1. Lord Russell

Going from Moore to Russell is taking a big leap - not philosophically perhaps but certainly in terms of World Historical Importance. While Moore was a professor of philosophy and a philosopher's philosopher, a figure of substance within the philosophical community, Russell was one of the great intellectual figures of the twentieth century, famous everywhere. He was important not only as a philosopher, but also as a social and political thinker and even, to some extent, as a political actor.

2. Russell's Biography

Russell lived for 98 years. Those years of his life were exceptionally rich in activities and experiences. In consequence, it is very difficult to write a brief biographical account of Russell. There is an outstanding recent (1996 and 2000) and thorough biography of Russell: but it is not brief: it occupies two substantial volumes. (Ray Monk, Bertrand Russell: the Spirit of Solitude 1872 - 1921 and Bertrand Russell: The Ghost of Madness: 1921 - 1972. Incidentally, Monk is a philosopher and so his biography is the most philosophically insightful of those that have been written about Russell.) Russell himself left many writings about his own life and times, including a three--volume autobiography. It is simple, and enjoyable, to read his accounts of his life (even if they are, as with all autobiographies, misleading in important respects.) So what I shall offer here is only the most skimpy of accounts - and much of that organized differently than is usual. If more is desired, there are the full accounts offered by Monk and Russell.

3. Family and Early Life

Russell came from an important British aristocratic and political family. His grandfather, Lord John Russell, later the first Earl Russell, had been politically significant since Napoleonic times and had been Prime Minister twice (1846-1852 and 1865-1866) during the reign of Queen Victoria. His wife, Russell's grandmother, was from another important aristocratic and political family. Their eldest son, Viscount Amberly, was Russell's father. Amberly and his wife were politically active in liberal causes (e.g. birth control). In fact, they succeeded in having John Stuart Mill become a secular version of godfather to their second son, Bertrand.

However, Lady Amberly and their third and youngest child died in 1874 when Bertrand was barely 3. In 1876, his father died also, from bronchitis and a broken heart. The guardians appointed by Bertrand's parents were not acceptable to the grandparents. In consequence, they had the will changed so that they became the official guardians. Bertrand and his older brother Frank were brought to live with the Russell's. The family had been given, by Queen Victoria, for their lifetime, a royal estate on the outskirts of London and Bertrand grew up there from 1876 until he went to Cambridge in 1890. Earl Russell lived only the first three years of Bertrand's residence there and so his grandmother was the central figure in his family life. Under her, it was a stuffy Victorian household, full of religion, moral earnestness, and aristocratic values - though with a deep liberal set of political views. Russell was strongly shaped by those features of his upbringing: though sometimes in negative ways. (He rather early became a religious disbeliever.)

He was educated at home (though his rebellious older brother was sent away to school). The young Bertrand had no friends and for much of his youth knew only one other boy - and their conversation was mostly concerned with sex, a growing field of interest for Bertrand. He first met a young man he shared some other interests with when he was 16 - and promptly fell in love with the boy's sister (the first of many such episodes.) He met his second love in 1889, and though he kept it a secret for some time, he eventually ended up marrying the girl.

It was decided that he would go up to Cambridge to study mathematics. He passed the entrance examination, the examiner being Alfred North Whitehead who would later become his collaborator on logic. Thus it came about that he enrolled at Cambridge in the fall of 1890.

4. Cambridge

Russell's life opened up at Cambridge. Whitehead had been impressed with his ability and passed the word to other top students, who quickly befriended him. Both intellectually and socially he had found a home and friends. He earned his mathematics degree in the normal three years but found himself more and more interested in philosophy, so stayed a fourth year to complete the required work. After that he won a Fellowship that gave him six years to more or less do what he pleased. It was during those six years that his philosophical career took off.

With that the biographical story becomes deeply enmeshed with the philosophical story and so it is best to proceed with an account of Russell's long, complicated, subsequent life in a different manner.

5. Marriages, Children and Lovers

Russell was married 4 times. (1) An American Quaker girl, Alys Pearsall Smith, whom he had met in 1889. They were married in 1894 (over his grandmother's protests), separated in 1911 (over Alys' protests) and divorced in 1921. (2) He married Dora Black in 1921 (just after the divorce from Alys and just before a baby was due.) Divorced 1935. Russell and Dora spent the rest of their lives fighting. (3) In 1936 he married Patricia (Peter) Spence. Divorced, with great bitterness, in 1952. (4) In 1952 at the age of 80 he married Edith Finch and lived happily ever after.

