- Length
- 144 pagesAustin, The Foundations of Arithmetic, Blackwell, 2nd rev. ed. 1953 - German with facing English
- Reading
- OpenWritten for a general reader and readable without preparation
- Availability
- In printA current standard edition is available
The Foundations of Arithmetic
A hundred and forty pages asking what a number is, which nobody read for twenty years and which is now taken to be the beginning of analytic philosophy. Its central insight is that numbers belong to concepts rather than to things, and its programme was destroyed by a letter.
- Date
- 1884
- Author
- Gottlob Frege
- Region
- Jena
- Language
- German
- 1950 English, J. L. Austin German with facing English; the standard, in copyright Attested
Where it stands in the argument
- Position
- Answered — Drew a named later reply, but was not itself a reply
- Form
- Treatise
- Method
- Analytic — Clarification of language and logical form
- Transmission
- Continuous — Copied and read without a break in its own language
- Standing
- Live — Still argued in current philosophical work
- Years to the reply
- 26 years
Sold badly, reviewed dismissively, and unread for twenty years; now regarded as the book in which analytic philosophy begins.
What the book says
A short book asking what a number is, written almost entirely in prose. Frege begins by clearing away the answers on offer. Numbers are not properties of external things, since the same pile of cards is one pack and fifty-two cards depending on how you take it. They are not collections of units, since units would have to be both identical and distinguishable. They are not mental images, since nothing that happens inside a head could be true for everyone or exist before anyone thought it. What a number attaches to, he argues, is a concept: to say there are four moons of Jupiter is to say something about the concept moon of Jupiter, namely how many things fall under it. From there he can define number without appealing to counting. Two concepts have the same number if their instances can be paired off one to one without remainder, which requires no counting at all; zero is the number belonging to the concept not identical with itself; and each number is defined from its predecessor. Threaded through is the demand that gives the book its method: never ask for the meaning of a word in isolation, but only in the context of a sentence.
What it argues
That arithmetic is derivable from logic and definitions alone, and that seeing this requires getting the question right first. Frege's negative arguments do most of the work: every account that makes number a property of objects, or a mental item, or an abstraction from counting, fails on cases he supplies. The positive definition works by equinumerosity - two concepts have the same number when their instances can be put in one-to-one correspondence - which allows number to be defined without presupposing it. Three principles carry the book: always separate the psychological from the logical, never ask after the meaning of a word except in the context of a sentence, and keep the distinction between concept and object firmly in view. The second of these, the context principle, has had a career of its own.
What answered it
Russell, by letter, in 1902. The paradox does not touch the Grundlagen directly - it derives from Basic Law V of the later formal system - but it destroyed the programme this book set out. Frege's appendix to the second Grundgesetze volume proposed a weakened version of the law; the repair does not work, as later logicians established. He abandoned logicism late in life and came to think arithmetic rested on geometry. The line from here runs to Russell and Whitehead's attempt, to Gödel's demonstration in 1931 that no such system can do everything asked of it, and to the neo-logicism of the last forty years, which revives Frege's approach using a principle he had already isolated.
How it argues
By elimination and then definition, in prose of exceptional clarity, with almost no symbolism - which is why this is the Frege that philosophers read and the Grundgesetze is the Frege that logicians read. The polemical sections against Mill and against psychologism are sharp and sometimes funny. The context principle is stated almost in passing and does an enormous amount of work: it licenses defining numbers by fixing the truth conditions of sentences containing number words, rather than by pointing at something a number is.
The moment
Frege spent his entire career at Jena, taught mathematics without distinction, published in obscure places, and was ignored by philosophers and mathematicians alike. The Begriffsschrift of 1879, which invented modern quantificational logic, was reviewed badly by people who did not understand it. He was fifty-four when Russell's letter arrived, forty-four when this book appeared, and he lived to see almost none of the influence he now has.
How it travelled
By being noticed by two people. Russell read Frege in 1900 and made him known in English; Wittgenstein went to Jena to see him and was sent on to Russell. Carnap attended his lectures. Through those three the work entered the tradition it founded, and the recovery accelerated after 1950 when Austin's translation and the Geach-Black selections made the writings available. He is now among the most studied philosophers of the nineteenth century, having been among the least read in it.
The strongest objection
That the programme cannot be completed, and that this is now a theorem rather than a suspicion. Russell's paradox showed that Frege's own system was inconsistent; Gödel showed in 1931 that any consistent formal system rich enough for arithmetic contains truths it cannot prove, which rules out the kind of complete derivation logicism wanted. Neo-logicists reply that a great deal survives - arithmetic can be derived from second-order logic plus a single principle about equinumerosity that Frege himself states - but whether that principle counts as logic, rather than as a substantial assumption about the existence of numbers, is precisely what is disputed. The programme's failure is not total and is not repairable in the form Frege wanted.
Why read it
Because it is short, clear and the best demonstration in philosophy of what it looks like to take a question nobody thought was hard and show that every available answer fails. The opening sections on what a number is not can be read in an hour and require no mathematics at all.
- First printed
- 1884
- First in English
- 1950
- Digitised
- Yes
The Grundlagen states the programme informally; the Grundgesetze der Arithmetik of 1893 and 1903 executes it in Frege's own notation, and it is the later work rather than this one that the paradox destroys.
Austin's 1950 English with facing German is the standard and is in copyright. The German is public domain.
Where it stands
Live. The definition of number by equinumerosity, the concept-object distinction, the context principle and the arguments against psychologism are all working parts of contemporary philosophy of mathematics and language, and neo-logicism is an active research programme.
On quoting this book
The German is public domain but the English translation is modern and in copyright, and Frege's terms are notoriously sensitive to rendering. Nothing is quoted.
Contested
Whether the context principle survives Frege's later distinction between sense and reference is a long-running interpretive dispute. Whether Frege was a platonist about numbers in the full sense usually attributed to him is argued. How much of the logicist programme survives the paradox divides neo-logicists from their critics, and turns on whether the equinumerosity principle is analytic. And his late political diary, published in 1994 and containing violently antisemitic entries, has raised a question about the man that the philosophical work does not settle and that his readers now have to hold alongside it.