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

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In summary, the author suggests that the traditional sequence of rigorous calculus, abstract algebra, real analysis, and introduction to algebraic topology should be replaced by a more abstract sequence that starts with modern algebra and then branches out into real analysis and complex analysis.
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mathwonk said:
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 )
 
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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.
 
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xristy said:
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? )
 
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yes birkhoff and maclane was used for the second year course and i took it from birkhoff.
 
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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.
 

1. What are Birkhoff-MacLane books?

Birkhoff-MacLane books are a series of textbooks that cover various topics in mathematics, written by mathematicians Garrett Birkhoff and Saunders MacLane. They are known for their clear and rigorous approach to mathematics, making them popular among students and professionals alike.

2. Who are the target audience for these books?

The Birkhoff-MacLane books are primarily aimed at 2nd year math students, although they can also be useful for advanced high school students and self-studiers. These books assume a solid foundation in mathematics, including calculus and basic algebra.

3. What topics do these books cover?

The Birkhoff-MacLane books cover a wide range of topics in mathematics, including abstract algebra, linear algebra, topology, and category theory. They also touch on areas such as group theory, ring theory, and homological algebra.

4. Are these books suitable for self-study?

Yes, these books can be used for self-study. They are written in a clear and concise manner, with many examples and exercises to help readers understand the material. However, it is recommended to have a strong mathematical background before attempting to use these books for self-study.

5. Are there any prerequisites for using these books?

As mentioned before, a strong foundation in mathematics is necessary for understanding these books. In addition, a basic understanding of proof techniques and mathematical notation will also be helpful. Familiarity with topics such as calculus, linear algebra, and basic abstract algebra will also be beneficial.

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