Pavel Pudlák.Logical Foundations of Mathematics and Computational Complexity: A Gentle Introduction. Springer Monographs in Mathematics. Springer, 2013. ISBN: 978-3-319-00118-0 (hbk); 978-3-319-00119-7 (ebook). Pp. xiv + 695

Abstract

This monograph by the outstanding Czech logician Pavel Pudlák provides a broad but also deep survey of work in logic and computer science relevant to foundational issues, interpreted in a wide sense. This kind of book used to be more common; as an example, the well known survey volume The Foundations of Mathematics: A Study in the Philosophy of Science1 by Evert W. Beth provided a philosophically sensitive survey of work in logic and foundations in the late 1950s. It is a most welcome development that a logician of Pudlák's stature has taken the time and trouble to produce a volume that should prove its value to readers interested in logic and foundations. The first chapter entitled ‘Mathematician's world’ is an elementary introduction to the notion of mathematical structures, the set-theoretical foundations of mathematics, the antinomies of set theory and the axiomatic method. It concludes with a discussion of the need for abstract concepts, illustrating this point with the mutilated-chessboard problem, and Wiles's proof of Fermat's Last Theorem. In both of these cases, an apparently simple problem only yielded to more abstract concepts; this theme prefigures the later chapter on proof complexity.