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.

Just as the motivation for these writings is irrelevant to the truth or falsity of the claims being made, the emotions expressed through them are irrelevant to their theoretical and descriptive content. It is also important, however, to dis­ tinguish two important aspects of the cognitive content of Berkeley's mathematical writings— one which may be called the 'technical' as opposed to the other which may be refered to as the 'philosophic aspect. The technical element involves his concern with the methods for the solutions of particular mathematical problems, e.g. the squaring of the circle, finding the tangent to a curve, etc., or with the proofs of specific theorems. Most dis­ cussions of Berkeley's specifically mathematical writings have concentrated on these points, either totally ignoring or at best de-emphasizing the philosophical content. In contrast to this approach,

I shall discuss the technical aspect only in so far as specific cases illuminate or serve as examples for the philosophical parts of Berkeley's mathematical thought.

I must here emphasize that when I refer to Berkeley's 'philosophy of mathematics', I am not suggesting that he was consciously writing on 8

the specific subject-matter that was first clearly

defined by Frege less than one hundred years ago.

As I have already indicated, a more accurate

description of my topic would be "the philosophical

aspect of Berkeley's mathematical thought." I

will show in the next chapter that mathematics

itself was still not a clearly defined discipline

at the beginning of the eighteenth century, and

it was often quite debatable whether one was doing

physics, meta-physics, theology, or mathematics. The

problem of the delineation of the subject-matter of mathematics is in fact one of the basic problems

of the philosophy of mathematics} it is the one with which Berkeley was most deeply concerned, and whose

solution he saw more clearly than any philosopher

prior to the twentieth century. I shall refer to

it simply as 'the basic question.'

Philosophers have found that there are a

number of questions related to the basic question,

some of which are t

1) What is the ontological status of mathe­ matical entities (e.g. numbers, figures, etc.)?

2) What is the epistemological status of these entities (i.e. what is the relation between the mathematician as knower and the entities as known)? 9

3) What is the nature of mathematical certainty and how is it different (if it is different) from other scientific truths, e.g. laws of physics? (In Kantian terms, are mathematical truths analytic or syntheic? apriori or aposteriori?)

Although not explicitly concerned with the answers to all of these questions, Berkeley does provide ample discussion of the basic question itself so that it is possible to formulate his answers to the related ones without becoming involved with conjectures and specu­ lations as to how he 'might' have answered them. But although his answers to these questions are stated explicitly in his various writings, they are not always obvious, and an accurate restatement of them must avoid a number of obstacles which have hindered earlier studies. I will point out some of these

•obstacles' in a brief examination of the errors prevalent in the few earlier accounts.

The incompleteness of previous studies has manifested itself in a variety of forms. One of the most prevalent 'sins of omission* has been the ignoring of the historical context of the mathematical aspect of Berkeley's writings— particularly the relevant elements of the intellectual environment of the period surrounding the 'invention' of the infinitesimal calculus, i.e. the second half of the seventeenth century. Berkeley was well-educated and very competant with regard to all aspects of the mathematics of this period. Its influence on his thought is revealed by his direct reference in his preliminary notebooks (i.e. the Philosophical

Commentaries) to the mathematical writings of most of the influential thinkers of the time, including

Wallis, Hobbes, Barrow, Newton, Locke, Leibniz,

Keill, and others. Berkeley's criticisms.of his predecessors cannot be understood without some knowledge of their positions as well. Likewise, and perhaps most important, the striking clarity and originality of Berkeley’s thought can be appreciated only when one is aware of the generally dominant atmosphere of confusion, befuddlement, and unthinking conformity surrounding all mathematical discussion at this time. Although several competant studies of the technical developments, of the period have been made (notably those of Cajori, Boyer, and

Whiteside), nothing has been published on the philosophical developments (or lack of development) in the field of mathematics. To fill this void, and to lay the necessary groundwork for a complete 11 discussion of Berkeley's contributions, I shall provide a summary of the philosophical attitudes towards mathematics in the second half of the seventeenth century in the next chapter.

The lack of secondary material on the philosophy of mathematics in the seventeenth century makes the ignoring of the historical context somewhat understandable, but there is no similar excuse for the equally common failure to examine all of Berkeley's writings before attempting a formulation of his theory. Most of the studies of this topic have been unjustifiably restricted to only some of the relevant materials, most commonly to one or two of the Philosophical

Commentaries. the Principles, the Essay, or The

Analyst. Not only is a careful analysis of all of these works necessary for a complete study, but a number of other writings contain important discussions of various related issues. These other relevant wc_-ks include Of infinites. De Motu,

The Theory of Vision— Vindicated and Explained.

Alciphron, and Siris.

A number of conflicting and confusing state­ ments have appeared in the partial studies made 12 to date. One example is found in William Kneale's discussion of Berkley in his Development of Logic.

Having found several passages in the Philosophical

Commentaries which, when taken out of the context of Berkeley's general philosophy, may appear to the twentieth century reader to contain suggestions of the modern theory of 'conventionalism', Kneale proceeds to make the following comment:

Being himself a mathematician who had made great discoveries, Leibniz could not accept the conventionalist theory of necessity, but his younger contemporary Berkeley thought that it was a natural consequence of his own empiricist philosophy,,. But these suggestions fin the Philosophical Commentaries"! were not published until the end of the nineteenth century, and the close alliance of conven­ tionalism and empiricism was not established until our own time.2

Ignoring for the moment the always risky business of applying twentieth century categories to eighteenth century theories, as well as Kneale's intimation that Berkeley was not a competant mathematician, it is important to note that Kneale's basic claim is simply false. Although it is true that the

Philosophical Commentaries was not published until

^William Kneale and Martha Kneale, The Development of Logic (Oxford: Oxford University Press, I962), p. 312. 18?1t the passages which Kneale refers to as

'conventionalistic' appeared in much more developed

form in Berkeley's original publications (for which

the Commentaries had served as a notebook)— the

Essay and the Principles— as well as in the later work Alciphron. Thus if Berkeley was a conven­

tionalist, as Kneale claims, and" if he was an

'empiricist', then he must be given credit for

being the first to establish "the close alliance

of conventionalism and empiricism." That this

is at least a questionable claim, based on an

inadequate examination of Berkeley's writings,

is made quite apparent by observing another claim based on an equally inadequate study. Carl Boyer, in his History of the Calculus and Its Conceptual development, grounds his interpretation almost entirely on a reading of Berkeley's essay The

Analyst: his conclusion is the exact contrary of

Kneale's.

Berkeley was unable to appreciate that math­ ematics was not concerned with a world of 'real' sense impressions. In much the same manner today some philosophers criticize the mathematical conceptions of infinity and the continuum, failing to realize that since mathematics deals with relations rather than with physical existence, its criterion of truth is inner consistency rather than 14

plausibility in the light of sense percep­ tion, or intuition.3

The juxtaposition of these two statements would

seem to indicate that Berkeley's theory underwent

considerable revision, if not total reversal, in

the twenty-five years between the Philosophical

Commentaries and The Analyst. As I shall show,

however, his position remained essentially un­

changed throughout his career, and these two works,

taken in conjunction with his other writings,

express a single coherent theory. In showing

this unity of his thought, I shall also try to

indicate the exact ways in which Kneale and Boyer are mistaken. In addition, it will follow directly that claims (such as the following by Boyer) to the effect that Berkeley's theory has no relevance to contemporary discussions will also have to be reconsidered, since they too are founded on an incomplete reading of the Berkeleyan corpus.

"...Those of Berkeley's arguments which are based on the inconceivability of the notions involved

^ C a r l b. Boyer, The History of the Calculus and its Conceptual Development (New Yorks Dover Publications, inc., 1959)* P« 227. 15 lose their force in the light of the modern view of the nature of mathematics.

