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)

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)

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.

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.

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)

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)

Esta extensa colección de ensayos en honor de Abel Lassalle Casanave refleja la amplitud de su compromiso filosófico con las ciencias formales. Cada ensayo, breve en extensión pero rico en la profundidad de su perspectiva, invita al lector a explorar una muestra de la investigación contemporánea en la historia y/o la filosofía de la lógica y la matemática. El efecto general es el de un anticipo de la diversidad de recientes áreas de investigación, así como de la variedad (...) de enfoques que compiten entre sí en su estudio. Al mismo tiempo, los ensayos, de manera colectiva, dan forma y constituyen un digno homenaje a la destacada influencia de Lassalle Casanave en la historia y filosofía de la matemática. (shrink)

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)

Dominic McIver Lopes’ Four Arts of Photography and Diarmuid Costello’s On Photography: A Philosophical Inquiry examine the state of the art in analytic philosophy of photography and present a new approach to the study of the medium. As opposed to the orthodox and prevalent view, which emphasizes its epistemic capacities, the new theory reconsiders the nature of photography, and redirects focus towards the aesthetic potential of the medium. This symposium comprises two papers that critically examine central questions addressed in the (...) two books, with responses by the two authors in defence of their respective positions. (shrink)

Este artículo analiza los aspectos fundamentales de la teoría del conocimiento expuestos en el curso dictado por fray Juan de Fuica (OFM), en 1689, en el Colegio San Diego de Alcalá (Santiago de Chile). Presentamos primero una breve descripción de los Comentarios filosóficos del fraile, volumen en el que se incluyen los Comentarios Acerca del alma sobre el que centraremos nuestro estudio. Luego, analizamos la doctrina del conocimiento ad mentem Scoti expuesta en este manuscrito inédito, testimonio exclusivo de la enseñanza (...) filosófica de la época. El principal aporte de este trabajo es visibilizar parte del patrimonio filosófico colonial del país, lo que permitirá futuras investigaciones en esta área de la disciplina. (shrink)

Ignacio Ellacuría reconstructs three aspects of Karl Rahner’s metaphysics and fundamental theology for a Latin American context. First, Ellacuría reframes Rahner’s focus on the metaphysics of being, arguing instead that historical reality is the proper object of a truly Latin American theology and philosophy. Second, Ellacuria builds upon and diverges from aspects of Rahner’s use of the hylomorphic theory and the role of the agent intellect in his theory of knowing, using Xavier Zubiri’s analysis of the role of sentient intelligence (...) in order to reconstruct Rahner’s theological epistemology. And third, Ellacuría appropriates and reframes Rahner’s supernatural existential, situating it within the larger horizon of historical reality, which he says the tradition asserts has been transformed by grace. (shrink)

The metaphysical question of what determines the contents of fictive utterances is closely related to the epistemological question of how audiences identify the.

The aim of this book is to provide a unified solution to a wide range of philosophical problems raised by fiction. While some of these problems have been the focus of extensive philosophical debate, others have received insufficient attention. In particular, the epistemology of fiction has not yet attracted the philosophical scrutiny it warrants. There has been considerable discussion of what determines the contents of works of fiction, but there have been few attempts to explain how audiences identify their contents, (...) or to identify the norms governing the correct understanding and interpretation of works of fiction. -/- This book answers a wide range of both metaphysical and epistemological questions concerning fiction in a way that clarifies the relations between them. The metaphysical questions include: What distinguishes works of fiction from works of non-fiction?; What is the nature of fictive utterances?; What determines the contents of works of fiction?; What kinds of fictive content are there?; How broad in scope is fictive content?; and What kinds of things are fictional entities? The epistemological questions include: How do audiences identify the contents of authors’ fictive utterances?; How does understanding a work of fiction differ from interpreting it?; and What role does thinking and talking about fiction from an external perspective play in enabling communication through fiction? It develops the first single theory that provides answers to all these questions. (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)

Although much has been written on Descartes’ thought on animals, not so much has originated in, or has taken full account of, Descartes’ views on emotions. I explore here the extent to which the latter can contribute to the debate on whether he embraced, and to which extent, the doctrine of the bête machine. I first try to show that Descartes’ views on emotions can help offer new support to the skeptical position without necessarily creating new tensions with other central (...) aspects of his philosophy. And second, I sketch the type of theory of animal passions which Descartes could have accepted. The general conclusion I draw is not that Descartes did not hold the view of the bête machine but rather that we can find within his thought a solid stream of ideas, which became stronger towards the end of his life, that points in the opposite direction. (shrink)

This paper outlines elements of a biblically informed theology for faith-based activists. First, it examines the historical reality of Jesus as the defining sign of the Word made flesh (Jn 1:14) and an important bridge to seventy years of biblical studies on the historical Jesus and his Palestinian context. Second, it argues that Jesus’s baptism “of repentance for the forgiveness of sins” (Mk 1:4b) functions as a highly symbolic public action defining his message, mission, and identity as God’s loving response (...) to the sinful decimation of Israel’s institutions. Third, it asserts that Jesus’s choice to accept responsibility for the sins of his people is the first step of a new exodus leading to the confrontation at the Temple in Jerusalem and the question every disciple must face: How is God moving and acting in history now, and how am I called to collaborate with those movements in my historical context? (shrink)

