DMCA. Copyrighted Work that you can Claim.
Base have 55 000 books.
Search: 


📙 Conceptual Roots of Mathematics (International Library of Philosophy) by J.R. Lucas — free epub


The price of this book appalls. Would someone please see to it that it remains in print as an affordable paperback edition? I begin by quoting from Lucas's Introduction: "To start from the absolutely incontestable is to start from nowhere... any actual starting point could be put into question by a skilful sceptic... a metaphor of Descartes metaphor is more appropriate: we should think of mathematics not as a building based on foundations, but as a tree grounded in the soil of our general conceptual structure, and growing both up and down." Thus the title of this book. Lucas, a fine British academic philosopher whose interests have ranged over mathematics, physics, classics, and political philosophy, is the author of a startling and very controversial 1961 article on the interface among philosophy, computer science, and the philosophy of mind: "Minds, Machines, and Godel." This article much influenced "Godel Escher Bach," and Howard DeLong's "Profile of Mathematical Logic." Dale Jacquette agrees with Lucas's thesis; Judson Webb does not. I share Lucas's skepticism re artificial intelligence. I think of Lucas's book as a recent and splendid addition to the saga begun by Russell's "Principles of Mathematics." Russell rightly believed that the philosophy of mathematics is fascinating, but his intellectual ethics left something to be desired. Frege and Brouwer fascinate as well, but are difficult. C S Peirce's contributions to logic and mathematics have only started to receive the detailed treatment they richly deserve, thanks to Geraldine Brady's 2000 book. Contemporary authors (e.g., Chihara, Hellman, Shapiro, Maddy, Crispin wright, Resnick, Detlefsen) are readable, but appreciating what they say requires knowing a lot about what their predecessors wrote. Such are the many pitfalls along the trail Russell blazed circa 100 years ago. Lucas' book is a major exception. Published in the author's 70th year, as he was ending a long and fruitful Oxford career, the book distills a lifetime of teaching and thinking about the philosophy of mathematics. Lucas advocates a chastened logicism, a stance with which I sympathize. But I admire this book not so much for its substantive conclusions but for its sound pedagogy and good humour. Unlike most books of its ilk, it contains diagrams, charts, tables, sidebar summaries, even cartoons, all to good pedagogic effect. Lucas rightly does not assume that every reader of his book is an experienced and polished intellectual. I invite the reader to peruse the "summaries" on pp. 435-41, and to ponder when she last saw something so pithy and useful in a philosophy text. I surmise that Lucas for many years taught undergrads, and these can leave something to be desired, even at Oxford! The references alone (footnotes only, alas no bibliography) make the book worthwhile. The result is a book written not to impress fellow dons, but to instruct the laity knowing some math and logic. Lucas tries hard to take the reader by the hand through Godel's famous results. He does not cite the simplified approach of George Boolos (reprinted in Reuben Hersh's "What Is Mathematics, Really?"). "Conceptual Foundations" was also finished just before David Berlinski published his related gentle treatment of Godel's results. Lucas inevitably slights some topics (e.g., model theory, Intuitionism) to which I would have given more airplay. Suppes's classic 1960 treatment of ZFC set theory is nowhere mentioned. Lucas argues that set theory is primarily transfinite arithmetic, which is not very relevant to mathematical practice; I concur. Indeed, only two infinities matter for nearly all of mathematics, aleph null, and the continuum. Lucas gives a convincing reason, drawn from Latin philology, for renaming the quantifiers as 'quotifiers.' He also proposes the notation 'V' for existential and 'A' for universal quantification. He did not know that, but for a minor detail, this is essentially the notation of Tarski and his Berkeley students! Lucas does discuss in some detail three topics I think insufficiently discussed elsewhere: 1. Godel proved that if Peano arithmetic (PA) is consistent, there are PA sentences that are true but unprovable, and that adding axioms does not alter this result. Hence PA (and all mathematics) is necessarily incomplete (and ipso facto undecidable). Now the only nontrivial PA axiom is a schema permitting induction over the naturals. Robinson showed in 1950 that a fragment of PA, whose axioms are (a) the usual recursive definitions of addition and multiplication, (b) "if x and y have equal successors, then x=y," and (c) "0 is the sole number that is not the successor of a number," is likewise incomplete. This fragment is elsewhere named Q but Lucas calls it "Sorites arithmetic." Q includes no schemata; hence the Godelian incompleteness of PA cannot be blamed on the principle of induction; 2. Lucas devotes a pleasant and discursive chapter to relations, culminating in the rich tree diagram on p. 270, and rightly arguing that transitive relations are central to mathematics. Add symmetry and reflexivity to obtain equivalence relations; these inturn give rise to groups, monoids, and category theory. Replace symmetry with antisymmetry and obtain partial orderings, giving rise to lattices, trees, logics, and mereology. Take reflexivity away and obtain strict orderings, which include the well-ordering beloved of set theorists. Let no one who has not mastered this material call himself an analytic philosopher! 3. Mereotopology, a marriage of geometry, mereology, and topology that began with Whitehead's "The Principles of Natural Knowledge" and "The Concept of Nature," and matured into the systems of Bowman Clarke and Casati & Varzi (1999). While the latter say much more about mereotopology than Lucas does, much of what he writes is novel because based on unpublished writing by David Bostock, his Merton College colleague and fellow classicist, whose ideas on logic and mathematics, like Lucas's, deserve more attention. Until Bostock publishes his own book on the foundations of mathematics, we will have to content ourselves with Lucas's version of Bostock's ideas. I close by quoting one of Lucas's last sentences: "...if these notes help towards informed disagreement, I shall be well content."