A third "sin of omission" has also appeared frequently in the previous studies, although it is more subtle and complex than the two already discussed. Basically, it involves one or both of the following mistakesi

1) The disproportionate over-emphasizing of a specific theme or element of the many different lines of thought which are in­ tricately interwoven throughout Berkeley's writings.

2) The examination of one theme in total isolation from all other aspects of his thought.

One possible example of the first mistake is A.A.

Luce's statement of the main purpose of the Philosophical

Commentaries. (I say 'possible* because Luce does not develop this point further, nor does he attempt to offer any other evidence in support of it. I assume that he did take this interpretation seriously, however, not only for the Commentaries but for

Berkeley's works as a whole, since he usually discusses Berkeley's mathematical thoughts in the context of their relevance to the immaterialist

^Ibid., p. 228. 16 hypothesis.) Luce states that

...the main doctrinal purpose of the Commentaries was the examination, exposition, and defence of immaterialism. Berkeley studied vision to show that we do not see matters he studied knowing to show that we do not perceive matters he studied physics to show that gravity is not proportional to matters he studied the mathematical problems to show that there is no infinite divisibility to be ascribed to matter.5

This claim is highly speculative, and though it may contain a great deal of truth, there is little or no concrete evidence to support its on the other hand, there is also little concrete evidence to refute it. Luce's error then (and one made frequently by others) is not making a wrong interpretation, but is in committing himself to a very narrow interpre­ tation with far from adequate justification. • There can be no doubt that Berkeley used mathematical arguments to support immaterialisms the relevant passages in the Principles speak for themselves on this issue. However, I must point out again that Berkeley was an extremely intelligent and original thinker, and that it is quite possible that his thoughts on mathematics, physics, and

^A.A. Luce, ed.. Berkeley's Philosophical Commentaries, editio diplomatics.. ("Edinburgh: Thomas Nelson & Sons, Ltd., 19^), p. xxxiv. other topics were originally conceived independently

of the immaterialist hypothesis, and were only later

applied to support it. It will later become

apparent that when his mathematical writings are

interpreted as having been intended solely as a

buttress for his immaterialism, the most important

and original ideas are easily overlooked. (I

suspect that Luce*s attitude was also prevalent

among Berkeley’s contemporaries and immediate

successors, thus explaining, in part at least,

their misunderstanding of his philosophy of mathe­ matics.) It is important to note that in fact

immaterialism neither supports nor contradicts

Berkeley's philosophy of science and mathematics, while the latter is interdependent with his theory

of signs.

I find it very difficult to account for

several other errors that have been made in the previous studies, although it would seem that the most likely explanation is the total disregard

of certain aspects of Berkeley's thought. E.W.

Strong apparently read most of Berkeley's writings, and was at least partly aware of his historical context, and still he bases his whole discussion on the following interpretation* 18

Toward the end of his first notebook, Berkeley jots down the following Axiom. "No reasoning about things whereof we have no idea." This is his empirical rule, test, or criterion for admitting an entity into mathematics as a meaningful "object" or else designating it as a meaningless "nothing". The rule stated, the conclusion follows: "Therefore no reasoning about Infinitesimals."6

Luce, in his note on this passage, emphasizes that

Berkeley rejected this 'axiom' before the end of the Commentaries ; but Strong assumes this to be the basis of all Berkeley's mathematical writings, thus forcing him into a rather simple-minded

'empiricist' position which is at best a caricature of his actual theory. Thus Strong concludes that

"He was more successful in forcing questions about philosophy of mathematics that needed to faced than he was in handling them in the way he wanted— the way common sense directed to the strictly sensible."7

Strong did recognize the relevance of the theory of meaning to the discussion of the philosophy

^E.W. Strong, "Mathematical Reasoning and its Object," University of California Publica­ tions in Philosophy (Berkeley and Los Angeles, 1957), P. ^9. ?Ibid., p. 88 19 of mathematics, but then proceeded to make the wrong interpretation of it.

Another element of Berkeley's thought relevant to his philosophy of mathematics is his

'pragmatism' or 'instrumentalism'. Actually, neither of these terms accurately describes the view he actually holds, but they are the best terms available in the present-day philosophical vocabulary and I shall have to use them from time to time. Whenever

I do use them I shall include a reminder that they are not necessarily to be taken in their modern technical sense. Others have recognized this aspect of Berkeley's account, but they have not been sure how to interpret it, and have thus come up with contradictory claims. Strong follows his basic emphasis on a simple-minded empiricism, and claims that the 'instrumentalist* aspect is totally secondary to the 'sensationalism*. "The domination of the empirical criterion over the criterion of use could have been avoided if Berkeley had followed up the lead of 'notions considered*. Instead,he stands ready to jettison instruments of mathematical analysis, no matter what their power to solve problems, if they suppose elements which sense 20 cannot perceive or involve operations which sense Q cannot follow."

Gerhard Stammler arrives at a similar conclusion, although he sees the ‘pragmatism* as being complementary to the 'empiricism'. "Arithmetik und Geometrie sind beides 'praktische Wissenschaften' von Bezeichnungen, die nich-t urn ihrer selbst willen betrachtet werden sollen, sondern bei der ersten auf Zahlen von Gegenstanden, bei der letzteren auf

Messungen von Linien und Proportionen bezogen werden mussen. Beide have nur so lange Wert, als sie mit

Dingen arbeiten, die perzipierbar sind.”9 My point is not that of determining which of the two is more I correct; it is rather that of showing both to be mistaken. Although both have detected some sort of

'pragmatic' element in Berkeley's thought, neither have attempted to examine and explicate it in sufficient detail, and thus neither is really certain what to do with it. Both escape this predicament by appealing again to the generally

8Ibid.., p. 81.

^Gerhard Stammler, "Berkeley's Philosophie der Mathematik,” Kant-Studien. Erganzunghefte. Nr. 55 (Berlin, 1921), p.33» 21 accepted clichd— "Berkeley is first and foremost an •empiricist'."— and then sweep this new feature

(i.e. the 'pragmatism') under the old carpet. Thus, rather than attempting to initiate an exploration of the uncharted territory of Berkeley's theory of meaning and its pragmatic overtones, such scholars have turned back at the boundries and remained in traditionally secure (and quite sterile) territory.

To discover Berkeley's complete philosophy.of mathematics it is necessary to re-examine these territories carefully, and with no preconceptions.

Otherwise, the only possible conclusion is one similar to that which Stammler gives in his discussion of Berkeley's relation to his contemporary, the

Scottish mathematician Colin Maclaurin.

Durch diese Gegenuberstellung, die wir spater noch zu Gelegenheit finden werden, wird die Stellung Berkeley's urn vieles Klarer. Die historische Bedeutung unseres irischen Denkers entbehrt nicht einer gewissen tragischen Ironie. Sie besteht darin, dass durch sein Werk einer der hervorragendsten Mathematiker veranlasst wurde, sich mit den Grundlagen seiner Lehre ausfuhrlich zu befassen; wobei er fand, wie wir spater sehen werden, wie unhaltbar das Berkeley'sche Denken ist, so scharfsinnig es auch genannt werden muss. Durch seine Theorie der Mathematik wurde die Vernichtung seiner Philosophic herausgefordert.10

10Ibid.. p.58 22

The true "tragic irony" is the fact that even

Berkeley's most sympathetic interpreters have failed to understand his philosophy of mathematics, and thus have not recognized that it is the mathema­ tician's position, rather than Berkley's, which is ultimately indefensible.

