Today marks the 150th birthday of Bertrand Russell and to honour the occasion one of our wonderful authors - Professor Bernard Linsky FRSC - was kind enough to write us a piece about the man, his mind, and his time at Trinity College. I am sharing it below and we will be recording it to add to our audio history of science in Cambridge.

Professor Linsky has done an amazing job of explaining some of the incredibly complicated ideas which Russell developed over the course of his long life, so if you've ever wondered just what he is so famous for, then read on!

**Bertrand Russell and Cambridge**

By Bernard Linsky, with contributions from Alex Klein and Kenneth Blackwell

May, 2022

The philosopher and mathematician Bertrand Arthur William Russell lived from 1872 to 1970, a life as a prominent logician and public philosopher that spanned the end of the Victorian era through the World Wars of the Twentieth through to the to the era of Nuclear threat in the Cold War in Vietnam. Bertrand's grandfather, Lord John Russell, had been Prime Minister under Queen Victoria on two occasions between 1846 and 1866. By age 4, Bertrand's father and mother had both died, and Lord John soon after, so Bertrand was raised by his grandmother. Little “Bertie” had an isolated but relatively unrestricted childhood roaming around Richmond Park where he lived with his older brother Frank in Pembroke Lodge, a cottage that had been a gift of Queen Victoria to his grandfather. As a result of this isolated upbringing, Russell’s first formal schooling began in 1890, reading for the Mathematical Tripos at Trinity College Cambridge. Although tutored at home and not socialized in the traditional manner of students of the great British public schools, Bertie obviously fit in quickly. The slim Russell was obviously talkative and sure of himself, and was suited to be cockswain in the Trinity College Eights. The boat did well, and Russell’s name appears on a trophy (image reproduced with thanks to the Bertrand Russell Peace Foundation).

Russell was already serious about mathematics and philosophy and after his degree in mathematics (as seventh Wrangler) in 1893, and then stayed on to win a fellowship in 1895 studying Philosophy. At Cambridge Russell came under the influence of the somewhat older Mathematician Alfred North Whitehead and a classics student of a similar age, G. E. Moore, who also turned to Philosophy. Together Russell and Moore led the rebellion against the established British Hegelian philosophy of F.H. Bradley and Bernard Bosanquet who were both associated with Oxford.

The Russell we know as Ban the Bomb activist and opponent of the U.S. led war in Vietnam, started his career as a social commentator with his first book *German Social Democracy *1896. Russell turned against the Boer War as the British Empire reached its zenith and settled into a pattern of opposing most of the wars of the next Century. In the following year, 1897, Russell published the first of his famous books on the logical foundations of mathematics with Cambridge University Press, *An Essay on the Foundations of Geometry*. Some themes from this early Hegelianism will be recognizable in the familiar projects of mathematical logic and the new Analytic Philosophy that Russell and Moore went on to found. One theme was the notion that knowledge of mathematics is based on a series of progressive stages, from the elementary counting of arithmetic, to the real numbers and lines of geometry, and on to other so-called “exact” sciences such as mathematical physics and the calculus. In the Hegelian picture these transitions were driven by *contradictions* at each level, and with some Hegelians, such as Karl Marx, this notion of contradictions in society became the basis of Communist ideology and class conflict. The contradictions that drove Russell’s thinking about mathematical theories were of a purely mathematical sort, however. He saw the progress from natural numbers to rational numbers, to real numbers and the mathematics of space and time as each adding a resolution to puzzles faced by earlier stages.

Russell did not simply drop the history of philosophy and change into a logician with his second book, *A Critical Exposition of the Philosophy of Leibniz*, based on lectures he gave at Cambridge, that was published by the University Press in 1900. Russell explained Leibniz’ notorious ontology of “monads” that reflected the whole state of the universe in splendid isolation from each other, as coming from relying on the traditional logic which did not allow for relations. The discovery of the logic of spatial relations, such as “*x* is next to *y*”, set Russell apart from Leibniz who held that such relations had to be reduced to statements of properties “*x* has the property of being-next-to-*y*”, for example. It was this denial of these so-called “external relations” that was the issue that Russell had with the “Absolute” Idealists, who, like Leibniz denied the existence of external relations, but unlike Leibniz saw the world as only one substance. All apparent diversity in the universe was absorbed into relational properties of the one Absolute subject. Russell’s thinking at this time was influenced by the new logic, but that was just a tool for continuing the battle with Idealism in metaphysics that he had begun with Moore after their undergraduate studies.