Along with the four wives he had several serious love affairs, the most significant being with Lady Ottoline Morrell, though there were also an (unconsummated) affair with Evelyn Whitehead, Whitehead's wife, and with Vivian Eliot, T.S. Eliot's wife. Along with these there were lesser affairs and unknown numbers of one-night stands.

Russell had three children: John (1921), Kate (1923), and Conrad (1937). He turned out to be a disaster as a parent and even managed to seriously mess up his grandchildren.

6. Acquaintances

Russell, because he came from such a famous family, because he was a brilliant thinker, because he was deeply involved in the political and social life of his country and then of the world, knew most of the world-historical figures of his times. Writers such as D.H. Lawrence, Joseph Conrad and T.S. Eliot, scientists such as Einstein, political figures such as Lenin and Trotsky, philosophers such as Wittgenstein and Bergson, and of course all the British philosophical, intellectual and political figures of a very important century in history.

A significant part of Russell's public stature, and his financial well-being, came from his writing about those famous persons whom he had known. The reading public was eager to know what Russell had to say about such people and he returned over and over to his personal knowledge of them. His writings remain a source of our knowledge of details of the lives and behavior of all those people he knew.

7. More of his Life

Because Russell's older brother had inherited the title and the substantial portion of the family money (what there was of it), Bertrand had to earn his own way. What he did best, brilliantly, was write. Part of his enormous output was thus for financial reasons. Even after he had become the third Earl Russell in 1931, he was still in continual need of income and so slowed down not one bit in his writing career.

Among other things that need to be known about his life in this very brief introduction: he was dismissed from his Cambridge post in during World War I (1916) for writing an anti-draft pamphlet. In 1918 he was jailed for six months for writing a pamphlet that, according to the government, "prejudiced His Majesty's relationship with the USA"; while in jail he read voluminously and wrote a major book. He opened a famous school, Beacon Hill, which he ran with his wife Dora between 1927 and 1932: it cost him a lot of money and his marriage.

To make money he often lectured in the US. As a result he was offered a position at the City College of New York in 1940. He was immediately attacked as immoral by American religious figures and New York officials. Though he was to teach logic, a judge, in a suit filed by the mother of a prospective student, voided his appointment on the grounds that he was an advocate of sexual immorality. The dismissal merely made Russell more famous.

He worked in the US during most of World War II, returning to England in 1944. The pendulum of public approval was changing in his favor (to Russell's consternation.) He was reelected to a fellowship at Cambridge. In 1950 he received the Nobel Prize: for literature not philosophy (since there is no Nobel for philosophy.) His letter of selection referred to his "many-sided and significant writings, in which he appeared as the champion of humanity and freedom of thought."

The end of his life was consumed with anti-nuclear and anti-Vietnam war activities. He was once more imprisoned, with his wife, though for only 7 days, for "inciting civil disobedience."

8. Character and Personality

Beatrice Webb was a very important radical English reformer. The following is from her diary in 1901 (thanks to Ray Monk's biography.) "Bertrand is a slight, dark-haired man, with prominent forehead, bright eyes, strong features except for a retreating chin, nervous hands and alert quick movements. In manner and dress and outward bearing he is most carefully trimmed, conventionally correct and punctiliously polite, and in speech he has an almost affectedly clear enunciation of words and preciseness of expression. In morals he is a puritan, in personal habits almost an ascetic, except that he lives for efficiency and therefore expects to be kept in the best physical condition. But intellectually he is audacious, an iconoclast, detesting religions or social convention, suspecting sentiment, believing only in the 'order of thought' and the order of things, in logic and in science. He indulges in the wildest paradox and in the broadest jokes, the latter always too abstrusely intellectual in their form to be vulgarly coarse. He is a delightful talker, especially in general conversation, when the intervention of other minds prevents him from tearing his subject to pieces with fine chopping logic. He looks at the world from a pinnacle of detachment, dissects persons and demolishes causes. What he lacks is sympathy and tolerance of other people's emotions, and, if you regard it as a virtue, Christian humility. The outline of both his intellect and his feelings are sharp, hard and permanent. He is a good hater. He is intolerant of blemishes and faults in himself and others. Bertrand is almost cruel in his desire to see cruelty revenged." [Monk, p. 139, Vol 1]

