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Comment author: SatvikBeri 13 August 2013 07:55:27PM *  2 points [-]

Subject: Commutative Algebra

Recommendation: Introduction to Commutative Algebra by Atiyah & MacDonald

Contenders: the introductory chapters of Commutative Algebra With a View Towards Algebraic Geometry by Eisenbud and the commutative algebra chapters of Algebra by Lang.

Atiyah & MacDonald is a short book that covers the essentials of Commutative Algebra, while most books cover significantly more material. So this review should be seen as comparing Atiyah & MacDonald to the corresponding chapters of other Commutative Algebra books. There are a few reasons why Introduction to Commutative Algebra is better than most other books:

  • Better abstractions. The abstractions Atiyah & MacDonald use (especially towards rings and ideals) are simply more broadly applicable and make several proofs simpler. Conversely other books tend to use an older set of abstractions which make the same proofs significantly more complex.

  • Exercise-driven approach. Atiyah & MacDonald's exercises are beautifully structured so that you build up important parts of the theory yourself. There's a very satisfying feeling of castle-buildng: each exercise draws upon your understanding of the previous problem, and they come together to form very nice results. Many books can give you the feeling of understanding Commutative Algebra, but this one helps you discover it, which is much more enjoyable and provides a much deeper understanding.

  • The right kind of conciseness. Atiyah & MacDonald's book is short because they cover a limited range of topics, but they do cover all the essential tools that are widely used. In contrast most books tend to bloat by trying to cover too many things, or tend to leave out critical parts of the theory.

Comment author: lukeprog 30 October 2014 07:54:55PM 1 point [-]

Thanks! Added.

Comment author: SatvikBeri 13 August 2013 07:41:42PM 3 points [-]

Subject: Introductory Real (Mathematical) Analysis:

Recommendation: Real Mathematical Analysis by Charles Pugh

The three introductory Analysis books I've read cover-to-cover are Lang's, Pugh's, and Rudin's.

What makes Pugh's book stand out is simply that he focuses on building up repeatedly useful machinery and concepts-a broad set of theorems that are clearly motivated and widely applicable to a lot of problems. Pugh's book is also chock-full of examples, which make understanding the material much faster. And finally, Pugh's book has a very large number of exercises of varying difficulty-Pugh has more than 500 exercises total.

In contrast, Rudin's book tends to focus on "magic." Rudin uses the shortest possible proofs for a given theorem. The problem is that the shortest proofs aren't necessarily the most instructive-while Baby Rudin is a beautiful work of Math qua Math, it's not a particularly good book to learn from.

Finally, Lang's book is frankly subpar. Lang leaves out critical details of some proofs (dismissing one 6 page proof as trivial!), is poorly motivated by examples, and has a number of mistakes.

If you want to really understand Mathematical Analysis and get to the point where you can use the concepts to create proofs and solve problems, Pugh is the best book on the topic. If you want a concise summary of undergraduate analysis to review, pick Rudin's book.

Comment author: lukeprog 30 October 2014 07:54:48PM 1 point [-]

Thanks! Added.

Comment author: orthonormal 24 April 2013 01:05:09AM 5 points [-]

Calculus: Spivak's Calculus over Thomas' Calculus and Stewart's Calculus. This is a bit of an unfair fight, because Spivak is an introduction to proof, rigor, and mathematical reasoning disguised as a calculus textbook; but unlike the other two, reading it is actually exciting and meaningful.

Analysis in R^n (not to be confused with Real Analysis and Measure Theory): Strichartz's The Way of Analysis over Rudin's Principles of Mathematical Analysis, Kolmogorov and Fomin's Introduction to Real Analysis (yes, they used the wrong title; they wrote it decades ago). Rudin is a lot of fun if you already know analysis, but Strichartz is a much more intuitive way to learn it in the first place. And after more than a decade, I still have trouble reading Kolmogorov and Fomin.

Real Analysis and Measure Theory (not to be confused with Analysis in R^n): Stein and Shakarchi's Measure Theory, Integration, and Hilbert Spaces over Royden's Real Analysis and Rudin's Real and Complex Analysis. Again, I prefer the one that engages with heuristics and intuitions rather than just proofs.

Partial Differential Equations: Strauss' Partial Differential Equations over Evans' Partial Differential Equations and Hormander's Analysis of Partial Differential Operators. Do not read the Hormander book until you've had a full course in differential equations, and want to suffer; the proofs are of the form "Apply Theorem 3.5.1 to Equations (2.4.17) and (5.2.16)". Evans is better, but has a zealot's disdain of useful tools like the Fourier transform for reasons of intellectual purity, and eschews examples. By contrast, Strauss is all about learning tools, examining examples, and connecting to real-world intuitions.

Comment author: lukeprog 30 October 2014 07:54:41PM 1 point [-]

Thanks! Added.

In response to Podcasts?
Comment author: gwillen 26 October 2014 12:53:27AM 1 point [-]

I'm surprised to see Radiolab get top billing, and This American Life not even get a mention. I think of Radiolab as an excellent imitation of TAL. :-)

In response to comment by gwillen on Podcasts?
Comment author: lukeprog 26 October 2014 03:41:24AM 2 points [-]

Ira Glass seems to think Radiolab is better.

In response to Podcasts?
Comment author: lukeprog 26 October 2014 12:29:55AM 4 points [-]

I listen to Dan Carlin's Hardcore History, EconTalk, Meet the Composer, Planet Money, Radiolab, Serial, StartUp, and This American Life.

In response to Weekly LW Meetups
Comment author: lukeprog 25 October 2014 07:02:19PM 0 points [-]

Thanks again for continuing to do this!

Comment author: sbenthall 20 October 2014 05:23:50AM 0 points [-]

Thanks. That's very helpful.

I've been thinking about Stuart Russell lately, which reminds me...bounded rationality. Isn't there a bunch of literature on that?

http://en.wikipedia.org/wiki/Bounded_rationality

Have you ever looked into any connections there? Any luck with that?

Comment author: lukeprog 20 October 2014 06:22:11PM *  0 points [-]

You might say bounded rationality is our primary framework for thinking about AI agents, just like it is in AI textbooks like Russell & Norvig's. So that question sounds to me like it might sound to a biologist if she was asked whether her sub-area had any connections to that "Neo-Darwinism" thing. :)

Comment author: lukeprog 19 October 2014 03:27:07AM 3 points [-]

It's not much, but: see our brief footnote #3 in IE:EI and the comments and sources I give in What is intelligence?

Comment author: lukeprog 19 October 2014 03:19:02AM *  4 points [-]
Comment author: KatjaGrace 14 October 2014 03:24:45AM 5 points [-]

If people were ten times faster, how much faster would economic growth be?

Comment author: lukeprog 14 October 2014 04:00:16AM 2 points [-]

On this subject, I'd like to link Stanovich on intelligence amplification.

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