Bertrand Russell

How to use this course

Bertrand Russell lived 97 years and wrote on almost everything: logic, philosophy, education, marriage, religion, war and nuclear weapons. He received the Nobel Prize in Literature and went to prison twice. This course concentrates on the decade, roughly 1900 to 1910, in which he did his mathematical work. In those years he found the paradox that bears his name, invented the theory of types to escape it, analysed definite descriptions, and wrote Principia Mathematica with Alfred North Whitehead. Those works set the agenda for logic in the twentieth century, from Kurt Gödel’s incompleteness theorem to Alonzo Church’s typed λ-calculus and today’s proof assistants. His philosophy and public life appear more briefly in Part III.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. We write formulas in modern notation, with \(\to\), \(\wedge\), \(\forall\) and \(\exists\) where Principia uses \(\supset\), dots, \((x)\) and \((\exists x)\). Propositions of Principia are cited by their starred numbers, as in \(\ast54.43\). This course pairs with the courses on Gottlob Frege, whose system the paradox destroyed, and on Gödel and Church, who built on Principia.

Timeline

Pembroke Lodge, 1872–1890

Bertrand Arthur William Russell was born on 18 May 1872 at Ravenscroft, near Trellech in Monmouthshire, the younger son of Viscount Amberley and his wife Katharine, a daughter of Lord and Lady Stanley of Alderley. His parents were radicals: they supported birth control and women’s suffrage. John Stuart Mill, a friend of the family, stood as his godfather in a secular sense. His mother died of diphtheria in 1874, and his father died in January 1876. The father’s will had named two freethinkers as guardians. The grandparents had it set aside, and the boys went to live at Pembroke Lodge in Richmond Park with their grandfather, Lord John Russell, twice Prime Minister, and their grandmother. After his grandfather’s death in 1878, his grandmother, a strict Scottish Presbyterian, brought him up.

He was educated at home by governesses and tutors. At eleven his brother Frank introduced him to Euclid. He later described it as one of the great events of his life, and he recalled his disappointment that the axioms had to be accepted without proof. That disappointment, the wish to know why the starting points of mathematics should be believed, runs through all his later work. As a teenager he kept a secret notebook, written in Greek letters, in which he argued himself out of free will, immortality and finally belief in God. He was lonely, and he said later that mathematics kept him from despair.

Cambridge and the break with idealism, 1890–1900

Russell entered Trinity College, Cambridge, in 1890 to read mathematics. Alfred North Whitehead, eleven years older and a fellow of Trinity, had examined him for his entrance scholarship and recommended him to the Cambridge Apostles, the university’s secret discussion society. Russell was Seventh Wrangler in 1893. He found the Tripos teaching, with its tricks for examination problems, unrewarding, and turned to philosophy, taking a first in Part II of the Moral Sciences Tripos in 1894.

Philosophy in Cambridge then meant British idealism, after Hegel and F. H. Bradley. Russell became an idealist. In 1894 he married Alys Pearsall Smith, an American Quaker, against his family’s wishes. They spent time in Berlin, where he studied the socialist movement; German Social Democracy (1896) was his first book. His fellowship dissertation of 1895 became An Essay on the Foundations of Geometry (1897), a Kantian argument that projective geometry is known a priori while the choice of metric geometry is empirical. Einstein’s general relativity later made its central claim, that space must have constant curvature, untenable.

In 1898 Russell and his friend G. E. Moore rebelled against idealism. Moore argued that the objects of knowledge exist independently of being known. Russell concluded that relations are real and cannot be reduced to properties of their terms, as the idealists had held. Lectures on Leibniz, which became A Critical Exposition of the Philosophy of Leibniz (1900), convinced him that Gottfried Wilhelm Leibniz’s metaphysics followed from a faulty logic of subject and predicate. A better logic of relations was needed. He found its tools in Paris.

Paris, the paradox and the Principles, 1900–1903

In August 1900 Russell and Whitehead went to the International Congress of Philosophy in Paris. There Russell heard Giuseppe Peano and his school and was struck by the precision their symbolism gave to discussion. He read all of Peano’s work in the following weeks. In September he extended Peano’s notation to a logic of relations (Lesson 04), and by the end of the year he had drafted most of a book arguing that all of pure mathematics can be derived from a few logical notions. He called the autumn of 1900 the intellectual high point of his life.

