WRITERS AND CRITICS - BERTRAND RUSSELL. by John Watling.

Transcription

1 WRITERS AND CRITICS - BERTRAND RUSSELL. by John Watling. NOTE: The pagination of this text corresponds as closely as possible to the original 1970 edition of Oliver and Boyd, Edinburgh. 1 Biography Bertrand Russell was born in His mother died only two years afterwards, and his father a year and a half after that. His parents held advanced opinions on matters of social reform. They supported universal suffrage and rejected the view that birth control was immoral, and they were actively engaged in discussing and furthering these views. 1 They were acquainted with some of the leading progressive thinkers of the time, including John Stuart Mill. His father s will left both Russell and his brother in the care of two men one of them had been tutor to his brother who were both atheists. Russell s paternal grandparents did not share his parents progressive views. They had little difficulty in obtaining custody of the children, and Russell was brought up by his grandmother, for his grandfather died a few years later. This grandfather, Lord John Russell, was eighty-three when Russell s father died. He had been Prime Minister during two Liberal administrations, and before he became Prime Minister had introduced the First Reform Bill. He and his wife lived in Pembroke Lodge, a grace-and-favour house in Richmond Park. There are still such houses in Richmond Park, but Pembroke Lodge and its gardens are now open to the public. Russell spent his childhood in this house where his grandparents were visited by many eminent people, chiefly from political life. He was not sent to school until he was sixteen, when he went to an army crammer to prepare for a Cambridge scholarship. In his later childhood he was intensely interested in mathematics, as well as in history and poetry. When he was eighteen, he sat the scholarship examination for Trinity College, Cambridge, gained a scholarship, and went up to Trinity a year later. Before going up 1 The Autobiography of Bertrand Russell , pp

2 he read J. S. Mill s Autobiography, Political Economy and A System of Logic, Herbert Spencer s The Man Versus the State, and some works of Henry George. At Cambridge Russell obtained a First in Mathematics after three years and a First in Moral Sciences after another one. In his second year, he was elected to a semi-secret discussion society known to outsiders as The Apostles which included both undergraduates and dons and whose members were chosen as undergraduates, one or two a year, for their intellectual brilliance. Through this society Russell quickly became acquainted with people of ability and like interests, for A. N. Whitehead, who had examined for the scholarships, had been impressed by Russell s papers and had suggested to other members of the society that they should call on him. Besides Whitehead, he came to know Ellis McTaggart and, later, G. E. Moore. Between the summers of 1894 and 1895, Russell spent three months as an attaché at the British Embassy in Paris, got married, spent three months studying at a marxist socialist movement in Berlin, and wrote a dissertation on the foundations of geometry, on which he was awarded a Fellowship of Trinity. The Fellowship did not entail teaching duties and Russell and his wife spent only part of their time at Cambridge; however he lectured on Leibniz in In the summer of 1900, he attended an International Congress of Philosophy in Paris with Whitehead, and so became acquainted with the Italian mathematician G. Peano. Russell had been thinking about the foundations of mathematics since he was an undergraduate and Peano s ideas inspired him to start work on the book which became The Principles of Mathematics. He had completed the first draft by the end of the year, and felt some satisfaction at having brought the problem of the nature of mathematics so close to a solution before the end of the nineteenth century. It was also about this time that he became a friend of the French logician and philosopher Louis Couturat. Until 1910, Russell was concerned with the writing of The Principles of Mathematics and the Principia Mathematica. He undertook the immense labour of writing out the manuscript of the Principia single-handed, but still found time to write a few articles on other philosophical topics and to engage in some political activity. He made public 2

3 speeches on behalf of Free Trade and contested a by-election as a women s suffrage candidate in He was elected a member of the Royal Society in After the publication of the Principia, Russell sought again to enter politics. He was a member of the Fabian Society 2 but his views cannot have squared very well with those of its other members, for Russell has throughout his life been an opponent of state control. In 1910, he sought to become a Liberal candidate but was rejected because of his atheism. After this he turned to academic life and accepted a Lectureship at Trinity. Russell has said that the writing of the Principia damaged his capacity for abstract thought, but the period from 1910 until 1914 was, philosophically, one of his most productive. It was during this time that Wittgenstein came to Cambridge to study the foundations of mathematics under Russell. About this time Russell gave away nearly all the money he had inherited. Since then he has earned his living either by academic teaching, by writing books, or by journalism. Russell says that he had become a pacifist in 1901, having until then supported the Boer War. During the next decade he was dismayed by the movement towards war with Germany, and when war was declared he was horrified both by the immediate and the more distant consequences he believed it would bring. He became an active anti-war propagandist, advocating that individuals should refuse war service. He was an equally active opponent of conscription and of the treatment meted out to conscientious objectors. In 1916, he was fined 100 for statements likely to prejudice the recruiting and discipline of His Majesty s Forces. These were alleged to have been made in a leaflet issued by the No-Conscription Fellowship concerning a conscientious objector who had been sentenced to two years imprisonment. Immediately after this case he was dismissed from his lectureship at Trinity College. He was refused a passport to enable him to travel to the United States to lecture at Harvard University, and restrictions were placed on his freedom of movement in the British Isles. Another prosecution arose from an article Russell wrote in the No-Conscription Fellowship s weekly 2 Alan Wood, Bertrand Russell the Passionate Sceptic, 1957, p