The final error which I must mention is one which pervades not only the writings I have been discussing thus far, but also infects a large part of contemporary studies in the history of ideas. This tendency is that of 'pigeon-holing' theories formulated centuries ago in categories devised in recent times, e.g. 'empiricism', 'phenomen­ alism', 'idealism', 'pragmatism', etc. In addition to the danger of attributing ideas to a thinker which he perhaps never even considered (and in many cases which he could not have possibly been aware of), the over-use of most of these terms has resulted in their loss of descriptive content— and thus of all utility. A prime example of the indiscriminate application of these categories is the following statement by Boyer:

It is interesting to notice that just as the arch-materialist Hobbes, being unable to con­ ceive of lines without thickness, denied them to geometry, so also Berkeley, the extreme idealist, wished to exclude from mathematics 23 the * inconceivable' idea of instantaneous velocity. This is in keeping with Berkeley's early sensationalism, which led him to think of geometry as an applied science dealing with finite magnitudes which are composed of indivisible 'minima sensibilia'

The italicized terms in this passage add nothing to the attempted statement of Berkeley's position.

They do, however, place a wedge between Hobbes' and Berkeley’s theories which in fact is not there, and they also seem to have led Boyer into making totally wrong interpretations of all the relevant issues. This is but one example from a single study; there are similar examples which could be given from each of the other studies or discussions of Berkeley's philosophy of mathematics. It suffices for my purposes merely to point out this fact.

It would require a complete independent study to examine the uses of each of the many categories which have been applied to Berkley's writings and to analyse the ways in which each has resulted in misinterpretations.

It is especially important to notice with regard to the study of philosophy of mathematics that the basic categories of this discipline— e.g.

•^Boyer, oo.cit.. p. 227. (Italics mine.) 2^

'formalism', 'logicism', 'intuitionism*, etc.— were not defined until this century— that is, until after the development of non-Euclidean geometries, set theory, axiomatics, and other revolutionary innovations. Thus, the application of any of these terms to the writings of mathematicians and philoso­ phers before 1800 would result in the possibility of very serious misinterpretations of these early theories. It is for this reason that I shall refrain from using not only the vague and generally non-descriptive categories discussed in the previous paragraph, but also the extremely descriptive vocabulary of contemporary philosophy of mathe­ matics in my presentation of the thoughts of the seventeenth and eighteenth century writers.

In summary, my discussion of the philoso­ phical aspect of Berkel^'s mathematical writings

(or, as I shall usually refer to it— his philosophy of mathematics.) will have the following character­ istics which have been lacking in previous studies.

First, it will be complete in the sense that all of

Berkeley's relevant writings will be examined, and also because I shall consider all of Berkeley's related theories— not just his immaterialism.

A particularly unique aspect of my approach will be the examination of Berkeley's theories of meaning and truth in the context of his general

theory of signs. I shall also refrain from using

the usual philosophical 'pigeon-holes' in my discussion, and I shall not attempt to apply present-day technical terms to eighteenth century writings. Finally, I shall attempt to place

Berkeley's philosophy of mathematics in its proper

historical context, a task which will begin now with a presentation of the philosophy of mathematics as it stood in the second half of the seventeenth century. CHAPTER I

As I have indicated, an understanding of

at least certain features of the seventeenth

century writings relevant to the issues of the

philosophy of mathematics is essential for a

correct interpretation of Berkeley's contributions

to this subject-matter. Such a study would require

a careful examination of all the published and

unpublished works of the period which are con­

cerned in any way with mathematics— a task far beyond the scope of the present.discussion.

I shall only examine those features of the theories most relevant to Berkeley's theory, and to which

he refers explicitly in his writings. The basic attitudes of his predecessors towards these issues are related to my discussion of Berkeley's theory

in three ways: insofar as 1) they were the object

of his critical remarks, 2) they provided possible

sources for his own philosophy, and 3) they present

contrasts to his truly original contributions. For reasons which will become apparent in the course 26 27 of my discussion, it is most ‘natural' to divide

this part of the discussion into four sections, in which I shall discuss Thomas Hobbes, the 'Mathe­ maticians', John Locke, and Pierre Bayle,

I

Thomas Hobbes' mathematical writings have been as seriously mistreated by historians of ideas as those of Berkeley. Most discussions of Hobbes do not go beyond the fact of his 'discovery' of

Euclidean geometry at the age of forty, and his subsequent 'unfortunate* controversy with Wallis over his attempted squaring of the circle and doubling of the square. But those who have gone further have quickly run into disputes over the nature of his position on a number of important issues. As a case in point, Krieale has asserted that Hobbes' writings contain "the beginning of the conventionalist theory of necessary truth,"1 while J.W.N. Watkins argues that if we take into account certain other elements of his writings, we find that his statements of necessary truth

"may be open to a less conventionalist reading

^•Kneale, op. cit.. p.312. 28 than they at first seemed to call for."2 What is of basic importance to my present discussion is what appeared in his major published works that is related to Berkeley's theory in any of the three ways listed in the last paragraph. In this section

I shall present a series of quotations from Hobbes' basic works with which Berkeley was directly or indirectly acquainted, with the intention of relating them to Berkeley's theory in the later chapters. I shall make no effort to prove that this is Hobbes' ''real" theory.

Hobbes' discussions of mathematics are closely connected with his theories of meaning and truth. The basic principle of his theory of meaning is that all words, with the exception of the copula 'is', are either names of things or else meaningless. A second principle is that all names can function either as 'marks* or 'signs't marks enable us to "remember our own thoughts", whereas signs are tools by means of which we

"make our thoughts known to others". In general, signs (and probably also marks) may be either

'natural' or 'artificial.'

2 J.W.N. Watkins, Hobbes' System of Ideas (Londons Hutchinson & Co, Ltd,.1965)* P« 1^9• 29

• ••.Those things we call SIGNS are the antece­ dents of their consequents, and the consequents of their antecedents. as often as we observe them to go before or follow after in the same manner> For example, a thick cloud is a sign of rain to follow, and rain is a sign that a cloud has gone before,... And of signs some are natural, whereof I have already given an example, others are aribtrarv. namely, those we make choice of at our ov/n pleasure, as a bush hung up, signifies that wine is. to be sold there; a stone set in the ground signifies the bound of a field; and words so and so connected, signify the cogitations and motions of our mind.3

Names, however, are explicitly asserted to be arbitrary signs, the only connection between the sign and the thing signified being that contributed by the person doing the naming.

...I suppose the origin of names to be arbitrary, judging it a thing that may be assumed unquestion­ able. For considering that new names are daily made, and old ones laid aside; that diverse nations use different names, and how impossible it is to observe similitude, or make compari­ son be+wixt a name and a thing, how can any man imagine that the names of things were imposed from their natures? ... Nor had mathe­ maticians need to ask leave of any but them­ selves to name the figures they invented, parabolas. hyperbolas. cissocides. etc. or to call one magnitude A, another B.^

3Elements of Philosophy I, ii, 2-3, in Works. Vo 1,1, pp._14-16, (All references to Hobbes', writings will be to the volume and page numbers of The English Works of Thomas Hobbes, ed. by W, Molesworth, (Londons 1839-^-5^ • )

^Elements I, ii, 4 in Works I, p. 16. Hobbes* justification for this assertion may seem unnecessary to the modern reader, but it was clearly contrary to an opinion widely accepted at that time, namely the Biblical theory that language was given to Adam by God in its fully developed form. In one interesting passage Hobbes himself allows this claim, and then argues that all 'natural' language was destroyed by God at the Tower of Babel and that man then had to invent new languages by himself "in such manner, as need (the mother of all inventions) taught them"5 Not only are names arbitrary in the sense that they were originally assigned meaning in a non-necessary manner, but also in the sense that they can be used at any time as a 'false' sign when the thing-signified is not present. For example, verbal expressions of psychological states "are expressions, or voluntary significations of our passions: but certain signs they be not: because they may be used arbitrarily, whether they that use them, have such passions or not. The best signs of passions present are either

^Leviathan I, iv in Works III, p. 19. 31 in the countenance, motions of the body, actions.•

Natural signs are certain in their signification* words

are not. Man has no control over natural signs and

their significates, as he does have over his words

and their significates. The natural sign-relation

is immutably fixed as far as man is concerned,

although Hobbes leaves open but does not discuss

the possibility that God can change it.

