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Philosophy of Mathematics: Selected Writings Philosophy Peirce The philosophy of mathematics plays a hilosophy of Peirce’s most vital role in the mature philosophy of Charles s. Peirce. Peirce received rigorous P importa-nt mathematical training from his father and Ma-thema-tic his philosophy carries on in decidedly philosophy Selected Writings writings on the mathematical and symbolic veins. For S Peirce, math was a philosophical tool and philosophy of many of his most productive ideas rest firmly on the foundation of mathematical cha-rles S. Peirce ma-thema-tics principles. This volume collects Peirce’s most important writings on the subject, many appearing in print for the first time. “Focuses on the major Peirce’s determination to understand writings Peirce produced matter, the cosmos, and “the grand design” of the universe remain relevant that are of greatest for contemporary students of science, significance for a correct technology, and symbolic logic. of appreciation of his larger Charles s. PeirCe (1839–1914) Ma-thema-ti was one of america’s most prolific philosophical agenda.” philosophers. he is noted for his —JosePh W. Dauben, contributions to logic, mathematics, City university oF neW york science, and semiotics. MattheW e. Moore is associate Professor of Philosophy at brooklyn College. he is editor of New Essays on Peirce’s Mathematical Philosophy. c Selections from the Writings of Cha-rles S. Peirce s PeirCe Edition ProJeCt INDIANA University Press Bloomington & Indianapolis www.iupress.indiana.edu INDIANA Edited by Ma-tthew E. Moore 1-800-842-6796 Phil of Mathematics MECH.indd 1 6/30/10 10:04 AM Beings of Reason.book Page i Wednesday, June 2, 2010 6:06 PM Philosophy of Mathematics Beings of Reason.book Page ii Wednesday, June 2, 2010 6:06 PM Beings of Reason.book Page iii Wednesday, June 2, 2010 6:06 PM Philosophy of Mathematics Selected Writings Charles S. Peirce Edited by Matthew E. Moore INDIANA UNIVERSITY PRESS Bloomington and Indianapolis Beings of Reason.book Page iv Wednesday, June 2, 2010 6:06 PM This book is a publication of Indiana University Press 601 North Morton Street Bloomington, Indiana 47404-3797 USA www.iupress.indiana.edu Telephone orders 800-842-6796 Fax orders 812-855-7931 Orders by e-mail [email protected] © 2010 by Indiana University Press All rights reserved No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage and retrieval system, without permission in writ- ing from the publisher. The Association of American University Presses' Resolution on Permissions constitutes the only exception to this prohibition. The paper used in this publication meets the minimum requirements of the American National Standard for Information Sciences—Permanence of Paper for Printed Library Materials, ANSI Z39.48-1992. Manufactured in the United States of America Cataloging information is available from the Library of Congress. ISBN 978-0-253-35563-8 (cl.) ISBN 978-0-253-22265-7 (pbk.) 1 2 3 4 5 15 14 13 12 11 10 Beings of Reason.book Page v Wednesday, June 2, 2010 6:06 PM Contents Preface / vii Introduction / xv 1. [The Nature of Mathematics] (1895) 1 2. The Regenerated Logic (1896) 11 3. The Logic of Mathematics in Relation to Education (1898) 15 4. The Simplest Mathematics (1902) 23 5. The Essence of Reasoning (1893) 37 6. New Elements of Geometry (1894) 39 7. On the Logic of Quantity (1895) 43 8. Sketch of Dichotomic Mathematics (1903) 57 9. [Pragmatism and Mathematics] (1903) 67 10. Prolegomena to an Apology for Pragmaticism (1906) 79 11. [‘Collection’ in The Century Dictionary] (1888–1914) 85 12. [On Collections and Substantive Possibility] (1903) 89 13. [The Ontology of Collections] (1903) 99 14. The Logic of Quantity (1893) 107 15. Recreations in Reasoning (1897) 113 16. Topical Geometry (1904) 119 Beings of Reason.book Page vi Wednesday, June 2, 2010 6:06 PM vi | Contents 17. A Geometrico-Logical Discussion (1906) 129 18. [‘Continuity’ in The Century Dictionary] (1888–1914) 135 19. The Law of Mind (1892) 141 20. [Scientific Fallibilism] (1893) 155 21. On Quantity [The Continuity of Time and Space] (1896) 159 22. Detached Ideas Continued and the Dispute between Nominalists and Realists (1898) 165 23. The Logic of Continuity (1898) 179 24. [On Multitudes] (1897) 189 25. Infinitesimals (1900) 201 26. The Bed-Rock beneath Pragmaticism (1905) 207 27. [Note and Addendum on Continuity] (1908) 211 28. Addition [on Continuity] (1908) 217 29. Supplement [on Continuity] (1908) 221 Notes / 227 Bibliography / 263 Index / 281 Beings of Reason.book Page vii Wednesday, June 2, 2010 6:06 PM Preface The purpose of this book is to make Peirce’s philosophy of mathematics more readily available to contemporary workers in the field, and to students of his thought. Peirce’s philosophical writings on mathematics are reason- ably