4 paper. The article was held to be defamatory of the British and American armed forces, and Russell was sentences to six months imprisonment. Because of the influence of his friends and relations he experienced little of the extreme mental and physical hardship of other prisoners at the time. His cell was furnished by his brother s wife and provided with fresh flowers, and he was able to devote his whole time to study. He wrote his book Introduction to Mathematical Philosophy. He published no philosophical work in 1916 and In 1919, Russell accepted an offer from Trinity to reinstate him, but he applied for a year s leave of absence in order to lecture at Peking University and, on his return, resigned his lectureship. He travelled to Russia as well as to China and wrote books about both countries. Soon after his return he twice stood as Labour candidate for Chelsea, but was not elected. In 1927, he started Beacon Hill School with his second wife, Dora. Russell s ideas on education fell significantly short of those of the most advanced thinkers of the time, such as Homer Lane and A. S. Neill, but the school allowed and encouraged great freedom of expression. It does not seem to have been a great success, perhaps because neither Russell nor his wife gave it their full attention, and perhaps because they lacked the ability to deal with the difficult children which a progressive school is likely to attract. Dora Russell continued to run the school after their marriage broke up in Russell became the 3 rd Earl Russell upon the death of his brother in He seems to have inherited little but liabilities from his brother and at that time it was impossible to renounce a title. In 1938, Russell went to the United States and he lived there until He held the post of Visiting Professor at the University of Chicago in 1938, and at the University of California in While in California he accepted a post as Professor of Philosophy at New York City College, but before he took up the appointment an action was brought against the College by a New York tax-payer to annul it. Russell was not a party to the proceedings and was not allowed to take any part in them. The action was successful and the appointment was annulled on three grounds: first, that Russell was not an American; second, that he had not obtained the post by competitive examination; and third, 4

5 that he had put forward immoral and salacious doctrines in his books. Harvard University resisted pressure to cancel an invitation to give the William James Lectures in These lectures were published as An Enquiry into Meaning and Truth. From 1940 until 1943, Russell lectured on the history of philosophy at the Barnes Foundation in Pennsylvania; he had a five-year contract but was dismissed at short notice and successfully sued for wrongful dismissal. His lectures at the Barnes Foundation formed the basis of The History of Western Philosophy, which figured several times in the United States bestsellers lists. In 1944, Trinity College offered him a Fellowship and Lectureship and he returned to England. He gave lectures at Trinity which he published as Human Knowledge, its Scope and Limits. This was his last large-scale contribution to philosophy. In contrast to his reputation in America, Russell became a respected figure in Britain, giving the Reith Lectures for the B.B.C. in 1948, and receiving the Order of Merit in In 1950, he received the Nobel Prize for Literature. In 1940, Russell renounced his pacifism, although not without misgivings. In 1914, he had thought the domination of Europe by the Kaiser s Germany preferable to war; in 1939, he thought the domination of Europe by Nazi Germany worse than war. On similar grounds, he supported British and American rearmament after the war, going so far as to advocate that Russia should be compelled by the threat of force to accept the provisions of the Baruch Plan for the internationalisation of atomic energy. 3 He continued in this opinion throughout the Korean War, but towards the middle of the fifties he began to speak against the possession of atomic weapons by any state. He argued that it was possible to produce scientific evidence to prove that in a future war no nation could be the victor, so that no nation could achieve its aims by war. He proposed an international conference of scientists to report on the probable results of a nuclear war. 4 This proposal was sponsored by an American, Cyrus Eaton, and the conference was held on his estate at Pugwash, Nova Scotia. This was the first Pugwash Conference 3 Christopher Driver, The Disarmers, 1964, p Portraits from Memory, pp

