This book introduces readers to the narrative structure of mathematical proofs and why mathematicians communicate that way, drawing examples from classic literature and employing metaphors and imagery.

In his book Gabriele Lolli discusses the notion of proof, which is, according to him, the most important and at the same time the least studied aspect of mathematics. According to Lolli, a theorem is a conditional sentence of the form ‘if T then A’ such that A is a logical consequence of T, where A is a sentence and T is a sentence or a conjunction or set of sentences. Verifying that A is a consequence of (...) T generally involves considering infinitely many interpretations; so it is something which is impossible to do in finite terms. Proofs may serve as ‘shortcuts’ in this respect. A proof is defined by Lolli as any finite argument certifying that A is a consequence of T. A proof is a shortcut in the sense that it spares us considering infinitely many interpretations.The reason for such a very general definition of proof is Lolli's strong belief that mathematics is not a rigid system of explicit rules, but rather a set of tools; as a consequence, there is no prescription as to what a proof should or should not be. Actually, mathematics is historically situated and not timeless, and the history of mathematics is the …. (shrink)

Au Congrès de philosophie scientifique de 1935 comme lors du lancement de l’Encyclopédie internationale de la science unifiée, Federigo Enriques était reconnu par les néo-positivistes comme un de leurs pères fondateurs, sans qu’il fût tout à fait d’accord. À Paris, Enriques représentait le groupe des philosophes des sciences italiens et son nom était lié au journal Scientia, ouvert aux contributions des positivistes logiques. Ces derniers, désireux de constituer un front commun pour lutter contre les philosophies idéalistes et métaphysiques alors dominantes, (...) surestimaient sans doute la force du groupe italien, comme ils sous-estimaient les critiques d’Enriques sur l’usage de la logique formelle ou sur l’importance du langage. Dès les débuts de son implication en philosophie, il avait souligné la nécessité d’une fondation psychologique des concepts scientifiques, et en 1901, il avait donné un exemple de ce type d’analyse pour les postulats de la géométrie. Il n’appréciait guère les travaux logiques de Giuseppe Peano, affirmant que son formalisme manquait de bases psychologiques. Cependant, contre les mouvements anti-rationalistes de son temps, il défendait la valeur cognitive de la science et, dans ce climat d’appréciation mutuelle, les différences passaient au second plan. (shrink)

In a review of Russell’s Principles from 1904, Edwin B. Wilson pays great attention to Peano’s work and that of his collaborators. His purpose was to make this work known in the USA where it “unfortunately is very little known and still less appreciated”. Wilson expands Russell’s well-known acknowledgement of Peano’s influence on his own development, seeing in Peano’s logic more than a new “mathematical tool”, describing Peano as a kind of proto-logicist, and defending him from Poincaré’s criticisms. Especially in (...) geometry, he vindicates several priority issues for Peano’s school with respect to Hilbert in the philosophy of the axiomatic method. (shrink)

In a review of Russell’s Principles from 1904, Edwin B. Wilson pays great attention to Peano’s work and that of his collaborators. His purpose was to make this work known in the USA where it “unfortunately is very little known and still less appreciated”. Wilson expands Russell’s well-known acknowledgement of Peano’s influence on his own development, seeing in Peano’s logic more than a new “mathematical tool”, describing Peano as a kind of proto-logicist, and defending him from Poincaré’s criticisms. Especially in (...) geometry, he vindicates several priority issues for Peano’s school with respect to Hilbert in the philosophy of the axiomatic method. (shrink)

In the Tractatus, it is stated that questions about logical formatting cannot be meaningfully formulated, since it is precisely the application of logical rules which enables the formulation of a question whatsoever; analogously, Wittgenstein’s celebrated infinite regress argument on rule-following seems to undermine any explanation of deduction, as relying on a logical argument. On the other hand, some recent mathematical developments of the Curry-Howard bridge between proof theory and type theory address the issue of describing the “subjective” side of logic, (...) that is, the concrete manipulation of rules and proofs in space and time. It is advocated that such developments can shed some light on the question of logical formatting and its apparently unintelligible paradoxes, thus reconsidering Wittgenstein’s verdict. (shrink)

