Search Results for "truth-through-proof-a-formalist-foundation-for-mathematics"

Truth Through Proof

Truth Through Proof

A Formalist Foundation for Mathematics

  • Author: Alan Weir
  • Publisher: OUP Oxford
  • ISBN: 9780199541492
  • Category: Mathematics
  • Page: 296
  • View: 1244
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Truth Through Proof defends an anti-platonist philosophy of mathematics derived from game formalism. Alan Weir aims to develop a more satisfactory successor to game formalism utilising a widely accepted, broadly neo-Fregean framework, in which the proposition expressed by an utterance is a function of both sense and background circumstance.

An Introduction to the Philosophy of Mathematics

An Introduction to the Philosophy of Mathematics

  • Author: Mark Colyvan
  • Publisher: Cambridge University Press
  • ISBN: 1107377005
  • Category: Science
  • Page: N.A
  • View: 4256
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This introduction to the philosophy of mathematics focuses on contemporary debates in an important and central area of philosophy. The reader is taken on a fascinating and entertaining journey through some intriguing mathematical and philosophical territory, including such topics as the realism/anti-realism debate in mathematics, mathematical explanation, the limits of mathematics, the significance of mathematical notation, inconsistent mathematics and the applications of mathematics. Each chapter has a number of discussion questions and recommended further reading from both the contemporary literature and older sources. Very little mathematical background is assumed and all of the mathematics encountered is clearly introduced and explained using a wide variety of examples. The book is suitable for an undergraduate course in philosophy of mathematics and, more widely, for anyone interested in philosophy and mathematics.

Principia Mathematica.

Principia Mathematica.

  • Author: Alfred North Whitehead,Bertrand Russell
  • Publisher: N.A
  • ISBN: N.A
  • Category: Logic, Symbolic and mathematical
  • Page: 167
  • View: 4517
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Social Constructivism as a Philosophy of Mathematics

Social Constructivism as a Philosophy of Mathematics

  • Author: Paul Ernest
  • Publisher: SUNY Press
  • ISBN: 1438402112
  • Category: Philosophy
  • Page: 315
  • View: 4514
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Proposing social constructivism as a novel philosophy of mathematics, this book is inspired by current work in sociology of knowledge and social studies of science. It extends the ideas of social constructivism to the philosophy of mathematics, developing a whole set of new notions. The outcome is a powerful critique of traditional absolutist conceptions of mathematics, as well as of the field of philosophy of mathematics itself. Proposed are a reconceptualization of the philosophy of mathematics and a new set of adequacy criteria. The book offers novel analyses of the important but under-recognized contributions of Wittgenstein and Lakatos to the philosophy of mathematics. Building on their ideas, it develops a theory of mathematical knowledge and its relation to the social context. It offers an original theory of mathematical knowledge based on the concept of conversation, and develops the rhetoric of mathematics to account for proof in mathematics. Another novel feature is the account of the social construction of subjective knowledge, which relates the learning of mathematics to philosophy of mathematics via the development of the individual mathematician. It concludes by considering the values of mathematics and its social responsibility.

Truth in Mathematics

Truth in Mathematics

  • Author: Harold G. Dales,Gianluigi Oliveri
  • Publisher: Oxford University Press
  • ISBN: 9780198514763
  • Category: Mathematics
  • Page: 376
  • View: 8100
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The nature of truth in mathematics is a problem which has exercised the minds of thinkers from at least the time of the ancient Greeks. This book is an overview of the most recent work undertaken in this subject, and is unique in being the result of interactions between researchers from both philosophy and mathematics. The articles are written by world leaders in their respective fields and are of interest to researchers in both disciplines.

Outlines of a Formalist Philosophy of Mathematics

Outlines of a Formalist Philosophy of Mathematics

  • Author: N.A
  • Publisher: Elsevier
  • ISBN: 0444533680
  • Category: Electronic books
  • Page: N.A
  • View: 4794
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New Directions in the Philosophy of Mathematics

New Directions in the Philosophy of Mathematics

An Anthology

  • Author: Thomas Tymoczko
  • Publisher: Princeton University Press
  • ISBN: 9780691034980
  • Category: Mathematics
  • Page: 436
  • View: 6735
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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.