6 and, according to I.F. Stone s Weekly, not a single American newspaper published its report. 5 After this, Russell became more and more active in the movement to ban the testing of atomic weapons and, later, the weapons themselves. He was a sponsor of a number of organisations, including the Campaign for Nuclear Disarmament, of which he became the first president. When C.N.D. split on the issue of civil disobedience, he resigned his presidency and was immediately elected president of the more militant Committee of 100. He took part in the first of the Committee s sit-down demonstrations against the government s decision to provide facilities for the nuclear-missile-firing submarines of the United States. Later in 1961, he was summoned to court, together with other members of the Committee, and required to bind himself over to keep the peace for one year. Together with most of the others summoned he refused, and was sentenced to two months imprisonment. This was reduced to one week on medical evidence. At this time Russell was eighty-nine years old. Early in the sixties, Russell became convinced, not only that the United States cause in Vietnam was unjust, but that it was being furthered in a barbarous manner. He became actively engaged in an attempt to bring the facts about the war to the notice of the American public and the rest of the world. These efforts culminated in the first session of Russell s international War Crimes Tribunal in Stockholm in This Tribunal was based upon a speech made at the Nuremburg war crimes trials by the chief prosecutor, a judge of the U.S. Supreme Court: If certain acts and violations of treaties are crimes, they are crimes whether the United States does them or whether Germany does them. 6 The members of the tribunal were writers, scientists, politicians, and lawyers, many of them with an international reputation. The Tribunal heard evidence and examined the people who presented it; these included observers which it had itself sent to Vietnam. In the sixties, the prodigious stream of publications which Russell produced during the previous seventy years has 5 Christopher Driver, The Disarmers, p War Crimes in Vietnam, p

7 diminished, but still averages at least one a year. In 1967, 1968 and 1969, he published three volumes of autobiography. These take the form of an anthology of autobiographical pieces, written at different times and many of them reprinted verbatim from earlier works, together with a large number of letters written or received by Russell. They succeed by the direct, simple and witty fashion in which Russell tells many aspects of his life, his relations and his distinguished friends. Of all the various subjects on which Russell has written and these include politics, pacifism, marriage and education the present book considers only one: philosophy. It deals mainly with his writings between 1900 and 1920, concerning the philosophy of logic and mathematics and of our apprehension of reality. It is these books and articles which contain his truly original work. Note. This book was in the press when Russell died on 2 February 1970 at the age of 97. 7

8 2 Leibniz Russell s first book on philosophy was An Essay on the Foundations of Geometry, the published version of his Fellowship dissertation. This essay presents an account of the nature of space which is a modification of Kant s. I shall begin, however, with his second book, A Critical Exposition of the Philosophy of Leibniz. The book arose out of the course of lectures which Russell gave at Cambridge in 1899, and it was published a year later. It is important for the light it throws on Leibniz s philosophy and for the doctrines which Russell set against those doctrines of Leibniz he held to be mistaken. It is important also because of the influence which the study of Leibniz exerted on the development of Russell s own philosophy. This influence reveals itself in a number of details but also in one fundamental respect. Russell discovered that Leibniz s metaphysical doctrines had their foundation in logic: the philosophy which Russell himself developed had this same foundation. Russell tells, in the preface to his book, how Leibniz s Monadology seemed to him like a fairy tale: a fantastic, although coherent, picture of the world which there was no reason to accept, and which gave no clue as to Leibniz s reasons for proposing it. Russell found the reasons he was seeking in the Discourse on Metaphysics and in the letters to the theologian Arnauld. These works convinced him that the philosophy of the Monadology was deduced from a few doctrines concerning the nature of propositions; doctrines which, apart from the conclusions which Leibniz drew from them, might seem unexceptionable. What is more, Russell held that the deductions were, for the most part, valid, and that Leibniz s logical doctrines do have as a consequence a large part of the philosophy of the 8

9 Monadology. The logical doctrines themselves, however, Russell rejected. The fairy-tale nature of the Monadology can hardly fail to strike anyone reading it for the first time. Leibniz held that nature was composed of simple substances which he called monads. These monads were without parts, so that they did not have any extension and shape, and were indivisible. They did, however, have qualities. Indeed each had an infinite number of qualities, for no two monads were alike and there were an infinite number of them. In one way each monad was a world apart, for no change in a monad could be produced by the influence of any other; but in another way they formed a unity, for each monad reflected the state of every other, so that in the qualities on any one monad the truth about the whole universe could be read. Each monad developed by an internal principle, but the changes in all the monads fitted together into a perfect order so that the monads gave the appearance of influencing one another. This order was pre-established by one of the monads, God, who necessarily existed. Russell did not believe that the whole of this system followed from the logical doctrines of the Discourse on Metaphysics. Two non-logical principles were required, and apart from these, certain inconsistencies were present. Some arose from Leibniz s deference to the prevailing religious opinions. For example, Russell pointed out that Leibniz s doctrine that men pursue what appears to them to be the greatest good is inconsistent with the existence of sin, which Leibniz accepted. Russell indeed had a low opinion of Leibniz s honesty, holding that he chose his arguments for their persuasiveness rather than their validity. Other doctrines of the Monadology, for example the view that there is more than one substance, are equally consistent with Leibniz s first principles but were genuinely held. Nevertheless, Russell considered that the logical doctrines of the Discourse on Metaphysics formed the principle foundation of Leibniz s philosophy. That all sound philosophy should begin with an analysis of propositions, is a truth too evident, perhaps, to demand a proof. 9