In his book Gabriele Lolli discusses the notion of proof, which is, according to him, the most important and at the same time the least studied aspect of mathematics. According to Lolli, a theorem is a conditional sentence of the form ‘if T then A’ such that A is a logical consequence of T, where A is a sentence and T is a sentence or a conjunction or set of sentences. Verifying that A is a consequence of (...) T generally involves considering infinitely many interpretations; so it is something which is impossible to do in finite terms. Proofs may serve as ‘shortcuts’ in this respect. A proof is defined by Lolli as any finite argument certifying that A is a consequence of T. A proof is a shortcut in the sense that it spares us considering infinitely many interpretations.The reason for such a very general definition of proof is Lolli's strong belief that mathematics is not a rigid system of explicit rules, but rather a set of tools; as a consequence, there is no prescription as to what a proof should or should not be. Actually, mathematics is historically situated and not timeless, and the history of mathematics is the …. (shrink)

In two often neglected passages of the Critique of Pure Reason, Kant submits that the Critique is a 'treatise' or a 'doctrine of method'. These passages are puzzling because the Critique is only cursorily concerned with identifying adequate procedures of argument for philosophy. In this book, Gabriele Gava argues that these passages point out that the Critique is the doctrine of method of metaphysics. Doctrines of method have the task of showing that a given science is indeed a science (...) because it possesses 'architectonic unity' – which happens when it realizes the 'idea' of a science. According to Gava's novel approach, the Critique establishes that metaphysics is capable of this unity, and his reading of the Critique from this perspective not only illuminates the central role of the Transcendental Doctrine of Method within it, but also clarifies the relationship between the different parts of the work. (shrink)

The central theme of this important book is that we are paying the price of an arrogance that refuses to recognize mystery. The author invites the reader to enter into the argument that he holds with himself on a great number of problems. Written in the early 1950s, Marcel's discussion of these topics are remarkably contemporary, e.g.: * Our crisis is a metaphysical, not merely social, one. * What a man is depends partly on what he thinks he is, and (...) a materialistic philosophy turns men into things. * Can a man be free except in a free country? * Stoicism is no longer a workable philosophy because today pressure can be put on the mind as well as on the body. * Technical progress is not evil in itself, but a technique is a means that, regarded as an end, can become either an idol or an excuse for self-idolatry. State control of scientific research, leading to a concentration on new means of destruction, is a calamity. * Fanaticism is an opinion that refuses to argue, and so the fanatic is an enemy of truth. * The kind of unification that science is bringing about today is really an ironing out of differences, but the only valuable kind of unity is one that implies a respect for differences. * We must beware of thinking in terms of great numbers and so blinding ourselves to the reality of individual suffering. Our philosophical approach to being is made possible only by our practical approach to our neighbor. * We must encourage the spirit of fraternity and distrust the kind of egalitarianism that is based on envy and resentment. * No man however humble should feel that he cannot spread the light among his friends. No easy solution is offered, but the author conveys his own faith that ultimately love and intelligence will triumph. (shrink)

In this chapter, I investigate a problem for Kant’s claim that metaphysics can reach the status of science. The problem arises when one considers Kant’s account of the “architectonic unity” of metaphysics in the Architectonic of Pure Reason. Attaining architectonic unity is a condition for becoming a science for any body of cognitions that purports to be such. This is achieved when the cognitions belonging to a science are systematically organized according to the “idea of reason” which lies at the (...) basis of that science. However, Kant suggests that nobody can ever legitimately claim to have matched the idea of the philosopher, where this appears to imply that nobody can justifiably claim that she has fully grasped the idea of metaphysics. This generates a problem. I suggest that when Kant claims that the idea of the philosopher must remain an ideal or an archetype, he has a particular conception of philosophy in mind, that is, philosophy as a doctrine of wisdom. According to this understanding, philosophy must provide an example of how one can become virtuous. However, this is only a partial solution to our problem, for reasons that will be illustrated. (shrink)

This paper takes a close look at the thought that mereological relations on material objects mirror, and are mirrored by, parallel mereological relations on their exact locations. This hypothesis is made more precise by means of a battery of principles from which more substantive consequences are derived. Mereological harmony turns out to entail, for example, that atomistic space is an inhospitable environment for material gunk or that Whiteheadian space is not a hospitable environment for unextended material atoms.

