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Philosophical Essays for Alfred North Whitehead, February Fifteenth UNIVERSITY OF FLORIDA LIBRARIES COLLEGE LIBRARY Digitized by the Internet Archive in 2010 with funding from Lyrasis IVIembers and Sloan Foundation http://www.archive.org/details/philosophicalessOOwhit PHILOSOPHICAL ESSAYS FOR ALFRED NORTH WHITEHEAD PHILOSOPHICAL ESSAYS for ALFRED NORTH WHITEHEAD February Fifteenth Nineteen Hundred and Thirty-Six NEW YORK/RUSSELL ^ RUSSELL FIRST PUBLISHED IN 1 936 REISSUED, 1967, BY RUSSELL & RUSSELL A DIVISION OF ATHENEUM HOUSE, INC. L. C. CATALOG CARD NO: 66-24769 PRINTED IN THE UNITED STATES OF AMERICA FOREWORD The following essays were written for Alfred North White- head by a group of his students. The membership of this, as of any such group, is arbitrary. It might have been either larger or smaller. The individuals composing the group, however, represent many different tendencies in contempo- rary American philosophy, and the volume indicates some, at least, of the many directions in which the thought of Whitehead is being felt. The arrangement of the essays calls for a word. Three are distinctly more historical in approach than the others. They have been placed first, and are followed by the remaining ones, arranged according to their subjects, which fall roughly under the headings of logic and methodology, metaphysics and ethics. The essays express the interest of their writers. Each deals with one special topic, and at the same time suggests the general philosophical viewpoint of its author. Consequently, the content of the volume is extremely varied. Nevertheless, certain leading ideas appear in many of the papers, though in very different forms and contexts, and thus relate the papers both to each other and to the philosophy which has influenced them all. Whether there is anything of the spirit of that philosophy in them — and the hope that there is constitutes the sole excuse for the appearance of this volume — the reader must judge. CONTENTS PAGE The Mathematical Background and Content of Greek Philosophy i F. S. C. Northrop, Yale JJjiiversity The One and the Many in Plato 41 Raphael Demos, Harvard University An Introduction to the De Modis Significandi of Thomas OF Erfurt . 67 Scott Buchanan, University of Virgijiia Truth by Convention 90 Willard V. Quine, Harvard University Logical Positivism and Speculative Philosophy . 125 Henry S. Leonard, Harvard U?iiversity The Nature and Status of Time and Passage . 153 Paul Weiss, Bryn Mawr College Causality 174 S. Kerby-Miller, Reed College The Compound Individual 193 Charles Hartshorne, University of Chicago The Good 221 Otis H. Lee, Fomona College Index . 245 THE MATHEMATICAL BACKGROUND AND CONTENT OF GREEK PHILOSOPHY By F. S. C. Northrop The foundations of modern mathematics were laid by Georg Cantor. His work led to the systematic derivation of mathe- matics from logic which was accomplished by Russell and Whitehead in Principia Mathematica. This required an entirely new theory of logic. The new logic in turn had a profound and far-reaching effect upon philosophy. It is now know that Cantor's theory leads to contradictions. Moreover, these contradictions are so fundamental that they appear in the logic as well as the technical mathematics. The significance of this for contemporary logic, mathematical science and philosophy should be obvious : Traditional modern theories of these subjects cannot be trusted. New mathematical, logical, and philosophical theory must be constructed. The proper way to determine the character of this new theory seems evident. Cantor's basic assumption, upon which modern mathematical theory rests, should be examined to de- termine the fundamental problem with reference to which his theory is a particular answer. Once this problem is known we should be able to designate the other possible an- swers which it permits and thereby gain a clue to a possible way out of our present difficulties. A reading of Cantor's original papers will show that his basic assumption is the existence of an actual infinite. It will indicate also that he regarded his mathematical ideas not merely as providing a new and rigorously fornmlated theory of mathematics but also as presupposing and requiring an original answer to the metaphysical problem of the one and 2 PHILOSOPHICAL ESSAYS the many/ It appears, therefore, that the fundamental problem which we face in logic, mathematics and philosophy at the present moment is, at bottom, none other than this old metaphysical issue. If we are to avoid falling into old errors in our present attempts to provide it with a solution, it seems to be essential that we should know what previous answers to this problem were given, what mathematical considerations led to their origin, and what later mathematical discoveries caused their rejection. These alternatives to the modern theory of the foundations of mathematics are to be found in great part in the Elements of Euclid. In order to get them before us, it is necessary to