The seventeenth century saw dramatic advances in mathematical theory and practice. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, the rules of analytic geometry, geometry of indivisibles, arithmatic of infinites, and calculus were developed. Although many technical studies have been devoted to these innovations, Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Starting with (...) the Renaissance debates on the certainty of mathematics, Mancosu leads the reader through the foundational issues raised by the emergence of these new mathematical techniques, including the influence of the Aristotelian conception of science in Cavalieri and Guldin, the foundational relevance of Descartes' Geometrie, the relation between geometrical and epistemological theories of the infinite, and the Leibnizian calculus and the opposition to infinitesimalist procedures. In the process Mancosu draws a sophisticated picture of the subtle dependencies between technical development and philosophical reflection in seventeenth century mathematics. (shrink)

There is an urgent need in philosophy of mathematics for new approaches which pay closer attention to mathematical practice. This book will blaze the trail: it offers philosophical analyses of important characteristics of contemporary mathematics and of many aspects of mathematical activity which escape purely formal logical treatment.

Italian political thinker Paolo Virno argues that the category of "multitude" is a far better tool to analyze contemporary issues than the Hobbesian concept of "people." Globalization is forcing us to rethink some of the categories—such as "the people"—that traditionally have been associated with the now eroding state. Italian political thinker Paolo Virno argues that the category of "multitude," elaborated by Spinoza and for the most part left fallow since the seventeenth century, is a far better tool to (...) analyze contemporary issues than the Hobbesian concept of "people," favored by classical political philosophy. Hobbes, who detested the notion of multitude, defined it as shunning political unity, resisting authority, and never entering into lasting agreements. "When they rebel against the state," Hobbes wrote, "the citizens are the multitude against the people." But the multitude isn't just a negative notion, it is a rich concept that allows us to examine anew plural experiences and forms of nonrepresentative democracy. Drawing from philosophy of language, political economics, and ethics, Virno shows that being foreign, "not-feeling-at-home-anywhere," is a condition that forces the multitude to place its trust in the intellect. In conclusion, Virno suggests that the metamorphosis of the social systems in the West during the last twenty years is leading to a paradoxical "Communism of the Capital.". (shrink)

The period from 1900 to 1935 was particularly fruitful and important for the development of logic and logical metatheory. This survey is organized along eight "itineraries" concentrating on historically and conceptually linked strands in this development. Itinerary I deals with the evolution of conceptions of axiomatics. Itinerary II centers on the logical work of Bertrand Russell. Itinerary III presents the development of set theory from Zermelo onward. Itinerary IV discusses the contributions of the algebra of logic tradition, in particular, Löwenheim (...) and Skolem. Itinerary V surveys the work in logic connected to the Hilbert school, and itinerary V deals specifically with consistency proofs and metamathematics, including the incompleteness theorems. Itinerary VII traces the development of intuitionistic and many-valued logics. Itinerary VIII surveys the development of semantical notions from the early work on axiomatics up to Tarski's work on truth. (shrink)

The mnemonic arts and the idea of a universal language that would capture the essence of all things were originally associated with cryptology, mysticism, and other occult practices. And it is commonly held that these enigmatic efforts were abandoned with the development of formal logic in the seventeenth century and the beginning of the modern era. In his distinguished book, Logic and the Art of Memory Italian philosopher and historian Paolo Rossi argues that this view is belied by an (...) examination of the history of the idea of a universal language. Based on comprehensive analyses of original texts, Rossi traces the development of this idea from late medieval thinkers such as Ramon Lull through Bruno, Bacon, Descartes, and finally Leibniz in the seventeenth century. The search for a symbolic mode of communication that would be intelligible to everyone was not a mere vestige of magical thinking and occult sciences, but a fundamental component of Renaissance and Enlightenment thought. Seen from this perspective, modern science and combinatorial logic represent not a break from the past but rather its full maturity. Available for the first time in English, this book (originally titled Clavis Universalis ) remains one of the most important contributions to the history of ideas ever written. In addition to his eagerly anticipated translation, Steven Clucas offers a substantial introduction that places this book in the context of other recent works on this fascinating subject. A rich history and valuable sourcebook, Logic and the Art of Memory documents an essential chapter in the development of human reason. (shrink)

