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70-13,977 BAUM, Robert James, 1941- | GEORGE BERKELEY'S PHILOSOPHY OF MATHEMATICS. | ■'3 The Ohio State University, Ph.D., 1969 J Philosophy University Microfilms, Inc., Ann Arbor, Michigan © Copyright by Robert James Baum f 197o| THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED GEORGE BERKELEY'S PHILOSOPHY OF MATHEMATICS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By - Robert James Baum, B-»A< * w &w &w- m, & The Ohio State University 1969 .) Approved by PLEASE NOTE: Not original copy. Blurred and faint type on several pages. Filmed as received. UNIVERSITY MICROFILMS. ACKNOWLEDGMENTS I wish to thank the members of the Department of Philosophy, particularly Professors Turnbull, Hinshaw, Nelson, Rosen, and Kielkopf, for providing me with a solid foundation for doing technical philosophy v/hile at the same time communicating their intense enthusiasm for the subject. In particular, I would like to thank Professor Paul Clscamp for the innumerable ways in which he has contributed both to my general education and to the formulation and development of this dissertation. Without his continual encouragement, as well as his numerous constructive criticisms and suggestions, this dissertation would never have been completed in its present form. VITA October 19, 19^1 • • • Born - Chicago, Illinois 1963 ......... B.A., Northwestern University, Evanston, Illinois 1963-196^ ...... Teaching Assistant, Department of Philosophy, Northwestern University, Evanston, Illinois 196^-1965 ...... Teaching Assistant, Department of Philosophy, The Ohio State University, Columbus, Ohio 1965-1967 ...... Visiting Lecturer in Logic and Philosophy, Kiddle East Technical University, Ankara, Turkey I967-I969 Teaching Associate, Department of Fhilosophy, The Ohio State University, Columbus, Ohio 1969- ...... Assistant Professor, Department of Philosophy, Rensselaer Polytechnic Institute, Troy, New York TABLE OF CONTENTS Page ACKNOWLEDGMENTS ......................... ii YITA ................................... iii LIST OF ABBREVIATIONS ........ ...... v INTRODUCTION .................. 1 Chapter I. BERKELEY'S PREDECESSORS............. 26 Hobbes The Mathematicians Locke Bayle II. BERKELEY'S GENERAL THEORIES ....... 150 Signs Meaning Truth III. BERKELEY'S CRITICISMS OF HIS PREDECESSORS ............. 230 IV..BERKELEY'S PHILOSOPHY OF MATHEMATICS . 262 V. SUMMARY................................31^ BIBLIOGRAPHY .......................... 322 iv LIST OF ABBREVIATIONS All references to Berkeley*s works in the body of the text are to the chapter and/or paragraph numbers, with the exception of references to the Three Dialogues between Hylas and Philonous, where the page number of volume 2 of the Luce and Jessop edition of the Works is given. The following abbrevi­ ations are used: Principles - A Treatise Concerning the Principles of Human Knowledge. Commentaries - Philosophical Commentaries (Luce's editio diplomatica of' 19^-) • Essay - An Essay Towards a New Theory of Vision. Dialogues - Three Dialogues between Hylas and Philonous. Alcinhron - Alcinhron or the Minute Philosopher. Theory - The Theory of Vision . Vindicated and Explained. Defence - A Defence of Free-thinking in P, at hematics. v INTRODUCTION George Berkeley's theory of immaterialism is unquestionably one of the most original and influential contributions to modern philosophy. It is still being debated today. Many claims to its refutation have been made, but it continues to thrive, although sometimes under new names such® as "phenomenalism." It is perhaps because of this impact of the theory that until recently little notice has been taken by philosophers and histor­ ians of philosophy of Berkeley's prolific writings on a diversity of other issues. It is not unusual that a man who has one brilliant insight into a particular subject-matter will also have important insights into other problems. Newton, for example, not only discovered the fundamental theorem of the calculusi his formulation of the three laws of mechanics and his hypothesizing of a universal •gravitational' force, his studies in optics, and other accomplishments are of equal or-greater 1 importance. His arch-rival, Leibniz, likewise was not restricted to an isolated achievement such as the development of the differential cal­ culus, It is thus not surprising to find that contemporary scholars are finally discovering that the genius of George Berkeley was not restricted to his arguments for the theory of immaterialism. It should not even be considered strange that some are beginning to suggest that although the impact of the immaterialist hypothesis on the philosophical thought of the last two centuries is of the greatest importance, Berkeley's treatment of it is of secondary importance to present-day philosophy when compared with his discussions of and insights into other equally problematic issues. Among contemporary writers, C.M. Turbayne, K. Popper, P. Olscamp, and others believe that the key to the whole of the Berkeleyan philosophy (that is, not merely to the immaterialist hypothesis) is his