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The Search for Certainty A Philosophical Account of Foundations of Mathematics

The Search for Certainty   A Philosophical Account of Foundations of Mathematics Author Marcus Giaquinto
ISBN-10 9780191588174
Release 2002-06-06
Pages 298
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The nineteenth century saw a movement to make higher mathematics rigorous. This seemed to be on the brink of success when it was thrown into confusion by the discovery of the class paradoxes. That initiated a period of intense research into the foundations of mathematics, and with it the birth of mathematical logic and a new, sharper debate in the philosophy of mathematics. The Search for Certainty examines this foundational endeavour from the discovery of the paradoxes to the present. Focusing on Russell's logicist programme and Hilbert's finitist programme, Giaquinto investigates how successful they were and how successful they could be. These questions are set in the context of a clear, non-technical exposition and assessment of the most important discoveries in mathematical logic, above all G--ouml--;del's underivability theorems. More than six decades after those discoveries, Giaquinto asks what our present perspective should be on the question of certainty in mathematics. Taking recent developments into account, he gives reasons for a surprisingly positive response.



Mathematics

Mathematics Author Morris Kline
ISBN-10 0195030850
Release 1982
Pages 366
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Refuting the accepted belief that mathematics is exact and infallible, the author examines the development of conflicting concepts of mathematics and their implications for the physical, applied, social, and computer sciences



Wittgenstein s Lectures on the Foundations of Mathematics Cambridge 1939

Wittgenstein s Lectures on the Foundations of Mathematics  Cambridge  1939 Author Ludwig Wittgenstein
ISBN-10 9780226308609
Release 2015-05-14
Pages 300
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For several terms at Cambridge in 1939, Ludwig Wittgenstein lectured on the philosophical foundations of mathematics. A lecture class taught by Wittgenstein, however, hardly resembled a lecture. He sat on a chair in the middle of the room, with some of the class sitting in chairs, some on the floor. He never used notes. He paused frequently, sometimes for several minutes, while he puzzled out a problem. He often asked his listeners questions and reacted to their replies. Many meetings were largely conversation. These lectures were attended by, among others, D. A. T. Gasking, J. N. Findlay, Stephen Toulmin, Alan Turing, G. H. von Wright, R. G. Bosanquet, Norman Malcolm, Rush Rhees, and Yorick Smythies. Notes taken by these last four are the basis for the thirty-one lectures in this book. The lectures covered such topics as the nature of mathematics, the distinctions between mathematical and everyday languages, the truth of mathematical propositions, consistency and contradiction in formal systems, the logicism of Frege and Russell, Platonism, identity, negation, and necessary truth. The mathematical examples used are nearly always elementary.



Philosophy of Mathematics

Philosophy of Mathematics Author Paul Benacerraf
ISBN-10 9781107268135
Release 1984-01-27
Pages
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The twentieth century has witnessed an unprecedented 'crisis in the foundations of mathematics', featuring a world-famous paradox (Russell's Paradox), a challenge to 'classical' mathematics from a world-famous mathematician (the 'mathematical intuitionism' of Brouwer), a new foundational school (Hilbert's Formalism), and the profound incompleteness results of Kurt Gödel. In the same period, the cross-fertilization of mathematics and philosophy resulted in a new sort of 'mathematical philosophy', associated most notably (but in different ways) with Bertrand Russell, W. V. Quine, and Gödel himself, and which remains at the focus of Anglo-Saxon philosophical discussion. The present collection brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers. It is a substantially revised version of the edition first published in 1964 and includes a revised bibliography. The volume will be welcomed as a major work of reference at this level in the field.



Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century

Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century Author Paolo Mancosu
ISBN-10 9780195132441
Release 1999
Pages 275
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The seventeenth century saw dramatic advances in mathematical theory and practice than any era before or since. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, analytic geometry, the geometry of indivisibles, the arithmetic of infinites, and the calculus had been developed. Although many technical studies have been devoted to these innovations, Paolo Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Beginning with the Renaissance debates on the certainty of mathematics, Mancosu leads the reader through the foundational issues raised by the emergence of these new mathematical techniques, including the influence of the Aristotelian conception of science in Cavalieri and Guldin, the foundational relevance of Descartes' Geometrie, the relationship between empiricist epistemology and infinitistic theorems in geometry, and the debates concerning the foundations of the Leibnizian calculus In the process Mancosu draws a sophisticated picture of the subtle dependencies between technical development and philosophical reflection in seventeenth century mathematics.



In Search of Certainty

In Search of Certainty Author Mark Burgess
ISBN-10 9781491923375
Release 2015-04-09
Pages 472
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Quite soon, the world’s information infrastructure is going to reach a level of scale and complexity that will force scientists and engineers to approach it in an entirely new way. The familiar notions of command and control are being thwarted by realities of a faster, denser world of communication where choice, variety, and indeterminism rule. The myth of the machine that does exactly what we tell it has come to an end. What makes us think we can rely on all this technology? What keeps it together today, and how might it work tomorrow? Will we know how to build the next generation—or will we be lulled into a stupor of dependence brought about by its conveniences? In this book, Mark Burgess focuses on the impact of computers and information on our modern infrastructure by taking you from the roots of science to the principles behind system operation and design. To shape the future of technology, we need to understand how it works—or else what we don’t understand will end up shaping us. This book explores this subject in three parts: Part I, Stability: describes the fundamentals of predictability, and why we have to give up the idea of control in its classical meaning Part II, Certainty: describes the science of what we can know, when we don’t control everything, and how we make the best of life with only imperfect information Part III, Promises: explains how the concepts of stability and certainty may be combined to approach information infrastructure as a new kind of virtual material, restoring a continuity to human-computer systems so that society can rely on them.



Roads to Infinity

Roads to Infinity Author John C. Stillwell
ISBN-10 9781439865507
Release 2010-07-13
Pages 250
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Winner of a CHOICE Outstanding Academic Title Award for 2011! This book offers an introduction to modern ideas about infinity and their implications for mathematics. It unifies ideas from set theory and mathematical logic, and traces their effects on mainstream mathematical topics of today, such as number theory and combinatorics. The treatment is historical and partly informal, but with due attention to the subtleties of the subject. Ideas are shown to evolve from natural mathematical questions about the nature of infinity and the nature of proof, set against a background of broader questions and developments in mathematics. A particular aim of the book is to acknowledge some important but neglected figures in the history of infinity, such as Post and Gentzen, alongside the recognized giants Cantor and Gödel.



Mathematics in Western Culture

Mathematics in Western Culture Author Morris Kline
ISBN-10 0195345452
Release 1964-12-31
Pages 512
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This book gives a remarkably fine account of the influences mathematics has exerted on the development of philosophy, the physical sciences, religion, and the arts in Western life.



Philosophical Remarks

Philosophical Remarks Author Ludwig Wittgenstein
ISBN-10 0226904318
Release 1980-10-15
Pages 357
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When in May 1930, the Council of Trinity College, Cambridge, had to decide whether to renew Wittgenstein's research grant, it turned to Bertrand Russell for an assessment of the work Wittgenstein had been doing over the past year. His verdict: "The theories contained in this new work . . . are novel, very original and indubitably important. Whether they are true, I do not know. As a logician who likes simplicity, I should like to think that they are not, but from what I have read of them I am quite sure that he ought to have an opportunity to work them out, since, when completed, they may easily prove to constitute a whole new philosophy." "[Philosophical Remarks] contains the seeds of Wittgenstein's later philosophy of mind and of mathematics. Principally, he here discusses the role of indispensable in language, criticizing Russell's The Analysis of Mind. He modifies the Tractatus's picture theory of meaning by stressing that the connection between the proposition and reality is not found in the picture itself. He analyzes generality in and out of mathematics, and the notions of proof and experiment. He formulates a pain/private-language argument and discusses both behaviorism and the verifiability principle. The work is difficult but important, and it belongs in every philosophy collection."—Robert Hoffman, Philosophy "Any serious student of Wittgenstein's work will want to study his Philosophical Remarks as a transitional book between his two great masterpieces. The Remarks is thus indispensible for anyone who seeks a complete understanding of Wittgenstein's philosophy."—Leonard Linsky, American Philosophical Association



