general, abstract situation. On the other side, I know that graduate students and all mathematicians sometimes falter because their intuitive, ..

This paper attempts to show that mathematical knowledge does not grow by a simple process of accumulation and that it is possible to provide a quasi-empirical (in Lakatos's sense) account of mathematical theories. Arguments supporting the first thesis are based on the study of the changes occurred within Eudidean geometry from the time of Euclid to that of Hilbert; whereas those in favour of the second arise from reflections on the criteria for refutation of mathematical theories.

The present article aims at showing that it is possible to construct a realist philosophy of mathematics which commits one neither to dream the dreams of Platonism nor to reduce the word ''realism'' to mere noise. It is argued that mathematics is a science of patterns, where patterns are not objects (or properties of objects), but aspects, or aspects of aspects, etc. of objects. (The notion of aspect originates from ideas sketched by Wittgenstein in the Philosophical Investigations.).

The present paper aims at showing that there are times when set theoretical knowledge increases in a non-cumulative way. In other words, what we call ‘set theory’ is not one theory which grows by simple addition of a theorem after the other, but a finite sequence of theories T1, ..., Tn in which Ti+1, for 1 ≤ i < n, supersedes Ti. This thesis has a great philosophical significance because it implies that there is a sense in which mathematical theories, (...) like the theories belonging to the empirical sciences, are fallible and that, consequently, mathematical knowledge has a quasi-empirical nature. The way I have chosen to provide evidence in favour of the correctness of the main thesis of this article consists in arguing that Cantor–Zermelo set theory is a Lakatosian Mathematical Research Programme (MRP). (shrink)

This book contains seminal discussions of central issues in the philosophy of language, mathematics, mind, religion and time. Is common language conceptually prior to idiolectics? What is a theory of meaning? Does constructivism provide a satisfactory account of mathematics? What are indefinitely extensible concepts? Can we change the past? These are only some of the very important questions addressed here. Both the papers written by the contributors and Dummett's replies provide a great wealth of stimulating ideas for those who currently (...) do research in the respective areas touched upon without making the reading exceedingly tedious. This feature, common to most of the papers in this book, makes it possible to use the material presented in undergraduate courses at university level. (shrink)

The present collection of essays aims at casting light on important elements of Wittgenstein's thought and intellectual development. Some of Wittgenstein's ideas are here discussed using the «Tractatus» as a frame of reference, others by relating them to the views of G. Frege, G.E.M. Anscombe and D. Davidson.

Bertrand Russell's contributions to last century's philosophy and, in particular, to the philosophy of mathematics cannot be overestimated.Russell, besides being, with Frege and G.E. Moore, one of the founding fathers of analytical philosophy, played a major rôle in the development of logicism, one of the oldest and most resilient1 programmes in the foundations of mathematics.Among his many achievements, we need to mention the discovery of the paradox that bears his name and the identification of its logical nature; the generalization to (...) the whole of mathematics of Frege's idea that it is not possible to draw a demarcation line between logic and arithmetic; the programme, carried out with Whitehead, of derivation of mathematics from the logical system of Principia Mathematica ; and the ramified theory of types, devised by Russell to protect the system of PM from the known paradoxes.Although there is an ample literature on these topics, it is quite important to reconsider Russell's contributions to the foundations of mathematics at a time when, as a consequence of the crisis of the classical programmes in the foundations of mathematics, new trends are beginning to develop within the philosophy of mathematics. These are trends which move in a very different direction from that of logicism, intuitionism, and Hilbert's programme.To see this we need to consider that, in spite of profound disagreements on the nature of mathematical activity, on the relationship existing between logic and mathematics, on the causes of and therapies for the paradoxes, etc., logicism, intuitionism, and Hilbert's programme share an important metaphor: the idea that mathematics is an edifice built on unshakable foundations,2 an edifice which makes possible only a cumulative growth of mathematical knowledge.Such a metaphor—which, together with more specific theses belonging to these schools of thought, remained unsupported …. (shrink)

Available from UMI in association with The British Library. Requires signed TDF. ;The central position defended in this work is that Analytical philosophy is the best realization so far of Descartes' concern for the construction of a method of scientific investigation in philosophy. ;In Frege, the father of Analytical philosophy, such concern was embodied in his Conceptual Notation, which should have provided philosophy with a rigorous and powerful deductive system, and in the 'linguistic turn', i.e. the study of the structure (...) of thought conducted by means of an analysis of the structure of language. This 'linguistic turn' should have produced, in Frege's view, definitive conclusions about the limits of what is thinkable not by setting up arbitrary systems of Pure Reason, but through the controllable investigation of the logical structure of an entity--language--of which we have conceptual experience. ;Wittgenstein's later philosophy is seen here as an attack against the aims and tenets of Analytical philosophy. In fact, what Wittgenstein shows in Philosophical Investigations is, among other things, that by means of an investigation of language it is impossible to justify the assumption that language is an expression of thought. But, if this were true, it would follow that we were not justified in assuming that there exists an isomorphism between language and thought. That there is such an isomorphism is the presupposition which lies at the basis of Frege's 'linguistic turn'. ;In conclusion I attempt to prove that there is no solution to Wittgenstein's objections unless the original exclusively linguistic Analytical philosophy model is modified by a 'pragmatic turn'. The key element of the 'pragmatic turn' is that the truth of certain assumptions--and therefore the justification for their introduction--can be confirmed by the impact that the system based on them has on reality in controlling and transforming it successfully through techniques which are logically dependent on the system. (shrink)