10 That Leibniz s began with such an analysis, is less evident, but seems to be no less true. 7 The logical doctrines which, Russell held, constituted Leibniz s analysis of propositions were three: I. Every proposition has a subject and a predicate. II. A subject may have predicates which are qualities existing at various times. (Such a subject is called a substance.) III. True propositions not asserting existence at particular times are necessary and analytic, but such as assert existence at particular times are contingent and synthetic. The latter depend upon final causes. 8 Premise I needs further amplification. It is doubtful whether Premise II expresses Leibniz s most fundamental definition of what a substance is. There is considerable doubt whether Premise III correctly presents Leibniz s view. What is not in doubt is this: that each of these premises concerns a matter which is fundamental in Leibniz s thought and that his views on these three matters did, to a very large extent, determine his philosophy. Russell s identification of these fundamental problems was, in itself, an important contribution to the study of Leibniz. Russell had no difficulty in finding, in the Discourse on Metaphysics and Leibniz s letters to Arnauld, passages which contained Premise I. In his letter to Arnauld dated 14 July 1686 Leibniz wrote: Finally, I have given a decisive reason which, in my opinion, takes the place of a demonstration; that is, that always in every affirmative proposition whether veritable, necessary or contingent, universal or singular, the concept of the predicate is comprised in some sort in that of the subject. Either the predicate is in the subject or else I do not know what truth is. 9 7 P.L., p P.L., p Leibniz, Basic Writings, tr. G. R. Montogomery, 1962, p

11 Russell held that the importance of Premise I lay in its consequences for relational and for numerical propositions. Both of these sorts of proposition played a large part in Leibniz s philosophy; he did not deny their possibility but supposed that both were species of subject-predicate proposition and would appear as such if properly understood. Here again, Russell drew attention to a passage of considerable importance. The ratio of proportion between two lines L and M may be conceived three several ways; as a ratio of the greater L to the lesser M; as a ratio of the lesser M to the greater L; and lastly, as something abstracted from both, that is as the ratio between L and M, without considering which is the antecedent and which the consequent; which the subject, and which the object. In the first way of considering them, L is greater is the subject; in the second, M the lesser is the subject of that accident which philosophers call relation or ratio. But which of them is the subject in the third way of considering them? It cannot be said that both of them, L and M together, are the subject of such an accident; for if so, we should have an accident in two subjects, with one leg in one, and the other in the other; which is contrary to the notion of accidents. Therefore we must say that this relation, in this third way of considering it, is indeed out of the subjects; but being neither a substance, nor an accident, it must be a mere ideal thing, the consideration of which is nevertheless useful. 10 Leibniz insists on regarding the fact that L is longer than M as the fact that L has the predicate is longer than M, or that M has the predicate is shorter than L, but not as the fact that the relation is longer than stands between them. Russell produced no argument against this attitude until his next book The Principles of Mathematics. There he offers several arguments of which the following is the simplest, and perhaps the most cogent. No predicate of L which does not involve a reference to M can imply a relation between them, while anything which does involve a reference to M is not merely a predicate of L. 11 That Leibniz 10 Leibniz, The Leibniz-Clarke Correspondence, ed. H. G. Alexander, 1956, p P.O.M., p

12 regarded being longer than M as a mere predicate of L is clear, as Russell points out, from the passage quoted above. It is also implied by Leibniz s view that in the predicates of any one substance the whole universe is revealed. Once the predicates of any substance are given, then its relations to all other substances are given too. I have not found this particular point in Russell, although he quotes selections from the passage in the letter to Arnauld of 14 July 1686, in which it occurs. Russell was able to cite passages which established that Premise I led Leibniz to his doctrine of the identity of indiscernibles, but he could nowhere find the argument set out. He makes a suggestion as to how the argument ran, and it seems very likely that he is right. 12 Consider two substances, let us cal them A and B. Between A and B, the argument goes, there will hold a relation which, in the logician s phrase, implies diversity. For example, A may be longer than B, or it may be at a distance from B. Each of these relations implies diversity, for A cannot be longer than itself, nor at a distance from itself. Doubts about the assumption that at least one such relation must hold may be answered by pointing out that whatever relations between A and B are lacking, one at least must hold: that of not being the very same thing. This, in fact, is the relation to which Russell points, but the assumption that not being the very same thing is a relation may reasonably be regarded with suspicion. Premise I implies that a relation between A and B must be regarded as a predicate of one of them, or perhaps as a pair of predicates, one of each of them. The latter seems more reasonable, since the relation concerns each. If this is so, Premise I implies, for the relation we are discussing, that if A and B are two different things, then A lacks the predicate of being the very same thing as B, and B lacks the predicate of being the very same thing as A. Now A is the very same thing as itself, and this relation must be regarded as a predicate of A, so that A will possess the predicate of being the very same thing as A, and B the predicate of being the very same thing as B. Therefore the predicate of being the very same thing as A is a predicate which A has and B lacks, while the predicate of being the very same thing as B is a predicate which B has and A 12 P.L., p