Mathematics and the Physical World

Mathematics and the Physical World Author Morris Kline
ISBN-10 9780486136318
Release 2012-03-15
Pages 512
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Stimulating account of development of mathematics from arithmetic, algebra, geometry and trigonometry, to calculus, differential equations, and non-Euclidean geometries. Also describes how math is used in optics, astronomy, and other phenomena.



The Search for Certainty

The Search for Certainty Author
ISBN-10 9789814467834
Release
Pages
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The Search for Certainty has been writing in one form or another for most of life. You can find so many inspiration from The Search for Certainty also informative, and entertaining. Click DOWNLOAD or Read Online button to get full The Search for Certainty book for free.



Understanding Wittgenstein s On Certainty

Understanding Wittgenstein s On Certainty Author D. Moyal-Sharrock
ISBN-10 9780230504462
Release 2004-04-19
Pages 250
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This book sheds unprecedented light on Wittgenstein's third masterpiece, On Certainty , clarifying his thoughts on basic beliefs and rebuttal of scepticism. As an introduction and commentary on Wittgenstein's final major philosophical work, Moyal-Sharrock's book will prove an indispensable guide to the student, scholar and general reader.



Logicomix

Logicomix Author Apostolos Doxiadis
ISBN-10 9781632864802
Release 2015-07-28
Pages 352
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This exceptional graphic novel recounts the spiritual odyssey of philosopher Bertrand Russell. In his agonized search for absolute truth, Russell crosses paths with legendary thinkers like Gottlob Frege, David Hilbert, and Kurt Gödel, and finds a passionate student in the great Ludwig Wittgenstein. But his most ambitious goal-to establish unshakable logical foundations of mathematics-continues to loom before him. Through love and hate, peace and war, Russell persists in the dogged mission that threatens to claim both his career and his personal happiness, finally driving him to the brink of insanity. This story is at the same time a historical novel and an accessible explication of some of the biggest ideas of mathematics and modern philosophy. With rich characterizations and expressive, atmospheric artwork, the book spins the pursuit of these ideas into a highly satisfying tale. Probing and ingeniously layered, the book throws light on Russell's inner struggles while setting them in the context of the timeless questions he spent his life trying to answer. At its heart, Logicomix is a story about the conflict between an ideal rationality and the unchanging, flawed fabric of reality.



How Mathematicians Think

How Mathematicians Think Author William Byers
ISBN-10 0691145997
Release 2010-05-02
Pages 424
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To many outsiders, mathematicians appear to think like computers, grimly grinding away with a strict formal logic and moving methodically--even algorithmically--from one black-and-white deduction to another. Yet mathematicians often describe their most important breakthroughs as creative, intuitive responses to ambiguity, contradiction, and paradox. A unique examination of this less-familiar aspect of mathematics, How Mathematicians Think reveals that mathematics is a profoundly creative activity and not just a body of formalized rules and results. Nonlogical qualities, William Byers shows, play an essential role in mathematics. Ambiguities, contradictions, and paradoxes can arise when ideas developed in different contexts come into contact. Uncertainties and conflicts do not impede but rather spur the development of mathematics. Creativity often means bringing apparently incompatible perspectives together as complementary aspects of a new, more subtle theory. The secret of mathematics is not to be found only in its logical structure. The creative dimensions of mathematical work have great implications for our notions of mathematical and scientific truth, and How Mathematicians Think provides a novel approach to many fundamental questions. Is mathematics objectively true? Is it discovered or invented? And is there such a thing as a "final" scientific theory? Ultimately, How Mathematicians Think shows that the nature of mathematical thinking can teach us a great deal about the human condition itself.



