Remarks on the foundations of mathematics

Remarks on the Foundations of Mathematics is not a book Wittgenstein finished. It is a selection, made after his death, from notebooks and typescripts he wrote between 1937 and 1944, the years in which he was also drafting the Philosophical Investigations. The editors, G.H. von Wright, Rush Rhees and Elizabeth Anscombe, published it in 1956 with Anscombe's English translation facing the German text.

The subject is not mathematics itself but the pictures we make of it. Wittgenstein does not prove theorems or propose a new foundation. He asks what we are doing when we calculate, prove and follow a rule, and why it is so tempting to think that numbers live in a realm of their own, waiting to be discovered.

How the book came about

From the late 1930s Wittgenstein filled manuscript after manuscript with remarks on proof, calculation, contradiction and logical necessity. Much of the first part comes from a 1938 typescript that also contains an early version of the opening of the Investigations. In 1939 he lectured on the same themes in Cambridge, with Alan Turing among the students; those lectures, reconstructed from students' notes, appeared in 1976 as Lectures on the Foundations of Mathematics, edited by Cora Diamond.

The first edition of 1956 had five parts. A revised and expanded edition followed in the 1970s (the English version in 1978), with seven parts and passages the first editors had left out. The Stanford Encyclopedia entry on his philosophy of mathematics describes in detail which manuscripts were used and which were not.

What it says

A few ideas keep returning.

• Proof makes a new rule. A proof does not report a fact about an abstract object. It persuades us to adopt a new way of seeing, a new criterion for what counts as correct. After the proof, we use the result as a norm.
• Mathematics is invented, not discovered. Wittgenstein is an outspoken anti-Platonist. He treats mathematical propositions as rules of grammar rather than descriptions.
• Rules need practice. Many of the famous passages on rule-following first appear here: the pupil who continues the series "add 2" in an unexpected way after 1000, and the question of what makes one continuation the right one.
• Contradiction is less frightening than logicians think. He asks why a hidden contradiction in a calculus should be a disaster if nobody ever runs into it, a point he argued over with Turing in 1939.
• Suspicion of the infinite. He criticises set theory and Cantor's diagonal argument, and treats infinity as a feature of a rule, not a completed totality.

An appendix to the first part contains the remarks on Gödel's incompleteness theorem, without naming Gödel. They are the most controversial pages of the book.

How to read it

This is the hardest of the later books. The remarks are dense, repetitive and sometimes exploratory to the point of contradicting each other. You need some feel for elementary logic and arithmetic, and a lot of patience. Use the revised edition (Blackwell 1978, MIT Press paperback), whose bilingual layout lets you check the German.

A good way in is to read the 1939 Cambridge lectures first: they cover the same ground in conversational form, with Turing pushing back. Then read the first part of the Remarks, and return to the Philosophical Investigations §§185-242 to see where the material ended up.

Reception

The early reviews were harsh. Michael Dummett read Wittgenstein as a radical conventionalist, Paul Bernays objected to the treatment of proof, and Georg Kreisel dismissed the book as a weak product of a brilliant mind (he later softened his view). The Gödel remarks were widely taken as a plain misunderstanding. Since 2000, Juliet Floyd and Hilary Putnam have argued that Wittgenstein understood Gödel better than his critics thought; that reading remains disputed. Today the book is read mainly by philosophers, and its discussion of rules has had more influence than its claims about mathematics.

Verdict

Read this if you already know the later Wittgenstein's view of meaning and want to see it tested on the hardest case: the apparent necessity of 2 + 2 = 4. Skip it as a first book. Its insights are real, but you have to dig for them, and some of the claims about Gödel and set theory are, at best, open to serious doubt.