Godel

A Life Of Logic, The Mind, And Mathematics

Author: John L. Casti,Werner DePauli,L Casti

Publisher: Basic Books

ISBN: 9780786747603

Category: Science

Page: 224

View: 8229

Kurt Gödel was an intellectual giant. His Incompleteness Theorem turned not only mathematics but also the whole world of science and philosophy on its head. Shattering hopes that logic would, in the end, allow us a complete understanding of the universe, Gödel's theorem also raised many provocative questions: What are the limits of rational thought? Can we ever fully understand the machines we build? Or the inner workings of our own minds? How should mathematicians proceed in the absence of complete certainty about their results? Equally legendary were Gödel's eccentricities, his close friendship with Albert Einstein, and his paranoid fear of germs that eventually led to his death from self-starvation. Now, in the first book for a general audience on this strange and brilliant thinker, John Casti and Werner DePauli bring the legend to life.
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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

View: 303

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.
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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: 7208

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.
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Essays on Gödel’s Reception of Leibniz, Husserl, and Brouwer

Author: Mark van Atten

Publisher: Springer

ISBN: 3319100319

Category: Philosophy

Page: 328

View: 9182

This volume tackles Gödel's two-stage project of first using Husserl's transcendental phenomenology to reconstruct and develop Leibniz' monadology, and then founding classical mathematics on the metaphysics thus obtained. The author analyses the historical and systematic aspects of that project, and then evaluates it, with an emphasis on the second stage. The book is organised around Gödel's use of Leibniz, Husserl and Brouwer. Far from considering past philosophers irrelevant to actual systematic concerns, Gödel embraced the use of historical authors to frame his own philosophical perspective. The philosophies of Leibniz and Husserl define his project, while Brouwer's intuitionism is its principal foil: the close affinities between phenomenology and intuitionism set the bar for Gödel's attempt to go far beyond intuitionism. The four central essays are `Monads and sets', `On the philosophical development of Kurt Gödel', `Gödel and intuitionism', and `Construction and constitution in mathematics'. The first analyses and criticises Gödel's attempt to justify, by an argument from analogy with the monadology, the reflection principle in set theory. It also provides further support for Gödel's idea that the monadology needs to be reconstructed phenomenologically, by showing that the unsupplemented monadology is not able to found mathematics directly. The second studies Gödel's reading of Husserl, its relation to Leibniz' monadology, and its influence on his publishe d writings. The third discusses how on various occasions Brouwer's intuitionism actually inspired Gödel's work, in particular the Dialectica Interpretation. The fourth addresses the question whether classical mathematics admits of the phenomenological foundation that Gödel envisaged, and concludes that it does not. The remaining essays provide further context. The essays collected here were written and published over the last decade. Notes have been added to record further thoughts, changes of mind, connections between the essays, and updates of references.
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In Search of the Riemann Zeros

Strings, Fractal Membranes and Noncommutative Spacetimes

Author: Michel Laurent Lapidus

Publisher: American Mathematical Soc.

ISBN: 9780821842225

Category: Mathematics

Page: 558

View: 9640

Formulated in 1859, the Riemann Hypothesis is the most celebrated and multifaceted open problem in mathematics. In essence, it states that the primes are distributed as harmoniously as possible--or, equivalently, that the Riemann zeros are located on a single vertical line, called the critical line. In this book, the author proposes a new approach to understand and possibly solve the Riemann Hypothesis. His reformulation builds upon earlier (joint) work on complex fractal dimensions and the vibrations of fractal strings, combined with string theory and noncommutative geometry. Accordingly, it relies on the new notion of a fractal membrane or quantized fractal string, along with the modular flow on the associated moduli space of fractal membranes. Conjecturally, under the action of the modular flow, the spacetime geometries become increasingly symmetric and crystal-like, hence, arithmetic. Correspondingly, the zeros of the associated zeta functions eventually condense onto the critical line, towards which they are attracted, thereby explaining why the Riemann Hypothesis must be true. Written with a diverse audience in mind, this unique book is suitable for graduate students, experts and nonexperts alike, with an interest in number theory, analysis, dynamical systems, arithmetic, fractal or noncommutative geometry, and mathematical or theoretical physics.
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There's Something About Gdel