Mechanism, Mentalism and Metamathematics

Mechanism, Mentalism and Metamathematics

An Essay on Finitism

  • Author: J. Webb
  • Publisher: Springer Science & Business Media
  • ISBN: 940157653X
  • Category: Philosophy
  • Page: 285
  • View: 4304
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This book grew out of a graduate student paper [261] in which I set down some criticisms of J. R. Lucas' attempt to refute mechanism by means of G6del's theorem. I had made several such abortive attempts myself and had become familiar with their pitfalls, and especially with the double edged nature of incompleteness arguments. My original idea was to model the refutation of mechanism on the almost universally accepted G6delian refutation of Hilbert's formalism, but I kept getting stuck on questions of mathematical philosophy which I found myself having to beg. A thorough study of the foundational works of Hilbert and Bernays finally convinced me that I had all too naively and uncritically bought this refutation of formalism. I did indeed discover points of surprisingly close contact between formalism and mechanism, but also that it was possible to under mine certain strong arguments against these positions precisely by invok ing G6del's and related work. I also began to realize that the Church Turing thesis itself is the principal bastion protecting mechanism, and that G6del's work was perhaps the best thing that ever happened to both mechanism and formalism. I pushed these lines of argument in my dis sertation with the patient help of my readers, Raymond Nelson and Howard Stein. I would especially like to thank the latter for many valuable criticisms of my dissertation as well as some helpful suggestions for reor ganizing it in the direction of the present book.

The Mathematical Experience, Study Edition

The Mathematical Experience, Study Edition

  • Author: Philip Davis,Reuben Hersh,Elena Anne Marchisotto
  • Publisher: Springer Science & Business Media
  • ISBN: 0817682953
  • Category: Mathematics
  • Page: 500
  • View: 2416
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Winner of the 1983 National Book Award! "...a perfectly marvelous book about the Queen of Sciences, from which one will get a real feeling for what mathematicians do and who they are. The exposition is clear and full of wit and humor..." - The New Yorker (1983 National Book Award edition) Mathematics has been a human activity for thousands of years. Yet only a few people from the vast population of users are professional mathematicians, who create, teach, foster, and apply it in a variety of situations. The authors of this book believe that it should be possible for these professional mathematicians to explain to non-professionals what they do, what they say they are doing, and why the world should support them at it. They also believe that mathematics should be taught to non-mathematics majors in such a way as to instill an appreciation of the power and beauty of mathematics. Many people from around the world have told the authors that they have done precisely that with the first edition and they have encouraged publication of this revised edition complete with exercises for helping students to demonstrate their understanding. This edition of the book should find a new generation of general readers and students who would like to know what mathematics is all about. It will prove invaluable as a course text for a general mathematics appreciation course, one in which the student can combine an appreciation for the esthetics with some satisfying and revealing applications. The text is ideal for 1) a GE course for Liberal Arts students 2) a Capstone course for perspective teachers 3) a writing course for mathematics teachers. A wealth of customizable online course materials for the book can be obtained from Elena Anne Marchisotto ([email protected]) upon request.