Five hundred years after his death, Niccolò Machiavelli still draws an astonishing range of contradictory characterizations. Was he a friend of tyrants? An ardent republican loyal to Florence’s free institutions? The father of political realism? A revolutionary populist? A calculating rationalist? A Renaissance humanist? A prophet of Italian unification? A theorist of mixed government? A forerunner to authoritarianism? The master of the dark arts of intrigue? This book provides a vivid and engaging introduction to Machiavelli’s life and works that sheds (...) new light on his originality and relevance. Gabriele Pedullà—a leading Italian expert and acclaimed writer—offers fresh readings of the Florentine thinker’s most famous writings, The Prince and the Discourses on Livy, as well as lesser-known texts. A new and often surprising Machiavelli emerges: one closer to his time but also better suited to inform our own. Pedullà’s portrait of Machiavelli highlights his close attention to social and emotional bonds, staunch opposition to oligarchy, keen awareness of the economic side of power dynamics, and strong preference for history over philosophy as a guide for leaders. This book recovers the excitement Machiavelli roused in his first readers for a twenty-first-century audience, capturing his capacity to provoke, both then and now, with unconventional ideas and startling insights. (shrink)

The volume deals with the history of logic, the question of the nature of logic, the relation of logic and mathematics, modal or alternative logics (many-valued, relevant, paraconsistent logics) and their relations, including translatability, to classical logic in the Fregean and Russellian sense, and, more generally, the aim or aims of philosophy of logic and mathematics. Also explored are several problems concerning the concept of definition, non-designating terms, the interdependence of quantifiers, and the idea of an assertion sign. The contributions (...) concerned with Wittgenstein's investigations into the philosophy of logic and mathematics pursue issues relating to logical necessity, the undeniability of the law of the excluded middle, and the source of self-evidence, often characterized in the literature as the "rule-following considerations". Additionally, they examine Wittgenstein's attitudes towards the very idea of set-theory as a possible foundation for arithmetic. The volume also includes a number of contributions on specific issues concerning Wittgenstein's views on moral and religious judgements. (shrink)

Carnap’s seminal ‘Empiricism, Semantics and Ontology’ makes important use of the notion of a framework and the related distinction between internal and external questions. But what exactly is a framework? And what role does the internal/external distinction play in Carnap’s metaontology? In an influential series of papers, Matti Eklund has recently defended a bracingly straightforward interpretation: A Carnapian framework, Eklund says, is just a natural language. To ask an internal question, then, is just to ask a question in, say, English. (...) To try to ask an external question is to try, absurdly, to ask a question in no language at all. Finding that so trivial an I/e distinction can’t help to explain Carnap’s deflationary metaontology, Eklund is led to attribute to Carnap a view he calls ontological pluralism. In this paper, I show that Eklund misreads Carnap, and I argue that this misreading obscures fundamental features of Carnap’s philosophy. I then defend an account of frameworks as what Carnap called semantical systems, and I place this account in the context of Carnap’s philosophical program of explication. Finally, I discuss the role that frameworks and the I/e distinction play in ESO, showing that ESO provides no reason to attribute the doctrine of ontological pluralism to Carnap. (shrink)

The goal of this paper is to provide an account of the role played by logic in the context of what Husserl names the “crisis of European sciences.” Presupposing the analyses offered in the Krisis, I look at Formale und Transzendentale Logik to demonstrate that the crisis of logic stems from the deviation of its original meaning as a “theory of science” and from its restriction to a mere “theoretical technique.” Through a comparison between Aristotelian syllogistic and modern logic, I (...) show why the modern discovery of the purely formal dimension of knowledge which makes possible such a mathematical technization is a positive achievement that hinders at the same time the disclosure of the truly philosophical nature of logic. The correct appraisal of this ambiguous phenomenon will explain why the rise of modern logic represents a decisive challenge for the success of Husserl’s late phenomenological project. (shrink)

Many consider the nature of human consciousness to be one of the last great unsolved mysteries. Why should the light turn on, so to speak, in human beings at all? And how is the electrical storm of neurons under our skull connected with our consciousness? Is the self only our brain's user interface, a kind of stage on which a show is performed that we cannot freely direct? In this book, philosopher Markus Gabriel challenges an increasing trend in the sciences (...) towards neurocentrism, a notion which rests on the assumption that the self is identical to the brain. Gabriel raises serious doubts as to whether we can know ourselves in this way. In a sharp critique of this approach, he presents a new defense of the free will and provides a timely introduction to philosophical thought about the self – all with verve, humor, and surprising insights. Gabriel criticizes the scientific image of the world and takes us on an eclectic journey of self-reflection by way of such concepts as self, consciousness, and freedom, with the aid of Kant, Schopenhauer, and Nagel but also Dr. Who, The Walking Dead, and Fargo. (shrink)

Schelling entwirft in seiner Spätphilosophie einen neuen Philosophietypus, der in Konkurrenz zu Hegels Metaphysik tritt. Dabei greift er Hegel nicht nur auf dem Gebiet fundamentaler theoretischen Annahmen an, sondern sucht ihn insbesondere durch einen neuen Begriff von Religion zu überbieten. Die Arbeit untersucht dies im Ausgang von Schellings "Philosophie der Mythologie".