arrange the thirteen books of Euclid in the order of their temporal origin. When this is done and the Greek philosophers are located in their proper temporal positions with reference to this mathematical background, it happens that considerable new light is thrown upon the content and development of Greek philosophy. Several centuries of scientific investigation are at the basis of Greek philosophy. Unless the science is known, the philo- sophical theories are devoid of content and easily misunder- stood, and the source of their authority for classical and medieval minds is completely missed. Mathematics and astronomy were the mature and leading sciences in the Greek period. Furthermore, mathematics was considered as a natural science. The modern conception of it as a subjectively-created subject which deals only with the possibles had not arisen. Instead, it was regarded as the basis of the actual and the necessary in nature. This conception was most natural. The original notions in mathematics were magnitude and number. They had their basis and origin in the observed continuous extension of nature and the many diverse things within this one continuum. The continuity 1 Georg Cantor. Gesammelte Abhandlungen. 1932. Especially pp. 370-377. F. S. C. NORTHROP 3 of nature gave birth to geometry ; its diversity brought forth arithmetic. In reaching a theory of the relation between these two factors, the fundamental mathematical problem concerning the relation between geometry and arithmetic, and the corresponding and more general fundamental metaphysical problem of the one and the many, arose. Anaximander was the first to explicitly designate nature as a single continuum. He called it the "Boundless." However, its first scientific formulation appeared when Anaxagoras de- fined it in terms of the principle that "there is a smaller but never a smallest." ^ This denies the existence of extended segments or atoms which are not further divisible, and clearly puts the conception of nature as irreducibly atomic into op- position with the conception of it as a real continuum. The Anaxagorean theory had its difficulties. Strictly speaking, it provided mathematics with but one actual entity — namely, extended nature as a whole. But a science of mathematics is impossible unless there are elements upon which to operate. To meet this demand the infinitely divisible continuum was conceived as made up, in any finite segment, of an infinite number of points. An examination of the first two definitions of Book I of Euclid will show that Greek mathematics began with this conception. But Zeno raised the question : Do the points have extension, or are they without magnitude ? He had no difficulty in showing that either assumption reveals a contradiction in this first theory of the foundations of mathematics. If the points have extension then there can be only a finite number of them in a finite magnitude and the Anaxagorean principle that any magnitude is divisible without limit is invalid. If, on the other hand, the points have no extension, then even an infinite number of them will give no magnitude whatever, and the assumption that nature is an extended continuum is contra- dicted. This first instance of the demonstration of a contra- 2 E. Frank. Plato und die Sogenannten Pythagoreer. Halle. 1923. p. 47. 4 PHILOSOPHICAL ESSAYS diction in the foundations of mathematics constituted a "veritable scandal" in Greek science.^ The first attempt at a resolution of the paradox was made by Democritos. He drew the obvious conclusion. Since nature is extended, and this cannot be the case if it be con- stituted, in any segment, by an infinite number of points, it follows that there must be an end to division and any finite interval must be constituted of a finite number of extended in- divisible points. Luria has recently shown, in an exceedingly important article,* that the atoms of Democritos Vv^ere arith- metical, as well as physical, units and that his atomic theory arose as much to meet Zeno's attack upon the traditional theory of the foundations of mathematics, as it did to meet Parmenides' arguments concerning the invalidity of the tradi- tional concept of motion in physics. The evidence in support of this first attempt to reduce geometry to arithmetic is well known. ^ The Pythagorean study of music and of triangular, square, and rectilinear figures and numbers supported it, as did Democritos' impor- tant work in founding the experimental and mathematical science of acoustics.*^ The theory worked beautifully ex- cept for the discovery of incommensurable magnitudes. Give arithmetical unit any fixed magnitude one pleases the atomic ; — if it goes into the side of a square a definite number of times leaving nothing over, it will not do so in the diagonal. This discovery of what we term the irrational, produced the sec- ond revolution in the fundamental concepts of Greek mathe- matics. It made a tremendous impression upon the Greek 3 H. Hasse u. H. Scholz.
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