makers that, prima facie, cannot be characterized as non-causal in this sense. Mathematical principles always involve variables which have counterfactual relations between them. However, we cannot simply assume that these difference-makers are causal. They can also be found in paradigmatic non-causal explanations and therefore they must be characterized as non-causal in some sense. I argue that, despite appearances, Chirimuuta's view can be applied to these cases. The mentioned counterfactual relations presuppose the counterpossible conditionals that describe the modification of a relevant mathematical principle. If these conditionals are the hallmark of non-causal relations, then Chirimuuta’s criterion has the desired implication that variables in mathematical principles are non-causal difference-makers. (shrink)

John Venn and Charles L. Dodgson (Lewis Carroll) created systems of logic diagrams capable of representing classes (sets) and their relations in the form of propositions. Each is a proof method for syllogisms, and Carroll's is a sound and complete system. For a large number of sets, Carroll diagrams are easier to draw because of their self-similarity and algorithmic construction. This regularity makes it easier to locate and thereby to erase cells corresponding with classes destroyed by the premises of an (...) argument, a particularly difficult task in Venn diagrams for more than four sets. Carroll diagrams can represent existential propositions easily, so they are capable of clearly representing more complex problems than Venn's system can. Finally, both Carroll and Venn diagrams are maximal, in the sense that no additional logic information like inclusive disjunctions is able to be represented by them. Carroll's logic diagrams and logic trees constitute his visual logic system. (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 are presented as relations of degree rather than absolute differences. Much like our primitive ancestors, he contends, routine human behaviour is governed more by unconscious instinct than self-conscious rationality, and is generally resistant to any major changes in customary habits of life, until compelled in time to do so. However, in extreme and rapidly evolving circumstances, as with climate change, tardy responses may become so detached from reality as to constitute an actual threat to survival. (shrink)

"Verzeichnis der Schriften Hans Posers": p. [463]-480.

I defend Christopher Peacocke's and Robert Hopkins's experienced resemblance accounts of depiction against criticisms put forward by Gavin McIntosh in a recent article in this journal. I argue that, while there may be reasons for rejecting Peacocke's and Hopkins's accounts, McIntosh fails to provide any.

I argue that our everyday evaluations of architecture are primarily evaluations of spaces and, in particular, of their inhabitability— that is, whether they serve or can serve to the realization of our individual ideal of life. Inhabitability is not only a functional criterion but an aesthetic one as well. It is aesthetic insofar as the evaluations about inhabitability include evaluations about the quality of the experience of actually doing something in —or simply occupying—a particular space. This aesthetic aspect of our (...) everyday evaluations explains, on one hand, the distinctive features of our everyday preferences for some architectonic spaces over others—namely, their very personal nature; their noticeable value in our daily lives; their inseparability from the activities we plan to realize in those spaces; and the importance of the spaces which are contiguous to the one we are evaluating. And on the other, it both reveals architecture’s significance for the quality of our daily life and serves as a guide for us to choose spaces that improve that quality. (shrink)