Customer Reviews

Mathematics: Form and Function

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Born 1909, MacLane is one of the great living mathematicians, by virtue of having been the coinventor and chief propagator of category theory. Hence this is an overview of contemporary mathematics, at about the level of a fully competent BA graduate, by someone fully qualified to talk about math. Important points. (1) MacLane is an expert on algebra and gives many examples of the unifying power of abstract algebra. (2) The book includes passages that amount to a gentle introduction to the philosophy of mathematics. MacLane takes exception to what philosophers have written about math, in part because they don't know enough math. (3) This book is also a good introduction to MacLane's radical claim that set theory is an inadequate foundation for math, and that the category theory he coinvented is better for this purpose. (4) Includes many fascinating "road maps" or "flow charts" of the intricate connections among the many branches and topics of mathematics.
Slight disappointment: MacLane does not do justice to the lattice theory his frequent coauthor, Garret Birkhoff, did so much to advance.

This is a survey of the whole of mathematics, at the undergraduate level, which attempts to give the "big picture". If you read and understand it, you will have a better grasp of this big picture than most graduate students. MacLane has written a book which every mathematician (and perhaps philosopher) should read and savour.

There are few technical details in this book. It is not a textbook per se, but a beautiful exposition of mathematics as a whole. It is not for learning any specific topic from; rather it is about appreciating the structure of mathematics as a whole, so that you know how each specific topic stands within that structure.

Indeed, this is an excellent book. Too bad for fashion that it is out-of-print.

N.B. There is a more advanced and technical book by Jean Dieudonne along the same lines. You might want to read it after enjoying your Mac Lane.

For an elegant overview of much of modern mathematics it can't be beat. How I wish that I had had this book as a first and second year graduate student!

A clear and careful expositio