Brouwer's Cambridge Lectures on Intuitionism


Author: Luitzen Egbertus Jan Brouwer,D. van Dalen,Brouwer
Publisher: Cambridge University Press
ISBN: 9780521177368
Category: Mathematics
Page: 122
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Luitzen Egburtus Jan Brouwer founded a school of thought whose aim was to include mathematics within the framework of intuitionistic philosophy; mathematics was to be regarded as an essentially free development of the human mind. What emerged diverged considerably at some points from tradition, but intuitionism has survived well the struggle between contending schools in the foundations of mathematics and exact philosophy. Originally published in 1981, this monograph contains a series of lectures dealing with most of the fundamental topics such as choice sequences, the continuum, the fan theorem, order and well-order. Brouwer's own powerful style is evident throughout the work.

L.E.J. Brouwer – Topologist, Intuitionist, Philosopher

How Mathematics Is Rooted in Life
Author: Dirk van Dalen
Publisher: Springer Science & Business Media
ISBN: 1447146166
Category: Mathematics
Page: 875
View: 5824

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Dirk van Dalen’s biography studies the fascinating life of the famous Dutch mathematician and philosopher Luitzen Egbertus Jan Brouwer. Brouwer belonged to a special class of genius; complex and often controversial and gifted with a deep intuition, he had an unparalleled access to the secrets and intricacies of mathematics. Most mathematicians remember L.E.J. Brouwer from his scientific breakthroughs in the young subject of topology and for the famous Brouwer fixed point theorem. Brouwer’s main interest, however, was in the foundation of mathematics which led him to introduce, and then consolidate, constructive methods under the name ‘intuitionism’. This made him one of the main protagonists in the ‘foundation crisis’ of mathematics. As a confirmed internationalist, he also got entangled in the interbellum struggle for the ending of the boycott of German and Austrian scientists. This time during the twentieth century was turbulent; nationalist resentment and friction between formalism and intuitionism led to the Mathematische Annalen conflict ('The war of the frogs and the mice'). It was here that Brouwer played a pivotal role. The present biography is an updated revision of the earlier two volume biography in one single book. It appeals to mathematicians and anybody interested in the history of mathematics in the first half of the twentieth century.

One Hundred Years of Intuitionism (1907-2007)

The Cerisy Conference
Author: Mark van Atten,Pascal Boldini,Michel Bourdeau,Gerhard Heinzmann
Publisher: Springer Science & Business Media
ISBN: 3764386533
Category: Science
Page: 422
View: 6877

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Intuitionism is one of the main foundations for mathematics proposed in the twentieth century and its views on logic have also notably become important with the development of theoretical computer science. This book reviews and completes the historical account of intuitionism. It also presents recent philosophical work on intuitionism and gives examples of new technical advances and applications. It brings together 21 contributions from today's leading authors on intuitionism.

Encyclopedia of Philosophy: Abbagnano - Byzantine philosophy


Author: Donald M. Borchert
Publisher: Thomson Gale/MacMillan Reference USA
ISBN: 9780028657813
Category: Philosophy
Page: 10
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Containing material from hundreds of highly distinguished contributors representing the world's top universities and institutions, the second edition has a truly global perspective. It contains more than 2,100 entries -- including more than 450 new articles. Among the many topics covered are African, Islamic, Jewish, Russian, Chinese, and Buddhist philosophies; bioethics and biomedical ethics; art and aesthetics; epistemology; metaphysics; peace and war; social and political philosophy; the Holocaust; feminist thought; and much more. Additionally, the second edition also features 1,000 biographical entries on major figures in philosophical thought throughout history.

Wittgenstein's Lectures on the Foundations of Mathematics, Cambridge, 1939


Author: Ludwig Wittgenstein
Publisher: University of Chicago Press
ISBN: 022630860X
Category: Philosophy
Page: 300
View: 9153

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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.