In May 1901, while studying Georg Cantor’s proof that there is no largest cardinal number, he found the contradiction now called Russell’s paradox (Lesson 05). At first he thought it a trivial error. It was not. In June 1902 he wrote to Gottlob Frege, whose Grundgesetze he had just read with admiration, and showed that the paradox could be derived in Frege’s system. Frege replied within a week that his foundations had been shaken. Russell later wrote that he knew of nothing to equal Frege’s dedication to truth in that reply.

The Principles of Mathematics appeared in 1903. It stated the logicist thesis, that mathematics and logic are one, discussed the paradox openly in its tenth chapter, added an appendix on Frege’s doctrines and a second sketching a theory of types. It did not solve the paradox. That took most of the next five years. The Principles was announced as volume I; volume II, the formal derivations, grew into Principia Mathematica, written with Whitehead from about 1902 to 1910.

The logic of relations

For the general reader. “Greater than”, “ancestor of” and “between” are relations, links between two or more things. Traditional logic treated them awkwardly, as properties of one of the things. Russell argued that relations are basic, and he gave them an algebra: converse, composition, domain, field. With it, order, series, continuity and the ordinal numbers could all be defined in logical terms.

Operations on relations

Following Peirce and Ernst Schröder in content and Peano in notation, Russell treated a binary relation \(R\) extensionally as the class of pairs \((x,y)\) with \(xRy\). The basic operations are the converse \(\breve R\) (\(y\breve Rx\iff xRy\)), the relative product \(R|S\) (\(x\,(R|S)\,z\iff\exists y\,(xRy\wedge ySz)\)), the domain, converse domain and field. These appear in Principia in sections \(\ast31\) to \(\ast38\).

Russell’s argument against reducing relations to properties ran as follows. If “\(a\) is greater than \(b\)” meant that \(a\) and \(b\) have magnitudes \(m_a,m_b\), one still needs the relation “\(m_a\) exceeds \(m_b\)” between the magnitudes. Symmetric relations such as “same colour as” can be reduced to a common property. Asymmetric relations cannot, and order is built from asymmetric relations.

The paradox

For the general reader. Some collections contain themselves: the collection of all collections with more than three members has more than three members. Most do not. Consider the collection of all collections that do not contain themselves. Does it contain itself? If it does, it does not; if it does not, it does. Russell found this in 1901, and it showed that the obvious principle “every property defines a collection” is false.

Proof. Take \(\varphi(x)\) to be \(x\notin x\). Comprehension provides \(r\) with \(\forall x\,(x\in r\leftrightarrow x\notin x)\). Instantiating \(x:=r\) gives \(r\in r\leftrightarrow r\notin r\), which is contradictory in classical, and even intuitionistic, logic.

Where it came from

Cantor had proved that every set \(A\) has more subsets than elements. If there were a universal class \(V\), its subsets would be elements of \(V\), so \(\mathcal P(V)\subseteq V\), and \(V\) would have at least as many elements as subsets. Russell tried to find the flaw in Cantor’s proof in this case. Running the diagonal argument with the identity map \(V\to\mathcal P(V)\), \(x\mapsto x\), produces the class of \(x\) with \(x\notin x\). That is the paradox.

Ernst Zermelo had found the same contradiction independently, probably a little earlier, and told Hilbert and Edmund Husserl in Göttingen, but did not publish it. The Burali-Forti paradox (1897) of the largest ordinal and Cantor’s own paradox of the largest cardinal belong to the same family. Russell also gathered paradoxes that involve language or truth: the liar, Berry’s paradox of “the least integer not nameable in fewer than nineteen syllables”, which he credited to G. G. Berry of the Bodleian Library, and the paradoxes of Jules Richard and Kurt Grelling. He insisted that one solution should cover them all.

