Mathematics Form and Function

by Saunders MacLane

This book records my efforts over the past four years to capture in words a description of the form and function of Mathematics, as a background for the Philosophy of Mathematics. My efforts have been encouraged by lec­ tures that I have given at Heidelberg under the auspices of the Alexander von Humboldt Stiftung, at the University of Chicago, and at the University of Minnesota, the latter under the auspices of the Institute for Mathematics and Its Applications. Jean Benabou has carefully read the entire manuscript and has offered incisive comments. George Glauberman, Car­ los Kenig, Christopher Mulvey, R. Narasimhan, and Dieter Puppe have provided similar comments on chosen chapters. Fred Linton has pointed out places requiring a more exact choice of wording. Many conversations with George Mackey have given me important insights on the nature of Mathematics. I have had similar help from Alfred Aeppli, John Gray, Jay Goldman, Peter Johnstone, Bill Lawvere, and Roger Lyndon. Over the years, I have profited from discussions of general issues with my colleagues Felix Browder and Melvin Rothenberg. Ideas from Tammo Tom Dieck, Albrecht Dold, Richard Lashof, and Ib Madsen have assisted in my study of geometry. Jerry Bona and B. L. Foster have helped with my examina­ tion of mechanics. My observations about logic have been subject to con­ structive scrutiny by Gert Miiller, Marian Boykan Pour-El, Ted Slaman, R. Voreadou, Volker Weispfennig, and Hugh Woodin.

Genres: Mathematics, Science, Philosophy, Nonfiction, Logic, Popular Science

476 pages, Paperback

First published December 19, 1985

Reviews

[Julian · Rating 4 out of 5]

The most important thing I took from this book was the unification and inevitability of our mathematics. Nothing else I've read has made such a strong point on this side of this philosophy of mathematics debate. MacLane's writing is engaging and often passionate, eager to demonstrate.

There are other good surveys of all of mathematics, but this one is the only one I've found that works hard to show the connections between ideas in different areas of mathematics, rather than leaning on history as a framework for exposition.

This would get five stars, but some of the later chapters get a bit muddy and seem less well-edited (many more typos, less brilliant writing, in my edition). Also, a newer book with the same style would be highly attractive as this dates from the 80s.

Readers without a strong mathematical background will find the introductory chapters interesting but are likely to soon become lost as the book assumes a lot of the reader.

[William Schram · Rating 5 out of 5]

In this book, Saunders Mac Lane attempts to answer some questions about mathematics; it's origins, how it is organized, whether or not mathematics has formalisms that come from facts, how mathematics develops, how does someone evaluate the depth and importance of mathematical research, what should be the standards of rigor and finally, why is mathematics effective?

That final question goes a lot deeper than some would think. It ties itself right into epistemology, and how we know things that we know. At the same time, it covers a great deal of mathematical ideas. Starting at counting it goes into Peano axioms and other things. It structures itself differently from other books that survey mathematics in that it seems to organize the information by subject rather than chronologically.

At the end of the book, Mac Lane answers the questions he put forth in the introduction of the book. All in all it was a satisfying and interesting read.

[Jared Tobin · Rating 5 out of 5]

This would be my recommendation to anyone who was going to read a single book on maths. Mac Lane gives a tour of mathematics, motivating it in terms of the questions: what does mathematics do, and what is it? I found this approach quite refreshing; the book reads more like a narrative than the typical theorem/proof reference.

Mac Lane is careful to put the question of 'what does mathematics do?' in terms of human activity. Counting, rearranging, measuring, moving, estimating, choosing, etc. - each of these is a human activity that Mac Lane then describes in terms of orders, groups, motion, topology, etc. At the same time, he develops the topics in a strangely 'natural' sequence from a pedagogical perspective - starting with sets and numbers and eventually concluding with categories and the nature of formal systems.

Readable and re-readable, and highly recommended.

[Fredrik · Rating 5 out of 5]

Anbefales! Se anmeldelse: http://cube.fredrikmeyer.net/?p=26

[Lulu]

the more I learn, the more I laugh at my previous pathetically naive ideas of what math is "all about". But if you want a glimpse, try these books * more advanced