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5The Nature of Mathematical ObjectsIn Bharath Sriraman (ed.), Handbook of the History and Philosophy of Mathematical Practice, Springer. pp. 35-61. 2024.A traditional question in the philosophy of mathematics is to give an answer to the question: What is the nature of mathematical objects? This chapter considers the main answers that have been given to this question, specifically those according to which mathematical objects are independently existing entities, or abstractions, or logical objects, or simplifications, or mental constructions, or structures, or fictions, or idealizations of sensible things, or idealizations of operations. The chap…Read more
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11Rethinking Knowledge: The Heuristic ViewSpringer. 2017.This monograph addresses the question of the increasing irrelevance of philosophy, which has seen scientists as well as philosophers concluding that philosophy is dead and has dissolved into the sciences. It seeks to answer the question of whether or not philosophy can still be fruitful and what kind of philosophy can be such. The author argues that from its very beginning philosophy has focused on knowledge and methods for acquiring knowledge. This view, however, has generally been abandoned in…Read more
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49. Reason and LogicIn Maria Cristina Amoretti & Nicla Vassallo (eds.), Reason and Rationality, Ontos Verlag. pp. 199-218. 2012.
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25This volume examines the limitations of mathematical logic and proposes a new approach to logic intended to overcome them. To this end, the book compares mathematical logic with earlier views of logic, both in the ancient and in the modern age, including those of Plato, Aristotle, Bacon, Descartes, Leibniz, and Kant. From the comparison it is apparent that a basic limitation of mathematical logic is that it narrows down the scope of logic confining it to the study of deduction, without providing…Read more
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15Mathematics and ExperienceFoundations of Science 1-15. forthcoming.The question of whether mathematics depends on experience, including experience of the external world, is problematic because, while it is clear that natural sciences depend on experience, it is not clear that mathematics depends on experience. Indeed, several mathematicians and philosophers think that mathematics does not depend on experience, and this is also the view of mainstream philosophy of mathematics. However, this view has had a deleterious effect on the philosophy of mathematics. This…Read more
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Logiche moderne: aspetti storici, filosofici e matematici della logica moderna e delle sue applicazioni; [a cura di Evandro Agazzi, Carlo Cellucci] (edited book)Istituto della Enciclopedia italiana. 1981.
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17The Making of Mathematics: Heuristic Philosophy of MathematicsSpringer. 2022.Mainstream philosophy of mathematics, namely the philosophy of mathematics that has prevailed for the past century, claims that the philosophy of mathematics cannot concern itself with the making of mathematics, in particular discovery, but only with finished mathematics, namely mathematics presented in finished form. On this basis, mainstream philosophy of mathematics argues that mathematics is theorem proving by the axiomatic method. This, however, is untenable because it is incompatible with …Read more
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68Definition in mathematicsEuropean Journal for Philosophy of Science 8 (3): 605-629. 2018.In the past century the received view of definition in mathematics has been the stipulative conception, according to which a definition merely stipulates the meaning of a term in other terms which are supposed to be already well known. The stipulative conception has been so absolutely dominant and accepted as unproblematic that the nature of definition has not been much discussed, yet it is inadequate. This paper examines its shortcomings and proposes an alternative, the heuristic conception.
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Il ruolo del principio di non contraddizione nelle teorie scientificheVerifiche: Rivista Trimestrale di Scienze Umane 10 (1-3): 129-160. 1981.
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Varieties of Maverick Philosophy of MathematicsIn B. Sriraman (ed.), Humanizing Mathematics and its Philosophy, Birkhäuser. pp. 223-251. 2017.Reuben Hersh is a champion of maverick philosophy of mathematics. He maintains that mathematics is a human activity, intelligible only in a social context; it is the subject where statements are capable in principle of being proved or disproved, and where proof or disproof bring unanimous agreement by all qualified experts; mathematicians' proof is deduction from established mathematics; mathematical objects exist only in the shared consciousness of human beings. In this paper I describe my seve…Read more
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32Reconnecting Logic with DiscoveryTopoi 39 (4): 869-880. 2020.According to a view going back to Plato, the aim of philosophy is to acquire knowledge and there is a method to acquire knowledge, namely a method of discovery. In the last century, however, this view has been completely abandoned, the attempt to give a rational account of discovery has been given up, and logic has been disconnected from discovery. This paper outlines a way of reconnecting logic with discovery.
