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Realizing Reason a Narrative of Truth and Knowing Comp. by: PG4144 Stage : Revises1 ChapterID: 0002074919 Date:9/1/14 Time:02:56:58 Filepath:d:/womat-filecopy/0002074919.3D3 OUP UNCORRECTED PROOF – REVISES, 9/1/2014, SPi Realizing Reason A Narrative of Truth and Knowing Danielle Macbeth 1 Comp. by: PG4144 Stage : Revises1 ChapterID: 0002074919 Date:9/1/14 Time:02:56:58 Filepath:d:/womat-filecopy/0002074919.3D4 OUP UNCORRECTED PROOF – REVISES, 9/1/2014, SPi 3 Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries # Danielle Macbeth 2014 The moral rights of the author have been asserted First Edition published in 2014 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 0000000000 ISBN 978–0–19–870475–1 As printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work. Comp. by: PG4144 Stage : Revises1 ChapterID: 0002074919 Date:9/1/14 Time:02:56:58 Filepath:d:/womat-filecopy/0002074919.3D5 OUP UNCORRECTED PROOF – REVISES, 9/1/2014, SPi The world of mathematics is not a world apart. We will not have an adequate account of the physical world and our knowledge of it until we understand better than we presently do the role played by mathematics in our accounts of physical phenomena. And it is not likely that we will satisfy ourselves on that score until we have produced accounts of knowledge, truth, and reality that deal adequately with pure mathematics as well. Benacerraf and Putnam Comp. by: PG4144 Stage : Revises1 ChapterID: 0002074919 Date:9/1/14 Time:02:56:58 Filepath:d:/womat-filecopy/0002074919.3D7 OUP UNCORRECTED PROOF – REVISES, 9/1/2014, SPi Preface When I began this project it had a different title and a more modest ambition. Only very recently did it become apparent to me that not only was I tracing developments in the practice of mathematics I was tracking also the realization of reason as a power of knowing. In retrospect, I think that it had to be this way: only after reason is realized as a power of knowing is it possible to recognize the process of its realization as such. And having become clear on the end of project, the book I was actually writing, I became clear also about its beginning, where, how, and when this project seems actually to have started, namely, I now think, with my dissertation in the philosophy of mind written under John Haugeland (and inspired by his reading of Heidegger) at the University of Pittsburgh in the 1980s. I could not have anticipated then that to understand our intentional directedness on objective reality would require, first, a radically new reading of Frege’s Begriffsschrift notation (the reading I develop in Frege’s Logic), and building on that reading, the account of the whole of the history of Western mathematics that I offer here. Nevertheless, that, so it seems to me, is what has happened. Mathematics and philosophy have of course been intimately related throughout their history in the West. For Plato, the study of mathematics was a propaedeutic to the philosophical inquiry he called dialectic. For Descartes, method in metaphysics was modeled on his new method in mathematics. Leibniz, Kant, and Frege all held that one way or another the practice of mathematics was central to the practice of philosophy. And so it is here. To understand the nature of mathematics is, on our account, integral to understanding ourselves as the rational beings we are. This is due in part to the fact that mathematics, like philosophy, is a paradigm of rational activity. But it depends also on the fact that in mathematics, and only in mathematics, can a language be developed that, like natural language, serves as a medium of our cognitive commerce with reality. (This is not to say that one needs to have special training in mathematics to comprehend this work. One does not.) Understanding how a mathematical language such as Frege’s concept-script can serve as a medium of our cognitive involvements in the world is the first step in understanding even our everyday involvements in the world. The most technically challenging material here is not, then, in the chapters on mathematical practice but instead in Chapter 7 on Frege’s notation. I have aimed, in that chapter, to go into detail sufficient to satisfy anyone who wants actually to