9. The Voluminous Lord Russell

It is extremely difficult to set out the views of Russell even at length - and in a short introduction such as this it is impossible. The first reason for the difficulty is that he had such a long and productive life. He died at the age of 98 and remained until the end intellectually active. Over the course of that long life, he was extravagantly productive. "The quantity of writing that Russell produced in his lifetime almost defies belief. His published output is extraordinary enough (the recently completed Bibliography of Bertrand Russell lists over three thousand publications), but the huge archive of papers and letters he left behind is, if anything, still more remarkable. The Russell Archives estimate that they have over 40,000 of his letters. This is in addition to the vast number of journals, manuscripts and other documents in their collection. Rarely can Russell have passed a day in his long lifetime without writing, in one form or another, two or three thousand words." (Monk, Bertrand Russell: The Spirit of Solitude 1972 - 1921, p. xvii.) [Note: the bibliography which Monk mentions, ed. by K. Blackwell and H. Ruja, is a three volume work itself!]

What he wrote and thought about is enormously various. Technical philosophy, philosophical views on a very wide range of social, political and scientific topics, magazine and newspaper articles as well as lectures intended for the general public, literary works, autobiographical writings, personal letters. Lastly, to add one final complication to the difficulty of telling the story of Russell's thought is the fact that his views on many topics changed over the years. A finished account of such a huge output over so many years on such a wide range of topics about which he did not hold the same views all his life cannot be accomplished. So, in this brief introductory account of his views, I will do as other writers and confine myself to his strictly and narrowly philosophical works.

10. Russell as Writer

Russell was one of the finest intellectual essayists of the twentieth-century, one of the best writers of English prose. He was very clear and also exceptionally witty. It was no accident that he was capable of making a living from his writing, even though his work was intellectually challenging and required thought from readers.

11. Russell's Major Works Categorized

To get some idea of Russell's writings and of the breadth of his interests, it is worth looking at an organized list of his major books.

           Major Philosophical Works

              1897  Foundations of Geometry

              1900  A Critical Exposition of the Philosophy of Leibniz

              1903  Principles of Mathematics

              1910  Principia Mathematica, Vol 1

              1912  Principia Mathematica, Vol 2

              1913  Principia Mathematica, Vol 3

              1918  The Philosophy of Logical Atomism

              1919  Introduction to Mathematical Philosophy

          Other Significant Philosophical Works

               1914  Our Knowledge of the External World

               1921  The Analysis of Mind

               1927  The Analysis of Matter

               1940  An Inquiry into Meaning and Truth

           Major Social and Political Works

              1896  German Social Democracy

              1916  Principles of Social Reconstruction

              1918  Roads to Freedom

              1929  Marriage and Morals

              1930  The Conquest of Happiness

              1934  Freedom and Organization

              1938  Power: A New Social Analysis

              1949  Authority and the Individual

              1951  The Impact of Science on Society

          Popular Science

              1923  The ABC of Atoms

              1925  The ABC of Relativity

              1931  The Scientific Outlook

           Collections of Essays

              1910  Philosophical Essays

              1918  Mysticism and Other Essays

              1925  What I Believe

              1928  Sceptical Essays

              1935  In Praise of Idleness and Other Essays

              1950  Unpopular Essays

              1957  Why I am not a Christian and Other Essays

          Popular Philosophical Works

              1912  The Problems of Philosophy

              1927  An Outline of Philosophy

              1945  A History of Western Philosophy

              1948  Human Knowledge: Its Scope and Limits


               1953  Satan in the Suburbs and Other Stories

               1954  Nightmares of Eminent Persons and Other Stories

           Personal Writings

               1956  Portraits from Memory and Other Essays

               1959  My Philosophical Development

               1967, 68 and 69 Autobiography of Bertrand Russell (3 vols)

12. Social and Political Thought

Because an examination of Russell's thought on social and political issues would take too long here, I will settle for announcing that there is an excellent book which examines, with philosophical sense, both Russell's thinking about those issues as well as his activism: Bertrand Russell: A Political Life by Alan Ryan (1988). Ryan is an English political philosopher and so knows how to fit Russell into English history and politics.

