Our understanding of how the human brain performs mathematical calculations is far from complete, but in recent years there have been many exciting breakthroughs by scientists all over the world. Now, in The Number Sense, Stanislas Dehaene offers a fascinating look at this recent research, in an enlightening exploration of the mathematical mind. Dehaene begins with the eye-opening discovery that animals--including rats, pigeons, raccoons, and chimpanzees--can perform simple mathematical calculations, and that human infants also have a rudimentary number sense. Dehaene suggests that this rudimentary number sense is as basic to the way the brain understands the world as our perception of color or of objects in space, and, like these other abilities, our number sense is wired into the brain. These are but a few of the wealth of fascinating observations contained here. We also discover, for example, that because Chinese names for numbers are so short, Chinese people can remember up to nine or ten digits at a time--English-speaking people can only remember seven. The book also explores the unique abilities of idiot savants and mathematical geniuses, and we meet people whose minute brain lesions render their mathematical ability useless. This new and completely updated edition includes all of the most recent scientific data on how numbers are encoded by single neurons, and which brain areas activate when we perform calculations. Perhaps most important, The NumberSense reaches many provocative conclusions that will intrigue anyone interested in learning, mathematics, or the mind.
"A delight."
--Ian Stewart, New Scientist
"Read The Number Sense for its rich insights into matters as varying as the cuneiform depiction of numbers, why Jean Piaget's theory of stages in infant learning is wrong, and to discover the brain regions involved in the number sense."
--The New York Times Book Review
"Dehaene weaves the latest technical research into a remarkably lucid and engrossing investigation. Even readers normally indifferent to mathematics will find themselves marveling at the wonder of minds making numbers."
--Booklist

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.

Category theory is a branch of abstract algebra with incredibly diverse applications. This text and reference book is aimed not only at mathematicians, but also researchers and students of computer science, logic, linguistics, cognitive science, philosophy, and any of the other fields in which the ideas are being applied. Containing clear definitions of the essential concepts, illuminated with numerous accessible examples, and providing full proofs of all important propositions and theorems, this book aims to make the basic ideas, theorems, and methods of category theory understandable to this broad readership. Although assuming few mathematical pre-requisites, the standard of mathematical rigour is not compromised. The material covered includes the standard core of categories; functors; natural transformations; equivalence; limits and colimits; functor categories; representables; Yoneda's lemma; adjoints; monads. An extra topic of cartesian closed categories and the lambda-calculus is also provided - a must for computer scientists, logicians and linguists! This Second Edition contains numerous revisions to the original text, including expanding the exposition, revising and elaborating the proofs, providing additional diagrams, correcting typographical errors and, finally, adding an entirely new section on monoidal categories. Nearly a hundred new exercises have also been added, many with solutions, to make the book more useful as a course text and for self-study.

Not all scientific explanations work by describing causal connections between events or the world's overall causal structure. Some mathematical proofs explain why the theorems being proved hold. In this book, Marc Lange proposes philosophical accounts of many kinds of non-causal explanations in science and mathematics. These topics have been unjustly neglected in the philosophy of science and mathematics. One important kind of non-causal scientific explanation is termed explanation by constraint. These explanations work by providing information about what makes certain facts especially inevitable - more necessary than the ordinary laws of nature connecting causes to their effects. Facts explained in this way transcend the hurly-burly of cause and effect. Many physicists have regarded the laws of kinematics, the great conservation laws, the coordinate transformations, and the parallelogram of forces as having explanations by constraint. This book presents an original account of explanations by constraint, concentrating on a variety of examples from classical physics and special relativity. This book also offers original accounts of several other varieties of non-causal scientific explanation. Dimensional explanations work by showing how some law of nature arises merely from the dimensional relations among the quantities involved. Really statistical explanations include explanations that appeal to regression toward the mean and other canonical manifestations of chance. Lange provides an original account of what makes certain mathematical proofs but not others explain what they prove. Mathematical explanation connects to a host of other important mathematical ideas, including coincidences in mathematics, the significance of giving multiple proofs of the same result, and natural properties in mathematics. Introducing many examples drawn from actual science and mathematics, with extended discussions of examples from Lagrange, Desargues, Thomson, Sylvester, Maxwell, Rayleigh, Einstein, and Feynman, Because Without Cause's proposals and examples should set the agenda for future work on non-causal explanation.

The mysterious beauty, harmony, and consistency of mathematics once caused philosopher Hilary Putnam to term its existence a "miracle." Now, advances in the understanding of physics suggest that the foundations of mathematics are encompassed by the laws of nature, an idea that sheds new light on both mathematics and physics.

