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Towards A Philosophy Of Real Mathematics

Author: David Corfield
Publisher: Cambridge University Press
ISBN: 9781139436397
Size: 35.14 MB
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In this ambitious study, David Corfield attacks the widely held view that it is the nature of mathematical knowledge which has shaped the way in which mathematics is treated philosophically and claims that contingent factors have brought us to the present thematically limited discipline. Illustrating his discussion with a wealth of examples, he sets out a variety of approaches to new thinking about the philosophy of mathematics, ranging from an exploration of whether computers producing mathematical proofs or conjectures are doing real mathematics, to the use of analogy, the prospects for a Bayesian confirmation theory, the notion of a mathematical research programme and the ways in which new concepts are justified. His inspiring book challenges both philosophers and mathematicians to develop the broadest and richest philosophical resources for work in their disciplines and points clearly to the ways in which this can be done.

Explanation And Proof In Mathematics

Author: Gila Hanna
Publisher: Springer Science & Business Media
ISBN: 9781441905765
Size: 44.82 MB
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In the four decades since Imre Lakatos declared mathematics a "quasi-empirical science," increasing attention has been paid to the process of proof and argumentation in the field -- a development paralleled by the rise of computer technology and the mounting interest in the logical underpinnings of mathematics. Explanantion and Proof in Mathematics assembles perspectives from mathematics education and from the philosophy and history of mathematics to strengthen mutual awareness and share recent findings and advances in their interrelated fields. With examples ranging from the geometrists of the 17th century and ancient Chinese algorithms to cognitive psychology and current educational practice, contributors explore the role of refutation in generating proofs, the varied links between experiment and deduction, the use of diagrammatic thinking in addition to pure logic, and the uses of proof in mathematics education (including a critique of "authoritative" versus "authoritarian" teaching styles). A sampling of the coverage: The conjoint origins of proof and theoretical physics in ancient Greece. Proof as bearers of mathematical knowledge. Bridging knowing and proving in mathematical reasoning. The role of mathematics in long-term cognitive development of reasoning. Proof as experiment in the work of Wittgenstein. Relationships between mathematical proof, problem-solving, and explanation. Explanation and Proof in Mathematics is certain to attract a wide range of readers, including mathematicians, mathematics education professionals, researchers, students, and philosophers and historians of mathematics.

Towards A Philosophy Of Critical Mathematics Education

Author: Ole Skovsmose
Publisher: Springer Science & Business Media
ISBN: 9401735565
Size: 36.73 MB
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In Nineteen Eighty-Four George Orwell gives a description of different forms of suppression. We learn about the telescreens placed everywhere, through which it is possible for Big-Brother to watch the inhabitants of Oceania. However, it is not only important to control the activities of the inhabitants, it is important as well to control their thoughts, and the Thought Police are on guard. This is a very direct form of monitoring and control, but Orwell also outlines a more imperceptible and calculated line of thought control. In the Appendix to Nineteen Eighty-Four Orwell explains some struc tures of 'Newspeak', which is going to become the official language of Oceania. Newspeak is being developed by the Ministry of Truth, and this language has to substitute 'Oldspeak' (similar to standard English). Newspeak should fit with the official politics of Oceania ruled by the Ingsoc party: "The purpose of Newspeak was not only to provide a medium of expression for the world-view and mental habits proper to the devotees of Ingsoc, but to make all other modes of thought impos sible. It was intended that when Newspeak had been adopted once and for all and Oldspeak forgotten, a heretical thought - that is, a thought diverging from the principles of Ingsoc - should be literally unthink able, at least as far as thought is dependent on words.