general theory of signs. The arguments for immaterialism are explicitly stated, among his published works, only in the Principles and Three Dialogues. whereas the theory of signs is discussed in these as well as in most of his other works, including An Essay towards a New Theory of Vision. Alciphron. De Motu, and Siris. It has been shown that the theory of signs is basic to his ethical theory and his arguments for the existence and nature of God (by Olscamp), his philosophy of science (by Popper), as well as to his optics and the theory of immaterialism (by Turbayne). It has also been shown that the theory of signs is not only basic to the rest of Berkeley's system, but that it has a great deal of merit on its own and is anticipatory of theories developed in the twentieth century, such as those of Morris and Stevenson. Whatever the reasons may be for the neglect of this aspect of Berkeley's thought, its recent rediscovery has forced both a re-evaluation of the traditional attitudes towards Berkeley and a more careful analysis of hitherto neglected segments of his writings. In line with these recent developments in the study of Berkeley's philosophy — the new emphasis on and explication of the theory of signs, and the investigation of other elements of his thought — I have examined in detail Berkeley's writings on the philosophy of mathematics. The resulting discoveries are quite surprising and must be considered of major importance for the history of ideas. Only a few partial examinations have been made of this component of Berkeley's philosophy, and none have been made since the recent explications of his theory of signs. In addition to providing the first complete statement and analysis of Berkeley's philosophy of mathematics, and the first interpretation of it in relation to his theory of signs, it is my intention to place it in its proper historical perspective — another difficult task which up to this time has not been satisfactorily performed. These tasks are mutually interdependent; one cannot adequately understand and explicate Berkeley's theory without understanding the historical context in which it was originally formulated, and vice versa. However, once the theory has been restated it will be both possible and profitable to examine it out of the historical context and to evaluate it on its own merits as an independent theory. How does one explain the long neglect of Berkeley's writings on mathematics? That they were totally misunderstood by most of his less- astute contemporaries is clear, although the answer to the question "Why?" is far from clear. It is possible that the less sophisticated and more explicit writings of his immediate successors such, as Hume and Kant on the same topic may have dis­ tracted readers who had ready access to only a few of his relevant works. It is even more difficult, however, to account for the lack of attention paid to this element of Berkeley’s thought by philosophers of the present century. All, or almost all, of the relevant source material has been readily available since Fraser's edition of the Collected Works in 1871— and, what is more important, the quantity of material is such as to make it impossible for it to be overlooked. More than one-third of what is generally taken to be Berkeley's 'philosophical' writings are directly or indirectly concerned with mathematical problems, yet only a minute fraction of the articles and books written on Berkeley in the last century have dealt with this aspect of his thought.^ Nothing has ever been published in English on more than isolated elements of his full-blown theory, and the non-English works, in addition to sharing the general inadequacies, are virtually •^Concrete evidence for this claim can be found in Colin Murray Turbayne and Robert Ware, "Bibliography of George Berkeley, 1933-1962," Journal of Philosophy, LX (1963), PP» 93-112. inaccessible to the casual student. One possible explanation for this is that Philosophy of Mathematics is only now beginning to come into its own as a quasi­ independent discipline. The subject-matter is still being defined and thus research into the historical backgrounds of these questions is just beginning. (That it has begun is evidenced by the recent publication of several works including Anders Wedburg's Plato's Philosophy of Mathematics and Hippocrates Apostle's Aristotle's Philosophy of Mathematics.) It is generally recognized that our understanding and appreciation of any subject of human knowledge is always increased by an understanding of its historical development. This is certainly the case with the Philosophy of Mathematics, and as I will show, George Berkeley stands out as one of the most original thinkers in its history. As with many of Berkeley's works-, his writings on mathematics often have strong aplogetic and polemical overtones. Since this 'emotive' content is of psychological rather than philosophical interest, I shall not discuss it further, except in those places where it is so strong as to obscure the theoretical content of a particular passage.

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