What Survives the Notation
Bertrand Russell's hardest books are written in symbols that almost no one uses now. What a reader keeps from them is a skill he learned the same way his readers do: by arguing with other thinkers and changing course when they were right.
---
I have three books by Bertrand Russell, each in a different state. The Principles of Mathematics (1903) I put down. The Analysis of Mind (1921) I am reading for the second time, and on this pass I can follow it with some confidence. The Analysis of Matter (1927) is about a third read, with notes. In itself that is an ordinary reading record. It is worth writing about because of the question underneath it. When a book is this hard, and its technical vocabulary has gone out of date, what is the reader actually learning?
Start with what Russell was attempting. Born in 1872 and trained as a mathematician at Cambridge, he set out around 1900 to show that all of pure mathematics follows from a small number of logical ideas, such as class, relation and implication. The view has a name, logicism: the claim that mathematics is logic carried far enough. Principles is the first long statement of it in English, and it was not written for beginners. It assumes the mathematics of its day, argues with philosophers most readers have never heard of, and changes its mind in its own appendices. A reader who arrives without that background meets page after page that gives up almost nothing until something else has been read first. Some writers hand the reader conclusions. Russell, in this period, hands over a problem and a set of tools, and every small gain has to be earned.
Putting such a book down is a reasonable decision. The difficulty of Principles is partly the difficulty of the subject and partly the difficulty of thinking that was still in progress, and the second kind repays a reader far less than the first. Russell rebuilt much of the book himself within a decade. Difficulty is not evidence of depth, and a reader's hours are limited.
Yet the book came out of people reading one another, and that is where an outsider can get in. In August 1900 Russell went to an international congress of philosophy in Paris. He later called it "a turning point in my intellectual life, because I there met Peano." Giuseppe Peano, an Italian mathematician, had devised a symbolic notation for logic: a way of writing arguments in signs so that every step could be checked by anyone who knew the signs. Russell had seen it before and, by his own account, "had not taken the trouble to master" it. At the congress he watched Peano come out ahead in discussion after discussion because he was more precise than anyone else in the room. Russell went home, read Peano's writings, and spent that September extending the method to the logic of relations. He took up the notation only after he had watched it settle live arguments, and he treated learning it as the cost of that precision.
The second episode cost someone else far more. Gottlob Frege, a German mathematician, had spent two decades building a system meant to derive arithmetic from logic, the same goal Russell was pursuing. Few people read him; his notation was unusual and his work was neglected. Russell did read him, and on 16 June 1902 he wrote to Frege to raise one point of difficulty. Frege's system allowed a class to be formed from any condition whatever. Russell had found a condition that breaks it: the class of all classes that are not members of themselves. If that class is a member of itself, then it is not one; if it is not a member, then it is. A system that can prove both a statement and its opposite can prove anything, and so its proofs are worth nothing. This contradiction is now called Russell's paradox. Frege replied that the discovery had caused him "the greatest surprise". The second volume of his life's work was going to press, and he added an appendix that begins by admitting that "hardly anything more unwelcome can befall a scientific writer than that one of the foundations of his edifice be shaken" once the building is complete. Russell's Principles appeared the following year with its own appendix on Frege's doctrines, one of the first serious accounts of them in English.
The episode is usually told as a story about a paradox. It is also a story about reading. Russell found the flaw because he had worked through a notation that others had not bothered with, in a system built to do exactly what his own was trying to do. The most useful criticism tends to come from a reader who has learned your language well enough to use it against you.
Russell received the same treatment ten years later, from his own student. Ludwig Wittgenstein arrived in Cambridge in 1911 to study with him. By 1913 Russell was writing a book on the theory of knowledge, and Wittgenstein objected to its account of judgment, the act of believing that something is the case. The objection was technical, and philosophers still argue about exactly what it was. Its effect is not in dispute: Russell abandoned the manuscript. In 1916 he wrote to Lady Ottoline Morrell that the criticism "was an event of first-rate importance in my life" and that "I saw he was right, and I saw that I could not hope ever again to do fundamental work in philosophy." He was wrong about the second part. He went on to write The Analysis of Mind, The Analysis of Matter and a great deal else. The episode shows a man taking an objection seriously enough to stop work, which is rarer than taking it seriously enough to write a reply.