The philosophical relationship between mathematics and the natural sciences is the subject of "Converging Realities," the latest work by one of the leading thinkers on the subject. Based on a simple but powerful idea, it shows that the axioms needed for the mathematics used in physics can also generate practically every field of contemporary pure mathematics. It also provides a foundation for current investigations in string theory and other areas of physics.

This approach to the nature of mathematics is not really new, but it became overshadowed by formalism near the end of nineteenth century. The debate turned eventually into an exclusive dialogue between mathematicians and philosophers, as if physics and nature did not exist. This unsatisfactory situation was enforced by the uncertain standing of physical reality in quantum mechanics.

The recent advances in the interpretation of quantum mechanics (as described in "Quantum Philosophy," also by Omnes) have now reconciled the foundations of physics with objectivity and common sense. In Converging Realities, Roland Omnes is among the first scholars to consider the connection of natural laws with mathematics."

Why bother to praise mathematics when you claim, as Alain Badiou does, that philosophy is first and foremost a metaphysics of happiness, or else it s not worth an hour of trouble? What possible relationship can there be between mathematics and happiness? That is precisely the issue at stake in this dialogue, which serves as a very accessible introduction to what mathematics is and an exploration of the crucial influence it has always exerted on the greatest philosophers. Far from the thankless, pointless exercises they are often thought to be, mathematics and logic are indispensable guides to ridding ourselves of dominant opinions and making possible an access to truths, or to a human experience of the utmost value. That is why mathematics may well be the shortest path to the true life, which, when it exists, is characterized by an incomparable happiness.

Building on the seminal work of Kit Fine in the 1980s, Leon Horsten here develops a new theory of arbitrary entities. He connects this theory to issues and debates in metaphysics, logic, and contemporary philosophy of mathematics, investigating the relation between specific and arbitrary objects and between specific and arbitrary systems of objects. His book shows how this innovative theory is highly applicable to problems in the philosophy of arithmetic, and explores in particular how arbitrary objects can engage with the nineteenth-century concept of variable mathematical quantities, how they are relevant for debates around mathematical structuralism, and how they can help our understanding of the concept of random variables in statistics. This fully worked through theory will open up new avenues within philosophy of mathematics, bringing in the work of other philosophers such as Saul Kripke, and providing new insights into the development of the foundations of mathematics from the eighteenth century to the present day.

A guide to the practical art of plausible reasoning, this book has relevance in every field of intellectual activity. Professor Polya, a world-famous mathematician from Stanford University, uses mathematics to show how hunches and guesses play an important part in even the most rigorously deductive science. He explains how solutions to problems can be guessed at; good guessing is often more important than rigorous deduction in finding correct solutions. Vol. I, on Induction and Analogy in Mathematics, covers a wide variety of mathematical problems, revealing the trains of thought that lead to solutions, pointing out false bypaths, discussing techniques of searching for proofs. Problems and examples challenge curiosity, judgment, and power of invention.

The traditional debate among philosophers of mathematics is whether there is an external mathematical reality, something out there to be discovered, or whether mathematics is the product of the human mind. This provocative book, now available in a revised and expanded paperback edition, goes beyond foundationalist questions to offer what has been called a "postmodern" assessment of the philosophy of mathematics--one that addresses issues of theoretical importance in terms of mathematical experience. By bringing together essays of leading philosophers, mathematicians, logicians, and computer scientists, Thomas Tymoczko reveals an evolving effort to account for the nature of mathematics in relation to other human activities. These accounts include such topics as the history of mathematics as a field of study, predictions about how computers will influence the future organization of mathematics, and what processes a proof undergoes before it reaches publishable form.

This expanded edition now contains essays by Penelope Maddy, Michael D. Resnik, and William P. Thurston that address the nature of mathematical proofs. The editor has provided a new afterword and a supplemental bibliography of recent work.

"Casti Tours offers the most spectacular vistas of modern applied mathematics."— Nature

Mathematical modeling is about rules—the rules of reality. Reality Rules explores the syntax and semantics of the language in which these rules are written, the language of mathematics. Characterized by the clarity and vision typical of the author's previous books, Reality Rules is a window onto the competing dialects of this language—in the form of mathematical models of real-world phenomena—that researchers use today to frame their views of reality.

Moving from the irreducible basics of modeling to the upper reaches of scientific and philosophical speculation, Volumes 1 and 2, The Fundamentals and The Frontier, are ideal complements, equally matched in difficulty, yet unique in their coverage of issues central to the contemporary modeling of complex systems.

