The Euclidean Programme
The Euclidean Programme

Author: A. C. Paseau

Publisher: Cambridge University Press

Published: 2024-02-14

Total Pages: 82

ISBN-13: 1009221973

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The Euclidean Programme embodies a traditional sort of epistemological foundationalism, according to which knowledge – especially mathematical knowledge – is obtained by deduction from self-evident axioms or first principles. Epistemologists have examined foundationalism extensively, but neglected its historically dominant Euclidean form. By contrast, this book offers a detailed examination of Euclidean foundationalism, which, following Lakatos, the authors call the Euclidean Programme. The book rationally reconstructs the programme's key principles, showing it to be an epistemological interpretation of the axiomatic method. It then compares the reconstructed programme with select historical sources: Euclid's Elements, Aristotle's Posterior Analytics, Descartes's Discourse on Method, Pascal's On the Geometric Mind and a twentieth-century account of axiomatisation. The second half of the book philosophically assesses the programme, exploring whether various areas of contemporary mathematics conform to it. The book concludes by outlining a replacement for the Euclidean Programme.

Proofs and Refutations
Proofs and Refutations

Author: Imre Lakatos

Publisher: Cambridge University Press

Published: 1976

Total Pages: 190

ISBN-13: 9780521290388

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Proofs and Refutations is for those interested in the methodology, philosophy and history of mathematics.

The Arguments of Time
The Arguments of Time

Author: British Academy

Publisher: Oxford University Press

Published: 2006-03-09

Total Pages: 278

ISBN-13: 9780197263464

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These nine essays, commissioned on the initiative of the Philosophy section of the British Academy, address fundamental questions about time in philosophy, physics, linguistics, and psychology. Are there facts about the future? Could we affect the past? In physics, general relativity and quantum theory give contradictory treatments of time. So in the current search for a theory of quantum gravity, which should give way: general relativity or quantum theory? In linguistics and psychology, how does our language represent time, and how do our minds keep track of it?

Order and Organism
Order and Organism

Author: Murray Code

Publisher: SUNY Press

Published: 1985-01-01

Total Pages: 280

ISBN-13: 9780873959513

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What is now needed is a way of thinking about the physical that is realistic in outlook but which departs radically from the mechanistic post-Galilean tradition. Since it seems clear that we can no longer take for granted the certainty and absolute objectivity of scientific knowledge, any alternative view must be able to do full justice to subjective modes of knowing. Order and Organism shows how Alfred North Whitehead's thought can reconcile some of the most insistent demands of common sense with the esoteric results of modern physics and mathematics. Whitehead shows a way to resolve the perennial puzzle of why mathematics works. Under his view, it is possible to account for the necessity and uniqueness of mathematical theories without denying the fact that such theories often arise from the mathematician's essentially aesthetic interest in various kinds of pattern.

Mathematics, Science and Epistemology: Volume 2, Philosophical Papers
Mathematics, Science and Epistemology: Volume 2, Philosophical Papers

Author: Imre Lakatos

Publisher: Cambridge University Press

Published: 1980-10-16

Total Pages: 302

ISBN-13: 9780521280303

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Volume I brings together his very influential but scattered papers on the philosophy of the physical sciences, and includes one important unpublished essay on the effect of Newton's scientific achievement. Volume 2 presents his work on the philosophy of mathematics together with some critical essays on contemporary philosophers of science.

Appraising Lakatos
Appraising Lakatos

Author: György Kampis

Publisher: Springer Science & Business Media

Published: 2013-06-29

Total Pages: 388

ISBN-13: 9401707693

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Imre Lakatos (1922-1974) was one of the protagonists in shaping the "new philosophy of science". More than 25 years after his untimely death, it is time for a critical re-evaluation of his ideas. His main theme of locating rationality within the scientific process appears even more compelling today, after many historical case studies have revealed the cultural and societal elements within scientific practices. Recently there has been, above all, an increasing interest in Lakatos' philosophy of mathematics, which emphasises heuristics and mathematical practice over logical justification. But suitable modifications of his approach are called for in order to make it applicable to modern axiomatised theories. Pioneering historical research in England and Hungary has unearthed hitherto unknown facts about Lakatos' personal life, his wartime activities and his involvement in the political developments of post-war Europe. From a communist activist committed to Györgyi Lukács' thinking, Lakatos developed into a staunch anti-Marxist who found his intellectual background in Popper's critical rationalism. The volume also publishes for the first time a part of his Debrecen Ph.D. thesis and it is concluded by a bibliography of his Hungarian writings.

Investigating Notions of Proof
Investigating Notions of Proof

Author: Keir Finlow-Bates

Publisher: Lulu.com

Published: 2009-10

Total Pages: 202

ISBN-13: 9529262922

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Although proof is seen by most mathematicians as lying at the heart of mathematics, it is rarely explicitly taught at any point in the mathematics curriculum. This is compounded by the fact that within the mathematics and education communities there is no clear definition of or consensus on what actually constitutes proof. In this book a fallibilist approach based on the work of Imre Lakatos is adopted, and proof and proving are set within the context of a form of social knowledge in order to gain insight into the proof-activities of degree level mathematics students.

Automata, Languages and Programming
Automata, Languages and Programming

Author: Peter Widmayer

Publisher: Springer

Published: 2003-08-03

Total Pages: 1089

ISBN-13: 3540454659

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This book constitutes the refereed proceedings of the 29th International Colloquium on Automata, Languages and Programming, ICALP 2002, held in Malaga, Spain, in July 2002.The 83 revised full papers presented together with 7 invited papers were carefully reviewed and selected from a total of 269 submissions. All current aspects of theoretical computer science are addressed and major new results are presented.

Notes on Geometry
Notes on Geometry

Author: Elmer G. Rees

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 119

ISBN-13: 3642617778

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In recent years, geometry has played a lesser role in undergraduate courses than it has ever done. Nevertheless, it still plays a leading role in mathematics at a higher level. Its central role in the history of mathematics has never been disputed. It is important, therefore, to introduce some geometry into university syllabuses. There are several ways of doing this, it can be incorporated into existing courses that are primarily devoted to other topics, it can be taught at a first year level or it can be taught in higher level courses devoted to differential geometry or to more classical topics. These notes are intended to fill a rather obvious gap in the literature. It treats the classical topics of Euclidean, projective and hyperbolic geometry but uses the material commonly taught to undergraduates: linear algebra, group theory, metric spaces and complex analysis. The notes are based on a course whose aim was two fold, firstly, to introduce the students to some geometry and secondly to deepen their understanding of topics that they have already met. What is required from the earlier material is a familiarity with the main ideas, specific topics that are used are usually redone.