We discuss the problem of defining a logic for analogical reasoning, and sketch a solution in the style of the semantics for Counterfactual Conditionals, Preferential Structures, etc.

Ontology is the name of the philosophical discipline that provides answers about what there is. The view laid out in the paper, i.e. austere realism, is realistic in that it defends the existence of a thought and language independent world. It is also inclined towards austerity in that it does not take this world to be as richly ontologically populated with entities as common sense initially presupposes. Yet it is a view that results from common sense taking a reflexive attitude about its ontological commitments. Despite its austere consequences, according to this view, many thoughts and sentences expressed by common sense are true, provided that truth is not considered as direct correspondence, i.e. not as the ultimate ontological correspondence to the world.

La question fondamentale d’Albert Lautman concerne la nature du réel et la capacité de l’esprit de l’appréhender. C’est pourquoi elle convoque les données de la physique, leur expression en concepts mathématiques et leur interprétation métaphysique qui doit préciser le rapport de la pensée humaine à la réalité du monde. L’examen des théories mathématiques les plus sophistiquées de son temps (surface de Riemann, loi de réciprocité quadratique, théorie du corps de classes) est destinée à montrer l’affi nité de la genèse des concepts mathématiques avec une Dialectique supérieure, qui met en jeu les Idées, comprises en un sens dérivé de Platon. Mon but est d’expliquer comment Lautman comprend les termes « Dialectique », « Idée », « genèse », simultanément sur le plan métaphysique et dans leur incarnation mathématique.

Contrary to popular misconception, the question in the title is far from simple. It involves sets of numbers on the first level, sets of sets of numbers on the second level, and so on, endlessly. The infinite hierarchy of the levels involved distinguishes the concept of “definable number” from such notions as “natural number”, “rational number”, “algebraic number”, “computable number” etc. (Explanatory essay for non-experts.)

El tema del presente artículo es la recepción y radicalización heideggeriana de la aporía del tiempo y el alma en Aristóteles en los textos del periodo conocido como la década fenomenológica. La cuestión acerca de si habría o no tiempo en caso de que no existiera el alma (al final de Physica ∆14) conduce a una difícil aporía que queda sin respuesta en el corpus aristotelicum. Heidegger piensa que el planteamiento aporético muestra que Aristóteles intuye ya (si bien no llega a ver con total claridad) que el Dasein mismo debe considerarse fundamentalmente tiempo. Puesto que el tiempo es la determinación básica de la existencia humana, el Dasein sería imposible sin el tiempo. En otras palabras, no habría Dasein sin tiempo.

One of the central tasks of Badioursquo;s Being and Event is to elaborate a theory of the subject in the wake of an axiomatic identification of ontology with mathematics, or, to be precise, with classical Zermelo-Fraenkel set theory. The subject, for Badiou, is essentially a free project that originates in an event, and subtracts itself from both being qua being, as well as the linguistic and epistemic apparatuses that govern the situation. The subjective project is, itself, conceived as the temporal unfolding of a lsquo;truthrsquo;. Originating in an event and unfolding in time, the subject cannot, for Badiou, be adequately understood in strictly ontological, i.e. set-theoretical, terms, insofar as neither the event nor time have any place in classical set theory. Badiou nevertheless seeks to articulate the ontological infrastructure of the subject within set theory, and for this he fastens onto Cohenrsquo;s concepts of genericity and forcing: the former gives us the set-theoretic structure of the truth to which the subject aspires, the latter gives us the immanent logic of the subjective procedure, the ldquo;law of the subjectrdquo;.

This article explores the following methodological principle for theory construction in physics: if an ontological theory predicts two scenarios that are ontologically distinct but empirically indiscernible, then this theory should be rejected and replaced by one relative to which the scenarios are ontologically the same. I defend the thesis that this methodological principle was first articulated by Leibniz as a version of his principle of the identity of indiscernibles, and that it was applied repeatedly to great effect by Einstein in his development of the special and general theories of relativity.

The embracing of actual infinity in mathematics leads naturally to the question of comparing the sizes of infinite collections. The basic dilemma is that the Cantor Principle (CP), according to which two sets have the same size if there is a one-to-one correspondence between their elements, and the Part-Whole Principle (PW), according to which the whole is greater than its part, are inconsistent for infinite collections. Contemporary axiomatic set-theoretic systems, for instance, ZFC, are based on CP. PW is not valid for infinite sets. In the last two decades, the topic of sizes of infinite sets has resurfaced again in a number of papers.

Este artículo constituye una detallada exploración de la profunda transformación experimentada por el concepto de “historia” a la luz del salto intentado por Martin Heidegger desde la metafísica hacia el ámbito del pensamiento (del Ser). En un primer momento, la exposición sigue de cerca la lógica interna del pensar metafísico para demostrar luego con ello el carácter necesariamente metafísico de sus dos maneras de “procesar” la historia: la filosofía de la historia y la historiografía. Desde ahí, el artículo avanza hacia un modo radicalmente nuevo de entender la naturaleza de la historia: aquello que Heidegger llama “el otro inicio”.

Mathematical proofs should be paired with formal proofs, whenever feasible.