13. The Break in Russell's Philosophical Life

Even pursuing Russell's strictly philosophical work would be a massive task as it is spread out over a very long life. There is, however, something that, for most writers, puts constraints on what needs to be covered in a brief philosophical survey.

Russell's philosophical work after 1920 is not nearly of the same quality, and philosophical influence, as it was before then. By and large, when philosophers admire Russell as a superb philosophical mind, what they are thinking of and talking about is what he did before 1920, especially up through the Lectures on Logical Atomism that he gave in 1918. He occasionally turned to philosophy in the last fifty years of his life but none of it has anywhere near the stature of what he accomplished by 1920. Why that happened to cause the break can be fitted into the narrative later.

14. Russell and Mathematics

To understand the origins and direction of Russell's philosophical life, it is necessary to go back to his childhood. When he was 11 his brother decided to teach him Euclidean geometry. Russell said of his response to geometry (a reaction very like that of Thomas Hobbes 250 years earlier) "I had not imagined that there was anything so delicious in the world....[It was] as dazzling as first love." (Monk, Vol 1, p.23) What was so wonderful about geometry? There are genuine proofs in geometry: its theorems are established as true beyond any doubt.

Monk guesses (intelligently): "Previously, alone in the garden at Pembroke Lodge, Russell had speculated a good deal about things he did not know, and could not know: what had has parents been like? was his mother really wicked? why did his grandmother tremble at any mention of insanity? were his grandmother's religious beliefs true? On some of these questions he might have opinions, on others he could only weave fantasies, but to none of them could he know for certain the right answer; he could only choose to accept or reject what he was told, and in the case of conflicting answers from different people, could only decide on the basis of who to support, who to give his loyalty to. The beauty of geometry was that the truth of a proposition was not just asserted, it was proved; it did not await any kind of denial or counter-assertion, and who said it or what they felt about it were neither here nor there. The idea that something - anything - could be known with certainty in this way was delightful, intoxicating." (Monk, Vol 1, p26)

Russell's philosophical life was centered on the search for certainty and it was directed by the idea that in mathematics (and only mathematics) could certainty be had. There was a problem, however. The theorems in Euclidean geometry are proven - they are deduced from the axioms. But why accept the axioms? Russell's brother did not know any answer to that and convinced Russell that if he was to proceed with his learning he simply had to accept them. The need for an answer to that provided Russell with his initial philosophical project.

Russell and his family decided that he would go to Cambridge, his father's school, to study mathematics. He studied hard, passed the entrance examination and was awarded a minor scholarship. In the autumn of 1890, he began his studies there. Mathematics at Cambridge, however, was not to his liking. It turns out that Cambridge was a mathematical backwater, not attuned to the latest developments then going on in Germany. The curriculum was old and stale, mostly concerned not with pure mathematics but with the application of mathematics to physical problems. And even then what Russell was taught seemed to him tricks to enable someone to solve problems rather than to understand. In his third year, he passed his examinations, winning (only) seventh place in the ranking system. And then: "When I had finished my [examinations], I sold my mathematics books and made a vow that I would never look at a mathematical book again." (Monk, Vol 1, p. 51.)

He turned to philosophy.

15. The Beginnings of Russell's Philosophical Career

In the mid-1890's idealism was the dominant view in English academic philosophy. It was not quite as strong at Cambridge as it was at Oxford. However, one of the major English idealists was on the philosophy faculty at Cambridge: J.M.E. McTaggart. McTaggart was not only one of Moore and Russell's teachers, formally speaking, but was also just a little older than they were, also a member of the Apostle's, and so philosophically and socially involved with them outside the lecture hall. All those things taken together, it is no wonder that both Russell and Moore began their philosophical development in the mid 1890's as idealists.

Russell was perhaps less interested in the idealists' claim that reality is mental than he was in the notion of the Absolute, the idea that reality is one: despite appearances, there is really only one thing, the Absolute. Moreover and consequently, every description of reality that is only partial, which does not connect with every other description, will involve contradictions.