13 lacks. From the assumption that they are not the same thing we have deduced that they are not exactly alike. Against this argument Russell contends that, whatever relations can be reduced to difference of predicates, the relation of numerical diversity cannot: For the numerical diversity of the substances is logically prior to their diversity as to predicates: there can be no question of their differing in respect of predicates, unless they first differ numerically. 13 He concludes that if there exist two things, then there will be at least one relation between them which is not reducible to predicates, so that Leibniz s doctrine that there are no relational propositions should have led him to the conclusion that there is only one substance. When Russell argues that numerical diversity is logically prior to diversity as to predicates, his point must be that having different predicates and having the same predicates both imply diversity, and hence that the diversity of two things cannot be due to their possession of different predicates. This argument of Russell s is no more than a denial of Leibniz s conclusion. It is true to say that two things have the same predicates implies that they are two, but that does not establish that their being two is consistent with their having the same predicates. It is this consistency which Leibniz denies. In the light of the logic which Russell s later work did so much to establish, the doctrine that all propositions have a subject and a predicate is refuted by the existence of universal propositions, such as that all men are mortal. In that logic this proposition has the import that everything is either not a man or mortal. This proposition does not, on the face of it, have a subject and a predicate. However, in the present work Russell is prepared to regard it having as subject the property of being a man, and as being the proposition that this property implies, or involves, the property of being mortal. 14 It seems that Leibniz believed that his general view that all propositions had a subject and a predicate left room for further enquiry as to the nature of such propositions. Couturat later published a number of papers by Leibniz in which various approaches to the logic of universal 13 P.L., pp P.L., p

14 propositions are sketched. On the whole, however, it seems that the view that Leibniz most often expressed and most wished to hold was the one which Russell attributed to him. Russell s main failure lies in his interpretation of Leibniz s views about propositions concerning individual people and things. Russell himself wrote in his preface to the second edition of The Philosophy of Leibniz: Wherever my interpretation of Leibniz differed from that of previous commentators, Couturat s work afforded conclusive confirmation But Couturat carried inorthodoxy further than I had done, and where his interpretation differed from mine, he was able to cite passages which seemed conclusive. 15 The trouble arises over Premise III, which is amplified by Russell in a number of passages. He takes it as equivalent to the view that a proposition to the effect that being a triangle implies having angles adding up to 180 degrees is an analytic and necessary proposition, while one to the effect that Caesar invaded Britain in 55 B.C. is synthetic and contingent. While he is developing an objection to Leibniz s doctrine of substance he says: Moreover, if this were the case, predications concerning actual substances would be just as analytic as those concerning essences of species 16 contrary, Russell implies, to Leibniz s actual views. Now Leibniz often speaks of predicates of individual subjects as being within the subject, and Russell regarded a proposition whose predicate is within the subject as an analytic proposition. 17 Moreover, not only does Leibniz, in passages quoted by Russell, speak of true propositions about individuals as being those, and only those, where the predicate is contained within the subject, but at least one of the deductions by which, according to Russell, Leibniz obtained his metaphysical conclusions explicitly involves this premise. I have in mind the passage where Russell outlines the deduction of the conclusion that each monad, or simple substance, is a world apart, in which no change could be produced by any other. Consider the proposition that Caesar invaded Britain in 55 B.C. If this proposition is true, this can only 15 P.L., p. v. 16 P.L., p P.L., p

15 be because the notion of invading Britain in 55 B.C. is contained within the notion of the subject, Caesar. But if that is the explanation of its truth, then the explanation cannot lie in the influence of any other substance upon Caesar. As Russell puts it, closely following a passage in the letter to Arnauld dated 14 July 1686, from which I quoted earlier: Every predicate, necessary or contingent, past, present or future, is comprised in the notion of the subject. From this proposition it follows, says Leibniz, that every soul is a world apart; for every soul, as a subject, has eternally, as predicates, all the states which time will bring it; and thus these states follow from its notion alone, without the need of action from without. 18 This passage shows that Russell himself doubted that Premise III correctly represented Leibniz s views. Yet in other places he firmly attributes to Leibniz the view of Premise III. Russell assumed and it is an assumption which there is some evidence for but more against that Leibniz identified being necessary with being analytic, and hence being contingent with being synthetic. He held that Leibniz took the propositions of logic and mathematics to be analytic, but those of physics to be synthetic. Russell says: The discovery which determined his views on this point was, that the laws of motion, and indeed all causal laws are synthetic, and therefore in his system, also contingent. 19 However, the passage which Russell cites expresses only the opinion that causal laws are contingent. It says nothing about whether they are synthetic: [Dynamics] is to a great extent the foundation of my system; for there we learn the different between the truths whose necessity is brute and geometric, and truth which have their source in fitness and final causes. 20 Couturat s view that Leibniz did not identify necessity with analyticity but, believing all true propositions to be analytic, saw 18 P.L., pp P.L., p Leibniz, quoted in P.L., p