The Search for Mathematical Roots 1870 1940

The Search for Mathematical Roots  1870 1940 Author I. Grattan-Guinness
ISBN-10 1400824044
Release 2011-02-11
Pages 624
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While many books have been written about Bertrand Russell's philosophy and some on his logic, I. Grattan-Guinness has written the first comprehensive history of the mathematical background, content, and impact of the mathematical logic and philosophy of mathematics that Russell developed with A. N. Whitehead in their Principia mathematica (1910-1913). ? This definitive history of a critical period in mathematics includes detailed accounts of the two principal influences upon Russell around 1900: the set theory of Cantor and the mathematical logic of Peano and his followers. Substantial surveys are provided of many related topics and figures of the late nineteenth century: the foundations of mathematical analysis under Weierstrass; the creation of algebraic logic by De Morgan, Boole, Peirce, Schröder, and Jevons; the contributions of Dedekind and Frege; the phenomenology of Husserl; and the proof theory of Hilbert. The many-sided story of the reception is recorded up to 1940, including the rise of logic in Poland and the impact on Vienna Circle philosophers Carnap and Gödel. A strong American theme runs though the story, beginning with the mathematician E. H. Moore and the philosopher Josiah Royce, and stretching through the emergence of Church and Quine, and the 1930s immigration of Carnap and GödeI. Grattan-Guinness draws on around fifty manuscript collections, including the Russell Archives, as well as many original reviews. The bibliography comprises around 1,900 items, bringing to light a wealth of primary materials. Written for mathematicians, logicians, historians, and philosophers--especially those interested in the historical interaction between these disciplines--this authoritative account tells an important story from its most neglected point of view. Whitehead and Russell hoped to show that (much of) mathematics was expressible within their logic; they failed in various ways, but no definitive alternative position emerged then or since.



Kepler s Philosophy and the New Astronomy

Kepler s Philosophy and the New Astronomy Author Rhonda Martens
ISBN-10 9781400831098
Release 2009-08-16
Pages 216
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Johannes Kepler contributed importantly to every field he addressed. He changed the face of astronomy by abandoning principles that had been in place for two millennia, made important discoveries in optics and mathematics, and was an uncommonly good philosopher. Generally, however, Kepler's philosophical ideas have been dismissed as irrelevant and even detrimental to his legacy of scientific accomplishment. Here, Rhonda Martens offers the first extended study of Kepler's philosophical views and shows how those views helped him construct and justify the new astronomy. Martens notes that since Kepler became a Copernican before any empirical evidence supported Copernicus over the entrenched Ptolemaic system, his initial reasons for preferring Copernicanism were not telescope observations but rather methodological and metaphysical commitments. Further, she shows that Kepler's metaphysics supported the strikingly modern view of astronomical method that led him to discover the three laws of planetary motion and to wed physics and astronomy--a key development in the scientific revolution. By tracing the evolution of Kepler's thought in his astronomical, metaphysical, and epistemological works, Martens explores the complex interplay between changes in his philosophical views and the status of his astronomical discoveries. She shows how Kepler's philosophy paved the way for the discovery of elliptical orbits and provided a defense of physical astronomy's methodological soundness. In doing so, Martens demonstrates how an empirical discipline was inspired and profoundly shaped by philosophical assumptions.



Introduction to Mathematical Philosophy

Introduction to Mathematical Philosophy Author Bertrand Russell
ISBN-10 UOM:39015075979883
Release 1920
Pages 208
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Introduction to Mathematical Philosophy has been writing in one form or another for most of life. You can find so many inspiration from Introduction to Mathematical Philosophy also informative, and entertaining. Click DOWNLOAD or Read Online button to get full Introduction to Mathematical Philosophy book for free.