An Introduction To The Philosophy Of Mathematics

Author: Mark Colyvan
Publisher: Cambridge University Press
ISBN: 0521826020
Size: 24.12 MB
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This introduction to the philosophy of mathematics focuses on contemporary debates in an important and central area of philosophy. The reader is taken on a fascinating and entertaining journey through some intriguing mathematical and philosophical territory, including such topics as the realism/anti-realism debate in mathematics, mathematical explanation, the limits of mathematics, the significance of mathematical notation, inconsistent mathematics and the applications of mathematics. Each chapter has a number of discussion questions and recommended further reading from both the contemporary literature and older sources. Very little mathematical background is assumed and all of the mathematics encountered is clearly introduced and explained using a wide variety of examples. The book is suitable for an undergraduate course in philosophy of mathematics and, more widely, for anyone interested in philosophy and mathematics.

Philosophy Of Mathematics

Author: Paul Benacerraf
Publisher: Cambridge University Press
ISBN: 9780521296489
Size: 17.70 MB
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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.

New Directions In The Philosophy Of Mathematics

Author: Thomas Tymoczko
Publisher: Princeton University Press
ISBN: 9780691034980
Size: 45.84 MB
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The traditional debate among philosophers of mathematics is whether there is an external mathematical reality, something out there to be discovered, or whether mathematics is the product of the human mind. This provocative book, now available in a revised and expanded paperback edition, goes beyond foundationalist questions to offer what has been called a "postmodern" assessment of the philosophy of mathematics--one that addresses issues of theoretical importance in terms of mathematical experience. By bringing together essays of leading philosophers, mathematicians, logicians, and computer scientists, Thomas Tymoczko reveals an evolving effort to account for the nature of mathematics in relation to other human activities. These accounts include such topics as the history of mathematics as a field of study, predictions about how computers will influence the future organization of mathematics, and what processes a proof undergoes before it reaches publishable form. This expanded edition now contains essays by Penelope Maddy, Michael D. Resnik, and William P. Thurston that address the nature of mathematical proofs. The editor has provided a new afterword and a supplemental bibliography of recent work.

What Is Mathematics Really

Author: Reuben Hersh
Publisher: Oxford University Press, USA
ISBN: 9780195130874
Size: 32.50 MB
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Most philosophers of mathematics treat it as isolated, timeless, ahistorical, inhuman. Reuben Hersh argues the contrary, that mathematics must be understood as a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context. Hersh pulls the screen back to reveal mathematics as seen by professionals, debunking many mathematical myths, and demonstrating how the "humanist" idea of the nature of mathematics more closely resembles how mathematicians actually work. At the heart of his book is a fascinating historical account of the mainstream of philosophy--ranging from Pythagoras, Descartes, and Spinoza, to Bertrand Russell, David Hilbert, and Rudolph Carnap--followed by the mavericks who saw mathematics as a human artifact, including Aristotle, Locke, Hume, Mill, and Lakatos. What is Mathematics, Really? reflects an insider's view of mathematical life, and will be hotly debated by anyone with an interest in mathematics or the philosophy of science.

Introduction To Mathematical Philosophy

Author: Bertrand Russell
Publisher: Charles River Editors via PublishDrive
ISBN: 1632955911
Size: 40.27 MB
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Bertrand Russell (18721970) was a British philosopher, mathematician, social critic, and political activist.Russell is considered to be one of the founders of analytic philosophy and one of the most important mathematicians and logicians in the 20th century.This edition of Russells Introduction to Mathematical Philosophy includes a table of contents.

Mathematics In Philosophy

Author: Charles Parsons
Publisher: Cornell University Press
ISBN: 9780801489815
Size: 51.64 MB
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This important book by a major American philosopher brings together eleven essays treating problems in logic and the philosophy of mathematics. A common point of view, that mathematical thought is central to our thought in general, underlies the essays. In his introduction, Parsons articulates that point of view and relates it to past and recent discussions of the foundations of mathematics.Mathematics in Philosophy is divided into three parts. Ontology—the question of the nature and extent of existence assumptions in mathematics—is the subject of Part One and recurs elsewhere. Part Two consists of essays on two important historical figures, Kant and Frege, and one contemporary, W. V. Quine. Part Three contains essays on the three interrelated notions of set, class, and truth.