Marcus Giaquinto tells the story of one of the great intellectual adventures of the modern era -- the attempt to find firm foundations for mathematics. From the late nineteenth century to the present day, this project has stimulated some of the most original and influential work in logic and philosophy.

Presenter l'analyse de base en suivant grosso modo l'ordre suivant laquelle elle a ete decouverte, voici le fil conducteur de cet ouvrage. Complete par un grand nombre de dessins, d'exemples et de contre-exemples, cet ouvrage est redige avec un veritable souci de pedagogie. Il est truffe de remarques historiques et de commentaires explicitant la motivation profonde des developpements exposes.

This distinctive anthology includes many of the most important recent contributions to the philosophy of mathematics. The featured papers are organized thematically, rather than chronologically, to provide the best overview of philosophical issues connected with mathematics and the development of mathematical knowledge. Coverage ranges from general topics in mathematical explanation and the concept of number, to specialized investigations of the ontology of mathematical entities and the nature of mathematical truth, models and methods of mathematical proof, intuitionistic mathematics, and philosophical foundations of set theory.

This volume explores the central problems and exposes intriguing new directions in the philosophy of mathematics, making it an essential teaching resource, reference work, and research guide.

The book complements "Philosophy of Logic: An Anthology" and "A Companion to Philosophical Logic, "also edited by Dale Jacquette (Blackwell 2001).

Charles Chihara gives a thorough critical exposition of modal realism, the philosophical doctrine that there exist many possible worlds of which the actual world--the universe in which we live--is just one. The striking success of possible-worlds semantics in modal logic has made this ontological doctrine attractive. Modal realists maintain that philosophers must accept the existence of possible worlds if they wish to have the benefit of using possible-worlds semantics to assess modal arguments and explain modal principles. Chihara challenges this claim, and argues instead for modality without worlds; he offers a new account of the role of interpretations or structures of the formal languages of logic.

In this book, Balaguer demonstrates that there are no good arguments for or against mathematical platonism. He does this by establishing that both platonism and anti-platonism are defensible views. Introducing a form of platonism ("full-blooded platonism") that solves all problems traditionally associated with the view, he proceeds to defend anti-platonism (in particular, mathematical fictionalism) against various attacks, most notably the Quine-Putnam indispensability attack. He concludes by arguing that it is not simply that we do not currently have any good argument for or against platonism, but that we could never have such an argument and, indeed, that there is no fact of the matter as to whether platonism is correct.

"Geschichte der Analysis" ist von einem internationalen Expertenteam geschrieben und stellt die gegenwartig umfassendste Darstellung der Herausbildung und Entwicklung dieser mathematischen Kerndisziplin dar. Der tiefgreifende begriffliche Wandel, den die Analysis im Laufe der Zeit durchgemacht hat, wird ebenso dargestellt, wie auch der Einfluss, den vor allem physikalische Probleme gehabt haben. Biographische und philosophische Hintergrunde werden ausgeleuchtet und ihre Relevanz fur die Theorieentwicklung gezeigt. Neben der eigentlichen Geschichte der Analysis bis ungefahr 1900 enthalt das Buch Spezialkapitel uber die Entwicklung der analytischen Mechanik im 18. Jahrhundert, Randwertprobleme der mathematischen Physik im 19. Jahrhundert, die Theorie der komplexen Funktionen, die Grundlagenkrise sowie historische Uberblicke uber die Variationsrechnung, Differentialgleichungen und Funktionalanalysis."

Naturalism in Mathematics investigates how the most fundamental assumptions of mathematics can be justified. One prevalent philosophical approach to the problem--realism--is examined and rejected in favour of another approach--naturalism--which attends more closely to practical considerations drawn from within mathematics itself. Penelope Maddy defines naturalism, explains the motivation for it, and shows how it can be successfully applied in set theory. Her clear, original discussion is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both disciplines.

This publication includes an unabridged and annotated translation of two works by Johann Heinrich Lambert (1728-1777) written in the 1760s: Vorlaufige Kenntnisse fur die, so die Quadratur und Rectification des Circuls suchen and Memoire sur quelques proprietes remarquables des quantites transcendentes circulaires et logarithmiques. The translations are accompanied by a contextualised study of each of these works and provide an overview of Lambert's contributions, showing both the background and the influence of his work. In addition, by adopting a biographical approach, it allows readers to better get to know the scientist himself. Lambert was a highly relevant scientist and polymath in his time, admired by the likes of Kant, who despite having made a wide variety of contributions to different branches of knowledge, later faded into an undeserved secondary place with respect to other scientists of the eighteenth century. In mathematics, in particular, he is famous for his research on non-Euclidean geometries, although he is likely best known for having been the first who proved the irrationality of pi. In his Memoire, he conducted one of the first studies on hyperbolic functions, offered a surprisingly rigorous proof of the irrationality of pi, established for the first time the modern distinction between algebraic and transcendental numbers, and based on such distinction, he conjectured the transcendence of pi and therefore the impossibility of squaring the circle.

