What Is a Calculation?

Abstract

At the second International Congress of Mathematicians (Paris, 1900), David Hilbert (1862-1943) presented a list of 23 problems, which he hoped would occupy mathematicians in the 20th century. We shall only talk about three of these problems here, as they concern the foundations of mathematics.

• 1. Prove or disprove the Continuum Hypothesis (Chapter 12).

• 2. Show that arithmetic, described as an axiomatic system, is consistent, that is, that it does not admit a proof that 0 = 1. (As Paul Erdös would say, if such a proof were ever to be discovered, the universe would vanish.)

• 3. Find an effective method or ‘algorithm’, as it is now called, for deciding whether a given polynomial Diophantine equation (with integer coefficients) is solvable (in integers). (For the origin of the word ‘algorithm’, see Part I, Chapter 22.)