The influential Italian thinker offers three essays in the political philosophy of language. Multitude between Innovation and Negation by Paolo Virno translated by James Cascaito. The publication of Paolo Virno's first book in English, Grammar of the Multitude, by Semiotext in 2004 was an event within the field of radical political thought and introduced post-'68 currents in Italy to American readers. Multitude between Innovation and Negation, written several years later, offers three essays that take the reader on a (...) journey through the political philosophy of language. “Wit and Innovative Action” explores the ambivalence inevitably arising when the semiotic and the semantic, grammar and experience, rule and regularity, and right and fact intersect. Virno unravels the infinite potential and wonders of everyday linguistic praxis and ambiguity. Wit, he argues, is a public performance, and its modus operandi characterizes human action in a state of emergency; it is a reaction, an articulate response, and a possible solution to a state of crisis. “Mirror Neurons, Linguistic Negation, and Mutual Recognition” examines the relationship of language and intersubjective empathy: without language, would human beings be able to recognize other members of their species? And finally, in “Multitude and Evil,” Virno challenges the distinction between the state of nature and civil society and argues for a political institution that resembles language in its ability to be at once nature and history. Few thinkers take the risks required by innovation. Like a philosophical entrepreneur, Virno is engaged in no less than rewriting the dictionary of political theory, an urgent and ambitious project when language, caught in a permanent state of emergency impossible to sustain, desperately needs to articulate and enact new practices of freedom for the multitude. Paolo Virno is the author of several books, including A Grammar of the Multitude, 2004). (shrink)

Summary This article presents three hitherto unpublished letters by Frank Plumpton Ramsey on the foundations of mathematics with commentary. One of the letters was sent to Abraham Fraenkel and the other two letters to Heinrich Behmann. The transcription of the letters is preceded by an account that details the extent of Ramsey's known contacts with mathematical logicians on the Continent.

In the last two decades there has been renewed interest in visualization in logic and mathematics. Visualization is usually understood in different ways but for the purposes of this article I will take a rather broad conception of visualization to include both visualization by means of mental images as well as visualizations by means of computer generated images or images drawn on paper, e.g. diagrams etc. These different types of visualization can differ substantially but I am interested in offering a (...) characterization of visualization that is as broad as possible. The article describes and explains (1) the way in which visual thinking fell into disrepute, (2) the renaissance of visual thinking in mathematics over recent decades, (3) the ways in which visual thinking has been rehabilitated in epistemology of mathematics and logic. (shrink)

Recently discovered correspondence from Oskar Becker to Hermann Weyl sheds new light on Weyl's engagement with Husserlian transcendental phenomenology in 1918-1927. Here the last two of these letters, dated July and August, 1926, dealing with issues in the philosophy of mathematics are presented, together with background and a detailed commentary. The letters provide an instructive context for re-assessing the connection between intuitionism and phenomenology in Weyl's foundational thought, and for understanding Weyl's term ‘symbolic construction’ as marking his own considered position (...) in the foundational controversy of the 1920s. In addition, they reveal Weyl's hitherto unknown objections to Becker's detailed attempt (Mathematische Existenz, 1927) to ground the transfinite phenomenologically. (shrink)

At the same time, the book is a contribution to recent philosophical debates, in particular on the prospects for a successful nominalist reconstruction of .

The aim of the paper is to discuss the influence exercised by Russell's thought inGöttingen in the period leading to the formulation of Hilbert's program in theearly twenties. I show that after a period of intense foundational work, culminatingwith the departure from Göttingen of Zermelo and Grelling in 1910 we witnessa reemergence of interest in foundations of mathematics towards the end of 1914. Itis this second period of foundational work that is my specific interest. Through theuse of unpublished archival sources (...) I will describe how Hilbert, Behmann, and Bernays,among others, were influenced by and reacted to the technical and philosophical thesespresented in Principia Mathematica. I also argue that there are some elements of continuity between Russell's approach and Hilbert's program as it was presented inthe early twenties. (shrink)