A number of answers have been given to

the question, "For Hobbes, what is the nature of

the thing signified by a name?" The two most

frequent answers are 'ideas' and 'material objects'; however, no conclusive argument has yet been presented in favor of either interpretation.

Nevertheless, it is. apparent that the explicit statements on this issue in Hobbes* basic writings do assert that names, at least in their function as signs, are signs of ideas, and not of experienced causes of the ideas, (Whether this is consistent with other aspects of Hobbes' philosophy or not is another question which is irrelevant to my discussion.)

^Leviathan I, vi in Works III, p, 5°» 32

But seeing names ordered in speech are signs of our conceptions, it is manifest they are not signs of the things themselves* for that the sound of this word stone should be the sign of a stone, cannot be understood in any sense but this, that he that hears it collects that he that pronounces it thinks of a stone.’'

Remembering that words are ’signs' for Hobbes only when they are being used to communicate or remember ideas, it is clear that a spoken word is a sign to the hearer that the speaker has a certain idea.

However, even though words are signs of ideas.

(or 'conceptions' or ’phantasms'), these ideas must be in some way related to sensations of physical objects, and thus to the physical objects themselves,

, . . Singly, [the thoughts of man] are every one a representation or appearance, of some quality, or other accident of a body without us* which is commonly called an object. Which object works on the eyes, ears, and other parts of man's body* and by diversity of working, produces diversity of appearances. The original of them all, is that which we call sense * (For there is no conception in a man's mind, which has not at first, totally, or by parts, been begotten upon the organs of sense.) The rest are derived from the original . , . The cause g of sense is the external body, or object . • .

It is beyond the scope of this study to investigate

^Elements I, ii, 5 in Works I, p. 17.

8Leviathan I, i in Works III, p. 1. the problems that arise from this view, particularly

with regard to the theory of perception implicit in

it. It is sufficient to recognize that this

entails a rather simple version of the correspon-.

dence theory of meaning that was accepted, with

variations, by many seventeenth century thinkers,

particularly by some of those referred to today

as 'empiricists'. It can be stated in brief ass

To every meaningful word there corresponds an

idea which in turn corresponds to some entity which

is the cause of the idea. 'Correspond' need not

be interpreted as a strict copying or imaging,

but rather as merely some sort of one-one or one-

many relation, depending on the particular version

of the theory.

Hobbes recognizes a number of consequences which follow directly from the assumption of this

theory of meaning. One of the most important

results is that since each word ultimately corre­

sponds to some sensible object (obviously not in

the sense of 'copies' or 'images'), and since

only particulars are sensed, universal names must

ultimately correspond only to particulars. If someone questions this reduction, Hobbes points

out that the question is confused and is mistaking

the sign itself for the thing-signified. His

argument runs as follows*

■ « • Some [namesJ are common to many things; as man, horse, tree, every of which, though but one name, is nevertheless the name of diverse particular things; in respect of all which together, it is called an universal; there being nothing in the world universal but names; for the things named, are every- . one of them individual and singular. One universal name is imposed on many things, for their similitude in some quality,- or other . accident: and whereas a proper name brings to mind one thing only: universals recall any one of those many.9

The universality of one name to many things, hath been the cause that men think the things are themselves universal . . • deceiving them­ selves, by taking the universal, or general appelation, for the thing it signifieth: for if one should desire the painter to make him the picture of a man, which is as much as to say, of a man in general: he meaneth no more, but that the painter should choose what man . he pleaseth to draw, which must needs be of some of them that are, or have been, or may be, none of which are universal . . .10

One might wish to argue here that Hobbes has left the concept out of his account, and that the possibility remains that there are universal ideas

^Leviathan I, iv in Works III, p. 21.

^ Human Nature V, 6 in Works IV, p. 22. See also Elements I, ii, 9 in Works I, pp. 19-20. constructed by the mind out of the original sensa­ tions, which are the objects signified by universal names. This possibility is ruled out by Hobbes, however, in so far as he maintains that ideas are merely "decaying sense" and that the mind has no power to form any idea which is not like an object 1 1 or possible object of sense experience. It is important to observe in the previous quotation that ideas are nothing more than pictures.

Hobbes goes to great pains to deny the existence of abstract ideas, and also the power of the mind to abstract. Since ideas are literally pictures, and since there can be no picture of

'redness' which does not have a particular shape and size, etc., it follows that there can be no abstract idea of redness. Although Hobbes' account of the nature of things signified by abstract names

(whose existence he has no desire to deny), is very confused and perhaps even self-contradictory when examined in detail, one passage in which.he discusses this question is of considerable impor­ tance in so far as it is a clear anticipation of

Berkeley's later theory.

Leviathan I, ii in Works III, p. 5* 36

Now in all matters that concern this life, but chiefly in philosophy, there is both great use and great abuse of abstract names: and the use consists in this, that without them we cannot, for the most part, either reason, or compute the properties of bodies • . • But the abuse proceeds from this, that some men seeing they can consider, that is bring into account the increasings and decreasings of quantity, heat, and other accidents, with­ out considering their bodies or subjects (which they call abstracting, or making to exist apart by themselves) they speak of accidents, as if they might be separated from all bodies , • • From the same fountain spring those insignificantwords, abstract substance. separated essence, and the like \ \

Thus Hobbes makes no attempt to deny the existence

or utility of abstract names, but he has gone to

great effort to emphasize that in dealing with

them, one must not confuse the sign with the thing

signified and talk as if there were such entities

as abstract ideas or objects.

Arithmetic for Hobbes is essentially the art of computing "increases and decreases of quantity" and thus, as indicated in the above passage, involves the use of abstract.names• It can also be seen that for him quantity must be essentially an accident of bodies, and thus arithmetic

is concerned, at least indirectly, with bodies. I shall later show that this was far from being an

•^Elements I, iii, 4 in Works I, pp. 33~3^» unusual attitude in the seventeenth century, and

that Hobbes was no more confused on this point

than any of the mathematicians of the period. In

order to understand Hobbes' theory of the nature of

the significates of number-words, it is necessary

to recognize that he wishes to limit the number

of types of objects of sensation, and thus the number of 'nameables', to four. "Now, all things to which we give names, may be reduced to these four kinds, namely bodies. accidents. phantasms. and names themselves . • ."13 He is inconsistent with regard to the significates of mathematical words, but it is unimportant for my purposes whether they are ultimately the accidents of the bodies themselves, or merely the phantasms of the accidents.

Like Kant later on, he relates our mathe­ matical ideas to our perceptions or intuitions in space and time, although in a rather confused way. His account is dependent on his notion of

•abstraction' as simply the concentration of the mind on a single specific aspect of a 'picture'.

For example, one can notice the color of a particular

^Elements v* 2 in Works I, pp..57-58. 38 object without noticing its shape or size or any of its other properties, even though all of these other properties are present in the same object.

Thus 'space* and 'time' are abstract names, and signify our ideas of particular objects in so far as we concentrate our attention on some specific property of them.