L'opera, pubblicata, anche per questa edizione, come Supplemento alla rivista LETTERA MATEMATICA, e frutto del convegno 'Matematica e Cultura' organizzato a Venezia nel Marzo 1998. Il convegno, giunto nel Marzo 1998 alla sua seconda edizione, si propone come un ponte tra i diversi aspetti del sapere umano. Pur avendo come punto di riferimento la matematica, si rivolge a tutti coloro che hanno curiosita e interessi culturali anche e soprattutto al di fuori della matematica. Nel volume si parla pertanto di musica, cinema, di arte, di filosofia, di letteratura, di internet e mass-media.

Leibniz's dispute with Newton over the physico-mathematical theories expounded in the Principia Mathematica (1687) have long been identified as a crucial episode in the history of science. Dr Bertoloni Meli examines several hitherto unpublished manuscripts in Leibniz's own hand illustrating his first reading of and reaction to Newton's Principia. Six of the most important manuscripts are here edited for the first time. Contrary to Leibniz's own claims, this new evidence shows that he had studied Newton's masterpiece before publishing An Essay on the Causes of Celestial Motions. This article, representing his response to Newton, is included here in English translation. "Bertoloni's book provides a very detailed and deep analysis of Leibniz's calculus and dynamics by focusing on a consistent and important set of previously unknown manuscripts..." Niccolo Guicciardini, Universita de Bologna "...Equivalence and Priority is a major contribution to our understanding of the development of mathematical physics in the late seventeenth and early eighteenth century." Daniel Garber, University of Chigago

We live an information-soaked existence - information pours into our lives through television, radio, books, and of course, the Internet. Some say we suffer from 'infoglut'. But what is information? The concept of 'information' is a profound one, rooted in mathematics, central to whole branches of science, yet with implications on every aspect of our everyday lives: DNA provides the information to create us; we learn through the information fed to us; we relate to each other through information transfer - gossip, lectures, reading. Information is not only a mathematically powerful concept, but its critical role in society raises wider ethical issues: who owns information? Who controls its dissemination? Who has access to information? Luciano Floridi, a philosopher of information, cuts across many subjects, from a brief look at the mathematical roots of information - its definition and measurement in 'bits'- to its role in genetics (we are information), and its social meaning and value. He ends by considering the ethics of information, including issues of ownership, privacy, and accessibility; copyright and open source. For those unfamiliar with its precise meaning and wide applicability as a philosophical concept, 'information' may seem a bland or mundane topic. Those who have studied some science or philosophy or sociology will already be aware of its centrality and richness. But for all readers, whether from the humanities or sciences, Floridi gives a fascinating and inspirational introduction to this most fundamental of ideas. ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.

Hellman here presents a detailed interpretation of mathematics as the investigation of "structural possibilities," as opposed to absolute, Platonic objects. After treating the natural numbers and analysis, he extends the approach to set theory, where he demonstrates how to dispense with a fixed universe of sets. Finally, he addresses problems of application to the physical world.

People who learn to solve problems ‘on the job’ often have to do it differently from people who learn in theory. Practical knowledge and theoretical knowledge is different in some ways but similar in other ways - or else one would end up with wrong solutions to the problems. Mathematics is also like this. People who learn to calculate, for example, because they are involved in commerce frequently have a more practical way of doing mathematics than the way we are taught at school. This book is about the differences between what we call practical knowledge of mathematics - that is street mathematics - and mathematics learned in school, which is not learned in practice. The authors look at the differences between these two ways of solving mathematical problems and discuss their advantages and disadvantages. They also discuss ways of trying to put theory and practice together in mathematics teaching.

Die Schwierigkeit Mathematik zu lernen und zu lehren ist jedem bekannt, der einmal mit diesem Fach in Beruhrung gekommen ist. Begriffe wie "reelle oder komplexe Zahlen, Pi" sind zwar jedem gelaufig, aber nur wenige wissen, was sich wirklich dahinter verbirgt. Die Autoren dieses Bandes geben jedem, der mehr wissen will als nur die Hulle der Begriffe, eine meisterhafte Einfuhrung in die Magie der Mathematik und schlagen einzigartige Brucken fur Studenten.

Die Rezensenten der ersten beiden Auflagen uberschlugen sich."