Russell may have tossed his mathematics books but what he was philosophically interested in from the start of his philosophical career was the philosophy of mathematics. Being a Hegelian, an idealist, meant that he assumed that mathematics contained contradictions that could only be resolved by incorporating it into the full metaphysical system. Mathematics itself seemed to second that belief: some of the chief concepts of mathematics as Russell knew them - continuity, the infinitesmal, the infinite - appeared to be contradictory to thinkers of the time.

Russell, in the late 1890's, was slowly led out of those views. For one thing, he learned of the work of German mathematicians - Weierstrass, Dedekind, Cantor - none of whom had been included in the Cambridge mathematical program. He learned from them that the supposed contradictions in mathematics could be made to vanish by purely mathematical, not metaphysical, means.

Further he was slowly weaned from his adherence to idealism over those same years. Probably the big step was in the summer of 1898 when he and Moore worked together just as Moore was completing his own rejection of idealism (which was announced to the world in the 1899 paper 'The Nature of Judgment'.) Both the idealist's monism and the idea that reality is mental were the victims of that development.

Russell was now a realist and a pluralist. He, and Moore, also became something more. Notice that for the idealist, understanding is gained by putting ideas together, by connecting them. Each lower level of a description of reality cannot stand on its own and will involve contradictions. Those can only be done away by incorporating the partial description into a larger scheme of thought, where the final stage of the story is an account of the Absolute, of the whole of reality. Hence genuine understanding is accomplished by putting matters in larger and larger frameworks.

With Moore's and Russell's rejection of idealism that account of what it is to understand was replaced by a different model. Understanding will be reached only by a process of analysis, by breaking things down into smaller and smaller pieces until rock-bottom is reached. From that point on their philosophical writing emphasized the virtues of analysis. That was the birth of analytic philosophy.

When Russell gave up idealism, he gave up the idea that mathematics must contain contradictions. There remained, however, another step: it was not thereby demonstrated that mathematics wasn't contradictory, not thereby proved that mathematics did involve the kind of certainty that he had assumed before he became a philosopher. Trying to prove that was Russell's next project.

16. 1900, Paris and Peano

The first World Congress of Philosophy was held in Paris in the summer of 1900. Russell as a bright young philosopher of mathematics was invited to read a paper. However, what turned out to be crucially important for his philosophical development was his encounter there with Giuseppe Peano, an Italian mathematician. Peano and his disciples struck Russell as the most accomplished and sharpest of those discussing the issues he was interested in. He asked Peano for all his writings. When they arrived following the Congress, he studied them closely.

The first part of Peano's project was to create a symbolism for statements. Russell picked it up and, with variations, established the standard symbolism for doing logic. Students of logic today learn what is basically the Peano-Russell symbolism.

With the symbolic notation in hand, Peano had created an axiom system from which arithmetic/mathematics could be formally derived as theorems. That system used three primitive notions - that of number, zero (0) and successor of. The axioms were:

1. 0 is a number

2. The successor of any number is a number

3. If x and y are numbers and if their successors are equal, then x and y are equal (i.e. each number has only one successor)

4. O is not the successor of any number

5. If S is the set of numbers that contains 0 and if the successor of any number is also in S, then S contains all the numbers.

Those axioms, along with appropriate definitions (e.g. 'the successor of 0' =df '1') can be used to derive modern mathematics. (Further details are not relevant here.)

Peano, however, was up to something even larger than the axiomation of mathematics. He had begun to create a new system of logic. That was where Russell entered the fray.

17. The Aftermath of Paris

The remainder of 1900, following the contact with Peano and the study of his work was, for Russell, a frenzy of creative activity. What has not been sufficiently worked out by Russell scholars is how much of what he accomplished had started with Peano and how much he developed on his own. So while I attribute below new ideas to Russell, more accurate historical work will certainly give Peano some significant credit for initiating at least some of the developments.

Russell realized that it is possible to start further back than Peano had done, that Peano's axioms and primitive notions could themselves be derived from more still more basic principles and ideas. And he came to see that the more basic material was logic.

At the time Russell was working, logic was still Aristotelian logic. Yes, there had been niggling additions to it over the centuries (see W. and M. Kneale's huge The Development of Logic) but what was then thought of as logic was mostly what Aristotle had produced over two thousand years earlier. There had been some interesting developments earlier in the 19th century - Boolean algebra especially. But those had not displaced the standard Aristotelian logic.

