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.

From Brouwer To Hilbert: The Debate on the Foundations of Mathematics in the 1920s offers the first comprehensive introduction to the most exciting period in the foundation of mathematics in the twentieth century. The 1920s witnessed the seminal foundational work of Hilbert and Bernays in proof theory, Brouwer's refinement of intuitionistic mathematics, and Weyl's predicativist approach to the foundations of analysis. This impressive collection makes available the first English translations of twenty-five central articles by these important contributors and many others. (...) The articles have been translated for the first time from Dutch, French, and German, and the volume is divided into four sections devoted to (1) Brouwer, (2) Weyl, (3) Bernays and Hilbert, and (4) the emergence of intuitionistic logic. Each section opens with an introduction which provides the necessary historical and technical context for understanding the articles. Although most contemporary work in this field takes its start from the groundbreaking contributions of these major figures, a good, scholarly introduction to the area was not available until now. Unique and accessible, From Brouwer To Hilbert will serve as an ideal text for undergraduate and graduate courses in the philosophy of mathematics, and will also be an invaluable resource for philosophers, mathematicians, and interested non-specialists. (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)

Cantor’s theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one association between two sets. As is well known, all countable infinite sets have the same ‘size’ in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collectionAis properly included in a collectionBthen the ‘size’ ofAshould be less than the (...) ‘size’ ofB(part–whole principle). This second intuition was not developed mathematically in a satisfactory way until quite recently. In this article I begin by reviewing the contributions of some thinkers who argued in favor of the assignment of different sizes to infinite collections of natural numbers (Thabit ibn Qurra, Grosseteste, Maignan, Bolzano). Then, I review some recent mathematical developments that generalize the part–whole principle to infinite sets in a coherent fashion (Katz, Benci, Di Nasso, Forti). Finally, I show how these new developments are important for a proper evaluation of a number of positions in philosophy of mathematics which argue either for the inevitability of the Cantorian notion of infinite number (Gödel) or for the rational nature of the Cantorian generalization as opposed to that, based on the part–whole principle, envisaged by Bolzano (Kitcher). (shrink)

An Introduction to Proof Theory provides an accessible introduction to the theory of proofs, with details of proofs worked out and examples and exercises to aid the reader's understanding. It also serves as a companion to reading the original pathbreaking articles by Gerhard Gentzen. The first half covers topics in structural proof theory, including the Gödel-Gentzen translation of classical into intuitionistic logic, natural deduction and the normalization theorems, the sequent calculus, including cut-elimination and mid-sequent theorems, and various applications of these (...) results. The second half examines ordinal proof theory, specifically Gentzen's consistency proof for first-order Peano Arithmetic. The theory of ordinal notations and other elements of ordinal theory are developed from scratch, and no knowledge of set theory is presumed. The proof methods needed to establish proof-theoretic results, especially proof by induction, are introduced in stages throughout the text. Mancosu, Galvan, and Zach's introduction will provide a solid foundation for those looking to understand this central area of mathematical logic and the philosophy of mathematics. (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)

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

In previous work by Baaz and Iemhoff, a Gentzen calculus for intuitionistic logic with existence predicate is presented that satisfies partial cut elimination and Craig's interpolation property; it is also conjectured that interpolation fails for the implication-free fragment. In this paper an equivalent calculus is introduced that satisfies full cut elimination and allows a direct proof of interpolation via Maehara's lemma. In this way, it is possible to obtain much simpler interpolants and to better understand and overcome the failure of (...) interpolation for the implication-free fragment. (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)

Two mereological theories are presented based on a primitive apartness relation along with binary relations of mereological excess and weak excess, respectively. It is shown that both theories are acceptable from the standpoint of constructive reasoning while remaining faithful to the spirit of classical mereology. The two theories are then compared and assessed with regard to their extensional import.