. . . SPACE is the phantasm of a thing existing without the mind simply; that is to say, that phantasm, .in which we consider no other accident but only that it appears without us. • . . . . . TIME is the phantasm of before and after in motion: which agrees with this definition of Aristotle, time is the number of motion according to former and latter, or that number­ ing is an act of the mind , , . r

It is important to notice the statement that

"numbering is an act of the mind," and to see that this is closely connected to his general account of the significates of abstract names.

The concept of unity, 'unity* being another abstract name, is basic to Hobbes' theory of number.

For him, as for almost all the mathematicians of this period, all the other positive rationals can be defined in terms of relations of units. Hobbes explains the signification of the abstract name

'unity' in the following mannert

•^Elements II, vii, 2-3 in Works, pp. 9^-95• 39

When space or time is considered among other spaces or times, it is said to be ONE, namely one of them • . • The common definition of one, namely, that one is that which is undivided, is obnoxious to an absurd consequence; for it may thence be inferred, that whatsoever is divided is many things, that is, that every divided thing, is divided things, which is insignificant,^5

Again we have the implicit indication of the activity of the mind in the 'considering' of spaces and times in particular ways, that is, as divided or undivided. This is made more explicit in the following passage*

To make •parts, or to part or DIVIDE space or time. is nothing else but to consider one and another within the same; so that if any man divide space or time, the diverse conceptions he has are more, by one, than the parts he makes; for his first conception is of that which is to be divided, then of some part of it , , , By division, I do not mean the severing or pulling asunder of one space or time from another * * , but diversity of consideration . . .

Thus there is a very strong sense in which it may be asserted that for Hobbes, number is a creation of the mind, for once the mind has 'created' the concept of unity by means of selective consideration, it can then construct any other positive rational.

^ Elements IIt vii, 6 in Works I, p. 9 6 .

•^Elements II, vii, 5 in Works I, pp. 95" 96. "NUMBER is one and one, or one one and one, and so

forwards? namely, one and one make the number two. and one one and one the number three ; so are all numbers made; which is all one as if we should say, 17 number is unities."

Although the mind does play a creative role in the construction of numbers, it must not be forgotten that, for Hobbes, number-words in even their most 'abstract' sense still signify bodies or ideas of bodies being considered by the mind in a particular way. His theory, although involving a strong subjective element, is neverthe­ less objectively grounded in material objects.

Number (or quantity) is for Hobbes one of the basic properties of material objects; thus the elaborate explanation given in the previous paragraph is ultimately reducible to the claim that number- words signify a particular property of bodies, namely their quantity. However, it is important not to put too much stress on this objective element in his account, since he certainly would deny statements which would assert, for example,

1?Ibid. 41

that the number 3 is one and the same with a

collection of three stones. At most, he would allow that '3* is a name which signifies this

collection in so far as some mind is concentrating

on a specific property«of it.

Although Hobbes sometimes speaks as though

there were totally objective numbers existing in things (as when he states that "In magnitude bodies differ when one is greater than another, as a cubit long and two cubits long of two rounds weight, and of three pounds weight,"18), it is important to recognize that he is merely placing the emphasis on one aspect of his account rather than denying the role of the subjective element. Without the objective element, many "gross errors of writers of metaphysics" result, "for because quantity may be considered without considering body, they think also that quantity may be without body, and body without quantity . , ."19

Hobbes considers this account sufficient for the system of natural numbers and the positive

X 8 Elements II, xi, 2 in Works I, p. 133* ^ Elements I, iii, 4 in Works I p, 3^. rationals, which he interprets as the the ratios or proportions of various quantities (or numbers).

There remain, however, three more problematic and philosophically very interesting types of number for which Hobbes (and many others) attempted to account— infinity, zero, and the negative rationals.

It follows directly from his general theory of signs that since we have no sensations of infinite bodies the name ‘infinity' cannot signify any actually existing entity. The term is meaningful, however, and Hobbes gives the following account of what it does in fact signify.

Whatsoever we imagine, is finite. Therefore there is no idea, or conception of anything we call infinite. No man can have in his mind an image of infinite magnitude; nor conceive of infinite swiftness, infinite time, or infinite force, or infinite power. When we say anything is infinite, we signify only, that we are not able to conceive the ends, and bounds of the thing named; having no conception of the thing, but of our own inability.

It is possible to speak of the set of natural numbers as infinite in this sense, although any particular number is itself finite. In other terms,

Hobbes will not allow meaning to the term 'actual infinite'.

^^Leviathan I, iii in Works III, p. 17* When we say number is infinite, we mean only that no number is expressed; for when we speak of the numbers two, three. a thousand, &c, they are always finite. But when no more is said but this, number is infinite, it is to be understood as if it were saidj^this name number is an indefinite name.

This analysis of the infinitely large is carried over into his discussion of infinitesimals, where he denies the existence of infinitely small entities on similar grounds, and where he gives a similar analysis of the significate of the name

'infinitesimal'•• In his own words

. . • the force of that famous argument of Zeno against motion, consisted in this pro­ position, whatsoever may be divided into parts. infinite in number, the same is infinite;;which he, without doubt, thought to be true, yet nevertheless is false. For to be divided into infinite parts, is nothing else but to be divided into as many parts as any man will. But it is not necessary that a line should have parts infinite in number, or be be infinite, because I can divide or subdivide it as often as I please; for how many parts soever I make, yet their number is finite; but because he says parts, simply, without adding how many, does not limit any number, but leaves it to the determination of the hearer, therefore we say commonly, a line may be divided infinitely; which cannot be true in any other s e n s e . 22

This argument not only denies the existence of

21Elements II, vii, 11 in Works I, pp. 98- 99. 22Elements I, v, 13 in Works I, pp. 63-64. mathematical points or infinitesimals, but it also entails, when taken in conjunction with the rest of Hobbes* theory, that there can be no

•transcendental' numbers, such as 'pi'. This is in part the crux of his celebrated feud with

Wallis over the squaring of the circle--since all lengths are determined by successive divisions of other lines, no two lengths are ultimately incommensurable. This argument is important to my discussion, however, only in so far as it offers an alternative, plausible or implausible, to the assumption of real entities corresponding to the mathematical terms 'infinity' and 'infinitesi­ mal* , and it is thus irrelevant to attempt to determine here which view is ultimately correct.

A careful analysis of the above account reveals an apparent inconsistency with Hobbes' statement, quoted earlier, that "all things to which we give names, may be reduced to these four kinds, namely bodies, accidents, phantasms. and names themselves • • because the word 'infinite' does not refer to any of these four kinds of thing and yet it is meaningful, Hobbes recognizes this difficulty in another passage and proceeds to explain* 45

There may also be other names, called negative: which are notes to signify that a word is not the name of the thing in questionj as these words nothing, no man, infinite. indocible. three want four, and the like; which are nevertheless of use in reckoning, or in correcting of reckoning; and call to mind our past cogitations, though they be not names of anything; because they make us refuse to admit of names not rightly used.23

It is very important to notice that Hobbes' justifi­ cation for breaking with his basic axiom that all meaningful words must name things is that man finds it useful to maintain other words in his language.

Hobbes nowhere explicitly asserts that the meaning of words such as 'infinite' is their use, although this may appear to be implicit in passages such as the above. It can also be seen that this passage accounts for the admission of zero and the negative numbers into the language, for even though they are not the names of any things, they do perform a useful and even necessary function in everyday life. Hobbes gives the following analysis of

'nothing' and 'zero', and indicates how this could easily be extended to cover negative numbers.

