
332Philosophy at a Crossroads: Escaping from IrrelevanceSyzetesis (1): 1353. 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

329Analytic cut treesLogic Journal of the IGPL 8 (6): 733750. 2000.It has been maintained by Smullyan that the importance of cutfree proofs does not stem from cut elimination per se but rather from the fact that they satisfy the subformula property. In accordance with such a viewpoint in this paper we introduce analytic cut trees, a system from which cuts cannot be eliminated but satisfying the subformula property. Like tableaux analytic cut trees are a refutation system but unlike tableaux they have a single inference rule and several branch closure rules. Th…Read more

307From closed to open systemsIn J. Czermak (ed.), Philosophy of Mathematics, pp. 206220, Hölderpichlertempsky. 1993.While Gödel's (first) incompleteness theorem has been used to refute the main contentions of Hilbert's program, it does not seem to have been generally used to stress that a basic ingredient of that program, the concept of formal system as a closed system  as well as the underlying view, embodied in the axiomatic method, that mathematical theories are deductions from first principles must be abandoned. Indeed the logical community has generally failed to learn Gödel's lesson that Hilbert's conc…Read more

301Mathematical logic: What has it done for the philosophy of mathematics?In Piergiorgio Odifreddi (ed.), Kreiseliana. About and Around Georg Kreisel, pp. 365388, A K Peters. 1996.onl y to discuss some claims concerning the relationship between mathematical logic and the philosophy of mathematics that repeatedly occur in his writings. Although I do not know to what extent they are representative of his present position, they correspond to widespread views of the logical community and so seem worth discussing anyhow. Such claims will be used as reference to make some remarks about the present state of relations between mathematical logic and the philosophy of mathematics.

237The growth of mathematical knowledge: An open world viewIn Emily Grosholz & Herbert Breger (eds.), The Growth of Mathematical Knowledge, pp. 153176, Kluwer Academic Publishers. pp. 153176. 2000.In his book The Value of Science Poincaré criticizes a certain view on the growth of mathematical knowledge: “The advance of science is not comparable to the changes of a city, where old edifices are pitilessly torn down to give place to new ones, but to the continuous evolution of zoological types which develop ceaselessly and end by becoming unrecognizable to the common sight, but where an expert eye finds always traces of the prior work of the centuries past” (Poincaré 1958, p. 14). The view …Read more

230The scope of logic: deduction, abduction, analogyTheoria 64 (23): 217242. 1998.The present form of mathematical logic originated in the twenties and early thirties from the partial merging of two different traditions, the algebra of logic and the logicist tradition (see [27], [41]). This resulted in a new form of logic in which several features of the two earlier traditions coexist. Clearly neither the algebra of logic nor the logicist’s logic is identical to the present form of mathematical logic, yet some of their basic ideas can be distinctly recognized within it. One o…Read more

190Review of M. Giaquinto, The Search for Certainty (review)European Journal of Philosophy 11 (3): 420423. 2003.Giaquinto’s book is a philosophical examination of how the search for certainty was carried out within the philosophy of mathematics from the late nineteenth to roughly the midtwentieth century. It is also a good introduction to the philosophy of mathematics and the views expressed in the body of the book, in addition to being thorough and stimulating, seem generally undisputable. Some doubts, however, could be raised about the concluding remarks concerning the present situation in the philoso…Read more

167In the past few decades the question of the meaning of life has received renewed attention. However, much of the recent literature on the topic reduces the question of the meaning of life to the question of meaning in life. This raises the problem: How should we think about the meaning of life? The paper tries to give an answer to this problem

96Mathematical Beauty, Understanding, and DiscoveryFoundations of Science 20 (4): 339355. 2015.In a very influential paper Rota stresses the relevance of mathematical beauty to mathematical research, and claims that a piece of mathematics is beautiful when it is enlightening. He stops short, however, of explaining what he means by ‘enlightening’. This paper proposes an alternative approach, according to which a mathematical demonstration or theorem is beautiful when it provides understanding. Mathematical beauty thus considered can have a role in mathematical discovery because it can guid…Read more

57Definition in mathematicsEuropean Journal for Philosophy of Science 8 (3): 605629. 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.

