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THE ETHNOMETHODOLOGICAL FOUNDATIONS OF MATHEMATICS Studies in ethnomethodology Edited by Harold Garfinkel Professor of Sociology University of California, Los Angeles Theclhnomclhod~o~c~ foundations of mathematics Eric Livingston Routledge & Kegan Paul London, Boston and Henley Dedicated to my parents Herbert and Rosetta Livingston First published in 1986 by Routledge & Kegan Paul plc 14 Leicester Square, London WC2H 7PH, England 9 Park Street, Boston, Mass. 02108, USA and Broadway House, Newtown Road, Henley on Thames, Oxon RG9 lEN, England Set in Times, 10 on 11 pt by Hope Services of Abingdon and printed in Great Britain by Billing & Sons Ltd., Worcester Copyright© Eric Livingston 1986 No part of this book may be reproduced in any form without permission from the publisher, except for the quotation of brief passages in criticism Library of Congress Cataloging in Publication Data Livingston, Eric. The ethnomethodological foundations of mathematics. (Studies in ethnomethodology) Bibliography: p. 1. Mathematics -Philosophy. 2. Ethnomethodology. I. Title. II. Series. QA8.4.L48 1985 511.3 85-1898 British Library CIP data also available ISBN 0-7102-0335-7 Contents Preface ix Acknowledgments xii Introduction A Non-Technical Introduction to Ethnomethodological Investigations of the Foundations ofM athematics through the Use ofa Theorem ofE uclidean Geometry A Guide to the Reading of this Book 15 Part I 21 Introduction. The Phenomenon: The Existence of Classical Studies of Mathematicians' Work 23 A Review of the Classical Representation of Mathematicians' Work as Formal Logistic Systems 25 2 An Introduction to Godel's Incompleteness Theorems: Their Metamathematicallnterpretation Contrasted with the Proposal to Study Their Natural Accountability in and as the Lived-Work of Their Proofs 31 Part II A Descriptive Analysis of the Work of Proving Gooel's First Incompleteness Theorem 37 3 GOdel Numbering and Related Topics: Background Materials for a Proof of Godel's Theorem 39 v CONTENTS 4 The Double-Diagonalization/'Proof: Features of the Closing Argument of a Proof of Go del's Theorem as Lived-Work 45 5 A Technical Lemma: A Lemma Used in the Proof of Godel's Theorem; Its Origins as a Technical Residue of the Work of Proving Go del's Theorem within that Self-Same Work 51 6 Primitive Recursive Functions and Relations: An Initial Discussion of the Irremediable Connection between a Prover's Use of the Abbreviatory Practices/Practical Techniques of Working with Primitive Recursive Functions and Relations and the Natural Accountability of a Proof of GOdel's Theorem 57 7 A Schedule of Proofs: An Extended Analysis of the Lived-Work of Producing the Body of a Proof of Godel's Theorem 65 A A Schedule ofP roofs 65 B A Schedule ofP roofs as Lived-Work 69 1 As it is used in developing a schedule of proofs, a Godel numbering is not an abstractly defined correspondence between the symbols, expressions, and sequences of expressions of P and the natural numbers; it is a technique of proving 69 2 The schedule of proofs has a 'directed' character: it leads to and is organized so as to permit, the construction of G as a primitive recursive relation 76 3 The selection and arrangement of these-particular propositions as composing this-particular, intrinsically sequentialized order of proving is the situated achievement of the work of producing that schedule of proofs 80 (a)' Six themes concerning the lived-work of producing a schedule of proofs 81 (b )The construction of a schedule of proofs so as to provide an apparatus within itself for the analysis of the work of its own construction 100 (c) The correspondence between the propositions of a schedule of proofs and the syntax of formal number theory as an achievement of the schedule of proofs itself 117 (d)A review of the work of generalizing a schedule of proofs so as to elucidate the character of the development and organization of a schedule as a radical problem, for the prover, in the production of social order 125 vi CONTENTS 4 The work of providing a consistent notation for a schedule of proofs articulates that schedule as one coherent object 137 8 A Structure of Proving 149 A The Characterization Problem: The Problem of Specifying What Identifies a Proof of Code/'s Theorem as a Naturally Accountable Proof ofJ ust That Theorem; The Texture of the Characterization Problem and the Constraints on Its Solution; The Characterization Problem as the Foundational Problem 149 B Generalizing the Proof of Code/'s Theorem (As a Means of Gaining Technical Access to the Characterization Problem) 154 C A Structure ofP roving: The Availability to a Prover of the Proof of Code/'s Theorem as a Structure ofP ractices; The Proof as the Pair The-Proof/The-Practices-of-Proving-to- Which-That-Proofis-Irremediably-Tied i 69 Part III Conclusion 173 9 Summary and Directions for Further Study 175 A Classical Studies ofM athematical Practice: A Review of the Book's Argument 175 B Prospectus: Mathematicians' Work as Structure Building 177 Appendix 179 The Use of Ethnomethodological Investigations of Mathematicians' Work