Nor, indeed, is it at all necessary that every name should be the name of something . . . To conclude* this word nothing is a name, which yet cannot be the name of any thing* for when,

Leviathan I, iv in Works III, p.27. 46

for example, we subtract 2 and 3 from 5* and so nothing remaining, we would call that subtraction to mind, this speech nothing remains, and in it the word nothing, is not unuseful. And for the same reason we say truly, less than nothing remains, when we subtract more from less; for the mind feigns such remains as these for doctrine's sake . . .24

Before making any hasty claims, however, about Hobbes being the first 'pragmatist', it is necessary to recognize that the emphasis on utility is, after all, only secondary to his ultimate analysis. In fact, he concludes the above passage with the confusing comment that ". . . seeing every name has some relation to that which is named, though that which we name be not always a thing which has being in nature, yet it is lawful for doctrine's sake to apply the word thing to whatsoever we name."25 Thus he ultimately does want to hold to his correspondence theory of meaning where every meaningful word must be considered as a name of some thing. At best, then, his account is confused? at worst it is inconsistent and inade­ quate. Hobbes' most valuable contribution to this subject-matter was to raise a number of new questions

^ Elements I, ii, 6 in Works I, pp. 17-18,

2 5ibid. 4? and at least give some indication of how one could go about answering them.

Similarly, Kobbes* importance with regard to the problem of the nature of mathematical truth consists in his raising of critical questions and his suggestions (however inadequate) for their solution. He clearly recognizes that only propositions are capable of being true and false. With regard to propositions containing universal names, he asserts that they are, to use Kant’s terminology, analytic, where their truth is dependent on the inconceivability of their denial. " • • . When we make a general assertion, unless it be a true one, the possibility of it is unconceivable."26

Thus, if one were to assert that "This circle is round", the statement would be true by virtue of the fact, that a non-round circle is inconceivable. It follows directly from this theory of truth that all necessary universal truths must be definitions, since in effect they are true because we have assigned two different names to the same set of things, or to one set of things

A / Leviathan I, v in Works III, p. 32. and a subset of this set. Hobbes asserts that

"... the first truths were arbitrarily made

by those that first of all imposed names upon

things, or received them from the imposition of

others. For it is true (for example) that man

is a living creature, but it is for this reason,

that it pleased man to impose both those names

on the same thing."2''7 It is from statements such

as this that Kneale concluded that Hobbes is the

founder of the 'conventionalist' theory of necessary

truth. However, a close scrutiny of the passage

reveals that the claim of truth by 'convention'

can be made in only a very weak sense, if at all,

for Hobbes' theory. The arbitrariness lies only

in the first assignment of names to particular

objects, and once this sign-relation has been

established, the truths follow necessarily by the

nature of the things signified by the names since

their properties are related by natural necessity and cannot be changed by the mind alone. Even the

original naming is not entirely arbitrary because

it is implicitly assumed in Hobbes' theory that

things with different properties must be given

^ Elements I, iii, 8 in Works I, p. 36. 49 different names. The failure to follow this rule

is "the greatest, if not the only, cause of discord amongst philosophers, as may easily be perceived by their abusing and confounding the names of things that differ in their nature . . ."28 Thus

if one object has the property of being two feet long and another the property of being three feet long, we are not permitted by Hobbes' rule to name both of them 'two foot long object'.

All basic axioms of demonstrative science must be definitions, for these are the only universal statements which are necessarily true and cannot 29 be derived from any other propositions. It is unclear as to exactly what Hobbes wished to take as the 'first principles' of mathematics; passages such as the following would seem to indicate that he considered all true mathematical statements to be true by definition, and none to be dependent on demonstration.

• . . If it be propounded that two and three make five; and by calling to mind, that the order of numeral words is so appointed by

2 8 Decameron Physiologicum in Works VII, p. 84. 29 See Elements I, vi, 13 in Works I, pp. 81-83* 50

the common consent of them who are of the same language with us, (as it were, by a certain contract necessary for human society), that five shall be the name of so many unities as are contained in two and three taken together, a man assent that this is therefore true, because two and three together are the same with five s this assent shall be called know­ ledge, And to know this truth is nothing else, but to acknowledge that it is made by our­ selves, For by whose will and rule of speaking the number //is called two, /// is called three, and ///// is called five; by their will also it comes to pass that this proposi­ tion is true, two and three taken together make five,3°

Again, however, it must be emphasized that statements such as this mathematical example are not simply true by definition, Hobbes' example brings this out very clearly in its ultimate dependence on the empirical fact that // and /// when combined result in /////, Since all mathematical statements are assumed by Hobbes to be meaningful, he has been forced (as seen earlier) to admit that numbers are properties of material objects and numerals (or number-names) signify these properties. Thus, although he has made statements which to modern ears sound as though he held that mathematical truths are 'analytic' or 'true by convention', a more careful reading would seem to indicate that

-^Philosophical Rudiments XVII, 4 in Works II. P. 303. 51 they are closer to being empirical truths. Even

the statement "The sun is shining." is an arbitrary

truth according to Hobbes' criteria since the names 'sun' and 'shining* were originally arbitrarily

stipulated to be signs for certain phenomena, in the same way that 'two' was assigned to the pheno­ menon //. Taking the rest of Hobbes' theory which

is relevant into account, it appears that the most representative statement of his theory of truth would be that expressed in the following passage.

In every •proposition. be it affirmative or negative, the latter appellation either comprehendeth the former, as in this proposi­ tion, charity is a virtue, the name of virtue comprehendeth the name of charity, and many other virtues besides; and then is the proposi­ tion said to be true . . . Or else the latter appellation apprehendeth not the former . . . and the proposition is said to be false . , ,31

This statement is compatible with Hobbes' account of statements being 'true by definition', but it puts less emphasis on this aspect and reveals the extent to which he is still in the Aristotelean tradition, at least with regard to the theory of truth,

Hobbes' theory of 'demonstration' is also essentially Aristotelean. He summarizes his general

Human Nature V, 10 in Works IV pp. 23-24. 52 theory with two rules* "one, that the principles be true and evident definitions; the other, that the inferences be necessary".32 He denies that the

•analytical* technique, as used by Descartes and others in mathematical proofs, is really demonstra­ tion; the only proper form is the synthetic method.

The v/hole method, therefore, of demonstration, is synthetical, consisting of that order of speech which begins from primary or most uni­ versal propositions, which are manifest of themselves, and proceeds by a perpetual composi­ tion of propositions into syllogisms • . .33

Despite his emphasis, discussed in the previous paragraph, on the basic axioms as being definitions, he generally speaks of them as being in some sense empirical generalizations. For example "when a man reasoneth from •principles that are found indubitable by experience, all deceptions of sense and equivocations of words avoided, the conclusion he maketh is said to be according to right reason

, . . "3^ Reasoning is dependent on the meanings of the words in arguments, and no demonstration can

32gix Lessons in Works VII, p.,212.

^ Elements I, vi, 12 Works I, p. 81,

^ Human Nature V, 12, in Works IV, p. 2^. 53 be made when the words have no meaning.

The first cause of absurd conclusions I ascribe to the want of method; in that they begin not their ratiocination from definitions; that is from settled significations of their words; as if they could cast account, without knowing the value of the numeral words, one. two, and three..

This statement would seem to rule out the possibility of any account of mathematics as the manipulation of meaningless symbols according to arbitrarily chosen rules. On the surface it may even appear to be inconsistent with Hobbes' discussion of zero, infinity, and the negative rationals, since he does admit that these names do not signify any quantity present as a property of things; but as I pointed out earlier, he ultimately insists that there must be 'things' of some sort corresponding to the signs

0,09 , etc.