The Complete Guide to the Incompleteness Theorem

Author: Francesco Berto

Publisher: John Wiley & Sons

ISBN: 1444357611

Category: Philosophy

Page: 256

View: 3359

Berto’s highly readable and lucid guide introduces students and the interested reader to Gödel’s celebrated Incompleteness Theorem, and discusses some of the most famous - and infamous - claims arising from Gödel's arguments. Offers a clear understanding of this difficult subject by presenting each of the key steps of the Theorem in separate chapters Discusses interpretations of the Theorem made by celebrated contemporary thinkers Sheds light on the wider extra-mathematical and philosophical implications of Gödel’s theories Written in an accessible, non-technical style
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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: 8914

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."
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Logical Dilemmas

The Life and Work of Kurt Gödel

Author: John Dawson

Publisher: A K Peters/CRC Press

ISBN: 9781568812564

Category: Mathematics

Page: 376

View: 8433

This authoritative biography of Kurt Goedel relates the life of this most important logician of our time to the development of the field. Goedel's seminal achievements that changed the perception and foundations of mathematics are explained in the context of his life from the turn of the century Austria to the Institute for Advanced Study in Princeton.
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A World Without Time

The Forgotten Legacy of Godel and Einstein

Author: Palle Yourgrau

Publisher: Basic Books

ISBN: 078673700X

Category: Science

Page: 224

View: 9530

In 1942, the logician Kurt Godel and Albert Einstein became close friends; they walked to and from their offices every day, exchanging ideas about science, philosophy, politics, and the lost world of German science. By 1949, Godel had produced a remarkable proof: In any universe described by the Theory of Relativity, time cannot exist. Einstein endorsed this result reluctantly but he could find no way to refute it, since then, neither has anyone else. Yet cosmologists and philosophers alike have proceeded as if this discovery was never made. In A World Without Time, Palle Yourgrau sets out to restore Godel to his rightful place in history, telling the story of two magnificent minds put on the shelf by the scientific fashions of their day, and attempts to rescue the brilliant work they did together.
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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: 9019

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.
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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: 7601

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Gödel's Theorem

An Incomplete Guide to Its Use and Abuse

Author: Torkel Franzén

Publisher: A K Peters/CRC Press

ISBN: 9781568812380

Category: Mathematics

Page: 182

View: 6762

"Among the many expositions of Gödel's incompleteness theorems written for non-specialists, this book stands apart. With exceptional clarity, Franzén gives careful, non-technical explanations both of what those theorems say and, more importantly, what they do not. No other book aims, as his does, to address in detail the misunderstandings and abuses of the incompleteness theorems that are so rife in popular discussions of their significance. As an antidote to the many spurious appeals to incompleteness in theological, anti-mechanist and post-modernist debates, it is a valuable addition to the literature." --- John W. Dawson, author of Logical Dilemmas: The Life and Work of Kurt Gödel
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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: 1383

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Principia Mathematica

Author: Alfred North Whitehead,Bertrand Russell

Publisher: N.A

ISBN: N.A

Category: Logic, Symbolic and mathematical

Page: N.A

View: 733

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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: 8277

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.
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The Universal Computer

The Road from Leibniz to Turing, Third Edition

Author: Martin Davis

Publisher: CRC Press

ISBN: 1351384813

Category: Mathematics

Page: 222

View: 1210

The breathtakingly rapid pace of change in computing makes it easy to overlook the pioneers who began it all. Written by Martin Davis, respected logician and researcher in the theory of computation, The Universal Computer: The Road from Leibniz to Turing explores the fascinating lives, ideas, and discoveries of seven remarkable mathematicians. It tells the stories of the unsung heroes of the computer age – the logicians.
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Pursuit of Genius

Flexner, Einstein, and the Early Faculty at the Institute for Advanced Study

Author: Steve Batterson

Publisher: A K Peters/CRC Press

ISBN: 9781568812595

Category: Mathematics

Page: 314

View: 1587

"The Institute for Advanced Study occupies a unique position among institutions of higher learning. An account of its early years is long overdue, so the appearance of the present volume, during the 75th anniversary of the Institute's founding, is most welcome. Batterson has mined the Institute's archives to provide a detailed and unvarnished account of the backstage conflicts and intrigue that attended the Institute's growth and determined its future. Those unfamiliar with the Institute will learn how one man's vision shaped a couple's philanthropy and created a haven for scholars in the midst of the Great Depression. Equally, those who have had the privilege of Institute membership will enhance their appreciation of the intellectual leaders who made their own Institute experiences possible." ---John W. Dawson, Jr., author of Logical Dilemmas: The Life and Work of Kurt Gödel
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