Computerdenken

Computerdenken

Die Debatte um Künstliche Intelligenz, Bewusstsein und die Gesetze der Physik

  • Author: Roger Penrose
  • Publisher: Spektrum Akademischer Verlag
  • ISBN: 9783827413321
  • Category: Science
  • Page: 454
  • View: 6208
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In seinem Klassiker erläutert der international führende Mathematiker und Physiker, Sir Roger Penrose, seine These, dass die geistigen Fähigkeiten des menschlichen Gehirns nicht durch Berechnungen von Elektronengehirnen erreicht werden können - und provozierte eine neue KI-Debatte. ...des Kaisers neue Kleider - steht auf dem Buchumschlag. Der renommierte englische Physiker Penrose will damit sichtbar machen, daß die Vertreter der Künstlichen Intelligenz (KI) nackt dastehen. Mit einem 400 Seiten langen Exkurs versucht er, ihre Behauptung zu widerlegen, daß Maschinen ebenso intelligent sein können wie Menschen. bild der wissenschaft Roger Penrose (...) gelang das Kunststück, mit dem formelgespickten Wälzer "The Emperors's New Mind" (auf deutsch jetzt unter dem geistlosen Titel "Computerdenken" erschienen) auf den US-Bestsellerlisten zu landen, ungeachtet aller Quanten-Ket-Vektoren und Einsteinscher Krüümungstensoren, mit denen der Autor seine Leser plagt. DER SPIEGEL Das erklärte Ziel dieses Buches ist, den Standpunkt einiger KI-Enthusiasten zu widerlegen, daß Computer irgendwann all das können, was menschliche Gehirne können - und sogar mehr. Aber der Leser merkt bald, dass Pnerose vor allem das Ziel verfolgt, einen Wegzur großen Synthese von klassischer Physik, Quantenphysik und Neurowissenschaften aufzuzeigen. John Horgan in Scientific American Wer "Computerdenken" liest (oder durcharbeitet), sollte nicht auf Antwort hoffen, darf aber neue Sichtwiesen und überraschende Interpretationen erwarten. Ein nahrhaftes Geschenk für naturwissenschaftlich Interessierte. Die Zeit Trotz des mathematichen Themas wurde The Emperor's New Mind prompt ein Bestseller und sein Autor zum bestgehaßten Mann der KI-Szene (...) Als Anfang der neunziger Jahre in England die Fördermittel für KI-Projekte nicht mehr so reichlich flossen, orteten manche eine KI-feindliche Stimmung in der Öffentlichkeit, die Penrose verschuldet habe. Die Zeit

Problem-Solving and Selected Topics in Euclidean Geometry

Problem-Solving and Selected Topics in Euclidean Geometry

In the Spirit of the Mathematical Olympiads

  • Author: Sotirios E. Louridas,Michael Th. Rassias
  • Publisher: Springer Science & Business Media
  • ISBN: 1461472733
  • Category: Mathematics
  • Page: 235
  • View: 705
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"Problem-Solving and Selected Topics in Euclidean Geometry: in the Spirit of the Mathematical Olympiads" contains theorems which are of particular value for the solution of geometrical problems. Emphasis is given in the discussion of a variety of methods, which play a significant role for the solution of problems in Euclidean Geometry. Before the complete solution of every problem, a key idea is presented so that the reader will be able to provide the solution. Applications of the basic geometrical methods which include analysis, synthesis, construction and proof are given. Selected problems which have been given in mathematical olympiads or proposed in short lists in IMO's are discussed. In addition, a number of problems proposed by leading mathematicians in the subject are included here. The book also contains new problems with their solutions. The scope of the publication of the present book is to teach mathematical thinking through Geometry and to provide inspiration for both students and teachers to formulate "positive" conjectures and provide solutions.

For the Learning of Mathematics

For the Learning of Mathematics

  • Author: N.A
  • Publisher: N.A
  • ISBN: N.A
  • Category: Mathematics
  • Page: N.A
  • View: 2099
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Foundations for New Economic Thinking

Foundations for New Economic Thinking

A Collection of Essays

  • Author: S. Dow
  • Publisher: Springer
  • ISBN: 1137000724
  • Category: Political Science
  • Page: 263
  • View: 6543
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New economic thinking is in demand in the light of the recent economic crisis. This book equips the reader with a better understanding of current ways of thinking as well as an awareness of other possibilities, providing the foundations for debate in theory and methodology alongside practical implications for policy.

Constantin Carathéodory

Constantin Carathéodory

Mathematics and Politics in Turbulent Times

  • Author: Maria Georgiadou
  • Publisher: Springer Science & Business Media
  • ISBN: 9783540203520
  • Category: Mathematics
  • Page: 651
  • View: 5541
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Constantin Carathéodory - Mathematics and Politics in Turbulent Times is the biography of a mathematician, born in Berlin in 1873, who became famous during his life time, but has hitherto been ignored by historians for half a century since his death in 1950, in Munich. In a thought-provoking approach, Maria Georgiadou devotes to Constantin Carathéodory all the attention such a personality deserves. With breathtaking detail and the appropriate scrutiny she elucidates his oeuvre, life and turbulent political and historical surroundings. A descendant of the the Greek élite of Constantinople, Carathéodory graduated from the military school of Brussels, became engineer at the Assiout dam in Egypt and finally dedicated a life of effort to mathematics and education. He studied and embarked on an international academic career, haunted by wars, catastrophes and personal tragedies. Over the last years of his life, he stayed in Munich despite World War II, an ambiguous decision upon which the author sheds unprecedented light. Carathéodory's most significant mathematical contributions were to the calculus of variations, the theory of point set measure and the theory of functions of a real variable, pde's, also to complex function theory. The interdisciplinary nature of the text allows easy access for both scholars and readers with a general interest in mathematics, politics and history. The thoroughness of the author’s research and evaluations is certain to leave everyone impressed and more knowledgeable.