What Russell did was to create an entirely new system of logic. Instead of starting logic with an updating of Aristotelian logic, Russell developed what we now call the propositional calculus as the basic logical sub-system. Then on top of that and making use of it was what we today call quantificational logic or the predicate calculus: this was Russell's modernization of Aristotelian logic. For that updating Russell made central the notion of a 'class', his term for what Georg Cantor had called 'set'. There was then added as a further sub-system the logic of relations ('3 is the successor of 2' is a relation) - on that topic it was DeMorgan, Peirce and Schroeder rather than Peano who had anticipated Russell.

All of Peano's axioms and then the whole of mathematics could be derived - so Russell believed - from his new logic. Hence came his famous idea, the logistic thesis, that mathematics is nothing but logic. "All pure mathematics - Arithmetic, Analysis and Geometry - is built up by combinations of the primitive ideas of logic, and its propositions are deduced from the general axioms of logic." ('Mathematics and the Metaphysicans', July 1901, collected in Mysticism and Logic.)

Russell threw himself into re-writing a huge manuscript entitled Principles of Mathematics that he had been working on for some time. All these new, post-Paris and post-Peano, ideas and developments were embodied there and the book has become the first exposition of Russell's views on logic. He thought that it would be the first of two volumes and that in the second he would produce completely symbolic proofs of his logic and the logistic thesis. The expectation expressed there is that that second volume would be produced swiftly. That was not to be - it was ten years later before the first volume of what, by then, had become three additional volumes was published.

In fact, the Principles of Mathematics itself was not published for another three years. That was because a major problem in his system reared its head. As he said:  "The honeymoon could not last, and early in the following year intellectual sorrow descended upon me in full measure."

18. Russell's Paradox (1)

In the spring of 1901, with the manuscript seemingly completed, Russell discovered what looked to be a contradiction within his system of logic. He thought it a trivial problem that he could get around easily enough. However, he came to realize that it was a deep and serious problem – hence "the intellectual sorrow".

The discovery of a contradiction within a deductive system struck logicians and mathematicians as destructive of the entire system. He struggled for the next two years (and more in fact) to find a solution that would save his achievement. What Russell discovered has come to be known as Russell's Paradox. It is something to be known by all post-Russellian analytic philosophers and logicians.

19. Russell's Paradox (2)

Perhaps the easiest way to start grasping the paradox is with an analogy that Russell himself created in explanation.

Suppose that there is a village with a barber who shaves all and only those who don't shave themselves. Question: does the barber then shave himself?

Suppose that he does shave himself. Then since he shaves only those people who don't shave themselves, he must then not shave himself. Hence if he shaves himself, he doesn't shave himself. Suppose now the other possibility, that he doesn't shave himself. Then since he shaves all the people who don't shave themselves, he must shave himself. Hence if he doesn't shave himself, he does shave himself. So if he does, he doesn't and if he doesn't, he does. That is a contradiction.

As a preliminary to an exposition of the paradox, we need to be a little more clear about the notion of a class which was central to Russell's new logic.

A class, or a set to use Cantor's more familiar term, has members. The class/set is a mother has as its members all entities that are mothers, i.e. all things of which it is true that they are mothers, all things that satisfy the predicate 'is a mother'. Of course classes or sets can be formed by including in it arbitrarily selected objects: e.g. the set S has as its members (say) me, seven tomcats and the number 14. But the usual way of forming a set is to specify a predicate and the members of the set will then be all the things of which that predicate is true.

Russell's assumption was that any old predicate will determine a class/a set. All and only those objects of which it can truthfully be said 'is a cat' will be members of the class of cats. And so too for any other predicate: 'is the number after 11', 'has Napoleon as an ancestor', 'is a grain of sand'.

It is important to realize that some sets are themselves members of other sets: for example the predicate 'has more than 17 members' applies to many classes (more than 17) and so those classes too are members of the class/set has more than 17 members.

