Techniques of constructive analysis
It has been 120 years since Leopold Kronecker proclaimed his finitist views, 100 years since L. E. J. Brouwer exposed the limitations of classical mathematics and the fantastical power of the excluded middle, and 40 years since Errett Bishop demonstrated the feasibility of strictly constructive methods. Yet today misunderstandings persist -- constructive mathematics is often mistakenly thought to be a branch of formal logic, or to involve recursive function theory. The recent Bridges-Vîţă textbook is an exemplar of perseverance in efforts to correct this situation. For students at the graduate level it is an excellent introduction to constructive mathematics; for the more experienced reader it is a portal to some of the latest research using constructive methods. Chapter 1 sets out the motivation, goals, and methods of constructive mathematics; persuasive examples related to number theory and to computer science are included. Section 1.4 includes some of the more esoteric logical and set-theoretic problems recently studied by some constructivists; this section is a valuable survey of that avenue. Some readers may not wish to become entangled in the deeper complexities of intuitionistic logic and may, after sections 1.1--1.3, proceed more directly to the heart of the subject in Chapter 2 -- real numbers, metric spaces, and normed linear spaces. For a start, to savour the delights of constructive results with enhanced meaning, one need only remember that to claim ``\( P \text{ or } Q \)'', one must either prove \( P \) or prove \(Q\) -- it is not sufficient to prove merely that \( \neg ( \neg P \text{ and } \neg Q ) \). The succeeding chapters treat constructive Hilbert space, convexity, separation, the Hahn-Banach theorem, and operator theory. A constructive version of Baire's theorem is included, with applications to the open mapping, inverse mapping, and closed graph theorems. Many of the results are of relatively recent origin. This textbook presents a significant segment of that mathematics which, thus far, has been constructivized. The treatment follows Bishop's original conception of a `straightforward realistic approach', and the `disengagement of mathematics from formal logic', as proposed in Chapter 1, ``A constructivist manifesto'', of \textit{E. A. Bishop}'s 1967 masterpiece [Foundations of constructive analysis. Maidenhead, Berksh.: McGraw-Hill Publishing Company, Ltd. (1967; Zbl 0183.01503)]. At the very beginning, on page 1, the text refers to the ``field of constructive mathematics''. Errett Bishop, on the other hand, anticipated that constructive methods would eventually become the accepted norm, used in virtually every field of mathematics. Some workers still adhere to this goal, even when today it may appear as but the gleaming tip of a mast rising above the horizon.
scientific article; zbMATH DE number 5071163