16 the distinction between necessity and contingency as a distinction between two kinds of analytic proposition, fits much better with Leibniz s writings. Probably the implausibility of regarding an analytic proposition as contingent prevented Russell from attributing this view to Leibniz. Russell, indeed, found it implausible enough to regard necessary propositions as analytic. He saw a continuous line of development from Spinoza who regarded all fundamental truths as analytic, via Leibniz who insisted on the synthetic nature of the causal laws of science, to Kant who insisted on the synthetic nature of the propositions of geometry. Russell, as we shall see, himself carried this development to its end by insisting on the synthetic character of the laws of logic. Perhaps the strongest evidence for Couturat s view is that Leibniz himself recognises the implausibility of regarding an analytic proposition as contingent. In Section XIII of the Discourse on Metaphysics, there is a passage in which Leibniz insists that all true propositions are analytic and tries to reconcile this view with his view that propositions about particular individuals are contingent: We have said that the concept of an individual substance includes once and for all everything which can ever happen to it and that in considering this concept one will be able to see everything which can truly be said concerning the individual, just as we are able to see in the nature of a circle all the properties which can be derived from it. But does it not seem that in this way the difference between necessary and contingent truths will be destroyed, that there will be no place for human liberty, and that an absolute fatality will rule as well over all our actions as over all the rest of the events of the world? 21 Later, to remove this appearance that the difference between contingent and necessary truths will be destroyed, he says: In order to meet the objection completely, I say that the connection or sequence is of two kinds; the one, absolutely necessary, whose contrary implies contradiction, occurs in the eternal verities like the truths of geometry; the other is necessary 21 Leibniz, Basic Writings, p

17 only ex hypothesi, and so to speak by accident, and in itself it is contingent since the contrary is not implied. 22 This explanation seems to make matters worse rather than better, for how can a truth whose contrary does not imply contradiction be an analytic truth? Undoubtedly it is passages like this that led Russell to his view that Leibniz held contingent truths to be synthetic. Leibniz himself sees that this explanation raises the same difficulty again, and attempts another explanation, this new one relying on a distinction between what is necessary ex hypothesi and what is necessary in itself. This, presumably, is a distinction between If anyone were Caesar, then he would cross the Rubicon, which Leibniz admits to be necessary, and Caesar crossed the Rubicon, which he denies is necessary. However, the distinction does not exist if propositions about the individual Caesar are all analytic. In yet another explanation, Leibniz insists that the predicate of a true contingent proposition, as of a true necessary one, is contained in the subject, but points to a distinction in the manner in which it is contained. Couturat found passages which explained this difference of manner as arising from the simplicity of the concept of a triangle and the infinite complexity of the concept of Caesar. None of these explanations is satisfactory, but their failure should not have blinded Russell to the attempt that was being made to reconcile the analytic character of all propositions with the contingent character of some of them. Leibniz s view that the negation of a true contingent proposition is not selfcontradictory cannot be reconciled, as might perhaps be thought, with his view that in every true proposition the predicate is contained within the subject, by ceasing to regard this latter phrase as implying that all true propositions are analytic. In Section VIII of the Discourse on Metaphysics Leibniz writes: Thus the content of the subject must always include that of the predicate in such a way that if one understands perfectly the concept of the subject, he will know that the predicate appertains to it also Leibniz, Basic Writings, p Leibniz, Basic Writings, p

18 Russell s failure to grasp Leibniz s views about necessity and contingency led him to criticise Leibniz for views about subject-predicate propositions which he never held. Russell was right on a fundamental point, that Leibniz held nature to be made up of substances because all true propositions have subjects. Russell says: The ground for assuming substances and this is a very important point is purely and solely logical. What Science deals with are states of substances, and it is these only that can be given in experience. They are assumed to be states of substances, because they are held to be of the logical nature of predicates, and thus to demand subjects of which they may be predicated. 24 How right Russell is in this can easily be proved, for the passage from the Discourse on Metaphysics which I quoted above is followed immediately by this passage: This being so, we are able to say that this is the nature of an individual substance or of a complete being, namely, to afford a conception so complete that the concept shall be sufficient for the understanding of it and for the deduction of all the predicates of which the substance is or may become the subject. 25 However, since Russell could not credit Leibniz with the view that propositions about individual substances are analytic he did not do justice to Leibniz s notion of substance. The passage from The Philosophy of Leibniz which I quoted above continues: And this brings us back to the distinction, which we made in Chapter II, between two kinds of subject-predicate proposition. The kind which is appropriate to contingent truths, to predications concerning actual substances, is the kind which says This is a man, not man is rational. Here this must be supposed defined, not primarily by predicates, but simply as that substance which it is Thus the substance remains, apart from its predicates, wholly destitute of meaning. As to the way in 24 P.L., p Leibniz, Basic Writings, p