work through and master Frege’s proof of theorem 133 in Part III of his 1879 logic, Begriffsschrift. There is much in that chapter that the less committed reader may want to skim. Nonetheless, the details matter. Frege is doing what I claim he is doing but the only way to see that he is, is by gaining proficiency in the language. To understand everything that is claimed in Chapter 7, by having Comp. by: PG4144 Stage : Revises1 ChapterID: 0002074919 Date:9/1/14 Time:02:56:58 Filepath:d:/womat-filecopy/0002074919.3D8 OUP UNCORRECTED PROOF – REVISES, 9/1/2014, SPi viii PREFACE worked through all the details of Frege’s proof that are there discussed, is to have the literacy that is needed fully to understand Frege’s notation. A proof in mathematics is of course very different from the sort of narrative that is told here. A proof is a kind of argument, a giving of reasons, grounds, or justifica- tions, and a narrative is not. But perhaps this is too simple. After all, the proofs that mathematicians concern themselves with generally have, as stories do, a central idea on which the whole development turns. And there is development in a mathematical proof. Things happen in a (real) proof; proofs unfold and can have surprising turns. Mathematical proofs—good ones, interesting ones—are, in short, rather like narra- tives. And narratives can in their way provide reasons, grounds, and justifications insofar as they can help one to learn new ways of making sense of things, and to see how old ways of making sense do not in fact withstand critical scrutiny. If one is trying to change a reader’s conception of what makes sense at all, trying to change the space of possibilities within which a reader’s thought moves (as I aim to here), then a traditional philosophical argument is of little use. Arguments can be formulated only within established disciplinary boundaries, where the rules and acceptable starting points have already been agreed upon, at least in the main. What is needed, and what these pages provide, is a narrative of our intellectual maturation and growth, one that, if successful, will change a reader’s way of looking at things. DM Comp. by: PG4144 Stage : Revises1 ChapterID: 0002074919 Date:9/1/14 Time:02:56:58 Filepath:d:/womat-filecopy/0002074919.3D9 OUP UNCORRECTED PROOF – REVISES, 9/1/2014, SPi Acknowledgements In the years I have been working on this project I have had the very good fortune to lecture on the work to audiences around the world. And some of those lectures, or papers based on them, have been subsequently published. My thanks to the following for kind permission to use previously published material. John Wiley and Sons: “The Coin of the Intentional Realm” (based on work in my dissertation), Journal for the Theory of Social Behavior 24 (1994): 143–66, for material in section 1.3. College Publications: “Diagrammatic Reasoning in Euclid’s Elements”, Philosoph- ical Perspectives on Mathematical Practice, Texts in Philosophy, vol. 12, ed. Bart Van Kerkove, Jonas De Vuyst, and Jean Paul Van Bendegem (London: College Publica- tions, 2010), for much of Chapter 2. New School for Social Research: “Vie`te, Descartes, and the Emergence of Modern Mathematics”, Graduate Faculty Philosophy Journal 25 (2004): 87–117, sections 3.2 through 3.4. Oxford University Press: “Logic and the Foundations of Mathematics”, The Oxford Handbook of American Philosophy, ed. Cheryl Misak (Oxford: Oxford University Press, 2008), material in section 5.4. Institute of Logic and Cognition, Guangzhou: “The Problem of Mathematical Truth”, Studies in Logic 2 (2009): 1–17, material in section 5.4. Editions Kime´: “Proof and Understanding in Mathematical Practice”, Philosophia Scientiae 16 (2012): 29–54, section 6.2. National Center for Logical Investigation, Brussels: “Writing Reason”, Logique et Analyse (2013) 221: 25–44, parts of section 6.4. Editions Rodopi: “Striving for Truth in the Practice of Mathematics: Kant and Frege”, Grazer Philosophische Studien 75 (2007): 65–92, a first attempt at some themes in Chapters 7 and 8. Springer Science + Business Media: “Diagrammatic Reasoning in Frege’s Begriffs- schrift”, Synthese 186 (2012): 289–314, a second pass through themes of Chapters 7 and 8.
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