Engagingly written and handsomely illustrated, Reality Rules is a fascinating journey into the conceptual underpinnings of reality itself, one that examines the major themes in dynamical system theory and modeling and the issues related to mathematical models in the broader contexts of science and philosophy. Far-reaching and far-sighted, Reality Rules is destined to shape the insight and work of students, researchers, and scholars in mathematics, science, and the social sciences for generations to come.

Of related interest . . .

ALTERNATE REALITIES

Mathematical Models of Nature and Man

John L. Casti

A thoroughly modern account of the theory and practice of mathematical modeling with a treatment focusing on system-theoretic concepts such as complexity, self-organization, adaptation, bifurcation, resilience, surprise and uncertainty, and the mathematical structures needed to employ these in a formal system.

1989 0-471-61842-X 493pp.

How did we make reliable predictions before Pascal and Fermat's discovery of the mathematics of probability in 1654? What methods in law, science, commerce, philosophy, and logic helped us to get at the truth in cases where certainty was not attainable? In The Science of Conjecture, James Franklin examines how judges, witch inquisitors, and juries evaluated evidence; how scientists weighed reasons for and against scientific theories; and how merchants counted shipwrecks to determine insurance rates. The Science of Conjecture provides a history of rational methods of dealing with uncertainty and explores the coming to consciousness of the human understanding of risk.

In Cognition, Content, and the A Priori, Robert Hanna works out a unified contemporary Kantian theory of rational human cognition and knowledge. Along the way, he provides accounts of (i) intentionality and its contents, including non-conceptual content and conceptual content, (ii) sense perception and perceptual knowledge, including perceptual self-knowledge, (iii) the analytic-synthetic distinction, (iv) the nature of logic, and (v) a priori truth and knowledge in mathematics, logic, and philosophy. This book is specifically intended to reach out to two very different audiences: contemporary analytic philosophers of mind and knowledge on the one hand, and contemporary Kantian philosophers or Kant-scholars on the other. At the same time, it is also riding the crest of a wave of exciting and even revolutionary emerging new trends and new work in the philosophy of mind and epistemology, with a special concentration on the philosophy of perception. What is revolutionary in this new wave are its strong emphases on action, on cognitive phenomenology, on disjunctivist direct realism, on embodiment, and on sense perception as a primitive and proto-rational capacity for cognizing the world. Cognition, Content, and the A Priori makes a fundamental contribution to this philosophical revolution by giving it a specifically contemporary Kantian twist, and by pushing these new lines of investigation radically further.

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.

Dieser Band fuhrt 16 Aufsatze von Herbert Breger zusammen, die um Leibniz' Arbeiten zur Mathematik und Physik und ihre philosophischen Voraussetzungen kreisen. Drei interessante und ungewoehnliche Aspekte stehen hierbei im Vordergrund: Kontinuum, Analysis und Informales. Leibniz' Kontinuum und seine Analysis sind gerade wegen ihres Unterschieds zur heutigen Mathematik interessant. Anhand zahlreicher Beispiele wird ferner die Frage nach dem Verhaltnis zwischen der mathematischen Rationalitat und der Kunst gestellt und die nach den engen Beziehungen zwischen Mathematik und Philosophie bei Leibniz eroertert. Es wird gezeigt, dass der Leibniz zugeschriebene Brief zum Prinzip der kleinsten Wirkung, der Anlass zu einem Streit zwischen Maupertuis, Samuel Koenig und Voltaire wurde, eine Falschung war. Das Buch erscheint im Leibniz-Jahr 2016, in dem auch der X. Leibniz-Kongress stattfindet.

Dieses Buch bietet einen historisch orientierten Einstieg in die Algorithmik, also die Lehre von den Algorithmen, in Mathematik, Informatik und daruber hinaus. Besondere Merkmale und Zielsetzungen sind: Elementaritat und Anschaulichkeit, die Berucksichtigung der historischen Entwicklung, Motivation der Begriffe und Verfahren anhand konkreter, aussagekraftiger Beispiele unter Einbezug moderner Werkzeuge (Computeralgebrasysteme, Internet). Als Zusatzmedien werden computer- und internetspezifische Interaktions- und Visualisierungsmoeglichkeiten (kostenlos) zur Verfugung gestellt. Das Werk wendet sich an Studierende und Lehrende an Schulen und Hochschulen sowie an Nichtspezialisten, die an den Themen "Computer/Algorithmen/Programmierung" einschliesslich ihrer historischen und geisteswissenschaftlichen Dimension interessiert sind.