Henri Poincaré argued in 1906 that all the paradoxes come from impredicative definitions, which define an object by quantifying over a totality to which it belongs. Russell agreed and formulated the vicious-circle principle: no totality can contain members defined only in terms of that totality. Zermelo’s answer (1908) was different: keep impredicative definitions but restrict comprehension to subsets of an already given set. That became ZF set theory.

The theory of descriptions

For the general reader. “The present King of France is bald” seems to be about the present King of France. There is none. Is the sentence then meaningless, true, or false? In “On Denoting” (1905), Russell argued that phrases like “the present King of France” do not name anything. Correctly analysed, the sentence says that there is exactly one King of France and that he is bald. It is false. The analysis showed how grammatical form can mislead about logical form, and it became a model for twentieth-century philosophy.

The first line says that exactly one thing satisfies \(\varphi\) and it satisfies \(\psi\). The second line, read “the \(\varphi\) exists”, says that exactly one thing satisfies \(\varphi\). A description is an incomplete symbol: it has no meaning in isolation, but every sentence containing it has one.

Three puzzles

Excluded middle. Either “the King of France is bald” or “the King of France is not bald” should be true. Russell distinguishes two readings of the negation, according to the scope of the description. With the description having wide (Russell: primary) scope, \(\exists x\,(\forall y\,(K(y)\leftrightarrow y=x)\wedge\neg B(x))\) is false. With narrow (secondary) scope, \(\neg\exists x\,(\forall y\,(K(y)\leftrightarrow y=x)\wedge B(x))\) is true. Excluded middle holds for each reading.

Identity. George IV wished to know whether Scott was the author of Waverley. He did not wish to know whether Scott was Scott. If “the author of Waverley” were a name for Scott, substitution would turn the first into the second. On Russell’s analysis, “Scott is the author of Waverley” says that exactly one person wrote Waverley and that person is Scott. It contains no name for Scott other than “Scott”, and nothing licenses the substitution.

Non-existence. “The round square does not exist” is \(\neg E!\,\iota x\,(R(x)\wedge S(x))\), a true statement that refers to no round square. Russell thus escaped the realm of non-existent objects of Alexius Meinong, which he had accepted in the Principles. Frege had treated the same puzzles with sense and reference (see the Frege course). Russell rejected that solution; he thought the relation between a sense and its reference could not be stated coherently.

The same strategy, eliminating apparent names by contextual definition, is applied in Principia \(\ast20\) to classes: a statement about the class \(\hat x\,\psi(x)\) is defined as a statement about some predicative function coextensive with \(\psi\). Russell called this the “no-class” theory.

The theory of types

For the general reader. Russell’s cure for the paradox was to sort everything into levels. At the bottom are individuals; next, classes of individuals; then classes of those classes, and so on. A class can contain only things of the level just below, so “a class that contains itself” cannot even be said. To block the paradoxes of language as well, he subdivided the levels further, then needed a special axiom to make ordinary mathematics possible again.

Simple types

Assign type \(0\) to individuals and type \(n+1\) to classes of objects of type \(n\). Each variable carries a type, \(x^n\), and the atomic formulas are \(x^n\in y^{n+1}\) and \(x^n=y^n\). Comprehension holds level by level:

The formula \(x\in x\) is not well formed, so neither is Russell’s \(\varphi\). Every object occurs at one level, and numbers recur at every level: there is a number 2 of type 2, the class of pairs of individuals, another of type 3, and so on. Russell called this typical ambiguity and wrote formulas without type indices when any consistent assignment would do.

Ramified types

The semantic paradoxes, such as Berry’s and the liar, involve quantifying over all propositions or all properties. To block them, Russell also divided functions of a given type by order, according to the totalities their definitions quantify over. A function is predicative, written \(\varphi!\), if it is of the lowest order compatible with its arguments, that is, its definition quantifies only over objects below its arguments. A definition such as “\(x\) has every property of order 1 that a great general has” defines a property of order 2, not of order 1, so it cannot be one of the properties it quantifies over. The vicious circle is broken.

The price is high. Cantor’s and Dedekind’s analysis of the real numbers defines the least upper bound of a bounded class of reals by quantifying over all reals, an impredicative definition. In the ramified theory the least upper bound would be of higher order than the reals it bounds, and analysis falls apart.