The change in Russell’s thinking came with his attendance at the International Congress of Philosophy in Paris in 1900, where Russell encountered the leading logicians of his day. The biggest surprise was the school of Giuseppe Peano from Turin in Italy, who made use of an invented symbolic language, borrowed from mathematical symbols, to express the new symbolic logic. Russell was swept away. Already almost finished with a draft of *The Principles of Mathematics* on arithmetic and the real numbers, moving into mathematical physics of space and time, Russell studied the earlier symbolic logic of Gottlob Frege in Germany, and considered a second volume of the work in Peano’s symbolic notation.

**Russell's Paradox**. First, however, Russell had to resolve the famous contradiction of “Russell’s paradox” that he discovered in 1901 and which held back work on the book. Frege had written a massive work on *Basic Laws of Arithmetic* in 1893 and was about to complete the second volume when he received a devastating letter from Russell in June of 1902. Russell was worried about the notion of *set*, or “class”, in mathematics. His plan was to define natural numbers of classes of similar classes, and then other numbers as classes of natural numbers. (Just think of the class of all *pairs*, that is classes containing only the distinct objects *a* and *b*, where the notion of “distinct” is purely logical and does not require counting. That class just is the number *two*!) But Russell found a contradiction in the very notion of a class. Consider the class of apples. This class is not itself an apple. We can say that the class itself is therefore not a member of the class of apples. Nor is the class a pear, so it is also not a member of the class of pears. But, as there are to be real numbers that are classes of classes of other numbers, there should be classes of classes of all sorts. In particular, consider the class of all classes that do not belong to themselves. Call it the “Russell class”. Is the Russell class a member of itself? If it is then it is one of the classes that are not members of themselves, and so not a member of itself. But then it qualifies as a member of the Russell class and so is not a member of itself. There seems to be a contradiction lurking in the very notion of a class. And, indeed, Russell was suspicious of this paradox. It looked too much like the Hegelian or Marxist “contradictions” that he was trying to overcome. After a year of thinking of the problem while preparing his *The Principles of Mathematics* for the press, Russell wrote to Frege and suggested that Frege’s formal system would easily express the contradiction. Frege responded famously by acknowledging the significance of the paradox, saying that “mathematics totters”, and leaving a solution to others.

The publication of *The Principles of Mathematics* in 1903 made Russell’s reputation. It was reviewed in the *Times Literary Supplement* by Godfrey Hardy, the famous mathematician that one may have heard of from the “Hardy-Weinberg” equations from genetics, or from his role in bringing the Indian mathematician Srinivasa Ramanujan to Trinity College. Already in his review of September 1903, Hardy laments what will undoubtedly be the odd position of Russell’s book, as too philosophical for the mathematicians, and too mathematical for the philosophers. Nevertheless, Russell instantly became the leading figure in his field, being elected to the Royal Society (in 1908) and joining Peano in Italy and Frege in Germany as the founders of mathematical logic.

Russell found that his fellow mathematician at Trinity, Alfred North Whitehead, had a similar project of symbolizing the higher reaches of the *Principles* in geometry, and so the two of them set to work on a second, symbolic, volume of Principles. This work was to take almost ten years to complete, becoming a massive and unreadable masterpiece of symbolic logic, *Principia Mathematica*, published by Cambridge University Press in three volumes between 1910 and 1913. (I recommend that even students of logic begin their study of *Principia Mathematica* by first working through the *Principles of Mathematics*).

By the time the first volume appeared Russell had been appointed as a teaching fellow at Trinity, and Whitehead had resigned to move as a teacher of mathematics to the newly founded Imperial College in London. One can see the wit of Russell on the title page where the authors are identified by their positions in 1910. Russell was Lecturer and his fellowship had ended in 1901, and so he was a “Late Fellow” of Trinity College while Whitehead who remained a Fellow but was a “Late Lecturer”.

The next phase of Russell’s connection with Cambridge came with his appointment as a lecturer in Trinity College in 1910, just as the proofs of Volume I of *Principia Mathematica* were working their way through the printing rooms of the press. Reading through some of the correspondence preserved in the Bertrand Russell Archives at McMaster University in Canada, one learns that Russell was an outsider of the tight group of academics at Trinity College. It was perhaps only with the urging of Whitehead and Hardy that a place was found in the curriculum for lectures by Russell on this new field of Mathematical Philosophy. A look at the *Cambridge Reporter* for lectures under Moral Sciences and Mathematics for 1910 shows Russell’s lectures and even the fee of 10s. 6d. that would be charged to students. Russell was a regular university lecturer with a trial appointment to see what the students would think of this new subject.

