This is a long-awaited new edition of one of the best known Oxford Logic Guides. The book gives an introduction to intuitionistic mathematics, leading the reader gently through the fundamental mathematical and philosophical concepts. The treatment of various topics, for example Brouwer's proof of the Bar Theorem, valuation systems, and the completeness of intuitionistic first-order logic, have been completely revised.

Our first aim is to make the study of informal notions of proof plausible. Put differently, since the raison d'étre of anything like existing proof theory seems to rest on such notions, the aim is nothing else but to make a case for proof theory; ...

In this paper we discuss a particular example of the passage from the informal, but rigorous description of a concept to the axiomatic formulation of principles holding for the concept; in particular, we look at the principles of continuity and lawlike choice in the theory of lawless sequences. Our discussion also leads to a better understanding of the rôle of the so-called density axiom for lawless sequences.

It is argued that the tensed theory of the creative subject provides a natural formulation of the logic underlying Brouwer's notion of unextendable order and explains the link between that notion and virtual order. The tensed theory of the creative subject is also shown to be a useful tool for interpreting recent evidence about the stages of Brouwer's thinking concerning these two notions of order.