The paper deals with the art of cross-examination or probe (peirastiké) wich is, according to Aristotle, a part of dialectic and aims at revealing falle pretensions to knowledge. A detailed discussion of the relevant passages from the Sophistical Refutations shows that "peirastic" argumenta were not easily distinguishablc forro the arguments used by the sophists against the scientists in order to undermine their reputation. It is also shows that the difficulty of telling dialectic from sophistry is a theme Aristotle inherits frorn (...) Plato's Sophist. (shrink)

In Sophistical Refutations 8 Aristotle claims that every sophistical refutation depends on a false belief which is implicitly held by the victim of the fallacy and can normally be elicited from him as an explicit additional premiss. In this case the fallacious argument will be turned into a valid one, albeit with a false premiss. The paper discusses the nature of the FVP and tries to discover how it works when it tacitly causes the false appearance of a fallacious argument.

We present a sequent calculus for extensional mereology. It extends the classical first-order sequent calculus with identity by rules of inference corresponding to well-known mereological axioms. Structural rules, including cut, are admissible.

The paper examines Posterior Analytics II 11, 94a20-36 and makes three points. (1) The confusing formula ‘given what things, is it necessary for this to be’ [τίνων ὄντων ἀνάγκη τοῦτ᾿ εἶναι] at a21-22 introduces material cause, not syllogistic necessity. (2) When biological material necessitation is the only causal factor, Aristotle is reluctant to formalize it in syllogistic terms, and this helps to explain why, in II 11, he turns to geometry in order to illustrate a kind of material cause that (...) can be expressed as the middle term of an explanatory syllogism. (3) If geometrical proof is viewed as a complex construction built on simpler constructions, it can in effect be described as a case of purely material constitution. (shrink)

Tarski, Carnap and Quine spent the academic year 1940?1941 together at Harvard. In their autobiographies, both Carnap and Quine highlight the importance of the conversations that took place among them during the year. These conversations centred around semantical issues related to the analytic/synthetic distinction and on the project of a finitist/nominalist construction of mathematics and science. Carnap's Nachlaß in Pittsburgh contains a set of detailed notes, amounting to more than 80 typescripted pages, taken by Carnap while these discussions were taking (...) place. In my article, I present a survey of these notes with special emphasis on Tarski's rejection of the analytic/synthetic distinction, the passage from typed languages to first-order languages, Tarski's finitism/nominalism, and the construction of a finitist language for mathematics and science. (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)

The list of katēgoriai presented at the start of Top. I 9 was traditionally interpreted as a version of the canonical Aristotelian list of categories, and as largely equivalent to the list we find in Categories 4. Accordingly, its first item, the ‘what it is’, was identified with the category of substance. This interpretation has been challenged by several scholars, all sharing the view that the ‘what it is’ in Top. I 9 cannot be substance, since it collects items belonging (...) to all Aristotelian categories (e. g. human being, colour, length). Rather, they say, it is a manner of predication – i. e. essential predication – and can only determine an ontologically miscellaneous class of items. Against this family of proposals, I argue afresh that the traditional interpretation is almost entirely correct. To this purpose, I take advantage of the distinction between kinds of predicate and kinds of predication. (shrink)