Meanwhile the technical language Russell had built was being carried somewhere he had not planned. Principia Mathematica, the three volumes he wrote with Alfred North Whitehead between 1910 and 1913, is set out almost entirely in a dense symbolism descended from Peano's. To block the paradox he had found in Frege's system, Russell added a device called the theory of types, which sorts objects into levels so that a class can never contain itself. Three later readers took this machinery in three directions. In July 1922 the young German philosopher Rudolf Carnap, who did not yet own a copy of Principia, thanked Russell for writing out its main definitions for him; the list survives in Russell's papers and runs to thirty-five leaves. Carnap used that language in an attempt to build an account of all knowledge out of experience. In 1931 the Austrian logician Kurt Gödel published a paper whose title names the book: "On formally undecidable propositions of Principia Mathematica and related systems". It showed that any consistent system of that kind, strong enough to do arithmetic, contains true statements it cannot prove, and with that the hope of completing logicism in the form Russell intended came to an end. In 1940 the American logician Alonzo Church published a simplified theory of types. Its descendants sit underneath the proof-checking software that mathematicians now use to have a computer confirm every step of a proof.
Nobody writes mathematics in the notation of Principia today. Its symbols have been retired, its system was shown to be incomplete, and its central device survives mainly inside computer science, in a form Russell would need help to recognise. The language changed hands three times in thirty years. What passed from reader to reader each time was a way of working: state a claim so exactly that someone else can check it, and then let them try to break it.
This is the part a general reader can use. When the author is a first-rank synthesizer, someone who pulls several fields into a single picture, the technical detail on any given page is the least durable thing in the book. Vocabulary dates quickly. What lasts is the ability to find where an argument carries its weight, whom it is answering and what it gives up. A reader learns that ability first on difficult books, slowly, and usually without noticing that it is happening.
That is close to what happened on my second pass through The Analysis of Mind. The book grew out of lectures Russell gave in London and in Peking; the preface is dated from Peking in January 1921. It sets out to reconcile two tendencies, one in psychology and one in physics, and it adopts a view Russell credits to William James and the American new realists: that the "stuff" of the world is "neither mental nor material, but a 'neutral stuff,' out of which both are constructed." The name for this is neutral monism, the view that a single kind of material underlies both minds and physical things, arranged in different ways. On a first reading the book can feel like a list of positions. On a second, the positions sort themselves by whom they answer: the behaviourists, who wanted to reduce the mind to observable conduct; the psychologists, who trusted introspection; the physicists, who had dissolved solid matter into events. Russell thanks the behaviourist John B. Watson in the preface for reading his manuscript, which tells you whom he expected to have to convince. Once I could see the opponents, I could follow the argument, and I could also disagree with parts of it. My notes on Lecture VI, on introspection, contain a shorthand I could not have written the first time through: subject and object as the same material, taken at different times. The shorthand is mine. What changed between the two readings is that I had learned where Russell was standing when he wrote.
The Analysis of Matter is where I am now, about a third of the way in. Russell takes the physics of 1927, relativity and the new quantum theory, and asks what it allows us to know about the world beyond our own perceptions. My notes underline a few things. He holds that the world outside can be known, which I marked in the margin as a break with Hume and Kant, both of whom doubted that we reach anything beyond our own impressions. A thing, on his account, is a series of events, closer to a biography than to a lump of stuff. Geometry becomes physics only once its points are defined as something in the world. In one chapter he observes how much of Newtonian thinking rested on crude notions of force and contact, borrowed from the experience of pushing and pulling. Each of these is a claim about relations. A chair is a pattern of events related in a certain way, and a point is a set of events related in a certain way.