19 which a term wholly destitute of meaning can be logically employed, or can be valuable in Metaphysics, I confess that I share Locke s wonder. 26 As the passage from the Discourse on Metaphysics shows, Leibniz s substances are not destitute of meaning. Exactly the opposite is true the substances afforded a conception which enabled them to be understood. Russell is right in saying that Leibniz s doctrine that nature is made up of substances stems from his logical doctrine that every proposition has a subject, but, being wrong about Leibniz s conception of subject-predicate proposition, he is inevitably wrong about Leibniz s conception of substance. This same mistake led him to give Premise II as Leibniz s definition of substance rather than the definition quoted above from the Discourse on Metaphysics that an individual substance has, of its nature, a complete conception from which all its predicates may be deduced. There is no space here to discuss more than the fundamental attitudes of Russell s work on Leibniz. The insight that these metaphysical views were founded on logical ones, led Russell to suspect that the same was true of a metaphysician of his own time, F. H. Bradley. Indeed, he found that Bradley shared some of Leibniz s assumptions but was led by them to very different conclusions. However, this can be discussed better after we have considered the views which Russell formed of the nature of logic. 26 P.L., pp

20 3 Geometry and Logic Russell s next book, The Principles of Mathematics, is very different from his work on Leibniz. In the first place, it is not a work of criticism but treats directly of an important philosophical subject, the nature of mathematics. In the second place, it deals with the subject in a comprehensive manner, so that an enormous number of different problems are raised and discussed. Since the main thesis of the book is that mathematics is a part of logic, it is inevitable that among these problems are many which are central in logic. In order to establish that mathematics and logic are one, Russell sought to show that all the concepts of mathematics could be defined in terms of concepts which belong to logic, and that all the theses of mathematics could be deduced from principles which belong to logic. In particular, he offered definitions of the notions one, two, three and so on, as they occur in such propositions as There is one chocolate in this box, and of such notions as equality in number, as it occurs in such propositions as There is the same number of chocolates in this box as in that one. He offered definitions of such notions as the sum of and the product of, and he offered proofs of the theses of arithmetic such as The sum of two and two is equal to the product of two and two. Since he held that geometry, no less than arithmetic, is a part of logic, one might expect to find definitions of such notions as point, straight line, and triangle, as they occur in such propositions as These three points lie in a straight line and do not form a triangle, and proofs, from principles of logic, of such 20

21 theses as The sum of the angles of a triangle is 180 degrees. However, no such definitions and proofs are offered. Russell did not claim that spatial concepts could be defined in terms of those of logic, nor that theses such as Euclid s theorems followed from principles of logic: we are to remain in the region of pure mathematics: the mathematical entities discussed will have certain affinities to the space of the actual world, but they will be discussed without any logical dependence upon these affinities. 27 In fact, Russell did not recognise the subject which is ordinarily called geometry as part of mathematics; he held that it belonged to applied mathematics, together with such subjects as mechanics. Applied mathematics, according to Russell, was not a part of logic but belonged to natural science. Nevertheless he held that mathematicians did study a subject which was properly, if not ordinarily, called geometry and which was a part of logic. How is this subject related to what is ordinarily thought of as geometry? When Euclid presented his geometry he divided his theses into two kinds: those based upon other theses, which are now called theorems, and those not based upon other theses, which are now called axioms. (He employed another category of proposition, the definitions, but these can be regarded as theses and counted among the axioms.) If chosen correctly, the axioms would imply the whole of Euclid s geometry. These axioms, as we have seen, were not regarded by Russell as theses of geometry: geometry was not concerned with the question of whether these axioms were true, nor did any geometrical theory assert them to be true. He suggested that geometry was concerned, not with the question Are Euclid s axioms true?, but with questions of the form If Euclid s axioms were true, then would be true? Similarly geometry was not concerned, according to Russell, with the truth of any of Euclid s theorems but with the question of whether a theorem was implied by the axioms. 28 In just the same way someone might turn from considering the question Are all 27 P.O.M., p P.O.M., pp

