Intuitionism: An introduction

From the preface ---------------- In order to prevent the reader from wasting his time in useless attempts to solve supposed riddles, I warn him that the persons of the dialogue are not caricatures of living or deceased persons, much less their doubles. They are pegs to hang ideas on, and nothing else. To a certain extent this is even true for Int, who represents the position of intuitionism. For the sake of clearness I made him speak sometimes in a somewhat more absolute way than I should have done if I had freely expressed my own opinions. The discussion in strictly limited to intuitionism; other conceptions of mathematics are touched on only in so far as they lead to objections against intuitionism. I reject any reproach for incomplete exposition of other points of view. It was necessary to give proofs in great detail, even where they differ only by small additions from the well-known classical ones. There was no other way to indicate in which places these additions had to be made. In the course of the book, as the reader is supposed to develop a feeling for the specifically intuitionistic difficulties, I have gradually adopted a more condensed style. In many places of the book the reader will find old-fashioned reasonings which lack generality and which are more clumsy than the modern methods. This has different reasons. In the first place, the powerful methods often make an excessive use of indirect proof, so that it is almost impossible to introduce them in intuitionistic mathematics. In the second place, the very general modern theories proceed by the axiomatic method. Now this method can only work well, if some concrete theories exist, from which the axiomatic theory can be constructed by generalization. For instance, general topology could only be developed after the topology of euclidean spaces was known in some detail. As a matter of fact, almost no part of intuitionistic mathematics has been investigated deeply enough to admit the construction of a general axiomatic theory. Thus in this book I had to confine myself to the most elementary case of integration; when this will be better known than it is at present, it will become possible to construct an axiomatic theory on the subject. Even in the case of algebra, where axiomatization is possible at this moment, it seemed better to treat the concrete example of the real number field, in view of the fact that the book is meant as an introduction. Probably in some cases I used antiquated methods because I did not know the modern ones. One of the aims of the book is, to enable working mathematicians to decide, which of their results can be proved intuitionistically. Intuitionism can only flourish, if mathematicians, working in different fields, become actively interested in it and make contributions to it. In order to build up a definite branch of intuitionistic mathematics, it is necessary in the first place to have a thorough knowledge of the corresponding branch of classical mathematics, and in the second place to know by experience where the intuitionistic pitfalls lie. I try in this book to teach the latter; I hope that some of my readers will give a more satisfactory treatment of details than I could, or that they will treat other theories intuitionistically. The "reading suggestions" are intended to help them; they indicate the most important intuitionistic work on some special subjects.