Some classes, although it isn't common (the class of cats is not itself a cat), are even members of themselves (or so Russell assumed at the time): the class of all classes is a member of itself. That causes no problems. Russell, however, started to think of those sets that are not members of themselves. And, given his general principle that any expression, any predicate, determines a class, he came to think that there should be a well-formed class that consists of all those classes that satisfy the predicate 'is not a member of itself'.

We have now come to the paradox. Is the class of classes not members of itself a member of itself? If it is a member of itself, then it must satisfy the predicate 'is not a member of itself' - i.e. if it is a member of itself, then it can't be a member of itself. But if it isn't a member of itself, then it, since it is truly a class that isn't a member of itself, it must be a member of itself – i.e. if it isn't a member of itself, it must be a member of itself. Hence the expression 'is a class/set which isn't a member of itself' seemed a perfectly normal predicate - however, it leads to a contradiction.

Russell had abandoned his idealist days in which there were thought to be contradictions in mathematics and was working to found a new logic from which mathematics in all its beauty followed - and had ended up with a system which contained a contradiction. That was the bitter fruit of 1901 for him.

(For a formal and technical account of Russell's Paradox, see the article in the Stanford Encyclopedia of Philosophy

20. Finding a Solution: The Theory of Types

As he reworked the part of his manuscript in which the problem occurred, he continued to search for a solution. He was able to write out a brief account of what came to be his solution as the final chapter of Part I. By the end of 1902 he developed the idea further in an appendix although he presented it as not the full solution. (Russell and set theory kept returning to the issue of the solution to the contradiction for many years.)

He called his solution The Theory of Types. It held that things could only be members of classes higher in a hierarchy than itself. Hence it makes no sense to ask of a class whether or not it is a member of itself. The idea of a class of all classes vanishes in the scheme. The contradiction that was caused by assuming that a class could be a member of itself was avoided.

(For a more technical account, see the Stanford Encylopedia of Philosophy -

21. Discovering Frege

While trying to work out the solution to the paradox and in preparation for sending the manuscript of the Principles of Mathematics off to a publisher, Russell decided to review other writings. He started with a German mathematician, Gottlob Frege. Russell had heard of Frege before, had even looked into his work – but had not been able to understand what Frege was up to. Now however he understood – Frege been engaged in exactly the same project he was working on, namely creating a new logic from which mathematics could be derived. (Frege's very awkward symbolism was chiefly responsible for Russell's previous inability to see that they were doing the same thing.) In fact, their resulting systems of logic were largely the same. And Frege was more than 20 years ahead of Russell!

At this point it is useful to engage in a short digression on Frege, now recognized to be one of the founders, along with Moore and Russell, of the analytic tradition, though almost unknown when Russell looked into his work.

22. Gottlob Frege: 1848 - 1925

Frege, one of the major names in the history (and the present) of analytic philosophy, had a less uneventful life than even G.E. Moore. He was born in northern Germany to parents who ran a girl's school. In 1869 he went off to the university at Jena and two years later transferred to Gottingen where he received his PhD in 1873. He was a mathematician, his dissertation on a topic in geometry, although he did study a little physics and a little philosophy as well. Upon graduation, he applied for a teaching job with no salary at Jena. In support of his application he submitted a paper which made a nice little mathematical contribution and on the basis of which he was hired (despite a poor interview). From 1874 to until his retirement in 1918, 44 years, he was on the mathematics faculty at Jena. He taught and wrote mathematics. He also married and had a family, though all of his natural children died young, an adopted son being the only one to survive.

Five years after acquiring his position (i.e. in 1879), he published a short book (pamphlet rather) entitled Begriffschrift (Concept Writing) which was sufficient to win him a salary. The book, however, did not impress either mathematicians or logicians, and was compared unfavorably to a somewhat similar enterprise by the English mathematician George Boole. Another five years later (1884), he published a further small, though slightly larger, book entitled Grundlagen dur Arithmetik (The Foundations of Arithmetic), an informal setting out of his logicism thesis. (The Grundlagen was, in being an informal discussion of logic and logicism, the analog of Russell's (1903) The Principles of Mathematics). To avoid some of the criticisms of his first book, he wrote it in a different, more accessible, style. That didn't work: its reception was even worse than that of the first book. There were only three reviews, all hostile. It was ignored for decades.