well represented—despite the shortcomings detailed in Dauben (1996, 28–39)—in the Collected Papers, much more so in The New Elements of Mathematics. The Chronological Edition of Peirce’s Writings will surely set the gold standards, in this area as in others, of comprehensiveness and tex- tual scholarship; it will also provide the annotations and other auxiliary apparatus whose absence so greatly diminishes the usability of the earlier editions for all but the most seasoned Peirceans. But the Chronological Edi- tion has only just reached the last quarter century of Peirce’s life, when most of the central texts on mathematics (and nearly all of those in the present vol- ume) were written. And its very comprehensiveness will make it difficult for those with a particular interest in the philosophy of mathematics to find their way to what they really need. Hence this book. It does not pretend to be a comprehensive selection, even of Peirce’s most important philosophical writings on mathematics. It seeks rather to be a selection of major texts that is comprehensive enough to serve as a serious introduction to his philosophy of mathematics, sufficient in itself for those whose primary interests lie elsewhere, and a stepping-stone for specialists to more advanced investigations. If it helps to turn some of its readers into specialists, so much the better; for the secondary literature on the mathematical aspects of Peirce’s thought makes no more than a good beginning, and much remains to be done. Precious little of that literature is due to philosophers of mathematics who come to Peirce for insight into the living problems of their discipline. I will argue in the introduction that some of Peirce’s deepest insights into mathematics cannot be fully appreciated in isolation from his larger philo- sophical system (which is not to say that one must buy fully into the system in order to appropriate the insights). This creates something of a dilemma for a volume such as this one; for the system is complicated and unfinished. Anything like a thorough exposition would overwhelm the primary content, a redundancy for much of its intended audience, and a stumbling block for the rest. My imperfect solution has been to provide a very brief overview in Beings of Reason.book Page viii Wednesday, June 2, 2010 6:06 PM viii | Preface the introduction, to be supplemented by more detailed piecemeal explana- tions in the headnotes to the individual selections. Those who feel the need of a more thorough systematic survey will do well to consult Nathan Houser’s introductions to the two volumes of The Essential Peirce, and Cheryl Misak’s to the Cambridge Companion to Peirce. These works pro- vide convenient points of entry into Peirce’s own writings and into the sec- ondary literature. An overlapping, and no less intractable, dilemma is posed by the explanatory apparatus: the endnotes and headnotes. As just noted, the cur- rently available editions of these texts are distinctly lacking in such appara- tus. The general aims are clear enough: the headnotes should prepare the reader for Peirce’s own words by placing them in context, by outlining the argumentative structure, and by clearing away the most serious interpretive obstacles; the endnotes serve the same purposes, but on a smaller scale, smoothing out the residual bumps on what should already be a relatively well-marked road. These generalities do not settle the practical question of what to include and what to omit. Even under the most favorable of circumstances this is something of a dilemma; for excess and defect alike diminish the value of a collection such as this. The dilemma is particularly acute with Peirce. He is an exceptionally allusive writer, whose reading was so wide that it is natural to despair, as with Pound’s Cantos, of truly understanding him until one has done the impossible and read everything he did. Much of what he alludes to has now faded into obscurity; his mathematical references are often couched in the now unfamiliar technical language of his time, and even when updated will go over many readers’ heads. I have sought, as a rule, to provide substantive explanations of matters that are either (a) essential to a basic understanding of the text in question, and not common knowledge; or else (b) of the first importance for the reader who wants to go beyond a basic understanding. The second conjunct of dis- junct (a) is meant to exclude anything that a reader with a standard philo- sophical education, and a serviceable grasp of college mathematics (including elementary calculus), can reasonably be expected to know: I pre- sume, for example, that the reader knows who Descartes was and what the Pythagorean Theorem says. Disjunct (a) will as a rule evoke more detailed explanations than (b).
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