The present text resulted from lectures given by the authors at the Rijks Universiteit at Utrecht. These lectures were part of a series on 'History of Contemporary Mathematics'. The need for such an enterprise was generally felt, since the curriculum at many universities is designed to suit an efficient treatment of advanced subjects rather than to reflect the development of notions and techniques. As it is very likely that this trend will continue, we decided to offer lectures of a less technical nature to provide students and interested listeners with a survey of the history of topics in our present-day mathematics. We consider it very useful for a mathematician to have an acquaintance with the history of the development of his subject, especially in the nineteenth century where the germs of many of modern disciplines can be found. Our attention has therefore been mainly directed to relatively young developments. In the lectures we tried to stay clear of both oversimplification and extreme technicality. The result is a text, that should not cause difficulties to a reader with a working knowledge of mathematics. The developments sketched in this book are fundamental for many areas in mathematics and the notions considered are crucial almost everywhere. The book may be most useful, in particular, for those teaching mathematics.

Luitzen Egburtus Jan Brouwer founded a school of thought whose aim was to include mathematics within the framework of intuitionistic philosophy; mathematics was to be regarded as an essentially free development of the human mind. What emerged diverged considerably at some points from tradition, but intuitionism has survived well the struggle between contending schools in the foundations of mathematics and exact philosophy. Originally published in 1981, this monograph contains a series of lectures dealing with most of the fundamental topics such as choice sequences, the continuum, the fan theorem, order and well-order. Brouwer's own powerful style is evident throughout the work.

These two volumes cover the principal approaches to constructivism in mathematics. They present a thorough, up-to-date introduction to the metamathematics of constructive mathematics, paying special attention to Intuitionism, Markov's constructivism and Martin-Lof's type theory with its operational semantics. A detailed exposition of the basic features of constructive mathematics, with illustrations from analysis, algebra and topology, is provided, with due attention to the metamathematical aspects. Volume 1 is a self-contained introduction to the practice and foundations of constructivism, and does not require specialized knowledge beyond basic mathematical logic. Volume 2 contains mainly advanced topics of a proof-theoretical and semantical nature.

Studies in Logic and the Foundations of Mathematics, Volume 123: Constructivism in Mathematics: An Introduction, Vol. II focuses on various studies in mathematics and logic, including metric spaces, polynomial rings, and Heyting algebras. The publication first takes a look at the topology of metric spaces, algebra, and finite-type arithmetic and theories of operators. Discussions focus on intuitionistic finite-type arithmetic, theories of operators and classes, rings and modules, linear algebra, polynomial rings, fields and local rings, complete separable metric spaces, and located sets. The text then examines proof theory of intuitionistic logic, theory of types and constructive set theory, and choice sequences. The book elaborates on semantical completeness, sheaves, sites, and higher-order logic, and applications of sheaf models. Topics include a derived rule of local continuity, axiom of countable choice, forcing over sites, sheaf models for higher-order logic, and complete Heyting algebras. The publication is a valuable reference for mathematicians and researchers interested in mathematics and logic.