About book:

About file:

  • File size: 2 147 176
  • Format: pdf


Security code:
Download button

Similar books results


Quantum Field Theory and Gravity: Conceptual and Mathematical Advances in the Search for a Unified Framework
Quantum Field Theory and Gravity: Conceptual and Mathematical Advances in the Search for a Unified Framework pdf free by Felix Finster, Olaf Müller, Marc Nardmann, Jürgen Tolksdorf, Eberhard Zeidler

One of the most challenging problems of contemporary theoretical physics is the mathematically rigorous construction of a theory which describes gravitation and the other fundamental physical interactions within a common framework. The physical ideas whic...

Personal Identity and Self-Consciousness (International Library of Philosophy)
Personal Identity and Self-Consciousness (International Library of Philosophy) free download by Brian Garrett

Unlike narratives that either travel a history of dead ideas or sycophants trying an exegesis of one or more authors; Garrett provides a summary discussion of the themes of personal identity (not consciousness) and the language use of the words. The tho...

Descriptive Psychology (International Library of Philosophy)
Descriptive Psychology (International Library of Philosophy) free download by Franz Brentano

Descriptive Psychology represents a series of lectures given by Franz Brentano in 1890; they were the culmination of Brentano's work and the clearest statements of his mature thought. It was this later period which proved to be so important in the work of...

Metaphysicians of Meaning: Russell and Frege on Sense and Denotation (International Library of Philosophy)
Metaphysicians of Meaning: Russell and Frege on Sense and Denotation (International Library of Philosophy) download pdf by Gideon Makin

Russell's On Denoting and Frege's On Sense and Reference are now widely held to be two of the founding papers of twentieth century philosophy and form the heart of the famous "linguistic turn". The Metaphysicians of Meaning is the first book to challenge ...

Conceptual Modeling: 31st International Conference ER 2012, Florence, Italy, October 15-18, 2012. Proceedings
Conceptual Modeling: 31st International Conference ER 2012, Florence, Italy, October 15-18, 2012. Proceedings free pdf by Stefano Ceri, Emanuele Della Valle, Dino Pedreschi, Roberto Trasarti (auth.), Paolo Atzeni, David Cheung, Sudha Ram (eds.)

This book constitutes the refereed proceedings of the 31st International Conference on Conceptual Modeling, ER 2012, held in Florence, Italy, in October 2012. The 24 regular papers presented together with 13 short papers, 6 poster papers and 3 keynotes we...