Proceedings of the Section of Sciences


Author: Koninklijke Nederlandse Akademie van Wetenschappen. Afdeling Natuurkunde
Publisher: N.A
ISBN: N.A
Category: Science
Page: N.A
View: 4224

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Proceedings of the Section of Sciences


Author: Koninklijke Akademie van Wetenschappen. Afdeeling voor de Wis- en Natuurkundige Wetenschappen,Koninklijke Nederlandse Akademie van Wetenschappen. Afdeling Natuurkunde,Koninklijke Akademie van Wetenschappen (Netherlands). Afdeeling Natuurkunde,Koninklijke Nederlandse Akademie van Wetenschappen
Publisher: N.A
ISBN: N.A
Category: Science
Page: N.A
View: 5862

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Polynesian Paradox

Essays in Honour of Professor 'I. Futa Helu
Author: Ian Christopher Campbell,Eve Coxon,'I. Futa Helu
Publisher: N.A
ISBN: N.A
Category: Tonga
Page: 267
View: 4221

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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: 2749

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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.

Computability in Analysis and Physics


Author: Marian B. Pour-El,J. Ian Richards
Publisher: Cambridge University Press
ISBN: 1107168449
Category: Mathematics
Page: 218
View: 1973

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Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. In this volume, the first publication in the Perspectives in Logic series, Pour-El and Richards present the first graduate-level treatment of computable analysis within the tradition of classical mathematical reasoning. The book focuses on the computability or noncomputability of standard processes in analysis and physics. Topics include classical analysis, Hilbert and Banach spaces, bounded and unbounded linear operators, eigenvalues, eigenvectors, and equations of mathematical physics. The work is self-contained, and although it is intended primarily for logicians and analysts, it should also be of interest to researchers and graduate students in physics and computer science.

The Logical Basis of Metaphysics


Author: Michael A. E. Dummett
Publisher: Harvard University Press
ISBN: 9780674537866
Category: Philosophy
Page: 355
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Michael Dummett's new book is the greatly expanded and recently revised version of his distinguished William James Lectures, delivered in 1976. Dummett regards the construction of a satisfactory theory of meaning as the most pressing task of contemporary analytical philosophy. He believes that the successful completion of this difficult assignment will lead to a resolution of problems before which philosophy has been stalled, in some instances for centuries. These problems turn on the correctness or incorrectness of a realistic view of one or another realm--the physical world, the mind, the past, mathematical reality, and so forth. Rejection of realism amounts to adoption of a variant semantics, and often of a variant logic, for the statements in a certain sector of our language. Dummett does not assume the correctness of any one logical system but shows how the choice between different logics arises at the level of the theory of meaning and depends upon the choice of one or another general form of meaning-theory. In order to determine the correct shape for a meaning-theory, we must attain a clear conception of what a meaning-theory can be expected to do. Such a conception, says Dummett, will form "a base camp for an assault on the metaphysical peaks: I have no greater ambition in this book than to set up a base camp."

Lectures on the Curry-Howard Isomorphism


Author: Morten Heine Sørensen,Pawel Urzyczyn
Publisher: Elsevier
ISBN: 9780080478920
Category: Mathematics
Page: 456
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The Curry-Howard isomorphism states an amazing correspondence between systems of formal logic as encountered in proof theory and computational calculi as found in type theory. For instance, minimal propositional logic corresponds to simply typed lambda-calculus, first-order logic corresponds to dependent types, second-order logic corresponds to polymorphic types, sequent calculus is related to explicit substitution, etc. The isomorphism has many aspects, even at the syntactic level: formulas correspond to types, proofs correspond to terms, provability corresponds to inhabitation, proof normalization corresponds to term reduction, etc. But there is more to the isomorphism than this. For instance, it is an old idea---due to Brouwer, Kolmogorov, and Heyting---that a constructive proof of an implication is a procedure that transforms proofs of the antecedent into proofs of the succedent; the Curry-Howard isomorphism gives syntactic representations of such procedures. The Curry-Howard isomorphism also provides theoretical foundations for many modern proof-assistant systems (e.g. Coq). This book give an introduction to parts of proof theory and related aspects of type theory relevant for the Curry-Howard isomorphism. It can serve as an introduction to any or both of typed lambda-calculus and intuitionistic logic. Key features - The Curry-Howard Isomorphism treated as common theme - Reader-friendly introduction to two complementary subjects: Lambda-calculus and constructive logics - Thorough study of the connection between calculi and logics - Elaborate study of classical logics and control operators - Account of dialogue games for classical and intuitionistic logic - Theoretical foundations of computer-assisted reasoning · The Curry-Howard Isomorphism treated as the common theme. · Reader-friendly introduction to two complementary subjects: lambda-calculus and constructive logics · Thorough study of the connection between calculi and logics. · Elaborate study of classical logics and control operators. · Account of dialogue games for classical and intuitionistic logic. · Theoretical foundations of computer-assisted reasoning