It is reported that in reply to John Wisdom’s request in 1944 to provide a dictionary entry describing his philosophy, Wittgenstein wrote only one sentence: “He has concerned himself principally with questions about the foundations of mathematics”. However, an understanding of his philosophy of mathematics has long been a desideratum. This was the case, in particular, for the period stretching from the Tractatus Logico-Philosophicus to the so-called transitional phase. Marion’s book represents a giant leap forward in this direction. In the (...) preface, Marion provides a diagnosis for why it has taken such a long time to obtain an accurate picture of Wittgenstein’s ideas in the foundations of mathematics. When the Remarks on the Foundations of Mathematics came out in 1956 the reception by specialists, such as Kreisel, was negative. This led many Wittgenstein scholars to set aside the issues in the foundations of mathematics and go on with the business of the day, focusing on Wittgenstein’s philosophy of language and psychology. However, these early negative appraisals were severely limited by two facts. First of all, the nature of the Remarks was a hindrance to an understanding of what Wittgenstein was up to. The book was edited by Wittgenstein’s literary executors by collating together passages from several manuscripts dating from 1937 to 1944, cutting, however, many passages in between remarks. Second, many of the original manuscripts and typescripts of the transitional phase were not available and thus it was virtually impossible to obtain a balanced picture of Wittgenstein’s development. Finally, a stark contraposition between the early and late Wittgenstein did not encourage people to work seriously on the development of Wittgenstein’s position from the Tractatus through the transitional period to the later writings. (shrink)

We will discuss a mathematical proof found in Wittgenstein’s Nachlass, a constructive version of Euler’s proof of the infinity of prime numbers. Although it does not amount to much, this proof allows us to see that Wittgenstein had at least some mathematical skills. At the very last, the proof shows that Wittgenstein was concerned with mathematical practice and it also gives further evidence in support of the claim that, after all, he held a constructivist stance, at least during the transitional (...) period of his thought (1929-33). (shrink)

In this paper, I propose the idea that the French mathematician Michel Chasles developed a foundational programme for geometry in the period 1827–1837. The basic concept behind the programme was to show that projective geometry is the foundation of the whole of geometry. In particular, the metric properties can be reduced to specific graphic properties. In the attempt to prove the validity of his conception, Chasles made fundamental contributions to the theory of polarity and also understood that a satisfactory development (...) of projective geometry has to overcome the specificity of that theory itself. In this perspective, he developed his ideas on duality and homography, showing their dependence on the concept that he was going to pose at the basis of geometry: the anharmonic ratio, today called cross-ratio. The conceptual itinerary that starts from Chasles’ studies on polarity and ends with his results based on the cross-ratio represents the itinerary of his foundational programme. I will follow this complex and interesting line of thought, place it in the context of what was then called descriptive geometry, and outline a connection with the later great results of Von Staudt, Cayley, and Klein. (shrink)

Quine e Tarski sul nominalismo - Quine and Tarski were kindred philosophical spirits. They rejected the analytic/synthetic distinction and shared a strong distaste for modal notions. Moreover, they both had nominalistic leanings. In this article I trace the nominalistic engagement of these two giants of contemporary analytic philosophy. Using recently discovered material from the Quine archive at Harvard, I begin by describing Quine’s engagement with nominalism up to 1940. I then summarize the impact of the 1940-1941 discussions on nominalism between (...) Carnap, Quine and Tarski and mention their influence on Goodman. The third part of the article deals with Quine’s allegiance to nominalism and his subsequent reluctant acceptance of Platonism. The fourth part focuses on Tarski’s defence of nominalism in Amersfoort in 1953. In the conclusion, Quine’s and Tarski’s approaches to nominalism are compared with some contemporary nominalistic programs. (shrink)

I. Hardly any other mythical creature has enjoyed the ubiquity of the clown: “There are few other myths such as this one about which it can be affirmed without any doubt that they involve the most ancient modes of human expression,” wrote Paul Radin in his famous essay on the figure of the clown among primitive American Indians. “There are few myths which have retained their original characters with so few changes.” The character of the Fool, the Jester, the Joker (...) appears in a clearly identifiable fashion in the most primitive as well as the most highly cultivated groups: among the Greeks as well as the Chinese; in Japanese civilization as well as the Semite world. The medieval juggler is the embodiment of this figure of many aspects who perdures in the theater, especially puppet theater, and in the guise of the circus clown. (shrink)

