- Length
- 2000 pagesCambridge University Press, 2nd edition 1925-27; the abridged Principia Mathematica to *56 is the usual reading text
- Reading
- SpecialistWritten for a school and normally read with a commentary
- Availability
- In printA current standard edition is available
Principia Mathematica
Three volumes and a decade of work attempting to show that mathematics is logic, undertaken after a letter from one of its authors had destroyed the previous attempt. Twenty years later a young Austrian logician proved that no such project could succeed, and named this book in his title when he did it.
- Date
- 1910-1913
- Author
- Bertrand Russell and Alfred North Whitehead
- Region
- Cambridge
- Language
- English
Where it stands in the argument
- Position
- Exchange — Both: answers an earlier work and drew a later one
- Form
- Treatise
- Method
- Demonstration — Proof from premises the opponent already grants
- Transmission
- Continuous — Copied and read without a break in its own language
- Standing
- Superseded — Overtaken by later work rather than defeated
- Years to the reply
- 11 years
- Years back to what it answers
- 26 years
One of the most cited and least read books ever written; the notation was superseded, the programme was shown to be unachievable, and its influence on twentieth-century philosophy is nonetheless immense.
What the book says
Three volumes attempting to derive mathematics from logic, symbol by symbol, with almost nothing taken for granted. The project inherits Frege's programme and has to solve the problem that destroyed it. Russell's answer is the theory of types: expressions are stratified into levels, and a class cannot be a member of itself because the two occurrences would belong to different levels, so the sentence that generates the paradox cannot be formed. Around this is built an apparatus of primitive ideas, definitions and step-by-step proofs of a density that is hard to convey - the demonstration that one plus one equals two arrives several hundred pages into the first volume, with a note observing that the proposition is occasionally useful. The introduction sets out the philosophical case, including the theory of descriptions that had already changed how philosophers handled sentences apparently referring to things that do not exist. The later volumes work up through cardinal and ordinal arithmetic and into series and measurement, and stop before geometry.
What it argues
That the truths of mathematics are truths of logic, and that this can be shown by actually deriving them. The philosophical content is mostly in the introduction and in the machinery required to avoid contradiction. Types stratify the universe of expressions so that self-referential class formation is ungrammatical rather than false; the axiom of reducibility is then needed to recover ordinary mathematical reasoning that the stratification has blocked, and Russell concedes he can give no reason to think it a logical truth beyond the fact that mathematics needs it. The theory of descriptions, developed a few years earlier and used throughout, shows how a sentence about the present King of France can be false rather than meaningless, which became the model for a whole style of philosophical analysis.
What it answered
The Foundations of Arithmetic 1884Gottlob Frege
Frege's logicism, and the paradox Russell had found in it. But the wider answer is to the state of mathematics itself, which at the turn of the century had discovered inconsistencies in its foundations and had no agreed account of what its objects were. Principia is one of three great responses - alongside Hilbert's formalism and Brouwer's intuitionism - and its distinctive claim is the most ambitious: not that mathematics can be made consistent, but that it was never anything other than logic.
What answered it
How to Make Our Ideas Clear 1878Kurt Gödel
Gödel, decisively, in 1931. His incompleteness theorems establish that any consistent system rich enough to express arithmetic will contain truths it cannot prove, and cannot prove its own consistency from within. Wittgenstein had already attacked the philosophical framing in the Tractatus, arguing that logical truths say nothing and that the whole apparatus of types is unnecessary. Ramsey simplified the type theory and removed the need for reducibility in one direction; Quine reworked the whole approach in another.
How it argues
By formal derivation, almost entirely in symbols, with prose confined to the introduction and to occasional remarks. The book is famous for being unreadable and the reputation is deserved - the notation is idiosyncratic, the volumes are enormous, and the celebrated proof that one and one are two appears hundreds of pages in. What made it influential was not that people read it through but that it demonstrated something could be done at all: that a substantial body of mathematics could be derived from explicitly stated logical premises, in public, with every step checkable.
The moment
Cambridge in the decade before the First World War, with Russell in his thirties and Whitehead his former teacher. Russell described the work as having permanently damaged his intellect; he estimated they had spent ten years and lost money on the publication. Whitehead left for London and then Harvard and moved into metaphysics; Russell went on to popular writing, political campaigning, prison for opposing the war, and a Nobel Prize in literature.
How it travelled
As a demonstration rather than a text. Its notation was replaced within a generation, but the idea that philosophical problems could be attacked by formal analysis of language spread from it into everything that followed - Wittgenstein, the Vienna Circle, Carnap, Quine and the whole analytic tradition. It also had an unintended technical afterlife: Gödel's engagement with it produced results that fed directly into the theory of computation, so a book written to secure the foundations of arithmetic is in the ancestry of the computer.
The strongest objection
That the axiom of reducibility gives the game away. The theory of types blocks the paradox by making certain expressions ill-formed, but it also blocks legitimate mathematics, and reducibility is introduced to restore what the stratification removed. Russell admitted he could offer no argument that it is a truth of logic rather than a convenient assumption - and if it is not, then the derivation shows that mathematics follows from logic plus something else, which is not the thesis. Ramsey and others tried to remove it; the repairs work only for part of what Principia wanted. Gödel's results then closed off the possibility of the programme being completed in any form.
Why read it
Almost nobody should read it, and this entry says so plainly. The introduction to the first volume and the introduction to the second edition contain the philosophy and are readable; the rest is a technical monument. Read Russell's own Introduction to Mathematical Philosophy of 1919 instead, which he wrote in prison and which says what the big book was for.
- First printed
- 1910
- First in English
- 1910
- Digitised
- Partial
The second edition of 1925-27 adds a new introduction and appendices by Russell alone, attempting to do without the axiom of reducibility. The abridgement to *56 is what most readers use.
Written in English and in symbols. In copyright in most jurisdictions; the notation is largely obsolete.
Where it stands
Superseded and foundational at once. Nobody works in its system, its central programme is known to be unachievable in the form intended, and the questions it raised - what logic is, whether mathematics reduces to it, what a foundation could be - remain live and are still discussed with reference to it.
On quoting this book
Substantially symbolic, and the prose portions remain in copyright in most jurisdictions. Nothing is quoted.
Contested
How the labour was divided between the two authors is not fully known; Russell said the philosophical parts were his and the mathematical largely joint, and Whitehead's later distance from the work complicates the picture. Whether the axiom of reducibility can be dispensed with is a technical question with a partial answer. Whether Gödel's theorems refute logicism, or only the specific ambition of complete derivation, is argued by philosophers of mathematics in both directions. And whether the theory of types is a genuine solution to the paradox or an ad hoc prohibition was disputed from the start, including by Wittgenstein.