As further evidence of this anti-formalist tendency, it is interesting to note that Hobbes thinks that one of the most effective- criticisms of any mathematician is to charge him with not knowing what the mathematical terms or symbols that he is using signify. Thus he includes in his attack on Wallis the following charge: "The

-^Leviathan• I. v in Works III, p. 33* practice you may have, but so hath any man that hath learned the bare propositions by heartj but they are not fit to be professors either of geometry or of any other science that dependeth on it."36

However unjustified this charge may be with regard to Wallis' mathematical writings, it clearly reveals

Hobbes' attitude towards any type of formalism. This attitude also partly explains Hobbes* (and others') hostility toward the introduction of algebraic notation into mathematics, and particularly into geometry, since he considered the new symbols as an attempt on the part of their users to conceal what was 'actually* going on in their 'so-called demonstrations' from any who did not understand their meaning (that is, denotation).

• • • Your way of demonstration, by putting N for a great number of sides of an equilateral polygon, is not to be admitted; for, though you understand something by it, you demonstrate nothing to anybody but those who understand your symbolic tongue, which is a very narrow language.

A 'formalistic' interpretation of mathematics is also

-^Six Lessons, in Works VII, p. 219.

37Ibid., p. 248. not permissible in so far as it involves mistaking the study of signs for the study of the things- signified,3®

Although the 'objective' or 'empirical' element is often difficult to discern in Hobbes' account of arithmetic, it is blatant in his state­ ment of the nature of geometry; in this case it is more difficult to distinguish the 'subjective' aspect. The following passage shows both features of his theory, and also reveals his emphasis on the'empirical' nature of geometry.

• . .The science of every subject is derived from a precognition of the causes, generation, and construction of the same; and consequently where the causes are known, there is place for demonstration, but not where the causes are to seek for. Geometry therefore is demonstrable, for the lines and figures from which we reason are drawn and described by ourselves . . .39

The study of geometry, then, is the study of causes, and literally t>hvsical causes. A more detailed explantion of the sense in which Hobbes means this is given in this account: % It remains, that we enquire what motion begets such and such effects; as, what motion makes a

38Ibid.. p. 187.

39Ibid., p. I8h-. 56

straight line, and what a circular • • First we are to observe what effect a body moved produceth, when we consider nothing in it besides its motion; and we see presently that this makes a line, or length • • • from which kind of contemplation sprung that part of philosophy which is called geometry . . . And because all appearance of things to sense is determined, and made to be of such and such quality and quantity by compounded motions, every one of which has a certain degree of velocity, and a certain and determined way; therefore in the first place, we are to search out the ways of motion simply (in which geometry consists) , ,

Although the emphasis is clearly on the objective

grounding of geometry, the subjective aspect is revealed as being analogous to that in arithmetic

by phrases such as "when we consider nothing in

it besides, • .

Geometry is ultimately grounded in our percep­

tions of bodies in motion, but it becomes a general

science in so far as the mind disregards particularizing

features in the observed individual cases. All geometrical terms must name some thing, but as in

the case of number, this 'thing* can be a property

of particular bodies which cannot exist or even be

thought of independently of some body. The basic geometrical terms must be defined in the following manner:

^°Elements I, vi, 6 in Works I pp. 71-73. 57 Euclid in the definitions of a point, a line, and a superfices, did not intend that a point should be nothing, or a line without latitude, or a superfices without thicknesss for if he did, his petitions are not only unreasonable to be granted, but also impossible to be per­ formed, For lines are not drawn but by- motion, and motion is of body only. And therefore his meaning was, that the quantity of a point, the breadth of a line, and the thickness of a superfices were not to be considered, that is to say, not to be reckoned in the demonstration of any theorems concerning the quantity of bodies , , , 1

To unpack the last sentence, Hobbes seems to be

saying that, in the proof of a geometrical theorem, we are always dealing with points with dimension,

lines with breadth, and so on, but that in the proof

itself we only mention the length of the line, etc., and that the other properties, although present, are irrelevant.

This concept of 'abstraction by consideration*

should be quite clear by now, but because of its relevance for my later discussion of Berkeley, I shall include one further detailed explanation of and argument for this account. In his criticism of Wallis' interpretation of Euclid's definitions,

Hobbes asserts that

^3-Six Lessons in Works VII, p. 211. 5 8 The second definition is of a line • • . *a line is a length which hath no breadth;' and if candidly interpreted, sound enough, though rigorously not so , . • One path may he broader than another path, but not one mile longer than another mile; and it is not the path but the mile which is the way's length. If therefore a man have any ingenuity he will understand it thus, that a line is a body whose length is considered without its breadth, else we must say . . . untruly, that there be bodies which have length and yet no breadth . . «^2

Again, with regard to the analysis of the

nature of geometrical truths, Hobbes shows the same

confusion as with regard to arithmetical truths,

namely that in some sense they are true by defini­

tion while at the same time they are empirical

generalizations. As stated earlier, the statement

'All circles are round' is analytically true since

its denial is inconceivable, and only such statements

can be considered to be necessarily true. This

statement is-, ..however, also essentially trivial,

and no non-trivial statements are necessarily or

certainly true. Although Hobbes tries to use his

concept of 'abstraction by consideration' to explain

how non-trivial geometrical theorems can be apodic-

• tically certain, the move ultimately fails, as

can be seen in his discussion of the theorem that

k-2 Ibid.. p. 202. 59 the three angles of a triangle are equal to two right angles.

• • • He that has the use of words, when he observes that such equality was consequent, not to the length of the sides, nor to any other particular thing in the triangle; but only to this, that the sides were straight, and the angles three; will boldly conclude universally, that such equality of angles is in all triangles whatsoever; and register his invention in these general terms, every triangle has its three angles equal to two right angles. And thus the consequence, found in one particular, comes to be registered and remembered, as an universal rule . • • and makes that which was found true here.and now, to be true in all times and places.^3

It can be seen that there is no real 'demonstration* herb in the previously defined sense, and that this is similar to the case of the addition of 2 and 3 where Hobbes is actually making an inductive generalization from a specific instance. The important point is not whether Hobbes could reconcile this account with his theory of demonstration; it is rather that he in fact does not reconcile them in any of his basic published writings. It is even questionable that Hobbes himself was aware of this incompatibility between his theories of truth and demonstration.

In summary, I have given textual evidence that

^ Leviathan I, iv in Works III, p. 22. Hobbes' philosophy of mathematics is integrally

tied to his theories of meaning and truth. His theory of meaning, although it contains undeveloped hints of pragmatic and conventionalistic tendencies,

is essentially a traditional 'correspondence' theory,

i.e. the meaning of any word is some corresponding

idea which itself is ultimately related to a sensation of a material body. The meaning of abstract and universal words has two elements— the 'objective' or the body which caused some sensation, and the 'subjective' or the 'consideration' of some single property of a body by the mind.

Mathematical words, being both abstract and universal, have both types of meaning. I have tried to show that although he speaks of truths being made by men, Hobbes does not hold a 'conventionalist' theory of truth in any modern sense of this term.

His theory of truth in fact is closely tied to his theory of meaning, and is also basically a corres­ pondence theory; the proposition "2 + 3 = 5" is true if and only if the thing signified by '2 ' when combined with the thing signified by '3 ' results in the thing signified by '5'« Thus, although he sometimes speaks of the basic axioms 6l as true by definition# and although he discusses demonstration as the deduction of truths from definitions, his examples indicate that he considered mathematical propositions to be inductive generali­ sations.