In this paper we intend to argue that: the question ‘True V or not True V’ is central to both the philosophical and mathematical investigations of the foundations of mathematics; when posed within a framework in which set theory is seen as a science of objects, the question ‘True V or not True V’ generates a dilemma each horn of which turns out to be unacceptable; a plausible way out of the dilemma mentioned at is provided by an approach to (...) set theory according to which this is considered to be a science of structures. (shrink)

Peter Abelard (1079–1142 ce) was the most wide‐ranging philosopher of the twelfth century. He quickly established himself as a leading teacher of logic in and near Paris shortly after 1100. After his affair with Heloise, and his subsequent castration, Abelard became a monk, but he returned to teaching in the Paris schools until 1140, when his work was condemned by a Church Council at Sens. His logical writings were based around discussion of the “Old Logic”: Porphyry's Isagoge, aristotle'S Categories and (...) On Interpretation and boethius'S textbook on topical inference. They comprise a freestanding Dialectica (“Logic”; probably c.1116), a set of commentaries (known as the Logica [Ingredientibus], c. 1119) and a later (c. 1125) commentary on the Isagoge (Logica Nostrorum Petititoni Sociorum or Glossulae). In a work Abelard called his Theologia, issued in three main versions (between 1120 and c.1134), he attempted a logical analysis of trinitarian relations and explored the philosophical problems surrounding God's claims to omnipotence and omniscience. The Collationes (“Debates,” also known as “Dialogue between a Christian, a Philosopher and a Jew”; probably c.1130) present a rational investigation into the nature of the highest good, in which the Christian and the Philosopher (who seems to be modeled on a philosopher of pagan antiquity) are remarkably in agreement. The unfinished Scito teipsum (“Know thyself,” also known as the “Ethics”; c.1138) analyses moral action. (shrink)

1.1 ContextIn the period following the demise of logicism, formalism, and intuitionism, contributors to the philosophy of mathematics have been divided. On the one hand, there are those who tend to focus on such issues as: Do mathematical entities exist? If so, what type of entities are they and how do we know about them? If not, how can we account for the role that mathematics plays in our everyday and scientific lives? Contributors to this school—let us call it the (...) analytic school—do not, on the whole, concern themselves with careful analyses of important historical developments in mathematics. On the other hand, there are those who contribute to an historical school in the philosophy of mathematics. Contributors to this school tend to concern themselves almost exclusively with detailed historical analyses of important developments in mathematics. They are typically interested in answering questions concerning the growth of mathematical knowledge.In recent years, interest in the historical school has been growing, as has its influence on the analytic school. This book marks another stride in this direction. Oliveri aims to employ tools developed for use within the historical school to address one of the major issues investigated by the analytic school, i.e., whether we should be realists or anti-realists about mathematics. This is a laudable objective, since even a successful partial integration of these two schools would be valuable to contributors within both.1.2 Noteworthy ContributionsIn attempting a partial integration of these two schools, Oliveri makes several noteworthy contributions. The most significant is his development of a new type of argument for structural realism about mathematics. This argument exploits tools for theorizing about mathematics that were developed by Imre Lakatos [1978] in his work on scientific research programs . It focuses attention on the progressive mathematical research program that started with …. (shrink)

Despite society’s increasing sensitivity toward green production, companies often struggle to find effective communication strategies that induce consumers to buy green products or engage in other environmentally friendly behaviors. To add clarity to this situation, we investigated the effectiveness of negative versus positive message framing in promoting green products, whereby companies highlight the detrimental versus beneficial environmental consequences of choosing less versus more green options, respectively. Across four experiments, we show that negatively framed messages are more effective than positively framed (...) ones in prompting consumers to engage in pro-environmental behaviors. More importantly, we find that anticipated shame is the emotion responsible for this effect. Furthermore, both environmental concern and the type of product promoted serve as moderators; thus, the mediating role of anticipated shame is attenuated when environmental concern is low and the product is a luxury one. Finally, we discuss the theoretical and managerial implications of our work, along with its limitations and some directions for future research. (shrink)