for Reformulating the Problem of the Relationship between Mathematics and Theoretical Physics as a Real-World Researchable Problem in the Production of Social Order 181 Notes 190 Bibliography 237 vii Preface The early Greeks were both amazed and perplexed by mathematical proofs. On one hand, the objects of geometry were made available and described, and their properties were established, through the use of drawn figures. Yet the Greeks recognized that the geometric objects themselves had a curious, unexplicated relationship to their depiction. They were further puzzled by the fact that the mathematical propo sition was demonstrated not as a matter of rhetorical argumentation, merely to the satisfaction of mathematicians immediately present. The mathematical theorem was proved as something necessarily true, a fact anonymous as to its authorship, available for endless inspection, established for all time - and this as a required feature of the actual demonstration itself. When it was proved that the 'field' of construct able lengths contained incommensurable elements, the Greeks were unable to turn away from the evidentness of the mathematical demon stration even though it went against their deepest philosophical commitments. It was said that Pythagoras, on proving this fact, com mitted suicide by drowning. At the turn of this century great interest was again shown in the origins of mathematical truth and in the nature of mathematical proofs. This was stimulated, in part, by circumstances similar to the Pythagorean proof of the existence of irrational numbers: developments in set theory had led to the proved existence of a continuous, 'space-filling' curve; even more spectacularly (but received with much greater sus picion), Zermelo proved that any set could be well-ordered. As a particular case, it follows that there exists some partial ordering of the real numbers such that every nonempty subset of them has a first element. Zermelo's proof, however, did not show what that partial order is. At the same time as these developments, elegant proofs using set theoretic methods - such as the proof that the transcendental numbers ix PREFACE are more numerous than the algebraic ones - were given. But it was also shown, by ingenious but elementary reasoning, that the unexpli cated notion of a 'set' led to contradictions, the most famous of these being Russell's paradox. Hilbert's solution of the 'invariant problem' demonstrated the power of abstract methods but, for some, raised questions as to the sense in which the problem had actually been solved. The burgeoning development of the new geometries had invigorated philosophers' consideration of the empirical status of mathematical truth. And Hilbert's researches in geometry again showed the strength of the axiomatic method but exhibited as well flaws in the reasoning of Euclid's Elements. One consequence of these origins of early twentieth-century investi gations of the foundations of mathematics was regrettable. The predominating interest of those studies, at least in the received view, was to demonstrate that the methods used by practising mathematicians and the results established through ordinary mathematical practice were themselves free from criticism. The attempt was made to construct indubitable foundations for mathematical practice. In consequence, interest in the original question - what made up the evident and transcendental character of mathematical proofs? - shifted to the problem of demonstrating the incorrigibility of those same proofs. During this time, mathematical research continued unabated, little affected by the investigations directed specifically to mathematical foundations. Moreover, it was mathematicians' daily production of ordinary, naturally accountable proofs that supplied not only the promised object but the basis of the logician's research. Although foundational studies did have an indirect effect on mathematical practice, a consequence of their most celebrated achievement, Godel's incompleteness theorems (themselves proved in the style of ordinary mathematics), was to hasten the incorporation of mathematical logic into mainstream mathematical research, not to alter mathematical practice. The living foundations of mathematics - and, as a particular case, the origins of the adequacy of GOdel's own proofs - remained untouched and unexamined. These remarks set in contrast the direction taken in the present work. This book is a study of the foundations of mathematics, but in the original sense. It is a study of the genetic origins of mathematical rigor, examining the proofs of ordinary mathematics and investigating how the adequacy of such proofs, for the purposes of everyday mathe matical inquiry, is practically obtained. The book formulates and, in a certain sense, solves the problem of the foundations of mathematics as a problem in the local production of social order. It does this not by reviewing received philosophies of mathematics, not by proffering a theory of social action, not through an historical or cultural analysis, but by rediscovering and exhibiting the naturally accountable mathe- X