Art, Beauty and Imitation in Plato's Philosophy -- Art and Imitation in Aristotle -- Horace, Pseudo-Longinus and the Aesthetics of Literature in Hellenism -- Plotinus, Neo-Platonic and Christian Conception of Beauty -- The Middle Ages and Dante Alighieri -- The Heritage of Kantian Philosophy in Romanticism -- Moritz: Beyond the Concept of Imitation -- Theory of Poetry of Early German Romanticism -- Hegel: Art as a Form of the Absolute Spirit -- Schopenhauer: Art as Disinterestedness and Knowledge of Reality -- (...) Nietzsche: Knowledge and Art -- Symbolism and Aestheticism -- Benedetto Croce: Art and intuition -- Linguistics and Criticism -- Antonio Gramsci: The Role of Intellectuals in Culture -- Structuralism -- Martin Heidegger: The Work of Art and Truth -- Hans-Georg Gadamer: Poetry and Interpretation -- Critical Theory: A New Attitude towards Art and Society -- Perspectives of Post-Structuralism -- The Practice of Deconstruction -- Contemporary Schools and Traditions in Literary and Critical Theory -- Postmodern and the New Character of the Literary Work. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Beginning in the 20s of the last century, historical research into Eiximenis's life and writings has thrown into relief his contribution to the language and political ideas of the kingdoms and towns of the Catalan-Aragonese Crown. Of fundamental importance has been the work of medievalists from North America, and in particular that of Canadian scholars during the last decades of the twentieth century.More recently, a number of studies have (...) underlined the decisive contribution of Eiximenis – a representative of a European textual tradition among friars of the Minor Observance that extended from Oxford to Florence – to the establishment of a system of ethical values for the economic sphere and for trading practices that continued to be developed by members of the Franciscan Order throughout the high middle ages and early modern period.As a follow-up to these studies, I believe that a program of textual analysis and systematic investigation of Eximenis's work could provide further information about the range and significance of his political thinking, based as it was on applying, in typically Franciscan fashion, experimental observation to gain an understanding of the nature both of the societas and of the market, the latter being thought of as a defining expression of civic institutions in Western Europe. This same experimental approach can be found in Franciscan thinking from the catalan-aragonese realm not only in the work of Arnau de Vilanova and Ramon Llull, but also in that of Pere Tomas and Anselm Turmeda, two figures fundamental to the preparation and political engagement of the Friar of Girona.1. The concept of contractIn Eiximenian texts, the communitarian dimension is defined as one that is of necessity based on contract. It is expounded in the long opening section of the Dotzè del Crestià, where Eiximenis outlines the political value of the civitas, as well as setting out the elements that constitute civic life.Borrowing from Aristotle's discussion in the First Book of the Politics , the Franciscan friar strongly affirms that one of the principal reasons behind the civitas is the "necessitat de contractes," the need for contracts. Going far beyond the highly qualified analysis of money and coinage set out by the Greek philosopher, Eiximenis here maintains that the aims of the community is the profit that the citizenry realises by means of different forms of exchange, through buying and selling, through contracts, and through trading. Moreover, that the binding nature of trading-contracts is a matter of concern not of the city in the narrow sense, but of each and every kind of human social organization.Even before contract-law, or, better said, conventional or foral procedure, Eiximenis views the idea of the contract as an essential part of the ethical and organizational principles of living as a community, the essence of the civitas. And it is of great significance that when he is discussing the value and function of money, Eiximenis chooses to call the sorts of exchange that to-day we term economic or financial les civils conmutacions, or exchanges between citizens.These lexical hints allow us to capture the strength of meaning and value, the ethical perspective by which the Gerundense defines the inextricable union of the market and civic society. But it is not only an analytical perspective, significant as that is, that unites market and civic ethics; we are presented with a value judgment of contract and economic exchange as elements that determine the very idea of civilitas.The "bona civilitas," an expression that can be roughly translated as "good civic culture," is in fact founded on a concept of economic ethics that has as its distinguishing feature the logic of profitable exchange, of the distribution of that profit, of contractualization, and of the circulation of goods and money. It is characterised by redefining the anti-value of avaritia as a real.. (shrink)

Il volume, frutto della rielaborazione della tesi di dottorato, costituisce una retrospettiva sull'antropologia filosofica moderna attraverso due dei suoi massimi esponenti. Più specificamente mostra come la mitologia, la filosofia della storia e del linguaggio, la concezione vichiana dell'uomo abbiano avuto come costante punto di riferimento la psicologia d'impostazione cartesiana di Malebranche.

Reconociendo la insoslayable ambigüedad que afecta a la relación entre retórica y filosofía en la obra viquiana, este trabajo se preocupa por resaltar la continuidad entre el Vico retórico y el Vico filósofo, antes que por indagar las diferencias entre ambos. Semejante continuidad se percibe en la imaginación y en la persuasión. Ambos tópicos son tratados en profundidad en la Scienza Nuova, pero ponen de manifiesto consideraciones teóricas sobre la retórica que arrancan especialmente de las Institutiones Oratoriae.