String Figures as Mathematics?: An Anthropological Approach to String Figure-making in Oral Tradition Societies
String Figures as Mathematics?: An Anthropological Approach to String Figure-making in Oral Tradition Societies epub download by Eric Vandendriessche (auth.)

This book addresses the mathematical rationality contained in the making of string figures. It does so by using interdisciplinary methods borrowed from anthropology, mathematics, history and philosophy of mathematics. The practice of string figure-making ...

Future Tendencies in Computer Science, Control and Applied Mathematics: International Conference on the Occasion of the 25th Anniversary of INRIA Paris, France, December 8–11, 1992 Proceedings
Future Tendencies in Computer Science, Control and Applied Mathematics: International Conference on the Occasion of the 25th Anniversary of INRIA Paris, France, December 8–11, 1992 Proceedings free download by J. L. Lions (auth.), A. Bensoussan, J. -P. Verjus (eds.)

This volume contains the proceedings of the International Conference on Research in Computer Science and Control, held on the occasion of the 25th anniversary of INRIA in December 1992. The objective of this conference was to bring together a large number...

Regression Analysis: Concepts and Applications
Regression Analysis: Concepts and Applications download pdf by Franklin A. Graybill, Hariharan K. Iyer

Regression Analysis: Concepts and Applications focuses on thinking clearly about and solving practical statistical problems. The approach leads from the theoretical (meaning conceptual not mathematical) to the applied, the idea being that samples (using t...

A First Course in Logic: An Introduction to Model Theory, Proof Theory, Computability, and Complexity
A First Course in Logic: An Introduction to Model Theory, Proof Theory, Computability, and Complexity download pdf by Shawn Hedman

The ability to reason and think in a logical manner forms the basis of learning for most mathematics, computer science, philosophy and logic students. Based on the author's teaching notes at the University of Maryland and aimed at a broad audience, this t...

Steel Frame Design Examples
Steel Frame Design Examples free download by Ian Robb A.M.I.Struct.E. (auth.)

Steady-State Enzyme Kinetics
Steady-State Enzyme Kinetics pdf free by Stanley Ainsworth (auth.)

Steel Frame Design Examples
Steel Frame Design Examples free pdf by Ian Robb M.Sc., C.Eng., M.I.Struct.E. (auth.)

Stem Cells in Modeling Human Genetic Diseases
Stem Cells in Modeling Human Genetic Diseases epub download by Mayana Zatz, Oswaldo Keith Okamoto (eds.)

While most stem cell books focus on basic aspects and/or cell therapy, this book emphasizes the relevance of stem cells obtained from patients, the so-called “patients in a petri dish” as tools to investigate human genetic diseases for which there are no ...

Stem Cells and Cardiac Regeneration
Stem Cells and Cardiac Regeneration pdf free by Rosalinda Madonna (eds.)

This book provides an update of recent advances in the basic and clinical applications of cell-based therapies for myocardial repair and regeneration in ischemic heart disease (IHD) and heart failure (HF). The first sections of the book illuminate basic a...

Stevie Smith: Between the Lines

Stepfamilies
Stepfamilies download pdf by Graham Allan, Graham Crow, Sheila Hawker (auth.)

Stem Cell Processing
Stem Cell Processing epub download by Phuc Van Pham (eds.)

This invaluable resource delineates procedures for development and use of stem cells in the laboratory and explores the potential for clinical applications. The text discusses mesenchymal stem cell isolation, isolation of adipose derived stem cells, new t...

Stealing Books in Eighteenth-Century London
Stealing Books in Eighteenth-Century London pdf free by Richard Coulton, Matthew Mauger, Christopher Reid (auth.)

This study offers an authoritative and readable account of the hidden history of book theft in eighteenth-century London. It exploits a rich primary source, the compelling narratives of crime contained in the digitised Proceedings of the Old Bailey. The a...

  Stick with It: A Scientifically Proven Process for Changing Your Life-for Good
Stick with It: A Scientifically Proven Process for Changing Your Life-for Good free epub by Sean D. Young

An award-winning psychologist and director of the UCLA Center for Digital Behavior shows everyone how to make real, lasting change in their lives in this exciting work of popular psychology that goes beyond The Power of Habit with science and practical st...