Kurt Gödel and the Foundations of Mathematics

Horizons of Truth
Author: Matthias Baaz,Christos H. Papadimitriou,Hilary W. Putnam,Dana S. Scott,Charles L. Harper, Jr
Publisher: Cambridge University Press
ISBN: 1139498436
Category: Mathematics
Page: N.A
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This volume commemorates the life, work and foundational views of Kurt Gödel (1906–78), most famous for his hallmark works on the completeness of first-order logic, the incompleteness of number theory, and the consistency - with the other widely accepted axioms of set theory - of the axiom of choice and of the generalized continuum hypothesis. It explores current research, advances and ideas for future directions not only in the foundations of mathematics and logic, but also in the fields of computer science, artificial intelligence, physics, cosmology, philosophy, theology and the history of science. The discussion is supplemented by personal reflections from several scholars who knew Gödel personally, providing some interesting insights into his life. By putting his ideas and life's work into the context of current thinking and perceptions, this book will extend the impact of Gödel's fundamental work in mathematics, logic, philosophy and other disciplines for future generations of researchers.

Kurt Gödel

Essays for his Centennial
Author: Solomon Feferman,Charles Parsons,Stephen G. Simpson
Publisher: Cambridge University Press
ISBN: 1139487752
Category: Mathematics
Page: N.A
View: 4421
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Kurt Gödel (1906–1978) did groundbreaking work that transformed logic and other important aspects of our understanding of mathematics, especially his proof of the incompleteness of formalized arithmetic. This book on different aspects of his work and on subjects in which his ideas have contemporary resonance includes papers from a May 2006 symposium celebrating Gödel's centennial as well as papers from a 2004 symposium. Proof theory, set theory, philosophy of mathematics, and the editing of Gödel's writings are among the topics covered. Several chapters discuss his intellectual development and his relation to predecessors and contemporaries such as Hilbert, Carnap, and Herbrand. Others consider his views on justification in set theory in light of more recent work and contemporary echoes of his incompleteness theorems and the concept of constructible sets.

After Gödel

Platonism and Rationalism in Mathematics and Logic
Author: Richard L. Tieszen
Publisher: Oxford University Press
ISBN: 019960620X
Category: Mathematics
Page: 245
View: 5742
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Richard Tieszen analyzes, develops, and defends the writings of Kurt Gödel (1906-1978) on the philosophy and foundations of mathematics and logic. Gödel's relation to the work of Plato, Leibniz, Husserl, and Kant is examined, and a new type of platonic rationalism that requires rational intuition, called 'constituted platonism', is proposed.

Ist Gott ein Mathematiker?

Warum das Buch der Natur in der Sprache der Mathematik geschrieben ist
Author: Mario Livio
Publisher: C.H.Beck
ISBN: 3406616607
Category: Mathematics
Page: 366
View: 9153
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Von der Antike bis in unsere Tage haben sich Wissenschaftler und Philosophen darüber gewundert, dass eine so abstrakte Disziplin wie die Mathematik die Natur derart perfekt erklären kann. Sogar mehr als das: Mathematiker haben oft Voraussagen gemacht, etwa über die Existenz bestimmter Teilchen, die sich später als richtig erwiesen haben. Was ist es, das der Mathematik solch unglaubliche Macht verleiht? Oder, wie Einstein sich einst fragte: „Wie ist es möglich, dass die Mathematik, die doch ein von aller Erfahrung unabhängiges Produkt des menschlichen Denkens ist, auf die Gegenstände der Welt so vortrefflich passt?“ Werden mathematische Erkenntnisse also nicht erfunden, sondern entdeckt? Diese Fragestellung bildet den roten Faden, an dem der Astrophysiker Mario Livio mit großer Lebendigkeit durch die Geistesgeschichte des mathematischen und naturwissenschaftlichen Denkens führt. Zugleich wirft sein Buch einen frischen und unterhaltsamen Blick auf die Lebensgeschichte und die Erkenntnisse großer Denker wie Pythagoras, Platon, Newton und Einstein. „Eine lebendige und faszinierende Lektüre für ein breites Publikum“ – Nature