In diesem Band soll eine zusammenfassende Darstellung der ausseren Ent- wicklung der Mathematik an den deutschen Universitaten gegeben wer- den. Dazu gehoert insbesondere eine moeglichst vollstandige und verlassliche Aufstellung des Personalbestandes der mathematischen Lehrstuhle und In- stitute. Eine solche Zusammenfassung hat bisher nicht existiert, was die mathematik-historische Forschung in mancher Hinsicht erschwert hat. Der Schwerpunkt der Darstellung liegt auf der institutionellen Seite; der Band enthalt zwar viele biographische Daten, aber keine eigentlichen Biogra- phien. Vor und bei der Erstellung dieses Buches waren eine Reihe grundsatzli- cher Fragen und zahlreiche Detailprobleme zu klaren. Als erstes musste der behandelte Zeitraum festgelegt werden. Hier schien die Periode von 1800 bis 1945 eine naheliegende Wahl zu sein. Vor den Universitatsreformen zu Beginn des 19. Jahrhunderts war die Mathematik an den Universitaten ganz unbedeutend; praktisch alle Professoren aus jener Zeit sind heute ver- gessen. Tatsachlich gilt dies auch noch fur die ersten Jahrzehnte des 19. Jahrhunderts, und ohne wesentlichen Verlust hatte man auch etwa 1830 beginnen koennen. Der gewahlte Zeitraum hat jedoch den Vorteil, dass der grosse Aufschwung der Universitaten allgemein und der Mathematik spe- ziell in der ersten Halfte des letzten Jahrhunderts deutlicher wird. Das Jahr 1945 stellt andererseits eine so einschneidende Zasur dar, dass es na- hezu zwingend war, die Darstellung hier abzuschliessen. Der enorme Ausbau des Universitatssystems ab den spaten funfziger Jahren musste einer weite- ren Publikation vorbehalten bleiben.

The Pythagorean idea that number is the key to understanding reality inspired philosophers in the fourth and fifth centuries to develop theories in physics and metaphysics using mathematical models. These theories were to become influential in medieval and early modern philosophy, yet until now, they have not received the serious attention they deserve. This book marks a breakthrough in our understanding of the subject by examining two themes in conjunction for the first time: the figure of Pythagoras as interpreted by the Neoplatonist philosophers of the period, and the use of mathematical ideas in physics and metaphysics.

Mit den hier abgedruckten klassischen biographischen Texten der Autoren Dirichlet, Kummer, Hensel, Frobenius und Hilbert werden dem Leser Einblicke in Leben und Werk herausragender Wissenschaftler erAffnet. AuAerdem erhAlt er authentische Informationen A1/4ber den Wissenschaftsbereich des 19. Jahrhunderts. Fotos und bisher unverAffentlichte Archivalien komplettieren diesen von H. Reichardt, dem langjAhrigen Direktor an den Mathematischen Instituten der Humboldt-UniversitAt Berlin sowie der Akademie der Wissenschaften, herausgegebenen Band.

Major shifts in the field of model theory in the twentieth century have seen the development of new tools, methods, and motivations for mathematicians and philosophers. In this book, John T. Baldwin places the revolution in its historical context from the ancient Greeks to the last century, argues for local rather than global foundations for mathematics, and provides philosophical viewpoints on the importance of modern model theory for both understanding and undertaking mathematical practice. The volume also addresses the impact of model theory on contemporary algebraic geometry, number theory, combinatorics, and differential equations. This comprehensive and detailed book will interest logicians and mathematicians as well as those working on the history and philosophy of mathematics.

What did it mean to be reasonable in the Age of Reason? Classical probabilists from Jakob Bernouli through Pierre Simon Laplace intended their theory as an answer to this question--as "nothing more at bottom than good sense reduced to a calculus," in Laplace's words. In terms that can be easily grasped by nonmathematicians, Lorraine Daston demonstrates how this view profoundly shaped the internal development of probability theory and defined its applications.

Includes several classic essays from the first edition, a representative selection of the most influential work of the past twenty years, a substantial introduction, and an extended bibliography. Originally published by Prentice-Hall in 1964.

Mathematical Explorations follows on from the author's previous book, Creative Mathematics, in the same series, and gives the reader experience in working on problems requiring a little more mathematical maturity. The author's main aim is to show that problems are often solved by using mathematics that is not obviously connected to the problem, and readers are encouraged to consider as wide a variety of mathematical ideas as possible. In each case, the emphasis is placed on the important underlying ideas rather than on the solutions for their own sake. To enhance understanding of how mathematical research is conducted, each problem has been chosen not for its mathematical importance, but because it provides a good illustration of how arguments can be developed. While the reader does not require a deep mathematical background to tackle these problems, they will find their mathematical understanding is enriched by attempting to solve them.

