The Continuum: A Critical Examination of the Foundation of Analysis

By Hermann Weyl

Book Overview

"The hard won power ... to assess correctly the continuum of the natural numbers grew out of titanic struggles in the realm of mathematical logic in which Hermann Weyl took a leading part." -- John Archibald Wheeler
Hermann Weyl (1885-1955) ranks among the most important mathematicians and physicists of this century. Though Weyl was not primarily a philosopher, his wide-ranging philosophical reflections on the formal and empirical sciences remain extremely valuable. Besides indicating clearly which results of classical analysis are invalidated by an important family of "non-circular" (predicative) theories, The Continuum wrestles with the problem of applying constructive mathematical models to cases of concrete physical and perceptual continuity. This new English edition features a personal reminiscence of Weyl written by John Archibald Wheeler.
Originally published in German in 1918, the book consists of two chapters. Chapter One, entitled Set and Function, deals with property, relation and existence, the principles of the combination of judgments, logical inference, natural numbers, iteration of the mathematical process, and other topics. The main ideas are developed in this chapter in such a way that it forms a self-contained whole.
In Chapter Two, The Concept of Numbers & The Continuum, Weyl systematically begins the construction of analysis and carries through its initial stages, taking up such matters as natural numbers and cardinalities, fractions and rational numbers, real numbers, continuous functions, curves and surfaces, and more.
Written with Weyl's characteristic passion, lucidity, and wisdom, this advanced-level volume is a mathematical and philosophical landmark that will be welcomed by mathematicians, physicists, philosophers, and anyone interested in foundational analysis.

• Format: Paperback
• Language: English
• ISBN: 0486679829
• ISBN13: 9780486679822
• Release Date: March 1994
• Publisher: Dover Publications
• Length: 176 Pages
• Weight: 0.45 lbs.
• Dimensions: 0.6" x 5.4" x 8.4"

Customer Reviews

Correction

I feel obliged to place a 5-star review as compensation for the incompetent review by "GangstaLawya". He is a true amateur, the exact opposite of Hermann Weyl. "GangstaLawya" obviously couldn't understand the book and blames Weyl for this, trying to prevent other people from understanding it, how childish. "GangstaLawya" claims in his "review" that Weyl didn't contribute to mathematics. Look for the "Weyl character formula" on the internet to see that he is wrong and thus incompetent.

a fascinating detour

I first learned of this book from Eves and Newsom back in the early 1960's. It sounded fascinating but I couldn't read German. Now we're lucky to have it in Englsh translation with an introduction that relates Weyl's notation and terminology to the current one. (Or, if you're really out of date like me, you can use it in reverse to catch up on the modern field of foundations studies). Precise statement is the essence of the study of the foundations of mathematics and what follows won't rise to that level but I hope it won't be seriously misleading either. In real life definitions are often circular; dictionaries define words in terms of other words, etc. Ordinarily this is not a problem but vicious circles can happen. In 1872 Dedekind published a definition of real numbers in terms of sets of rational numbers. This fulfilled a long term dream of defining the reals without reference to geometric concepts. Encouraged by this Frege began his project of deriving all of mathematics from basic logical notions. He was largely successful but Russell found a contradiction within his system. It wasn't clear what caused this problem and Poincare suggested that it arose because Frege had allowed a certain kind of circular definition called 'impredicative'. While it was true that the contradiction could be eliminated by avoiding impredicative definitions, this solution was very drastic: it also barred Dedekind's defintion of the real. Most mathenaticians, including Whitehead and Russell, shrank from this step and proposed more moderate ways of fixing the foundations of mathematics. Working in the aftermath of World War I, Weyl was attracted to the more radical idea of trying to develop mathematics without using any impredicative definitions. He managed to derive some, but far from all, of analysis and the result was this book. Subsequently, Weyl was attracted to an even more radical critique of mathematical foundations proposed by Brouwer (you can read about this in Mancosu's great anthology "From Brouwer to Hilbert". At the same time Weyl remained passionately attached to mainstream mathematics. As far as I know, he never resolved his own conflicts about this. Naturally, anything by Weyl is brilliant and worth reading and this book is no exception.