Mathematics Form and Function 1986th Edition by Saunders MacLane (PDF)

Description

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.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Incredibly beautiful.One may not agree with the intelectual / conceptual development the author lays out, but it powerful and engaging with that issue is itself interesting.As a collection of interlinked evolving ideas covering intermediately accessible maths of many stripes it is simply beautiful.If you know the maths read it for the simple joy and new perspective.If you don’t know the maths read it to get a sense of what beautiful things exist and what’s out there to learn.

⭐Every math major should read this book to gain insight into how various concepts in mathematics evolved over time. The author is clear and concise. This is not, however, a text and should be treated as expository material. Read it while hanging at the beach or relaxing at the park or coffee shop. This does not imply that it the book is a “light” read, as it requires the reader to follow proofs and think abstractly about various concepts. An interesting side note, reading MacLane’s work in this book sheds some light into how he thought about mathematics, which may have been a major thrust behind him (and Eilenberg) to co-invent category theory.

⭐Is a nice review of various topics making emphasis in the need of the development of mathematics. Is more like story of maths well written and easy to read.

⭐Brilliant and clear description of wide areas of mathematics, from the man who saw abstract connections between areas of math thought to be unrelated.

⭐A tour de force by the master.

⭐As has already been duly noted, this survey-book should be required reading for students majoring in mathematics.I will take that one step further: this survey-book should also be required reading for students of physics. Happily, the book is lucidly written and neither subset of students (mathematics or physics) should face any difficult in perusing the exposition. Its efficacy for mathematics students accepted, I concentrate upon my reasons for suggesting this unique exposition for the later category of students. That is, physics students.(1) Begin with the fascinating ninth chapter, Mechanics; allow me to quote Saunders Maclane: “Understanding is not easy…it has taken me over fifty years to understand the derivation of Hamilton’s equations.” Learn of “the difficulty of getting to the bottom of it all.” References for this chapter include: Osgood’s 1937 Mechanics textbook, available from web-archive, Pars’ Treatise on Analytical Dynamics and Maclane’s self-penned journal article “Hamiltonian Mechanics and Geometry.” (American Mathematical Monthly,1970). As an adjunct to Maclane: the exposition presented in Burke’s Applied Differential Geometry (1985) is quite enlightening.(2) What do we learn of Cauchy’s Theorem ? Read: “that an integral of a holomorphic function is independent of path is a direct extension of the fundamental theorem of calculus, and is another representation of the idea that the whole is the sum of the parts.”(page 322). Meet Riemann surfaces, later germs and sheaves. Beautiful !(3) Much of the exposition focuses upon the concept of “function” (page 126). In one manner, or another, whether it be sets, categories, functors, the concept of function prominent throughout. Read: “the clarity which is achieved when all the relevant concepts are defined firmly and formally.” (page 413). Again, suggested reading for physics students !(4) A topic fast-approaching a favorite of mine is Projective Geometry. There is a discussion of that topic here (pages 234 and 240). For more enlightenment read Hilton & Griffith’s A Comprehensive Textbook of Classical Mathematics (1970, chapter 17) where they write: “A beautiful example of the growth of a theory, for theory of conics.” Another elementary exposition on the topic: Greenberg’s Euclidean and Non-Euclidean Geometries (1993, third edition).(5) Tensors: learn why tensor algebra is a “graded algebra,” in Maclane’s chapter (seven) of Linear Algebra.Differential Forms, read: “they need not be mysterious” (page 211). Learn more of the functorial viewpoint.A highlight: the section entitled “Collapse By Quotients,” a process for constructing linear maps. Beautiful !(6) Finally, I mention, as all others have done before me, the interesting and lucid exposition of categories and functors. The discussion serves as prelude to yet another text: Dodson’s Categories, Bundles and Spacetime Topology (1980). Dodson writes: “the right way to view it all is through the theory of categories, and increasingly the language of this theory is appearing in the literature of Physics.” (Robert Geroch, Mathematical Physics, is yet another source).(7) Hopefully I have convinced, through these examples, why this book is as useful for physics students as it is for mathematics students. The book is beautifully written and dense with information of continuing efficacy.Highly recommended !

⭐Category Theory, which Saunders had a strong hand in inventing has been dragged out of the dustbin of curious arcane thinking and thrust into the light as concepts of functional programming have grabbed IT by storm since 2010. MacLane’s book is a very strong overview to the philosophy of mathematics that speaks to the practical applied side. His book is lucid, has pictures of mathematical relationships that resonate today as strongly as they did when this book was published.The strongest testament to this timeless work is that nearly 40 years AFTER I purchased this and browsed it as a curiosity, I have now put it on my desk as a reference while delving into the construction of rule based systems for infrastructure design.

⭐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.

⭐This is most interesting “different take” on mathematics by Saunders MacLane, whom, together with Garrett Birkhoff, also wrote an influential book on modern algebra. Although now old, this book directs the readers attention to the myriad of sometime surprising connections that exist between prima facie seemingly different mathematical topics. An excellent book.

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