Die Gesammelten Abhandlungen von Ferdinand Georg Frobenius erscheinen in drei Banden. Band I enthalt in chronologischer Abfolge seine Veroeffentlichungen von 1870 bis 1880, Band II jene von 1880 bis 1896, und Band III die Artikel von 1896 bis 1917. Band II umfasst die Artikel Nr. 22 bis 52. R. Brauer: ...if the reader wants to get an idea about the importance of Frobenius work today, all he has to do is to look at books and papers on groups...

Die Gesammelten Abhandlungen von Ferdinand Georg Frobenius erscheinen in drei Banden. Band I enthalt in chronologischer Abfolge seine Veroeffentlichungen von 1870 bis 1880, Band II jene von 1880 bis 1896, und Band III die Artikel von 1896 bis 1917. Band III umfasst die Veroeffentlichungen Nr. 53 bis 107. R. Brauer: ...if the reader wants to get an idea about the importance of Frobenius work today, all he has to do is to look at books and papers on groups...

Aus dem Vorwort: "Die Ergebnisse, Methoden und Begriffe, die die mathematische Wissenschaft dem Forscher ISSAI SCHUR verdankt, haben ihre nachhaltige Wirkung bis in die Gegenwart hinein erwiesen und werden sie unverandert beibehalten. Immer wieder wird auf Unter suchungen von SCHUR zuruckgegriffen, werden Erkenntnisse von ihm benutzt oder fortgefuhrt und werden Vermutungen von ihm bestatigt... Die Besonderheit des mathematischen Schaffens von SCHUR hat einst MAX PLANCK, als Sekretar der physikalisch-mathematischen Klasse der Preussischen Akademie der Wissenschaften zu Berlin, gut gekennzeichnet. In seiner Erwiderung auf die Antrittsrede von SCHUR bei dessen Aufnahme als ordentliches Mitglied der Akademie am 29. Juni 1922 bezeugte er, dass SCHUR "wie nur wenige Mathematiker die grosse Abelsche Kunst ube, die Probleme richtig zu formulieren, passend umzuformen, geschickt zu teilen und dann einzeln zu bewaltigen"."Band III enthalt 28 von Issai Schur verfasste Artikel ab 1925 sowie u.a. Inhalte aus dem nicht veroeffentlichten Nachlass.

Originaltext und historischer und mathematischer Kommentar von Klaus VolkertDavid Hilberts "Festschrift" Grundlagen der Geometrie" aus dem Jahre 1899 wurde zu einem der einflussreichsten Texte der Mathematikgeschichte. Wie kein anderes Werk pragte es die Mathematik des 20. Jahrhunderts und ist auch heute noch von groesstem Interesse. Aus der Perspektive eines Mathematikhistorikers schildert der Herausgeber die Entwicklung einer Axiomatik der Geometrie, die spatestens mit Euklids "Elemente" (ca. 300 v. u. Z.) begann und erst durch Hilbert zu einem vollstandigen und handhabbaren System gefuhrt wurde. Nach einer ausfuhrlichen Erlauterung des Hilbertschen Textes wird seine Rezeption bis 1905 umfassend dargestellt und daran anschliessend viele der von ihm ausgehenden weiteren direkten und indirekten Entwicklungen skizziert. Die Faszination des Textes ist auch dem heutigen Leser direkt zuganglich, da Hilberts axiomatischer Ansatz ohne mengentheoretische Argumente oder formale Logik auskommt.

Der Band enthalt zum ersten Mal in deutscher Sprache grundlegende Themen der chinesischen und indischen Mathematik, die den Nahrboden fur spatere Fragestellungen bereiten. Die nicht zu uberschatzende Rolle, die islamische Gelehrte bei der Entwicklung der Algebra und der Verbreitung des Ziffernsystems gespielt haben, wird in exemplarischen Episoden veranschaulicht. Unterhaltsam wird geschildert, wie Fibonacci die orientalische Aufgabenkultur nach Italien bringt. Zahlreiche Beispiele demonstrieren das neue kaufmannische Rechnen, dessen Methoden sich in ganz Europa verbreiten. In Deutschland erwachst eine neue Generation von Rechenmeistern, die mit ihren erstmals im Druck verbreiteten Schriften eine ungeheure Popularisierung des Rechnens bewirken. UEberraschende Einblicke in die Historie bieten die Kapitel uber die Vermittlung mathematischen Wissens in Kloestern und Universitaten. Das Buch ist eine Fundgrube fur historisch Interessierte; zahlreiche Aufgaben bieten vergnuglichen Stoff fur Unterricht, Vorlesung und Selbststudium.

