Numbers in Presence and Absence: A Study of Husserl’s Philosophy of Mathematics PDF
Preview Numbers in Presence and Absence: A Study of Husserl’s Philosophy of Mathematics
NUMBERS IN PRESENCE AND ABSENCE: A STUDY OF HUSSERL'S PHILOSOPHY OF MATHEMATICS PHAENOMENOLOGICA COLLECTION FONDEE PAR H.L. VAN BREDA ET PUBLIEE SOUS LE PATRONAGE DES CENTRES D'ARCHIVES - HUSSERL 90 J. PHILIP MILLER NUMBERS IN PRESENCE AND ABSENCE: A Study of Husserl's Philosophy of Mathematics Comite de redaction de la collection: President: S. IJsseling (Leuven) Membres: M. Farbert (Buffalo), L. Landgrebe (Koln), W. Marx (Freiburg i. Br.), J.N. Mohanty (Oklahoma), P. Ricoeur (Paris), E. Stroker (Koln), J. Taminiaux (Louvain-La-Neuve), K.H. Volkmann-Schluckt (Ktiln) Secretaire: J. Taminiaux J. PHILIP MILLER NUMBERS IN PRESENCE AND ABSENCE: A Study of Husser/'s Philosophy of Mathematics • 1982 MARTINUS NIJHOFF PUBLISHERS THE HAGUE/BOSTON/LONDON Distributors: for the United States and Canada Kluwer, Boston, Inc. 190 Old Derby Street Hingham, MA 02043 USA for all other countries Kluwer Academic Publishers Group Distribution Center P.O. Box 322 3300 AH Dordrecht The Netherlands I.ibrary of Congress Cataloging in Publication Data Miller, J. Philip. Numbers in presence and absence. (Phaenomenologica ; 90) Bibliography: p. Includes index. 1. Husserl, Edmund, 1859-1938. 2. Mathematics- Philosophy. I. Title. II •. Series. QA8.4.H87M54 1982 512'.72 82-8044 ISBN-13: 978-94-009-7626-9 e-ISBN-I 3: 978-94-009-7624-5 001: 10.1007/978-94-009-7624-5 Copyright © 1982 by Martinus Nijhoff Publishers, The Hague. So/kover reprint of the hardcover 1s t edition 1982 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, mechanical, photocopying, recording, or otherwise, without the prior permission of the publishers, Martinus Nijhoff Publishers, P.O. Box 566, 2502 CN The Hage, The Netherlands. In memory of my father, EDWARD D. MILLER VII ACKNOWLEDGEMENlS A substantial portion of the research on which this study is based was carried out at the HusserI Archives in Cologne, West Germany. I am grateful to the staff of the Archives, and in particular to Professor Elisabeth Straker, the Director, for their assistance. My stay in Cologne was made possible by a grant from the Fulbright Com mission for Educational Exchange, which I hereby gratefully acknowledge. I also wish to express my special thanks to those who read and commented upon earlier drafts of this work, especially John Brough of Georgetown Univer sity and Thomas Prufer of The Catholic University of America. My deepest thanks however go to the Reverend Robert Sokolowski, also of The Catholic University of America; his stimulating suggestions and patient criticisms provided guidance for me at every stage of my work on this project. IX CONTENTS CHAPTER I: THE EMERGENCE AND DEVELOPMENT OF HUSSERL'S 'PHILOSOPHY OF ARITHMETIC' 1. Historical Background: Weierstrass and the Arith- metization of Analysis 2. HusserI's First Stage: Analysis as a Science of Number 2 3. HusserI's Second Stage: Analysis as a Formal Technique 10 4. Husserl'sThird Stage: Analysis as Manifold Theory 15 5. The Problem ofPsychologism in Husserl's Early Writings 19 CHAPTER II: HUSSERL AND THE CONCEPT OF NUMBER 31 1. The Definition of Number 31 2. The Origin of Number as a Phenomenological Problem 34 3. The Origin of Number in Husserl's Eearly Writings 37 CHAPTER III: THE PRESENCE OF NUMBER 45 1. Sensuous Groups 45 2. Explication 51 3. Comparison 55 CHAPTER IV: NUMBERS AS IDENTITIES IN PRESENCE AND ABSENCE 65 1. Intending Numbers in Their Absence 65 2. The Unity of Number 69 3. The Unity of Large Numbers 76 4. Sedirnented Number Meanings 79 x CHAPTER V: THE SENSE OF ARITHMETIC 89 1. Ideal Numbers 89 2. The Formal Character of the Concept of Number 98 3. Arithmetic as Formal Ontology 100 CHAPTER VI: THE SENSE OF ANALYSIS 109 1 . The Algebraization of Arithmetic 109 2. Theory Forms and Manifolds 113 3. Analysis as Manifold Theory 120 4. Husserl's Attempted Justification of Analysis 125 CONCLUSION 135 NOTE ON ABBREVIATIONS 139 BIBLIOGRAPHY 141 1 CHAPTER I THE EMERGENCE AND DEVELOPMENT OF HUSSERL'S 'PHILOSOPHY OF ARITHMETIC' 1. HISTORICAL BACKGROUND: WEIERSTRASS AND THE ARITHMETIZATION OF ANALYSIS In the preface to his earIy work Philosophie der Arithmetik, HusserI boldly an nounced his intention to make a contribution to that 'desideratum of centuries,' a 'true philosophy of the calculus' (PA 7).1 Despite the title of the work, it was not arithmetic in the usual sense, but rather calculus or 'analysis' itself - that great achievement of modern mathematics, and the field in which HusserI himself had received a doctorate several years earlier2 - which stood at the center of his philo sophical concern. The problems addressed in the work had occupied HusserI for several years prior to the publication of PA, and they would continue to be of in terest to him until the very end of his career. In