In Kant’s Critique of Pure Reason and the Method of Metaphysics (CUP 2023), I argue that the first Critique is not only a ‘propaedeutic’ to metaphysics, but actually already establishes parts of metaphysics. These parts belong to what Kant calls transcendental philosophy. Additionally, I also provide an account of Kant’s critique of dogmatism and Wolff as its main defender. In this paper, I take up Luigi Filieri’s and Davide Dalla Rosa’s invitation to further develop my characterization of transcendental philosophy and (...) I respond to Michael Walschots’s objections against my interpretation of Kant’s critique of Wolff’s dogmatism. (shrink)

This chapter analyzes two short texts in which Peirce sketches out an anti-skeptical argument inspired by Kant’s refutation of idealism. The chapter will first consider why Peirce found Kant’s argument interesting and promising, given that it is often regarded as problematic and unsuccessful. It will then briefly reconstruct Kant’s refutation, highlighting its most problematic passages. Moreover, since Peirce’s own version of the argument relies on Kant’s views regarding the temporal structure of consciousness, the chapter will explain how Peirce tackles this (...) issue in “The Law of Mind.” Finally, the chapter considers Peirce’s own anti-skeptical argument and examines whether and how it can be seen as appropriating Kant’s strategy. (shrink)

This discussion of pride, shame, and guilt centers on the beliefs involved in the experience of any of these emotions. Through a detailed study, the author demonstrates how these beliefs are alike--in that they are all directed towards the self--and how they differ. The experience of these three emotions are illustrated by examples taken from English literature. These concrete cases supply a context for study and indicate the complexity of the situations in which these emotions usually occur.

In this paper I examine the relation between modern transhumanism and Nietzsche’s philosophy of the superhuman. Following Loeb, I argue that transhumanists cannot claim affinity to Nietzsche’s philosophy until they incorporate the doctrine of eternal recurrence to their project of technological enhancement. This doctrine liberates us from resentment against time by teaching us reconciliation with time and something higher than all reconciliation. Unlike Loeb, however, I claim that this “something higher” is not a new skill (prospective memory), but rather a (...) love for the past in the form of loving that aspect of it that is still with us, namely, the will to power itself, which is the engine of all life. Love of the past is thus equivalent to love of life. Since human beings are conscious incarnations of the will to power, in our case, love of life manifests itself as love of our humanity or love for that aspect of ourselves that connects us to each other, for we recognize it to be the same in all of us. Thus, learning this kind of love enables us to joyfully coordinate our wills in the pursuit of Zarathustra’s superhuman ideal without turning it into a destructive mockery of itself. While learning this kind of love would facilitate a joyful version of transhumanism, I conclude by suggesting that it is unlikely to be achieved through technological interventions of the sort envisioned by transhumanists. Instead, it requires the kind of participatory pedagogical program that Nietzsche thought his Zarathustra would fulfill. (shrink)

O objetivo deste trabalho é investigar a natureza ontológica e epistemológica do conceito de informação, focalizando inicialmente a Filosofia e Ética da Informação (EI), propostas por Floridi (1999; 2002; 2004; 2011; 2016), cujo fundamento teórico servirá de suporte para reflexões acerca da inteligência artificial generativa (IAGen), mais especificamente, o ChatGPT. O conceito de informação será abordado a partir de três vertentes na perspectiva filosófico-interdisciplinar: (1) realismo informacional, (2) ecologia informacional e (3) semântica informacional, segundo Gonzalez, Nascimento & Haselager (2004). A (...) partir dessa abordagem, direcionamos nosso estudo para propostas de Floridi (2004), indicando os seus possíveis problemas. Procuraremos entender a Filosofia da Informação, a partir da perspectiva da Ética da Informação (EI), analisando como a EI pode contribuir para reflexões acerca do uso de IA generativa. Problematizamos a EI a fim de construir uma ponte com a Ética Intercultural da Informação (EII) proposta por Capurro (2010). (shrink)

Sexualpolitik ist eine Machttechnik. Sie reguliert Verhaltensweisen oder schliesst Gruppen aus - das Feld Sexualität ist dabei besonders skandalisierbar. Angeblich problematische Sexualitäten werden nicht nur mit Geschlecht, sondern auch mit Ethnizität und Religion verflochten. Gabriele Dietze diskutiert diesen Zusammenhang in historischer, theoretischer und gegenwartsanalytischer Perspektive von feministischen Orientalismen der Ersten Frauenbewegung bis hin zu den Ereignissen von Köln in der Silvesternacht 2015.