To repeat my earlier disclaimer, this brief presentation is intended as nothing more than a prolegomena to my discussion of Berkeley, and thus is based entirely on the explicit state­ ments in Hobbes' major works. Although modern scholarship may be able to reconcile some of the inconsistencies I have pointed out, and even though it may be able to uncover evidence from correspon­ dence and other unpublished writings that Hobbes was more of a 'pragmatist' or'conventionalist' than his major works indicate, this would not affect the points I have just made nor their relevance to Berkeley's thought. 62

II

Although several hooks have been written

on the development of specific mathematical concepts

in the seventeenth century, I have been unable to discover any general study of the philosophical concepts assumed or expressed by the mathematicians

of this period. D. T. Whiteside, probably the- world's foremost authority on seventeenth century mathematics, recently reported that he had once desired ."to probe into the philosophical basis of concepts, especially those of number, space, and limit • • but that he had not carried out the project for several reasons. One of his main reasons was that on surveying all of the available literature he found that in comparison with the

"great richness of material bearing on developments in technique" there is "a paucity of anything which can be interpreted as original comment on underlying structure or methods of proof.

The basic conclusion at which he arrived after his survey of the original texts was

^ D e r e k T. Whiteside, "Fatterns of Mathe­ matical Thought in the later Seventeenth Century," Archive for the History of Exact Sciences. I, 3 (1961), p. 179. 63 • • • that basic concepts were not investigated in the seventeenth century with any insight, but that an adequate basis for mathematics, accepted as.a matter of practice, did exist which was little different, if at all, from that explored in Greek and medieval times. The seventeenth century is, in mathematics, a period of rapid advance using valid but tenuously defined, concepts as a basis for a rich and varied technical achievement. The greatness of that achievement is to be evaluated by a detailed study not of what seventeenth century mathematicians thought but of the evolving pattern of what they did . . .^5

The "adequate basis for mathematics" of which Whiteside speaks is ultimately merely "a mixture of Euclidean axiomatics and Aristotelean syllogistic (in its developed scholastic f o r m s ) . "^6

This is essentially a methodological base and provides no real conceptual foundation for the construction of mathematical systems, I will show in this section that even the method of the syllogism was given only lip-service and that very few proofs used rigorous deduction. Likewise, there are very few (if any) examples of sound axiomatic systems in the literature of this period.

I want to suggest that it was the inability

^ Ibid.. p. 196. ^6ibid.. p. I85. of the seventeenth century mathematicians either to assimilate the available Greek and Medieval concepts or to provide new concepts with which to replace them that resulted in the general con­ fusion which pervades their writings. Briefly, the most important available concepts included the Platonic and Aristotelean accounts of the objects of mathematical knowledge (i.e. as "Pure

Forms" or "enmattered forms"), the denial of all connections between mathematics and the physical world, the 'solutions' to Zeno's Paradoxes, and the sharp dichotomy between Geometry (as the study of the continuous) and Arithmetic (as the study of the discrete). The mathematicians felt unable to accept any of these concepts without contra­ dicting other ideas which they considered essential to their new techniques. Thus, there were few, if any, genuine 'Platonists' among the mathematicians of the period, and a thorough-going Aristotelean

•empiricism' was not generally considered as an adequate grounding for mathematics as they practiced it.

Rather than a general agreement on the philos­ ophical basis of mathematical concepts fiwhich 65

Whiteside's statement seems to suggest), there was widespread confusion which led to debates among the leading mathematicians and philosophers which continued long after Berkeley gave the first

suggestions for an adequate philosophy of mathe­ matics (and which are still going on today in a modified form). Since it is this general con­ fusion which Berkeley was to react against and

in which he was able to discern a single cohesive thread, it is necessary to examine the attitudes of the mathematicians in some detail, particularly with regard to their concepts of demonstration, number, infinity, infinitesimals, and geometry.

This section will describe the attempts of the mathematicians either to reconcile the Greek concepts listed in the previous sentence or to replace the

Greek concepts with entirely new ones.

With regard to demonstration, the main criterion of the correctness of a mathematical

'proof seemed to be most often the correctness of its conclusion. Since the basic concepts were inadequately defined (when they were defined at all), and since Aristotelean syllogistic is not applicable to many mathematical arguments, the 66

derivations were far from rigorous in any modern

sense of the term. The texts abound with •axioms',

•postulates*, * lemmas*, and •theorems*, but few

deductions are actually carried out.

To one accustomed to the idea that the exact proof-trees shall be set down in rigorous mathematical argumentation very.few proofs of any kind in classical mathematics will be allowable, and certainly none was given in the seventeenth century on any but the most elementary numerical level. Rather, we would do well to criticize the form of a seventeenth century mathematical proof from the viewpoint that it is a psychologi­ cally satisfying sketch and no more. Such a proof does, in a very strong sense, prove a result which we find valuable and new . . . and in historical fact very often mirrors more adequately than a tight and rigorous modern form the thought-processes which led to its formulation. '

This lack of rigor is not as much a failure as it is a manifestation of the strong 'intuitive* grounding (although not in the Platonic sense) of all certain knowledge during this period. The basic criterion of certainty, accepted by most thinkers of the period in one form or-another, was that of 'clarity and distinctness', and although this criterion has since been shown to be excessively vague, if not totally meaningless, it was adequate in so far as the thinkers who used it were generally

Ibid.. p. 184. 6? able to distinguish 'correct* from 'incorrect' statements and arguments. As Whiteside points out,

We tend, too, to assume that mathematics has always been developed in abstraction from any model other than a logical one, forgetting, for example, that before the nineteenth century geometry was in part developed on the basis of conventional real space. . . . In fact, extramathematical ("psychological") considera­ tions still play a large role in seventeenth century mathematical procedures, but thereby compensate for the apparent lack of rigour or loose assumptions rather than invalidate the proof forms used.T8

This method of determining the truth of mathematical propositions by some faculty of 'intuition' is still assumed by a significant number of 20th century philosophers, among them G. E. Moore and W. D.

Ross.

Most of the 'theorems' of 17th century mathematics were thus essentially axioms in so far as their truth was determined by inspection or intuition rather than by strict deduction from a small set of axioms. Often general theorems were 'proved' by 'induction'— not that which is referred to today as mathematical induction, but rather something very similar to the empirical

^ 8Ibid.. p. 183. 68

induction espoused by Francis Bacon. Thus we find

John Wallis asserting that

The simplest method of investigation in this and in a certain number of problems which follow is to consider a certain number of individual cases, and to observe the emergent ratios, so that a universal proposition may by induction be established. °

This is a method of enumeration, little different

from that used by Galileo in his formulation of

the laws of freely falling bodies. It shows little

resemblance to either the Platonic or the Aristotelean

concepts of the method for arriving at ’first

principles*, and Wallis was not the only mathema­

tician to use it. "The logic of the Method of

Induction as used by Wallis was not flawless.

Like many other mathematicians of the period

he had fallen into the error of generalizing from

a number of isolated cases."5°

Even Newton used no clear form of demon­

stration in his mathematical treatises, relying again on the ability of his readers to somehow

intuitively grasp the truth of his assertions.

**9John Wallis, Arithmetica Infinitorium, Proposition 1, quoted in J. F. Scott, The Mathematical Works of John Wallis (Londons Taylor and Francis, Ltd., 1938), p. 3o 7

5°Scott, Wallis, p. 79. 6? Boyer points out that

. . . although the work of Newton contains the essential procedures of the calculus, the justification of these is not clear from the explanation he gave . . . His contribu­ tion was that of facilitating the operations, rather than clarifying the conceptions. As Newton himself admitted . . • his method is •shortly explained rather than accurately demonstrated.'51

Newton explicitly maintains in one of his last published works that in mathematics “the Investi­ gation of difficult Things by the Method of Analysis ought ever to precede the Method of Composition.

This Analysis consists in making Experiments and

Observations, and in drawing general