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28Existential instantiation and normalization in sequent natural deductionAnnals of Pure and Applied Logic 58 (2): 111-148. 1992.ellucci, C., Existential instantiation and normalization in sequent natural deduction, Annals of Pure and Applied Logic 58 111–148. A sequent conclusion natural deduction system is introduced in which classical logic is treated per se, not as a special case of intuitionistic logic. The system includes an existential instantiation rule and involves restrictions on the discharge rules. Contrary to the standard formula conclusion natural deduction systems for classical logic, its normal derivations…Read more
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22Is There a Scientific Method? The Analytic Model of ScienceIn Lorenzo Magnani & Claudia Casadio (eds.), Model Based Reasoning in Science and Technology. Logical, Epistemological, and Cognitive Issues, Springer Verlag. 2006.The nature of the scientific method has been a main concern of philosophy from Plato to Mill. In that period logic has been considered to be a part of the methodology of science. Since Mill, however, the situation has completely changed. Logic has ceased to be a part of the methodology of science, and no Discourse on method has been written. Both logic and the methodology of science have stopped dealing with the process of discovery, and generally with the actual process of scientific research. …Read more
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25The Role of Notations in MathematicsPhilosophia 48 (4): 1397-1412. 2020.The terms of a mathematical problem become precise and concise if they are expressed in an appropriate notation, therefore notations are useful to mathematics. But are notations only useful, or also essential? According to prevailing view, they are not essential. Contrary to this view, this paper argues that notations are essential to mathematics, because they may play a crucial role in mathematical discovery. Specifically, since notations may consist of symbolic notations, diagrammatic notation…Read more
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20Georg Kreisel and Gaisi Takeuti. Formally self-referential propositions for cut free analysis and related systems. Dissertationes mathematicae , no. 118, Polska Akademia Nauk, Instytut Matematyczny, Warsaw 1974, 50 pp. - Peter Päppinghaus. A version of the Σ1-reflection principle for CFA provable in PRA. Archiv für mathematische Logik und Grundlagenforschung, vol. 20 , pp. 27–40 (review)Journal of Symbolic Logic 50 (1): 244-246. 1985.
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48Diagrams in MathematicsFoundations of Science 24 (3): 583-604. 2019.In the last few decades there has been a revival of interest in diagrams in mathematics. But the revival, at least at its origin, has been motivated by adherence to the view that the method of mathematics is the axiomatic method, and specifically by the attempt to fit diagrams into the axiomatic method, translating particular diagrams into statements and inference rules of a formal system. This approach does not deal with diagrams qua diagrams, and is incapable of accounting for the role diagram…Read more
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466Philosophy at a Crossroads: Escaping from IrrelevanceSyzetesis (1): 13-53. 2018.Although there have never been so many professional philosophers as today, most of the questions discussed by today’s philosophers are of no interest to cultured people at large. Specifically, several scientists have maintained that philosophy has become an irrelevant subject. Thus philosophy is at a crossroads: either to continue on the present line, which relegates it into irrelevance, or to analyse the reasons of the irrelevance and seek an escape. This paper is an attempt to explore the seco…Read more
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75Is Mathematics Problem Solving or Theorem Proving?Foundations of Science 22 (1): 183-199. 2017.The question that is the subject of this article is not intended to be a sociological or statistical question about the practice of today’s mathematicians, but a philosophical question about the nature of mathematics, and specifically the method of mathematics. Since antiquity, saying that mathematics is problem solving has been an expression of the view that the method of mathematics is the analytic method, while saying that mathematics is theorem proving has been an expression of the view that…Read more
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Why should the logic of discovery be revived?In E. Ippoliti (ed.), Heuristic Reasoning, Springer. pp. 11-27. 2014.Three decades ago Laudan posed the challenge: Why should the logic of discovery be revived? This paper tries to answer this question arguing that the logic of discovery should be revived, on the one hand, because, by Gödel’s second incompleteness theorem, mathematical logic fails to be the logic of justification, and only reviving the logic of discovery logic may continue to have an important role. On the other hand, scientists use heuristic tools in their work, and it may be useful to study…Read more
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228Rethinking PhilosophyPhilosophia 42 (2): 271-288. 2014.Can philosophy still be fruitful, and what kind of philosophy can be such? In particular, what kind of philosophy can be legitimized in the face of sciences? The aim of this paper is to answer these questions, listing the characteristics philosophy should have to be fruitful and legitimized in the face of sciences. Since the characteristics in question demand that philosophy search for new knowledge and new rules of discovery, a philosophy with such characteristics may be called the ‘heuristic v…Read more
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Temi E Prospettive Della Logica E Della Filosofia Della Scienza Contemporanee Organizzato Dalla Società Italiana di Logica E Filosofia Delle Scienze : Cesena, 7-10 Gennaio 1987 (review)Cooperativa Libraria Universitaria Editrice Bologna. 1988.
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106Philosophy of mathematics: Making a fresh startStudies in History and Philosophy of Science Part A 44 (1): 32-42. 2013.The paper distinguishes between two kinds of mathematics, natural mathematics which is a result of biological evolution and artificial mathematics which is a result of cultural evolution. On this basis, it outlines an approach to the philosophy of mathematics which involves a new treatment of the method of mathematics, the notion of demonstration, the questions of discovery and justification, the nature of mathematical objects, the character of mathematical definition, the role of intuition, the…Read more
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48Is Philosophy a Humanistic Discipline?Philosophia 43 (2): 259-269. 2015.According to Bernard Williams, philosophy is a humanistic discipline essentially different from the sciences. While the sciences describe the world as it is in itself, independent of perspective, philosophy tries to make sense of ourselves and of our activities. Only the humanistic disciplines, in particular philosophy, can do this, the sciences have nothing to say about it. In this note I point out some limitations of Williams’ view and outline an alternative view