this study we will examine Hus serI's treatment of these problems and of other problems which developed out of them. Prior to beginning our study of HusserI's own ideas, however, it will prove helpful to glance briefly at the historical context in which they arose. The area of mathematics known as analysis originated in the seventeenth centu ry through the epoch-making discoveries of Newton and Leibniz. The calculating techniques they developed made it possible to deal successfully with a great many problems that had previously been insoluble. These techniques proved especially important, of course, in the development of the mathematical sciences of nature.3 It seems, however, that very little attention was paid at first to the theoretical foundations of the calculus. As one survey of the development of this discipline comments: It was natural that this wide and amazing applicability of the new subject should attract mathematical researchers of the day, and that papers should be turned out in great profusion with seemingly little concern regarding the very unsatisfactory foundations of the subject. It was much more exciting to apply the marvelous new tool than to examine its logical soundness, for, after all, the processes employed justified themselves to the researchers in view of the fact that they worked.4 See notes at the end of this chapter. 2 To be sure, the theoretical obscurities associated with the calculus did not go entirely unnoticed. The concept of the 'infinitesimal,' or of an infinitely small quantity, seemed especially troublesome to some, even though it also appeared to be essential to the calculus. Berkeley referred derisively to Newton's 'fluxions' as the 'ghosts of departed quantities,' while Voltaire described the calculus as an art of 'numbering and measuring exactly a thing whose existence cannot be con ceived.'s But through most of the eighteenth century, mathematicians themselves remained generally unconcerned by these obscurities. They seem rather to have taken to heart the advice which d' Alembert gave to a hesitant mathematical friend: 'Allez en avant, la foi vous viendra!'6 In the nineteenth century, however, a new attitude emerged. Mathematicians themselves grew concerned about the difficulties regarding the fundamental con cepts of the calculus and began to seek a firmer foundation for this important dis cipline. One of the most prominent of these mathematicians was the brilliant and influential Berlin professor Karl Weierstrass. Although major theoretical steps had been taken earlier in the century, it was Weierstrass who finally developed the reso lution which is accepted by mathematicians even today. What Weierstrass advocated has been described as 'a program wherein the real number system itself should first be rigorized; then all the basic concepts of analysis should be derived from this number system." It is this program which subsequently came to be known as the 'arithmetization of analysis.' Through arithmetization, it was felt, all of the obscuri ties associated with the calculus could be eliminated. If the differential, for exam ple, could be derived from the properties of the real number system, there would no longer be a need for the mysterious notion of an infmitely small quantity. Weier strass and his followers eventually showed that this could be done not only for the differential, but also for other concepts of analysis as well. And thus, in the words of one commentary, 'today it can be fairly said that classical analysis has been firmly established upon the real number system as a foundation.'8 The cornerstone of Weierstrass' program was of course the rigorous develop ment of the real number system itself. Mathematicians had spoken of 'rational,' 'irrational' and 'negative' numbers for some time prior to Weierstrass' work. But generally speaking, the understanding of real numbers had been based upon the intuitive notion of a number line. Weierstrass, however, sought to dispense with all such 'geometrical intuitions' and build up the system of real numbers in a purely formal manner. Beginning with a rigorous development of the system of positive whole numbers, he proceeded to generate the system of integers in general, the system of rational numbers and, finally, the system of real numbers itself, through a series of precisely formulated steps.9 In this way, it was thought, the number system which was fundamental to analysis as a whole could be made secure. We must assume that the young Husserl was thoroughly familiar with Weier strass' work on the foundations of analysis. He had studied mathematics in Berlin from 1878 until 1881. In each of the six semesters he spent there, he had attended Weierstrass' lectures and taken extensive notes.10 And after earning his doctorate in Vienna in 1883, he had returned to Berlin for a brief period to serve as an as-
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