The Continuum: A Critical Examination of the Foundation of Analysis

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.

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 (Or, if you're really out of date like me, you can use it in reverse to catch up on the modern field of foundations studies).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.