The strongest objection to Russell grows out of this book, and it arrived within about a year. By 1927 his position was that we know the physical world only through its structure: the pattern of relations among events, as distinct from what the events are in themselves. In 1928 the Cambridge mathematician Max Newman published a short paper in the journal Mind pointing out a difficulty. Given any collection of things with enough members, logic alone guarantees that some relation among them has whatever structure you care to name. So if all we know about the world is that its parts stand in some relation with a certain structure, we know little more than how many parts there are. To say anything further, we must say which relation, and knowing which relation is knowledge of something besides structure. Russell wrote back from Petersfield in April 1928. He accepted that his claim, as he had stated it, was "either false or trivial", and he defended other parts of his view in the same letter. The objection has never gone away. Philosophers who hold that science tells us mainly about structure, a family of positions called structural realism, still treat Newman's argument as the first thing any version must answer.
Newman's point also cuts against the account of reading I have been giving. If what lasts in a great synthesizer is the shape of his arguments, the reader has to ask what that shape is a shape of. Skill in following arguments can be practised on bad ones as easily as on good ones, and it does not by itself say which are true. The history above is also told from the side of the people who kept going. Frege did not. He spent his last years looking for a different foundation for arithmetic, and the flaw in his system, once found, was not repaired by any amount of careful reading. Russell's own record is one of repeated reversal, and a critic can fairly ask whether a course that changes direction this often deserves to be called a trajectory. The honest answer is that this skill keeps a reader moving and promises nothing about where the reader ends up. It is necessary for the growth of knowledge and far from sufficient; the arguments still have to meet the world.
There is a further complication. The claim that technical languages pass away while the skill of reading persists is too tidy. Some of the language persisted. The theory of types was Russell's patch for his own paradox, and versions of it now run inside proof-checking software. Gödel proved his result in a language built to carry Russell's programme. Later readers rebuild a technical language, and sometimes the rebuilt version does work the original could not. The skill and the language travel together, and each keeps altering the other.
Last week's Thinker, Norbert Wiener, worked in a different technical language, the language of feedback, signal and noise, and he studied with Russell at Cambridge in 1913. What drew me to Wiener was a stance he took about order emerging from disorder, one that does not depend on his equations. What draws me to Russell is similar. His symbols belong in a museum. His habit of setting out a claim so exactly that someone else could break it, and then changing course when someone did, is still in daily use.
So I count progress through these books differently now. Principles went back on the shelf, a fair decision about a book its own author rebuilt. The Analysis of Mind took two passes, and the second worked because by then I could see whom Russell was arguing with. The Analysis of Matter is a third read, and it already contains its own best objection, raised within about a year of publication. What a reader carries out of a hard book is not its vocabulary but the practice of finding where its argument has to hold, and that practice is what let Carnap, Gödel and Church take Russell's language somewhere he had not gone. The place to test it next is the objection Russell half-conceded in 1928: search for Newman's objection to structural realism, read Newman's short paper, then read the chapters of The Analysis of Matter it answers, and decide which of the two you think survives.
---
Sources — Sourced: Russell, Autobiography, vol. 1 (Paris congress and Peano) · Russell to Frege, 16 June 1902; Frege's reply and the Grundgesetze vol. II appendix (MacTutor History of Mathematics, St Andrews) · Russell to Ottoline Morrell, 1916 (*Dear Bertie*, McMaster University) · Carnap to Russell, 29 July 1922 and 27 February 1926 (BRACERS, McMaster; Vienna Carnap–Russell correspondence) · Linsky, The Evolution of Principia Mathematica (Cambridge, 2011), "List of definitions for Carnap" · Gödel, Monatshefte für Mathematik und Physik 38 (1931) · Church, "A Formulation of the Simple Theory of Types", Journal of Symbolic Logic 5 (1940) · Russell, The Analysis of Mind (1921), preface · Newman, "Mr. Russell's 'Causal Theory of Perception'", Mind 37 (1928): 137–148 · Russell to Newman, 24 April 1928 (Newman papers, St John's College, Cambridge; text in Russell, Autobiography, vol. 2). Observed: the editor's reading notes on all three books.