am 11. August 2007
This book is an introduction to constructive analysis - in its stricter intuitionistic form - covering real numbers, metric spaces and continuous functions. The author adopts a refreshingly down-to-earth approach, exemplified right from the start by his definition of real numbers as Cauchy sequences of decimal fractions. The book's most important feature is a reasonably straightforward proof of the celebrated but contentious Brouwer-Weyl theorem that any function between complete metric spaces is continuous. This is to be contrasted with Errett Bishop's view (stated on p. 76 of E. Bishop and D. Bridges, Constructive Analysis, Springer-Verlag) that "while reflection makes it extremely plausible that [all functions on the reals] are continuous, to accept Brouwer's arguments as a proof would be to destroy the character of mathematics."
Happily, this drastic latter eventuality is not occasioned by the author's arguments, which rest essentially on the identification of functions with precisely defined procedures of a certain kind. For a real function f this amounts to agreeing that, for any real number r (given as a decimal expansion), for any integer j, there is an integer k such that the jth entry in f(r) can be calculated from the first k entries of r. Of course this stipulation makes f continuous. As a result, continuity is - as it was up until the 19th century - built in to the idea of function. The author justly remarks that this theorem provides a mathematical embodiment of Leibniz's famous apothegm "natura non facit saltus".
The book also contains proofs (suitably generalized to metric spaces) of Brouwer's theorems that every function defined on a closed real interval is uniformly continuous, and that any pointwise convergent sequence of real functions defined on such intervals are uniformly convergent. In addition, the author provides constructive formulations and proofs of a number of familiar classical results, for example the boundedness of
uniformly continuous real-valued functions on totally bounded sets,and the intermediate value theorem for strictly monotone functions. This book can be recommended as a good short introduction to its subject. It is also graced by a frontispiece featuring a number of photographs of a beaming Hermann Weyl which the reviewer had not seen before.