Die Ergebnisse, Methoden und Begriffe, die die mathematische Wissenschaft dem Forscher ISSAr SCHUR verdankt, haben ihre nachhaltige Wirkung bis in die Gegenwart hinein erwiesen und werden sie unverandert beibehalten. Immer wieder wird auf Unter- suchungen von SCHUR zuriickgegriffen, werden Erkenntnisse von ihm benutzt oder fortgefiihrt und werden Vermutungen von ihm bestatigt. Daher ist es sehr zu begriifien, dafi sich der Springer-Verlag bereit erklart hat, die wissenschaftlichen Veroffentlichungen von I. SCHUR als Gesammelte Abhandlungen herauszugeben. Die Besonderheit des mathematischen Schaffens von SCHUR hat einst MAX PLANCK, als Sekretar der physikalisch-mathematischen Klasse der Preufiischen Akademie der Wissenschaften zu Berlin, gut gekennzeichnet. In seiner Erwiderung auf die Antrittsrede von SCHUR bei dessen Aufnahme als ordentliches Mitglied der Akademie am 29. Juni 1922 bezeugte er, dafi SCHUR wie nur wenige Mathematiker die grofie Abelsche Kunst iibe, die Probleme richtig zu formulieren, passend umzuformen, geschickt zu teilen und dann einzeln zu bewaltigen. Zum Gedacntnis an I. SCHUR gab die Schriftleitung der Mathematischen Zeitschrift 1955 einen Gedenkband heraus, aus dessen Vorrede wir folgendes entnehmen (Mathe- matische Zeitschrift 63, 1955/56): Aus Anlafi der 80. Wiederkehr des Tages, an dem Schur in Mohilew am Dnjepr geboren wurde, vereinen sich Freunde und Schiiler, urn sein Andenken mit diesem Bande der Zeitschrift zu ehren, die er selbst begriindet hat.

Frege's Theorem collects eleven essays by Richard G Heck, Jr, one of the world's leading authorities on Frege's philosophy. The Theorem is the central contribution of Gottlob Frege's formal work on arithmetic. It tells us that the axioms of arithmetic can be derived, purely logically, from a single principle: the number of these things is the same as the number of those things just in case these can be matched up one-to-one with those. But that principle seems so utterly fundamental to thought about number that it might almost count as a definition of number. If so, Frege's Theorem shows that arithmetic follows, purely logically, from a near definition. As Crispin Wright was the first to make clear, that means that Frege's logicism, long thought dead, might yet be viable. Heck probes the philosophical significance of the Theorem, using it to launch and then guide a wide-ranging exploration of historical, philosophical, and technical issues in the philosophy of mathematics and logic, and of their connections with metaphysics, epistemology, the philosophy of language and mind, and even developmental psychology. The book begins with an overview that introduces the Theorem and the issues surrounding it, and explores how the essays that follow contribute to our understanding of those issues. There are also new postscripts to five of the essays, which discuss changes of mind, respond to published criticisms, and advance the discussion yet further.

Hilberts algebraische Arbeiten "UEber die Theorie der algebraischen Formen" und "UEber die vollen Invariantensysteme" haben einen umwalzenden Einfluss auf das algebraische Denken gehabt. Sie ragen in Methode und Bedeutung uber den Bereich der Invariantentheorie weit hinaus. Ihr wesentlicher Kern besteht in der Anwendung arithmetischer Methoden auf algebraische Probleme. Indem Hilbert den Invariantenkoerper als Spezialfall eines Funktionenkoerpers betrachtet, steht er am Wendepunkt einer historischen Entwicklung, woraus spater die allgemeine Theorie der abstrakten Koerper, Ringe und Moduln erwuchs.Der Band enthalt daruber hinaus eine von Arnold Schmidt verfasste UEbersicht uber Hilberts geometrische Untersuchungen.

A survey of recent developments both in the classical and modern fields of the theory. Contents include: The complex analytic structure of the space of closed Riemann surfaces; Complex analysis on noncompact Riemann domains; Proof of the Teichmuller-Ahlfors theorem; The conformal mapping of Riemann surfaces; On certain coefficients of univalent functions; Compact analytic surfaces; On differentiable mappings; Deformations of complex analytic manifolds. Originally published in 1960. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.