Birkhoff-MacLane Books: Overview for 2nd Year Math Students

[wisvuze]

Mod note: thread split off of https://www.physicsforums.com/showthread.php?t=566908

In the old days, the progression was roughly: rigorous one variable (Spivak) calculus, Abstract algebra (Birkhoff and Maclane), rigorous advanced calculus (Loomis and Sternberg), introductory real and complex analysis via metric spaces as in Mackey's complex analysis book, general analysis as in Royden, (big) Rudin, or Halmos and Ahlfors, algebra as in Lang, and algebraic topology as in Spanier. Then you specialize.

In particular Spivak was written for a first semester freshman book.

Do you mean that the Birkhoff and MacLane book was used for second year math students? Were students taught about categories in their second year? ( sorry for getting off topic, just curious )

 

[xristy]

Maclane & Birkhoff "Algebra" is the category-theoretic experimental text. Birkhoff & Maclane "A Survey of Modern Algebra" is one of the mainstays of introductory algebra texts.

 

[wisvuze]

Maclane & Birkhoff "Algebra" is the category-theoretic experimental text. Birkhoff & Maclane "A Survey of Modern Algebra" is one of the mainstays of introductory algebra texts.

Thanks for clearing that up, I'm not too familiar with their other algebra book, so when I hear "Mac lane" I just think of the bigger Algebra book ( when do you think the bigger "Algebra" book would fit into a standard "pure" mathematics curriculum? )

 

[mathwonk]

yes birkhoff and maclane was used for the second year course and i took it from birkhoff.

 

[xristy]

Maclane & Birkhoff is a bit higher level than Birkhoff & Maclane owing to the intention to introduce modern levels of abstraction such as categories and modules at an earlier stage than would have been customary in the late 60's. As they indicate in the Preface to the First Edition:

This book proposes to present algebra for undergraduates on the basis of these new insights. In order to combine the standard material with the new, it seemed best to make a wholly new start. At the same time, just as in our Survey, we hold that the general and abstract ideas needed should grow naturally from concrete instances.

It's my impression that the book was not widely adopted, but I recall at the time the idea seemed cool.