Si tratta di una raccolta di saggi scritti tra il '65 e il '69 e pubblicati una prima volta nel 1971. Rivisti e parzialmente aggiornati vengono riproposti con la vecchia introduzione nella quale l'autore polemizza con le critiche della ragione scientifica che caratterizzarono alcuni aspetti della cultura del '68. Il primo saggio è dedicato al declino della astrologia all'inizio dell'età moderna; sottolineando il divario tra la concezione magica che caratterizza la visione astrologica del mondo e le posizione di quella che (...) sarebbe diventata la scienza, Paolo Rossi rivaluta gli scritti contro l'astrologia di Pico della Mirandola. Dopo un breve scritto su Leonardo, l'autore analizza i rapporti tra pensiero baconiano e cristianesimo. Segue un saggio che riesamina alcuni problemi di interpretazione delle lettere di Galileo a Piero Dini e a Cristina di Lorena. Nei capitoli successivi vengono affrontate varie questioni quali, il passaggio dalla concezione aristotelica a quella moderna, il peso di alcune ipotesi astronomiche e in particolare la crisi dell'antropocentrismo. Gli ultimi due capitoli dedicati a dispute di linguistica e di geologia sono particolarmente interessanti; infatti queste due discipline occupano una posizione anomala nella cultura del seicento. Se a quel tempo esistevano matematici, astronomi, medici, non si può certo parlare di linguisti o geologi propriamente detti, questo fatto rende particolarmente difficile e interessante ricostruire lo sviluppo di concetti appartenenti a branche della scienza che non erano ancora consce della loro autonomia. (shrink)

The interpretation of Parmenides’ Περί Φύσεως is a fascinating topic to which philosophers, historians of philosophy and scientists have dedicated many studies along the history of Western thought. The aim of this paper is to present the reading of Parmenides’s work offered by Federigo Enriques. It is based on several original theses: (1) Parmenides was the discoverer of abstract geometry; (2) his critics was addressed against the Pythagoreans rather than against Heraclitus; (3) Parmenides discovered and applied the contradiction and the (...) third excluded principles in the context of his research on foundation of geometry; (4) Parmenides’s metaphysical and physical conceptions have their bases in his speculation on geometry; (5) Parmenides used the principle of sufficient reason. Enriques’s reading is worth being expounded and discussed within the historical, philosophical and scientific context in which it is inserted. Since Enriques’s ideas are not widely known and discussed, my research has the purpose to fill this gap. The article also aims to provide elements to illustrate the discussion on Parmenides in the first half of the last century. (shrink)

Was liberalism really an outdated ideology in post-1945 European political systems, as claimed by some scholars? The great success of socialism on one side and various forms of Christian Democracy on the other could make that claim appear reasonable. In fact a closer view shows how postwar constitutions in some countries presented once again fundamental liberal values, reformulated in different words. One of the roots of that difference is the gap between the Anglo-Saxon approach to liberalism, which takes into account (...) history and communities, and the continental tradition of liberalism, linked to an abstract philosophical approach and highly dependent on long-standing struggles between Churches and the body politic. (shrink)

Christian Democracy is an evolving concept. This chapter discusses the long journey of the term in political debate. Considered at the beginning unacceptable by the Popes, as an offspring of the ‘liberty of critics’ toward authority, it was later accepted step by step as a mean of inserting Christians into the frame of modern constitutionalism. After a period in which the contrast with the liberal view was still retained, catholic political thought turned to a positive approach towards Western constitutionalism. After (...) 1945 political parties labelled as ‘Christian Democratic’ were protagonists of the rebuilding of some European countries and the Popes recognized that presence as positive and coherent with its orthodoxy. In a sense Christian Democracy became the new way of dealing with the problem of church and state relations. Matters were more complicated from the viewpoint of the Reformed Churches. In the more recent cases one may also find a ‘social doctrine’ even if it is quite different from the official Catholic attitude of the Vatican authorities towards social problems. (shrink)