Russell’s lectures were a great success. Though only as small number of students at Trinity attended, the lectures were enormously popular with others in Cambridge and with a series of post graduate students sent from around the world to learn the new mathematical logic. From Girton College the logician E.E. Constance Jones attended some of the lectures. Harvard University in America sent a new graduate Henry Sheffer, to attend the first lectures. His notes from those classes have survived in the Harvard University Archives and are just now being transcribed from the Pitman shorthand in which they were taken. (Sheffer’s discovery, the “Sheffer stroke” is now known as the basis for the *nand gate* (for ``not...and...'') in logic circuits which is the basis for all the “Logic” names for computing companies and keyboard manufacturers.)

The next year a young Engineering student from Austria, Ludwig Wittgenstein, was sent along from Manchester to study mathematical logic with Russell. In 1913 there was a visit from a young mathematician just finishing a PhD from Harvard, Norbert Wiener, who spend his career at the Massachusetts Institute of Technology and go on to fame as the creator of the early foundations of information studies called *cybernetics*. A young Frenchman Jean Nicod, visited Cambridge and joined Russell’s “school” of mathematical philosophy in 1913. A Girtonian, Dorothy Wrinch, met Russell in 1916. Wrinch assisted Russell while in Brixton by bringing books and carrying out his correspondence, and then, while a mathematics student under Hardy, with the second edition of *Principia Mathematica* in 1925–27 along with another famous Cambridge mathematician, Frank Ramsey, who died in 1930 at the age of 26. She went on to a career in mathematical biology in the mid 1920s. So, by 1930 the influence of Russell's "mathematical philosophy" had withered, to be replaced by the new center of Cambridge Philosophy, Ludwig Wittgenstein, just returned from his home in Vienna.

Russell's hope of founding of a school of students already ended with the Great War. Wiener went home in 1915, before America joined in the war, while Wittgenstein served in the Austro-Hungarian army, carrying the text of his *Tractatus Logico-Philosophicus* with him in his rucksack. Russell saw the war as the dismal result of the power politics of the imperialist powers of Europe. Already opposed to wars for the British Empire since the fighting in South Africa, Russell saw this Great War as a similarly pointless waste of life with a prescience about the terrible toll of trench warfare.

Russell was convicted under the “Defense of the Realm Act” for his statements in opposition to American involvement in the war in 1916 and later again in 1918, at which time he was sentenced to Brixton Prison for six months. The conviction and fine in 1916 were too much for his Trinity colleagues. The details of the story were described by Hardy, who left Cambridge for Oxford. Hardy left Oxford, however returning to Cambridge in 1931, and became once again a fellow of Trinity College, and Sadlerian Professor of Pure Mathematics until 1942. Hardy's memoir was privately published by CUP in 1942 as *Bertrand Russell and Trinity*.

That was the end of Russell’s initial closest connection with Cambridge University. After the dismissal, there was almost immediately attempts to mend the relationship, but Russell moved on to a new life after Brixton prison. His reinstatement as a Fellow in 1919 was followed by a leave of absence, so he never resumed the lectureship or residence in Cambridge.

In 1920 Russell spent a year in China lecturing in Beijing, and writing one of the series of books on social and political issues that marked the last 50 (!) years of his long career as a public intellectual. Russell returned to give the Tarner lectures in 1926, but the resulting book, *The Analysis of Matter*, was not published by Cambridge University Press. By that time Russell had a new life beyond the University with his second wife, Dora Russell, and their children first running his experimental Beacon Hill School and later as a public intellectual living in London and lecturing around the world. Russell was awarded a special, over-retirement age fellowship in 1944 which lasted until 1949 when he became a fellow for life; he had the use of Newton’s rooms and regularly taught large classes in an introduction to philosophy and smaller ones in non-deductive inference. Cambridge University did step in again and published his lecture *Physics and Experience* in 1946, *Philosophy and Politics* in 1947, and posthumously, for the Russell Archives, *My Own Philosophy *in 1972. There are brass plaques for Moore, Wittgenstein and Russell in the entry way of the Chapel of Trinity college that people might be told to find. Otherwise, there is little of a physical memorial to Bertrand Russell in Cambridge, outside of the bookshops.