The axiom restores impredicative reasoning, in extension, at each type. Russell admitted that it was not obviously a truth of logic; he accepted it because mathematics seemed to need it. Frank Ramsey argued in 1925 that once the semantic paradoxes are assigned to the theory of language, the ramification can be dropped, leaving the simple theory of types. Church’s 1940 formulation, on which the Church course builds, is a simple theory of types.

Principia Mathematica: classes and cardinals

For the general reader. Principia Mathematica, written by Whitehead and Russell between about 1902 and 1910, attempted to derive arithmetic, the theory of cardinal and ordinal numbers, and the beginnings of the theory of real numbers from logic, with every step written in symbols. It runs to nearly 2,000 pages in three volumes. The proposition from which \(1+1=2\) follows appears several hundred pages into the first volume, and the authors remark, in effect, that it is occasionally useful.

Structure

Part I, mathematical logic, begins with the propositional calculus (\(\ast1\)–\(\ast5\)), with negation and disjunction primitive, and quantification theory (\(\ast9\)–\(\ast11\)). It then sets out the theory of types and reducibility (\(\ast12\)), identity (\(\ast13\)), descriptions (\(\ast14\)), classes (\(\ast20\)) and relations (\(\ast21\)). Parts II to VI treat cardinal arithmetic, relation-arithmetic, series, and quantity; volume III ends with measurement and the real numbers. A fourth volume on geometry, to be written by Whitehead, was never completed.

Publication cost money. Cambridge University Press estimated a loss of £600 and agreed to bear half; the Royal Society contributed £200; the authors paid the last £100 between them.

Russell arrived at this definition independently of Frege in 1901, as he acknowledged in the Principles. Unlike Frege, he does not treat numbers as objects: in Principia a cardinal is a class of classes, a type-ambiguous logical construction.

Cardinal addition is defined by disjoint union: \(\mathrm{Nc}\,\alpha+_c\mathrm{Nc}\,\beta\) is the number of \(\alpha'\cup\beta'\) for disjoint copies \(\alpha'\sim\alpha\), \(\beta'\sim\beta\). Copies are made type-theoretically by pairing with distinct markers. The length of the road to \(1+1=2\) reflects the generality of the definitions, not the difficulty of the fact.

Infinity and choice

For the general reader. Logic alone cannot prove that there are infinitely many things, and in Russell’s type theory arithmetic breaks down without them. Nor can it prove that from infinitely many pairs of socks one can choose one sock from each pair. Russell isolated both assumptions, called them the axiom of infinity and the multiplicative axiom, and wrote them in Principia as explicit hypotheses rather than as laws of logic.

Infinity

Exercise 7.2 showed that with \(k\) individuals the numbers at a fixed type stop being distinct after \(k\). Principia defines a class to be inductive if its cardinal belongs to every class containing \(0\) and closed under \(+1\), and then defines the axiom of infinity, \(\ast120.03\), as the statement that no inductive cardinal is empty at the type of individuals: there are at least \(n\) individuals for every natural number \(n\). Theorems that depend on it carry it as an antecedent: “if the axiom of infinity holds, then …”. Russell wrote in 1919 that there is no logical reason for believing it.

Choice

Russell needed it to show that the product of infinitely many non-zero cardinals is non-zero, hence the name. Zermelo had used an equivalent principle in 1904 to prove that every set can be well-ordered. Russell formulated his axiom independently; in his 1906 paper “On some difficulties in the theory of transfinite numbers and order types” he discussed it alongside three possible ways of escaping the paradoxes: a “zigzag” theory, limitation of size, and the no-classes theory that became Principia.

After Principia

For the general reader. Principia was more admired than read. Its importance lay in showing that mathematics could be written in a single formal language, which turned questions about mathematics into mathematical questions. Others then asked whether its axioms are consistent and complete. Gödel’s answer of 1931 was addressed to “Principia Mathematica and related systems”.