Metamathematics, Machines and Gödel's Proof

Metamathematics, Machines and Gödel's Proof

  • Author: N. Shankar
  • Publisher: Cambridge University Press
  • ISBN: 9780521585330
  • Category: Computers
  • Page: 202
  • View: 1270
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Describes the use of computer programs to check several proofs in the foundations of mathematics.

Die Grundlagen der Mathematik

Die Grundlagen der Mathematik

  • Author: David Hilbert
  • Publisher: Springer-Verlag
  • ISBN: 3663161021
  • Category: Mathematics
  • Page: 29
  • View: 4099
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Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.

Collected Works: Philosophy and foundations of mathematics, edited by A. Heyting

Collected Works: Philosophy and foundations of mathematics, edited by A. Heyting

  • Author: Luitzen Egbertus Jan Brouwer,Arend Heyting
  • Publisher: N.A
  • ISBN: N.A
  • Category: Mathematics
  • Page: 628
  • View: 7514
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Mathematics and the Roots of Postmodern Thought

Mathematics and the Roots of Postmodern Thought

  • Author: Vladimir Tasic
  • Publisher: Oxford University Press
  • ISBN: 9780195349955
  • Category: Mathematics
  • Page: 200
  • View: 6483
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This is a charming and insightful contribution to an understanding of the "Science Wars" between postmodernist humanism and science, driving toward a resolution of the mutual misunderstanding that has driven the controversy. It traces the root of postmodern theory to a debate on the foundations of mathematics early in the 20th century, then compares developments in mathematics to what took place in the arts and humanities, discussing issues as diverse as literary theory, arts, and artificial intelligence. This is a straightforward, easily understood presentation of what can be difficult theoretical concepts It demonstrates that a pattern of misreading mathematics can be seen both on the part of science and on the part of postmodern thinking. This is a humorous, playful yet deeply serious look at the intellectual foundations of mathematics for those in the humanities and the perfect critical introduction to the bases of modernism and postmodernism for those in the sciences.

Was ist Mathematik?

Was ist Mathematik?

  • Author: Richard Courant,Herbert Robbins
  • Publisher: Springer-Verlag
  • ISBN: 3662000539
  • Category: Mathematics
  • Page: N.A
  • View: 960
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47 brauchen nur den Nenner n so groß zu wählen, daß das Intervall [0, IJn] kleiner wird als das fragliche Intervall [A, B], dann muß mindestens einer der Brüche m/n innerhalb des Intervalls liegen. Also kann es kein noch so kleines Intervall auf der Achse geben, das von rationalen Punkten frei wäre. Es folgt weiterhin, daß es in jedem Intervall unendlich viele rationale Punkte geben muß; denn wenn es nur eine endliche Anzahl gäbe, so könnte das Intervall zwischen zwei beliebigen benachbarten Punkten keine rationalen Punkte enthalten, was, wie wir eben sahen, unmöglich ist. § 2. Inkommensurable Strecken, irrationale Zahlen und der Grenzwertbegriff 1. Einleitung Vergleicht man zwei Strecken a und b hinsichtlich ihrer Größe, so kann es vor kommen, daß a in b genau r-mal enthalten ist, wobei r eine ganze Zahl darstellt. In diesem Fall können wir das Maß der Strecke b durch das von a ausdrücken, indem wir sagen, daß die Länge von b das r-fache der Länge von a ist.

Formalism in Ai and Computer Science

Formalism in Ai and Computer Science

  • Author: Philip Leith
  • Publisher: Ellis Horwood Limited
  • ISBN: 9780133255492
  • Category: Computers
  • Page: 225
  • View: 856
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