The Metaphysics of World Order

A Synthesis of Philosophy, Theology, and Politics
Author: Nicolas Laos
Publisher: Wipf and Stock Publishers
ISBN: 1498201024
Category: Philosophy
Page: 244
View: 3381
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In this book, Nicolas Laos studies the meaning of the terms "world" and "order," the moral dimensions of each world order model, and wider issues of meaning and interpretation generated by humanity's attempt to live in a meaningful world and to find the logos of the beings and things in the world. The aim of this book is to propose a unified theory of world order (i.e., a theory that combines philosophy, theology, and political theory). In this context, the author provides a thought-provoking (re)interpretation of classical philosophy (placing particular emphasis on Platonism), an in-depth inquiry into medieval philosophy and spirituality (placing particular emphasis on the cultural differences between the Greek East and the Latino-Frankish West), and an intellectually challenging review and evaluation of modern Western philosophy (including Descartes, Spinoza, Leibniz, Locke, Berkeley, Hume, Rousseau, Kant, Hegel, Kierkegaard, Husserl, and Heidegger) and of Nietzsche's and the postmodernists' revolt against modernity. He then elucidates the philosophical foundations and "pedigree" of each of the three basic political theories of modernity (i.e., Liberalism, Communism, and Fascism), and he studies the basic theoretical debates in International Relations, Geopolitics, and Noopolitics. Finally, Laos proposes a new, "fourth," political theory which he calls "metaphysical republicanism."

The Encyclopedia of Philosophy

Supplement
Author: Donald M. Borchert
Publisher: Macmillan Reference USA
ISBN: 9780028646299
Category: Philosophy
Page: 775
View: 1004
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The 357 articles address areas of thought that were overlooked by the original set or that have developed since the Encyclopedia's publication, and they do it in a precise manner that will allow all readers to fully understand every side of the topic.

Handbook of the History of Logic: The rise of modern logic : from Leibniz to Frege


Author: Dov M. Gabbay,John Hayden Woods
Publisher: N.A
ISBN: N.A
Category: Induction (Logic)
Page: 770
View: 2133
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With the publication of the present volume, the Handbook of the History of Logic turns its attention to the rise of modern logic. The period covered is 1685-1900, with this volume carving out the territory from Leibniz to Frege. What is striking about this period is the earliness and persistence of what could be called 'the mathematical turn in logic'. Virtually every working logician is aware that, after a centuries-long run, the logic that originated in antiquity came to be displaced by a new approach with a dominantly mathematical character. It is, however, a substantial error to suppose that the mathematization of logic was, in all essentials, Frege's accomplishment or, if not his alone, a development ensuing from the second half of the nineteenth century. The mathematical turn in logic, although given considerable torque by events of the nineteenth century, can with assurance be dated from the final quarter of the seventeenth century in the impressively prescient work of Leibniz. It is true that, in the three hundred year run-up to the Begriffsschrift, one does not see a smoothly continuous evolution of the mathematical turn, but the idea that logic is mathematics, albeit perhaps only the most general part of mathematics, is one that attracted some degree of support throughout the entire period in question. Still, as Alfred North Whitehead once noted, the relationship between mathematics and symbolic logic has been an "uneasy" one, as is the present-day association of mathematics with computing. Some of this unease has a philosophical texture. For example, those who equate mathematics and logic sometimes disagree about the directionality of the purported identity. Frege and Russell made themselves famous by insisting (though for different reasons) that logic was the senior partner. Indeed logicism is the view that mathematics can be re-expressed without relevant loss in a suitably framed symbolic logic. But for a number of thinkers who took an algebraic approach to logic, the dependency relation was reversed, with mathematics in some form emerging as the senior partner. This was the precursor of the modern view that, in its four main precincts (set theory, proof theory, model theory and recursion theory), logic is indeed a branch of pure mathematics. It would be a mistake to leave the impression that the mathematization of logic (or the logicization of mathematics) was the sole concern of the history of logic between 1665 and 1900. There are, in this long interval, aspects of the modern unfolding of logic that bear no stamp of the imperial designs of mathematicians, as the chapters on Kant and Hegcl make clear. Of the two, Hcgel's influence on logic is arguably the greater, serving as a spur to the unfolding of an idealist tradition in logic - a development that will be covered in a further volume, British Logic in the Nineteenth Century.