54Logic and Knowledge (edited book)Cambridge Scholar Publishing. 2011.Logic and Knowledge Editor: Carlo Cellucci, Emily Grosholz and Emiliano Ippoliti Date Of Publication: Aug 2011 Isbn13: 9781443830089 Isbn: 1443830089 The problematic relation between logic and knowledge has given rise to some of the most important works in the history of philosophy, from Books VI–VII of Plato’s Republic and Aristotle’s Prior and Posterior Analytics, to Kant’s Critique of Pure Reason and Mill’s A System of Logic, Ratiocinative and Inductive. It provides the title of an imp…Read more

40Frege on Thinking and Its Epistemic Significance (review)History and Philosophy of Logic 38 (1): 9295. 2017.Given the large literature on Frege, one might believe that it would be impossible to say anything essentially new on the subject. This book contradicts this belief, calling attention to Frege's in...

40Is There a Scientific Method? The Analytic Model of ScienceIn Lorenzo Magnani & Claudia Casadio (eds.), Studies in Applied Philosophy, Epistemology and Rational Ethics volume 25, Springer. pp. 489505. 2016.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

28On Quine's Approach to Natural Deduction'In Paolo Leonardi & Marco Santambrogio (eds.), On Quine: New Essays, Cambridge University Press. pp. 314335. 1995.This article examines Quine's original proposal for a natural deduction calculus including an existential specification rule, it argues that it introduces a new paradigm of natural deduction alternative to Gentzen's but has some substantial defects. As an alternative the article puts forward a system of sequent natural deduction.

18Reconnecting Logic with DiscoveryTopoi 112. 2017.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.

15Georg Kreisel and Gaisi Takeuti. Formally selfreferential 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 Σ1reflection principle for CFA provable in PRA. Archiv für mathematische Logik und Grundlagenforschung, vol. 20 , pp. 27–40 (review)Journal of Symbolic Logic 50 (1): 244246. 1985.

10Logic and Knowledge (edited book)Cambridge Scholars Publications. 2011.The problematic relation between logic and knowledge has given rise to some of the most important works in the history of philosophy, from Books VIVII of Platos Republic and Aristotles Prior and Posterior Analytics, to Kants Critique of Pure Reason and Mills A System of Logic, Ratiocinative and Inductive. It provides the title of an important collection of papers by Bertrand Russell. However, it has remained an underdeveloped theme in the last century, because logic has been treated as separate …Read more

6This 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

2The Question Hume Didn't Ask: Why Should We Accept Deductive Inferences?In Carlo Cellucci & Paolo Pecere (eds.), Demonstrative and NonDemonstrative Reasoning in Mathematics and Natural Science, Edizioni Dell'università Di Cassino. pp. 207235. 2006.This article examines the current justifications of deductive inferences, and finds them wanting. It argues that this depends on the fact that all such justification take no account of the role deductive inferences play in knowledge. Alternatively, the article argues that a justification of deductive inferences may be given in terms of the fact that they are nonampliative, in the sense that the content of the conclusion is merely a reformulation of the content of the premises. Some possible obj…Read more

1Mathematical Discourse vs. Mathematical IntuitionIn Carlo Cellucci & Donald Gillies (eds.), Mathematical Reasoning and Heuristics, College Publications. 2005.The aim of this article is to show that intuition plays no role in mathematics. That intuition plays a role in mathematics is mainly associated to the view that the method of mathematics is the axiomatic method. It is assumed that axioms are directly (Gödel) or indirectly (Hilbert) justified by intuition. This article argues that all attempts to justify axioms in terms of intuition fail. As an alternative, the article supports the view that the method of mathematics is the analytic method, a met…Read more

1Rethinking 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

1Book Reviews (review)History and Philosophy of Logic 9 (1): 107129. 1988.MEDIEVAL AND RENAISSANCE LOGICSMARK D. JOHNSTON, The spiritual logic of Ramon Llull. Oxford: Clarendon Press,1987. xi + 336 pp. £35.00E. J. ASHWORTH, Thomas Bricot: Tractatus Znsolubilium. Nijmegen: Ingenium, 1986. xxiii+ 155 pp. 44 Dfl.CYPRIANI REGNERI, Demonstratio logicae verae iuridica. Edited by G. Kalinowski. Bologna: Cooperativa Libraria Universitaria Editrice Bologna, 1986. xxviii + 167 pp. No price stated.GIROLAMO SACCHERI, Euclides vindicatus. Edited and translated by George Bruce Hals…Read more

The Most Urgent Task of Philosophy TodayBorderless Philosophy 2 4575. 2019.In this essay, first, I briefly describe the view that philosophy is acquisition of knowledge. Then second, I answer some objections against it and examine thealternative view of philosophy as a search for understanding put forward by PeterHacker in several books. Third and finally, I conclude that the view that philosophy is asearch for understanding is inadequate, and not a viablealternative to the view thatphilosophy is knowledge acquisition.