However badly received, it is now recognized that in those two books he founded modern logic, and made huge and original contributions to the philosophy of logic and mathematics.

Between 1884 and 1893, he wrote three papers that are today accepted as major and founding works in the modern philosophy of language. His aim in them was to analyze language as part of his logicist project.

The final effort in that line of work was to be his Grundgesetze dur Arithmetik (The Basic Laws of Arithmetic) which was completed in 1893. However, no publisher was willing to publish both volumes. An agreement was worked out to publish the first volume and if enough interest was generated the second would be issued. However, so little interest was given it in the next ten years that ultimately Frege himself had to pay for the 1903 publication of Volume 2. Volume 1, however, did earn him a senior professorship at Jena. It also fell into the hands of the Italian mathematician and logician Peano who brought it to Russell's attention with major results.

Frege had the second volume in press when he received a letter from Russell establishing that Axiom 5 of Grundgesetze made the entire system of The Basic Laws inconsistent. Frege, of course, was completely depressed. He tried some patching, weakening the offending axiom. Since Volume 2 was then in press, the best he could do was to insert an appendix in which he addressed Russell's Paradox and offered his replacement axiom. (That too makes the system inconsistent though it took Frege some time to realize it.)

He retired in 1918 and wrote some further papers and reviews (one of which had a significant influence on Husserl) - but at bottom he gave up on logicism and thought that his life's work was a failure. Well, he didn't entirely give up. There remains a letter to his adopted son, which he wrote six months before his death while making out his will, in which he left his unpublished writings to him: "Dear Alfred, Do not despise the pieces I have written. Even if all is not gold, there is gold in them. I believe there are things here which will one day be prized much more highly than they are now. Take care that nothing gets lost. Your loving father (It is a large part of myself which I bequeath to you herewith.)"

One final biographical note should be added. Michael Dummett, who is the leading expositor and defender of Frege as philosopher in contemporary philosophy, also has been deeply involved in opposing racism in Britain, in fact for some time taking a break from his academic pursuits to work for racial justice in the country. He concludes the Preface to one of his important books on Frege thusly: "There is some irony for me in the fact that the man about whose philosophical views I have devoted, over years, a great deal of time to thinking, was, at least at the end of his life, a virulent racist, specifically an anti-semite. This fact is revealed by a fragment of a diary which survives among Frege's Nachlass, but which was not published with the rest by Professor Hans Hermes in Freges nachgelasseme Schriften. The diary shows Frege to have been a man of extreme right-wing political opinions, bitterly opposed to the parliamentary system, democrats, liberals, Catholics, the French and, above all, Jews, who he thought ought to be deprived of political rights and preferably, expelled from Germany. When I first reread that diary, many years ago, I was deeply shocked, because I had revered Frege as an absolutely rational man, if, perhaps, not a very likeable one. I regret that the editors of Frege's Nachlass chose to suppress that particular item. From it I learned something about human beings which I should be sorry not to know; perhaps something about Europe, also." (Dummett, Frege: Philosophy of Language, p. xii.)

23. Russell and Frege

Russell responded excellently to his discovery of Frege and the fact that Frege was engaged in the same project that he thought of as his own - and was in fact many years ahead in the work. He said in the Preface to the Principles "Professor Frege's work, which largely anticipates my own, was for the most part unknown to me when the printing of the present work began. I had seen his Grundgesetze der Arithmetik, but, owing to the great difficulty of his symbolism, I had failed to grasp its importance or to understand its contents....If I had become acquainted sooner with the work of Professor Frege, I should have owed a great deal to him, but as it is I arrived independently at many results which he had already established." Russell added an Appendix to the Principles. 'The Logical and Arithmetic Doctrines of Frege' in which he gave the first sympathetic exposition of Frege's views.

However, he quickly spotted the fact that the contradiction which troubled his own system was formuable in Frege's logic. In June of 1902 he wrote a letter to Frege developing the Paradox within Frege's Grundgesetze. Frege immediately saw that Russell was correct. Unfortunately the second volume of that work was already in press and the best that Frege could do was try to patch things up in an appendix. Later it was discovered that the solution he provided there did not work.

The connection made between the two creators of modern logic, they then carried on a correspondence for some months - with considerable disagreement, especially on the notion of propositions, i.e. of truth-bearers.