22 men mortal? to the question Is it true that if all men are mortal, then nothing not mortal is a man? However, these modified geometrical questions involve spatial concepts such as point and straight line which do not figure in principles of logic, and which Russell did not regard as definable in terms of concepts belonging to logic. Russell interpreted these modified questions strictly, as questions about implication according to the principles of logic. The questions with which geometry is concerned always take the form, Do Euclid s axioms imply, according to the principles of logic, that? In the same way one might ask Does the proposition that all men are mortal imply, according to the principles of logic, that nothing not mortal is a man? One way of answering such a question would be to argue that since the concepts of man and mortality do not occur in the principles of logic, and cannot be defined in terms of concepts which do, then any principle of logic according to which the implication holds will be a principle according to which the implication would hold, no matter what concepts replace those of man and mortality. Therefore we need only consider the question Is it true that, whatever concepts X and Y may be, if all X are Y, then nothing which is not Y is X? Russell held that geometrical theses provided answers to such questions. Variables have replaced the spatial concepts and the theses are universal; they have consequences for propositions obtained by taking any concepts as the values of the variables. By these two steps, which might be called conditionalising and generalising, Russell obtained a subject which was indeed a branch of logic from the subject which is ordinarily thought of as geometry. 29 It is clear that this branch of logic is of great help in the study of what is ordinarily thought of as geometry, and which Russell called applied geometry. A knowledge of it will enable a scientist who is considering a body of theses of applied geometry, each of which he holds to be true, to tell that they follow from a group of axioms. He will know that the axioms embody everything which needs to be assumed in order to deduce the theorems. All assumptions, analytic or synthetic, a priori or a 29 P.O.M., pp. 7-8,

23 posteriori, trivial or important, obvious or doubtful, will be there. Russell believed that Euclid intended to produce such an axiomatisation of his theses but that, through lack of rigour, he failed in the attempt. 30 Again, this branch of logic enables a scientist who is considering a group of axioms to see what they imply, and so, perhaps, to decide their truth. It is equally clear how strange it is to give the name geometry to a subject which is not concerned with spatial concepts. This strangeness was concealed from Russell by his use of such words as point and straight line for concepts which do occur in his subject but which are not the concepts of point and straight line. How this came about needs to be explained. Consider the proposition All flesh is grass, but there is grass which is not flesh. This might be thought of as a proposition to the effect that the concepts flesh and grass are related in a particular way, that is, as species and genus. A logician who recognised the similarity between this proposition and others such as All squares are rectangles, but there are rectangles which are not squares might be said to have appreciated the species and genus relationship, and if he investigated what followed from the assertion that two concepts were related as species then he might be said to be investigating the species and genus relationship. In a similar way, Euclid s axioms may be thought of as a group of propositions to the effect that the concepts of point, straight line, and so on, stand in a particular relation. This same relation might be asserted to hold between other, non-spatial, groups of concepts. Russell characterised it as one which established a particular sort of two-dimensional order. A logician who recognised a similarity between Euclid s axioms and another group of axioms involving different concepts, perhaps concepts applying to complex numbers, may be said to have appreciated what it is for a group of concepts to establish a two-dimensional order of this sort, and if he investigated what followed from the assertion that a group of concepts did establish such an order, then he may be said to be investigating twodimensional orders of that particular sort. In short, just as the investigation of the species and genus relationship is not an 30 P.O.M., pp

24 investigation of what relationship holds between the concepts flesh and grass, but of one particular relationship which might be said to hold between them, so Russell s Euclidean geometry is not an investigation of what relationships hold between spatial concepts, but of one relation which might be said to hold between them. This is the force of Russell s definition of geometry as the study of orders of two or more dimensions. 31 Now in saying that the concepts flesh and grass are related by the species and genus relationship, we need to indicate whether we mean that all flesh is grass but there is grass which is not flesh, or whether we mean that all grass is flesh but there is flesh which is not grass. We indicate that the former is our meaning by specifying that flesh is the species and grass the genus. If we lacked the words species and genus we might use those for well-known examples of concepts which have been held to exhibit this relationship. For example, if we wished to say that the concepts square and rectangular were related as species and genus we might say that they are related as flesh and grass, meaning that they are related as the concepts flesh and grass are said to be related in the proverb. If this usage were adopted, a logician investigating the species and genus relationship would be said to be investigating the flesh and grass relationship, although all he is doing is to investigate a relation which, in a well-known proverb, is held to relate these two concepts. In the same way, when someone says of a group of concepts that they are related in the way in which Euclid s axioms assert spatial concepts to be related, he needs to specify which concept takes the place of point, which takes the place of straight line, and so on. It is natural for him, in the absence of other terminology, to use the words point and straight line to mean any two concepts which are related as the concepts point and straight line are said to be related in Euclid s axioms. It then seems as if he is discussing the spatial concepts of point and straight line although the truth is that he is discussing a relation which has been asserted to hold between them in a wellknown group of axioms. This usage explains Russell s assertion 32 that the concepts of geometry are definable in logical terms: the 31 P.O.M., p P.O.M., pp