Logic and Structure


Author: Dirk van Dalen
Publisher: Springer Science & Business Media
ISBN: 3662029626
Category: Mathematics
Page: 220
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New corrected printing of a well-established text on logic at the introductory level.

The Right and the Good


Author: William David Ross,Philip Stratton-Lake
Publisher: Oxford University Press
ISBN: 9780199252657
Category: Philosophy
Page: 183
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The Right and the Good, a classic of twentieth-century philosophy by the great scholar Sir David Ross, is now presented in a new edition with a substantial introduction by Philip Stratton-Lake, a leading expert on Ross. Ross's book is the pinnacle of ethical intuitionism, which was the dominant moral theory in British philosophy for much of the nineteenth and early twentieth century. Intuitionism is now enjoying a considerable revival, and Stratton-Lake provides the context for a proper understanding of Ross's great work today.

Frege

Philosophy of Language
Author: Michael Dummett
Publisher: Harvard University Press
ISBN: 9780674319318
Category: Mathematics
Page: 708
View: 1952

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No one has figured more prominently in the study of German philosopher Gottlob Frege than Michael Dummett. This highly acclaimed book is a major contribution to the philosophy of language as well as a systematic interpretation of Frege, indisputably the father of analytic philosophy. Frege: Philosophy of Language remains indispensable for an understanding of contemporary philosophy. Harvard University Press is pleased to reissue this classic book in paperback.

Mathematics and Art

A Cultural History
Author: Lynn Gamwell
Publisher: Princeton University Press
ISBN: 0691165289
Category: Art
Page: 576
View: 5719

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This is a cultural history of mathematics and art, from antiquity to the present. Mathematicians and artists have long been on a quest to understand the physical world they see before them and the abstract objects they know by thought alone. Taking readers on a tour of the practice of mathematics and the philosophical ideas that drive the discipline, Lynn Gamwell points out the important ways mathematical concepts have been expressed by artists. Sumptuous illustrations of artworks and cogent math diagrams are featured in Gamwell’s comprehensive exploration. Gamwell begins by describing mathematics from antiquity to the Enlightenment, including Greek, Islamic, and Asian mathematics. Then focusing on modern culture, Gamwell traces mathematicians’ search for the foundations of their science, such as David Hilbert’s conception of mathematics as an arrangement of meaning-free signs, as well as artists’ search for the essence of their craft, such as Aleksandr Rodchenko’s monochrome paintings. She shows that self-reflection is inherent to the practice of both modern mathematics and art, and that this introspection points to a deep resonance between the two fields: Kurt Gödel posed questions about the nature of mathematics in the language of mathematics and Jasper Johns asked “What is art?” in the vocabulary of art. Throughout, Gamwell describes the personalities and cultural environments of a multitude of mathematicians and artists, from Gottlob Frege and Benoît Mandelbrot to Max Bill and Xu Bing. Mathematics and Art demonstrates how mathematical ideas are embodied in the visual arts and will enlighten all who are interested in the complex intellectual pursuits, personalities, and cultural settings that connect these vast disciplines.

A Primer of Infinitesimal Analysis


Author: John L. Bell
Publisher: Cambridge University Press
ISBN: 0521887186
Category: Mathematics
Page: 124
View: 634

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A rigorous, axiomatically formulated presentation of the 'zero-square', or 'nilpotent' infinitesimal.