Die wissenschaftlichen Leistungen Richard Dedekinds (1831-1916), an dessen 150. Geburtstag dieser Gedenkband erinnern solI, sind jedem Mathematiker bekannt: Seine Begriindung der algebraischen Zahlen- theorie, die verbunden war mit der Ausarbeitung fundamentaler alge- braischer Begriffe, der Dedekindsche Schnitt, der die erste exakte Kon- sttuktion der reellen Zahlen und die Grundlegung der Analysis ermog- lichte, oder seine mit H. Weber entworfene Theorie der algebraischen Funktionenkorper gehoren zu den wichtigsten Fortschritten in der Mathematik des vorigen Jahrhunderts. 1m Zuge zunehmenden Interesses an geschichtlichen Entwicklungen und historischer Betrachtungsweise hat dariiber hinaus Dedekind in den letzten J ahren auch in besonderem Mage die Aufmerksamkeit der Mathematikhistoriker auf sich gezogen. Eine ganze Reihe von Arbeiten, die sich ausschlieglich oder wesentlich mit ihm und seinem Werk beschaftigen, sind in letzter Zeit erschienen. Dennoch ist unser Bild sowohl des Mathematikers als auch des Menschen Richard Dedekind bis heute unvollstandig und liickenhaft geblieben. Dies gilt vor allem rur den jungen Dedekind, der von 1854 bis 1871 fast nur kleinere Ge1egenheitsarbeiten publizierte, obwohl sich in diesen J ahren schon seine Hauptarbeitsgebiete und auch seine Auffassungen von der Mathematik und wie sie zu betreiben sei herausbildeten und festigten. Auch der bisher bekanntgewordene und publizierte Brief- wechsel stammt ganz iiberwiegend aus spaterer Zeit.

If numbers were objects, how could there be human knowledge of number? Numbers are not physical objects: must we conclude that we have a mysterious power of perceiving the abstract realm? Or should we instead conclude that numbers are fictions? This book argues that numbers are not objects: they are magnitude properties. Properties are not fictions and we certainly have scientific knowledge of them. Much is already known about magnitude properties such as inertial mass and electric charge, and much continues to be discovered. The book says the same is true of numbers. In the theory of magnitudes, the categorial distinction between quantity and individual is of central importance, for magnitudes are properties of quantities, not properties of individuals. Quantity entails divisibility, so the logic of quantity needs mereology, the a priori logic of part and whole. The three species of quantity are pluralities, continua and series, and the book presents three variants of mereology, one for each species of quantity. Given Euclid's axioms of equality, it is possible without the use of set theory to deduce the axioms of the natural, real and ordinal numbers from the respective mereologies of pluralities, continua and series. Knowledge and the Philosophy of Number carries out these deductions, arriving at a metaphysics of number that makes room for our a priori knowledge of mathematical reality.

Kaum jemals wird tin Werk eines Historikers einen so starken Reiz tiben und so tiefe Einblicke in das Wesen der Geschichte offnen wie Gedanken und Erinnerungen eines groBen Staatsmannes, welcher selbst ein langes Leben hindurch an fUhrender Stelle in die Geschicke der Welt eingegriffen hat und eine tiberlegene geistige Per- sonlichkeit mit der Kraft ktinstlerischer schriftstellerischer Gestaltung verbindet. Solchc Werke, schon fUr die politische Geschichte eine kostbare Seltenheit, sind fiir die Geschichte der exakten Wissenschaften bis- her wohl kaum geschrieben worden. Urn so notwendiger erschien es, als Felix Klein vor Jahresfrist starb, mit der Herausgabe seiner Vor- lesungen zur Geschichte der Mathematik und mathematischen Physik des 19. Jahrhunderts nicht zu zogern. Diese Vorlesungen sind die reife Frucht eines reichen Lebens in- mitten der wissenschaftlichen Ereignisse, der Ausdruck tiberlegener Weisheit und tiefen historischen Sinnes, einer hohen menschlichen Kultur und einer meisterhaften Gestaltungskraft; sie werden sicherlich auf aIle Mathematiker und Physiker und weit tiber diesen Kreis hin- aus eine groBe Wirkung austiben. In einer Zeit, wo der Blick der Menschen auch in der Wissenschaft allzusehr am Gegenwartigen hangt und das Einzelne in unnatiirlicher VergroBerung und iiber- triebener Bedeutung gegentiber dem Ganzen zu betrachten pflegt, kann das Kleinsche Werk vielen die Augen wieder offnen fUr die Zusammenhange und Entwicklungslinien unserer Wissenschaft im GroBen.