The Bulletin of Symbolic Logic


Author: N.A
Publisher: N.A
ISBN: N.A
Category: Logic, Symbolic and mathematical
Page: N.A
View: 6852
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Theories of types and proofs


Author: Masako Takahashi,Mitsuhiro Okada,Mariangiola Dezani-Ciancaglini,Nihon Sūgakkai
Publisher: N.A
ISBN: N.A
Category: Mathematics
Page: 295
View: 6272
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Grundzüge der theoretischen Logik


Author: David Hilbert,Wilhelm Ackermann
Publisher: Springer-Verlag
ISBN: 3662119323
Category: Mathematics
Page: 190
View: 620
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Die theoretische Logik, auch mathematische oder symbolische Logik genannt, ist eine Ausdehnung der formalen Methode der Mathematik auf das Gebiet der Logik. Sie wendet für die Logik eine ähnliche Formel sprache an, wie sie zum Ausdruck mathematischer Beziehungen schon seit langem gebräuchlich ist. In der Mathematik würde es heute als eine Utopie gelten, wollte man beim Aufbau einer mathematischen Disziplin sich nur der gewöhnlichen Sprache bedienen. Die großen Fortschritte, die in der Mathematik seit der Antike gemacht worden sind, sind zum wesentlichen Teil mit dadurch bedingt, daß es gelang, einen brauchbaren und leistungsfähigen Formalismus zu finden. - Was durch die Formel sprache in der Mathematik erreicht wird, das soll auch in der theoretischen Logik durch diese erzielt werden, nämlich eine exakte, wissenschaftliche Behandlung ihres Gegenstandes. Die logischen Sachverhalte, die zwischen Urteilen, Begriffen usw. bestehen, finden ihre Darstellung durch Formeln, deren Interpretation frei ist von den Unklarheiten, die beim sprachlichen Ausdruck leicht auftreten können. Der Übergang zu logischen Folgerungen, wie er durch das Schließen geschieht, wird in seine letzten Elemente zerlegt und erscheint als formale Umgestaltung der Ausgangsformeln nach gewissen Regeln, die den Rechenregeln in der Algebra analog sind; das logische Denken findet sein Abbild in einem Logikkalkül. Dieser Kalkül macht die erfolgreiche Inangriffnahme von Problemen möglich, bei denen das rein inhaltliche Denken prinzipiell versagt. Zu diesen gehört z. B.

Kurt Gödel

Jahrhundertmathematiker und großer Entdecker
Author: Rebecca Goldstein
Publisher: N.A
ISBN: 9783492249607
Category:
Page: 312
View: 8589
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Choice

Publication of the Association of College and Research Libraries, a Division of the American Library Association
Author: N.A
Publisher: N.A
ISBN: N.A
Category: Academic libraries
Page: N.A
View: 4418
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Set theory

techniques and applications : Curaçao 1995 and Barcelona 1996 conferences
Author: C. A. Di Prisco
Publisher: Kluwer Academic Pub
ISBN: N.A
Category: Mathematics
Page: 226
View: 2923
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During the past 25 years, set theory has developed in several interesting directions. The most outstanding results cover the application of sophisticated techniques to problems in analysis, topology, infinitary combinatorics and other areas of mathematics. This book contains a selection of contributions, some of which are expository in nature, embracing various aspects of the latest developments. Amongst topics treated are forcing axioms and their applications, combinatorial principles used to construct models, and a variety of other set theoretical tools including inner models, partitions and trees. Audience: This book will be of interest to graduate students and researchers in foundational problems of mathematics.

Principia Mathematica.


Author: Alfred North Whitehead,Bertrand Russell
Publisher: N.A
ISBN: N.A
Category: Logic, Symbolic and mathematical
Page: 167
View: 6041
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Shadows of Cavernous Shades

Charting the Chiaroscuro of Realistic Computing
Author: Erik Persson
Publisher: N.A
ISBN: N.A
Category: Component software
Page: 727
View: 9260
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American Book Publishing Record


Author: N.A
Publisher: N.A
ISBN: N.A
Category: American literature
Page: N.A
View: 2145
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Biographische Enzyklopädie deutschsprachiger Naturwissenschaftler


Author: Dietrich von Engelhardt
Publisher: N.A
ISBN: N.A
Category: Scientists
Page: 1320
View: 6741
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Georg Cantor

der Jahrhundertmathematiker und die Entdeckung des Unendlichen
Author: David Foster Wallace
Publisher: N.A
ISBN: 9783492048262
Category:
Page: 407
View: 9398
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Mathematical Reviews


Author: N.A
Publisher: N.A
ISBN: N.A
Category: Mathematics
Page: N.A
View: 1178
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Vorlesungen Über die Zahlentheorie der Quaternionen


Author: Adolf Hurwitz
Publisher: Springer-Verlag
ISBN: 3642475361
Category: Mathematics
Page: 76
View: 4795
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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.