The second edition

For the second edition (1925–27) Russell wrote a new introduction and three appendices, under the influence of Wittgenstein and Ramsey. Following Henry Sheffer (1913), he made a single connective, incompatibility, the only propositional primitive, and Jean Nicod, Russell’s student, reduced the propositional axioms to one (1917). Russell also explored dropping reducibility in favour of an extensional view in which functions occur only through their values; in Appendix B he tried to show that induction survives, with a proof whose gap Gödel pointed out in 1944.

The logicist goal of Principia, a complete derivation of mathematics from logic, was thereby shown to be unreachable for any single formal system. The Gödel course gives the proofs. In his essay “Russell’s mathematical logic” (1944), Gödel praised Russell’s analysis of the paradoxes while arguing that the vicious-circle principle is false for a realist about classes.

Types today

Church’s simple theory of types (1940) is the logic of the HOL proof assistants. Typed λ-calculi, Per Martin-Löf’s intuitionistic type theory and the calculus of constructions underlie Coq, Lean and Agda. In these systems types serve Russell’s original purpose, preventing self-application, and a new one: by the Curry–Howard correspondence, a proof is a typed program.

Philosopher and dissenter, 1910–1944

With Principia finished, Russell became a lecturer at Trinity in 1910 and turned to the theory of knowledge. The Problems of Philosophy (1912) introduced his distinction between knowledge by acquaintance and knowledge by description. In 1911 a young Austrian engineer, Ludwig Wittgenstein, arrived on Frege’s advice; within a year Russell regarded him as his successor in logic, and Wittgenstein’s criticisms in 1913 discouraged Russell from finishing a book on the theory of knowledge. Norbert Wiener studied with him in 1913–14. In 1914 he lectured at Harvard; the Lowell Lectures of that year became Our Knowledge of the External World.

The First World War turned him into a public figure. He opposed the war and worked with the No-Conscription Fellowship. In 1916 he was fined £100 for a leaflet defending a conscientious objector, and Trinity removed him from his lectureship. He was refused a passport to take up an invitation from Harvard. In 1918 an article suggesting that American troops might be used against British strikers brought a sentence of six months in Brixton prison, where he wrote Introduction to Mathematical Philosophy (1919), still the clearest short account of his logic. He gave the lectures “The Philosophy of Logical Atomism” the same year, published 1918–19.

In 1920 he visited Soviet Russia, met Lenin, and wrote a critical book about Bolshevism. From autumn 1920 to 1921 he lectured at Peking University. He divorced Alys and in 1921 married Dora Black; they had two children, and in 1927 founded Beacon Hill School, an experiment in progressive education that he left in 1932. He lived by writing and lecturing: The ABC of Relativity (1925), The Analysis of Matter (1927), Marriage and Morals (1929). In 1931 he succeeded his brother as 3rd Earl Russell. He married Patricia Spence in 1936; their son was born in 1937.

He moved to the United States in 1938 and taught at Chicago and at the University of California, Los Angeles. In 1940 his appointment at the City College of New York was annulled after a lawsuit claimed that his writings on sex and marriage made him unfit to teach; the case became a cause for academic freedom, and Einstein publicly defended him. He then lectured at the Barnes Foundation in Pennsylvania; those lectures became A History of Western Philosophy (1945), the book that made him financially secure.

The last decades and legacy, 1944–1970

Russell returned to Trinity as a fellow in 1944. In 1948 he gave the first BBC Reith Lectures, published as Authority and the Individual, and Human Knowledge: Its Scope and Limits appeared the same year. He received the Order of Merit in 1949 and the Nobel Prize in Literature in 1950, for writings championing humanitarian ideals and freedom of thought. In 1952 he married Edith Finch and settled at Plas Penrhyn, in north Wales.

He devoted his last two decades to the danger of nuclear war. The Russell–Einstein Manifesto, released in July 1955 and signed by Einstein shortly before his death, called on scientists to assess the dangers of weapons of mass destruction. It led to the first Pugwash Conference in 1957. Russell was founding president of the Campaign for Nuclear Disarmament in 1958, left it to support civil disobedience through the Committee of 100, and in 1961, aged 89, spent a week in prison after a protest. During the Cuban missile crisis of 1962 he wrote to Kennedy and Khrushchev. He opposed the Vietnam War and with Jean-Paul Sartre convened the International War Crimes Tribunal in 1966–67. His Autobiography appeared in three volumes between 1967 and 1969. He died at Plas Penrhyn, Penrhyndeudraeth, on 2 February 1970, aged 97.