I explore one distinctive aspect of our relation to spaces which is particularly significant in our everyday aesthetic evaluations of cities (urban spaces): the frequent use of linguistic expressions referring to the sense of touch. We say that a space is oppressive or expansive, or warm or cold, or (more indirectly) cozy or desolate. Touching, as involved in these expressions, seems to refer, rather than to the contact of a physical object with our skin, to a sort of touching with (...) our life. To understand this, I try to make explicit both (a) what is being touched, i.e., the view of space behind these expressions; and (b) how and in which sense our life touches it. I propose to think about space as a plenum made of qualities (e.g., colors, smells, sounds, textures), some of which are penetrable and some of which are not. And as to how we touch spaces, I defend that we do so as we immerse ourselves in them. Once immersed in them, we are constantly both evaluating how well we fit in them as well as looking for a better-quality fitting. This is how we evaluate spaces aesthetically. Our embodied life fits better in a space when (a) we can realize more significant (for us, according to our ideal of life) possibilities in it, and (b) the felt quality of realizing those possibilities is enhanced by that space (“it feels better to read in this café”). (shrink)

Schrödinger's definition of life needs a slight modification to absorb the criticism of it. It is the comparison of the entropy level of a system before and after a process which makes one view it as living: we consider the stability of the deviation from the probable a sign of life. This explains why we do not hesitate to consider as remnants of living systems skeletons and fossils anywhere and physical culture on any archeological site.

For some years now the proliferation of new religious movements in Africa and the search by individuals for new meanings in belief have held the interest of scholars of religion. But their interpretations of the significance of these ‘religious flowerings’ raise a number of questions, in particular questions about the meaning and applicability of the word ‘new’ in religion. Instead of taking it literally, we should understand this religious ‘innovation’ on two planes of transaction with the sacred, the horizontal and (...) the vertical. First, on the horizontal plane this religious innovation may be discerned in the ongoing adjustments made by traditional African religions as they encounter, whether peaceably or violently, religions originating in the West or Asia; these encounters have given rise to neo-traditional and prophetic movements and to independent Churches. We shall touch later on the two-way borrowings and adaptations that have taken place in some prophetic religions as regards their organization, rites, liturgy and charismatic communication of religious truth. On the other hand, religious renewal is discernible in the adjustments and innovative processes of elaboration adopted by such ‘travelling religions’ as Pentecostalism and Charismatic Renewal in relation to the basic process of making sense of the life of the individual and the community of the faithful. (shrink)

El presente trabajo se propone recuperar los principales aportes de Hegel sobre el problema del delito y la pena en el esquema que presenta en la Filosofía del Derecho, haciendo un continuo esfuerzo por enmarcarlos en su compleja propuesta filosófica más general, es decir, en el marco del movimiento dialéctico con el que piensa el continuo hacerse de las sociedades y los sujetos. Buscaremos principalmente mostrar el modo en que Hegel rechaza las concepciones utilitaristas sobre la pena para proponer una (...) idea del castigo en tanto retribución, pero donde retribución adquiere un sentido peculiar que, consideramos, sólo puede ser leído a la luz de su dialéctica y de su singular concepción del derecho. (shrink)

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.

Although it has been argued that mechanistic explanation is compatible with abstraction, there are still doubts about whether mechanism can account for the explanatory power of significant abstract models in computational neuroscience. Chirimuuta has recently claimed that models describing canonical neural computations must be evaluated using a non-mechanistic framework. I defend two claims regarding these models. First, I argue that their prevailing neurocognitive interpretation is mechanistic. Additionally, a criterion recently proposed by Levy and Bechtel to legitimize mechanistic abstract models, and (...) also a criterion proposed by Chirimuuta herself aimed to distinguish between causal and non-causal explanation, can be employed to show why these models are explanatory only under this interpretation. Second, I argue that mechanism is able to account for the special epistemic achievement implied by CNC models. Canonical neural components contribute to an integrated understanding of different cognitive functions. They make it possible for us to explain these functions by describing different mechanisms constituted by common basic components arranged in different ways. (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)

Business Ethics: A European Review, EarlyView.

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)

It is generally recognised that an adequate resemblance-based account of depiction must specify some standard of correctness which explains how a picture’s content differs from the content we would attribute to it purely on the basis of resemblance. For example, an adequate standard should explain why stick figure drawings do not depict emaciated beings with gargantuan heads. Most attempts to specify a standard of correctness appeal to the intentions of the picture’s maker. However, I argue that the most detailed such (...) attempt to date is incomplete. I argue that an adequate standard can be given only if one construes a picture’s content as being pictorially implicated, in a way analogous to that in which Grice explains an utterance’s meaning as being conversationally implicated. I propose a theory of pictorial implicature and use it as the basis for an intention-based standard of correctness. I show how this standard is able to explain both the ways in which the content of pictures differs from the content we would attribute to them solely on the basis of resemblance, and how we are able to apply an intention-based standard of correctness even though we lack any independent knowledge of the intentions of pictures’ makers. (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.