Essays on the Foundations of Mathematics by Moritz Pasch THE WESTERN ONTARIO SERIES IN PHILOSOPHY OF SCIENCE A SERIES OF BOOKS IN PHILOSOPHY OF SCIENCE, METHODOLOGY, EPISTEMOLOGY, LOGIC, HISTORY OF SCIENCE, AND RELATED FIELDS Managing Editor WILLIAM DEMOPOLOS Department of Philosophy, niversity of Western Ontario, Canada Managing Editor 1980–1997 ROBERT E. BTTS Late, Department of Philosophy, niversity of Western Ontario, Canada Editorial Board JOHN L. BELL, niversity of Western Ontario JEFFREY BB, niversity of Maryland PETER CLARK, St Andrews niversity DAID DEIDI, niversity of Waterloo ROBERT DiSALLE, niversity of Western Ontario MICHAEL FRIEDMAN, Stanford niversity MICHAEL HALLETT, McGill niversity WILLIAM HARPER, niversity of Western Ontario CLIFFORD A. HOOKER, niversity of Newcastle ASONIO MARRAS, niversity of Western Ontario JÜRGEN MITTELSTRASS, niversität Konstanz WAYNE C. MYROLD, niversity of Western Ontario THOMAS EBFL, niversity of Manchester † ITAMAR PITOWSKY , Hebrew niversity OLME 83 Stephen Pollard Editor Essays on the Foundations of Mathematics by Moritz Pasch 123 Editor Prof. Stephen Pollard Truman State niversity Dept. Philosophy & Religion 63501 Kirksville Missouri SA [email protected] ISBN 978-90-481-9415-5 e-ISBN 978-90-481-9416-2 DOI 10.1007/978-90-481-9416-2 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2010929951 c Springer Science+Business Media B.. 2010 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specificall for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface Moritz Pasch (1843–1930) has a secure reputation as a key figur in the history of axiomatic geometry. Less well known are his contributions to other areas of founda- tional research. This volume features English translations of fourteen papers Pasch published in the decade 1917–1926 during the surge in productivity after his retire- ment from the niversity of Giessen (Justus-Liebig-niversit at¨ Gießen). In them, Pasch argues that geometry and number theory are branches of empirical science; he provides axioms for the combinatorial reasoning essential to Hilbert’s program of consistency proofs; he explores implicit definitio (a generalization of definitio by abstraction) and indicates how this technique yields an empiricist reconstruc- tion of set theory; he argues that we cannot fully understand the logical structure of mathematics without clearly distinguishing between decidable and undecidable properties; he offers a rare glimpse into the mind of a master of axiomatics, sur- veying in detail the thought experiments he employed as he struggled to identify fundamental mathematical principles; and much more. The fourteen papers in this volume present Pasch’s most mature positions on key foundational issues that had occupied him for decades. They will: • Introduce English speakers to an important body of work from a turbulent and pivotal period in the history of mathematics. • Help us look beyond the familiar triad of formalism, intuitionism, and logicism. • Show how deeply we can see with the help of a guide determined to present fundamental mathematical ideas in ways that match our human capacities. The book should interest researchers in logic and the foundations of mathematics, including historians, mathematicians, and philosophers. In translating Pasch’s German into English, my goal has been to produce a text that might have been composed by a mathematically and philosophically competent writer whose f rst language is English. I mention this so readers will be aware that my errors will generally not be in the direction of excessive literalness. I have made no effort to produce a text from which the original German can be mechanically reproduced. The result of such an effort is unlikely to be idiomatic English. Readers should also be aware that I have silently corrected some obvious misprints. When it v vi Preface is clear beyond any shadow of a doubt that, for example, Pasch meant l rather than λ , I do not interrupt the narrative with an explanatory note. I have occasionally provided descriptive section titles where Pasch has only numbers or only generic headings that give no hint about the contents. One last liberty: I have rendered in English the titles of German works cited by Pasch. Readers who turn to this volume because they have little or no German may fin this helpful. For a period of weeks, my colleague David Gillette was subjected to daily, if not hourly, questions about Pasch’s German. I am grateful for his help. I also owe special thanks to Florence Emily Pollard who “wanted to see me write a book.” Kirksville, Missouri, SA Stephen Pollard February 2010 Contents Translator’s Introduction ........................................... 1 0.1 Pasch of Giessen ........................................... 1 0.2 Chains and Lines ........................................... 3 0.3 Existence of Lines .......................................... 4 0.4 Extracting Lines from Lines .................................. 6 0.5 Justifying Induction ........................................ 8 0.6 Initial Segments ............................................ 9 0.7 Conformity and Abstraction .................................. 11 0.8 Finite Ordinals ............................................. 12 0.9 Addition and Multiplication .................................. 15 0.10 How Much Arithmetic? ..................................... 18 0.11 To Justify the Ways of Peano to Men .......................... 23 0.12 Empiricist Arithmetic? ...................................... 25 0.13 Ideal Divisors .............................................. 30 0.14 Implicit Definitio .......................................... 36 0.15 Sets ...................................................... 39 References ..................................................... 42 1 Fundamental Questions of Geometry ............................. 45 1.1 Deductive Presentation of Geometry ........................... 45 1.2 Applicability of Geometry ................................... 46 1.3 Empiricist Geometry ........................................ 46 1.4 The Levels of Concept Formation ............................. 47 1.5 Proof Procedure ............................................ 48 1.6 Core Propositions for Straight Lines and Planes ................. 49 References ..................................................... 49 2 The Decidability Requirement ................................... 51 2.1 Rigid Mathematics ......................................... 51 2.2 Kronecker’s Requirement .................................... 52 2.3 Core Concepts and Propositions .............................. 53 References ..................................................... 54 vii viii Contents 3 The Origin of the Concept of Number ............................ 55 Introduction .................................................... 55 I Preliminary Facts .............................................. 58 3.1 Things and Proper Names ................................... 58 3.2 Specification and Collective Names .......................... 59 3.3 Earlier and Later ........................................... 59 3.4 First and Last .............................................. 60 3.5 Inferences ................................................. 61 3.6 Between .................................................. 62 3.7 Immediate Succession....................................... 63 3.8 Immediate Precedence ...................................... 64 3.9 The Possibility of Specification .............................. 65 3.10 Chains of Events ........................................... 67 3.11 Lines of Things ............................................ 68 3.12 Neighbor-Lines ............................................ 69 3.13 Pacing Off a Line .......................................... 71 3.14 Application to Collective Names .............................. 72 3.15 Proof by Pacing Off ........................................ 73 3.16 Collections of Things ....................................... 75 3.17 Implicit Definitio .......................................... 76 3.18 Consequences of Implicit Definitio ........................... 77 3.19 Applications of Proof by Pacing Off ........................... 78 3.20 Backwards Pacing .......................................... 79 II Summary of the Preceding Results ............................... 79 3.21 Summary of 3.1 ............................................ 79 3.22 Summary of 3.2 ............................................ 80 3.23 Summary of 3.3 ............................................ 80 3.24 Summary of 3.4 ............................................ 81 3.25 Summary of 3.5 ............................................ 81 3.26 Summary of 3.6 ............................................ 81 3.27 Summary of 3.7 ............................................ 81 3.28 Summary of 3.8 ............................................ 82 3.29 Summary of 3.9 ............................................ 82 3.30 Summary of 3.10 ........................................... 82 3.31 Summary of 3.11 ........................................... 83 3.32 Summary of 3.12