Legacy

• Logic and set theory: the paradox ended naive set theory and prompted both type theory and Zermelo’s axioms; the multiplicative axiom and the axiom of infinity were isolated as non-logical assumptions.
• Formal mathematics: Principia showed that mathematics can be written in one formal language, the precondition for Gödel’s theorems and for proof assistants.
• Type theory: through Ramsey and Church, Russell’s hierarchy became simple type theory, then the type systems of programming languages and of Coq, Lean and Agda.
• Philosophy: the theory of descriptions became a paradigm of logical analysis; with Moore, Frege and Wittgenstein, Russell is counted among the founders of analytic philosophy.
• Honours: Fellow of the Royal Society (1908), De Morgan Medal (1932), Sylvester Medal (1934), Order of Merit (1949), Nobel Prize in Literature (1950), Kalinga Prize (1957).

Connections

People linked to Bertrand Russell across this collection. A name without a link has no page here yet. See Bertrand Russell in the network of all 337 lives →

Teachers and advisors

• Frederick Soddy

Students

• Norbert Wiener 1913-1914
• Jean Nicod
• Ludwig Wittgenstein 1911-1913

Influenced by

• Charles Sanders Peirce
• Georg Cantor
• George Boole
• Gottfried Wilhelm Leibniz
• Giuseppe Peano 1900

Influenced

• Alonzo Church
• Kurt Gödel
• Norbert Wiener 1913-1914
• Frank Ramsey

Colleagues and collaborators

• Albert Einstein 1940-1955
• Gottlob Frege 1902-1912
• Alfred North Whitehead 1900-1913
• G. E. Moore

Rivals

• Henri Poincaré
• Ernst Zermelo

Friends

• G. H. Hardy

Also mentioned in this life

• Augustus De Morgan
• Richard Dedekind
• Euclid
• David Hilbert
• Alfred Tarski

Fields

• mathematical logic 10 lives
• foundations of mathematics 10 lives
• philosophy 12 lives

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Bertrand Russell, MacTutor History of Mathematics.
2. Bertrand Russell, Wikipedia.
3. A. D. Irvine, Bertrand Russell, Stanford Encyclopedia of Philosophy.
4. Bertrand Russell, biographical, NobelPrize.org.
5. B. Russell, The Autobiography of Bertrand Russell, 3 vols., Allen & Unwin, 1967–69.
6. R. Monk, Bertrand Russell, 2 vols., Jonathan Cape, 1996 and 2000.

Logic and mathematics

1. B. Russell, The Principles of Mathematics, Cambridge University Press, 1903.
2. B. Russell, On denoting, Mind 14 (1905) 479–493.
3. B. Russell, Mathematical logic as based on the theory of types, American Journal of Mathematics 30 (1908) 222–262.
4. A. N. Whitehead and B. Russell, Principia Mathematica, 3 vols., Cambridge University Press, 1910–1913; 2nd ed. 1925–27.
5. B. Russell, Introduction to Mathematical Philosophy, Allen & Unwin, 1919.
6. B. Russell, Letter to Frege (1902), in J. van Heijenoort (ed.), From Frege to Gödel, Harvard, 1967.
7. A. D. Irvine and H. Deutsch, Russell’s paradox, Stanford Encyclopedia of Philosophy.
8. B. Linsky and A. D. Irvine, Principia Mathematica, Stanford Encyclopedia of Philosophy.
9. K. Gödel, Russell’s mathematical logic, in P. A. Schilpp (ed.), The Philosophy of Bertrand Russell, Northwestern University, 1944.
10. F. P. Ramsey, The foundations of mathematics, Proc. London Math. Soc. (2) 25 (1926) 338–384.
11. M. J. Dominus, A reading of ∗54.43, The Universe of Discourse.