This paper investigates some of the main problems concerning the normativity of aesthetic judgments in Kant’s Critique of Judgment. Kant’s attempt to provide a transcendental foundation to aesthetic judgments clashes with the impossibility to find a determined conceptual rule under which they could be subsumed. This paper aims to discuss and clarify the set of problems derived from this attempt, focusing especially on the artistic perspective. In order to do that, it will be necessary to deal with the following notions: (...) eautonomy, sensus communis aestheticus and genius. From this analysis it will emerge the importance of exemplarity as concept that allow us to highlight some implicit and usually neglected aspects of Kant’s aesthetic theory. (shrink)

The focus of the book, which consists in three studies, can be described in the following aspects: Considerations on Aristotle's universals, reconstruction of Aristotle's critics to Plato' s ideas in Aristotle's lost work “On Ideas”, analysis of Aristotle's substance in the works Categories, Metaphysics, On the Soul, Posterior Analytics, Physics. My point of view is that Aristotle refuses every aspect of Plato's ideas in a radical way. I analyze Aristotle's conditions for a synonymy of predication and compare them with the (...) condition for a not-homonymy of predication in the Argument from Relatives of “On Ideas”. My reflections on substance plead for the presence of a plurality of values of substance in the works of Aristotle: substance can be, for instance, the individual biological entity or the essence/nature/form of a biological entity; a coexistence of both values can be noticed in the different works of Aristotle. (shrink)

Uno studio filosofico sull’etica è sempre di grande attualità. E in questo «dappertutto vi circola la passione per la verità e per il bene» (dalla Presentazione di Carmelo Vigna). Mai come oggi si parla di salute e di benessere, probabilmente perché si percepisce la mancanza di entrambi. In Occidente è così. L’etica si affaccia alla riflessione umana rivestita con i panni di un’etica filosofica, prima ancora che ulteriormente aggettivata, essendo una terminologia che rimanda, come è risaputo, ad Aristotele. Con «etica» (...) si intende oggi quel confine oltre cui non si può più andare, discutere, transigere, indagare: per questo – e correttamente – si parla rispettivamente di «confini etici», «indole etica » o non, «bioetica», «decisione etica», «comportamento etico» e così via. In qualche modo con «etica» si intende ciò che si impone ancor più della scienza o della decisione democratica. L’«etica» pone in questione il baluardo sollevato proprio da queste ultime due. Sanando la questione. Il modo, talvolta, sfugge, diventando, così, ambito proprio della filosofia e della sua visione interale. Questa è esattamente l’etica filosofica. Questo libro affronta temi di grande importanza e fascinosi, quali il bene, il male, la bontà, la crudeltà, la ragione, il sentimento, il «senso morale», la coscienza, la responsabilità, la Stato e la nazione. (shrink)

Il volume si presenta come un manuale di riferimento sul pensiero di Umberto Galimberti, una miscellanea di studi cui ha contribuito un alto numero di studiosi di diverse discipline. Questo libro indaga il lungo arco del lavoro galimbertiano: dal pensiero simbolico, dalla fenomenologia alla psichiatria, fino ad approdare alla riflessione sull'età della tecnica e alla consulenza filosofica.

We reconsider the pragmatic interpretation of intuitionistic logic [21] regarded as a logic of assertions and their justi cations and its relations with classical logic. We recall an extension of this approach to a logic dealing with assertions and obligations, related by a notion of causal implication [14, 45]. We focus on the extension to co-intuitionistic logic, seen as a logic of hypotheses [8, 9, 13] and on polarized bi-intuitionistic logic as a logic of assertions and conjectures: looking at the (...) S4 modal translation, we give a de nition of a system AHL of bi-intuitionistic logic that correctly represents the duality between intuitionistic and co-intuitionistic logic, correcting a mistake in previous work [7, 10]. A computational interpretation of cointuitionism as a distributed calculus of coroutines is then used to give an operational interpretation of subtraction.Work on linear co-intuitionism is then recalled, a linear calculus of co-intuitionistic coroutines is de ned and a probabilistic interpretation of linear co-intuitionism is given as in [9]. Also we remark that by extending the language of intuitionistic logic we can express the notion of expectation, an assertion that in all situations the truth of p is possible and that in a logic of expectations the law of double negation holds. Similarly, extending co-intuitionistic logic, we can express the notion of conjecture that p, de ned as a hypothesis that in some situation the truth of p is epistemically necessary. (shrink)

Different phases of the restoration of Michelangelo's Sistine Chapel frescoes are discussed, including the techniques used to remove foreign matter deposited or applied in the course of time. Huge scientific and technical support is used for this project in contrast to the age?old operations of previous restorers and the simplicity of their equipment.

This essay explores the connection between post-truth and fear, arguing that the latter may be used as a tool to control information and thus manipulate public opinion. Furthermore, it discusses the role of law in regulating post-truth (among others, by a better regulation of Information Society providers). Eventually, some solutions are proposed.