[mathrat]

In mathematics, my experience falls emphatically on this explanation: "Hard texts are that much better than anything else." Books like Rudin's Principles of Mathematical Analysis are mathematics; their more accessible counterparts are more "about mathematics." As to why successful people don't read more difficult books, well, being a great mathematician just isn't that important to success. Reading Rudin sets you on the path to being very good at mathematics, but that isn't most people's goal. If your goal is material success then yes, hard books are probably overrated.

The programmers here may appreciate this analogy: reading Rudin is like reading the source code, whereas more gentle texts read more like documentation or UML. That's not to say good doc can't be enlightening, but in the end only the source code matters. The doc can be wrong, or misleading, or incomplete. If you want to really understand what's going on, you need to read the source. It's the same way in mathematics.

[yaketysax]

PoMA presents a mathematical approach that's done in countless other analysis textbooks. What people (students) hate is that the author makes leaps in his proofs. But that's not something unique to Rudin's style. And who says these students wouldn't be capable of making these leaps later on? Please explain to me how "Rudin sets you on the path to being very good at mathematics" (more so than any of the "easier" books), if you agree with my claim that there is in fact nothing unique about his book. If you don't agree, please tell me what is unique.

"As to why successful people don't read more difficult books, well, being a great mathematician just isn't that important to success."

???

"Rudin is like reading the source code"

Rudin's book is incomplete to a beginner. The analogy doesn't work at all.

[mathrat]

Ah, I think we are responding to two separate issues. I was referring to Rudin as a mathematical treatment of calculus, in the definition-theorem-proof style, to be contrasted with the more popular and easy heuristic approach. I contend that the Rudin style is a qualitatively better way to learn mathematics. It is also undoubtedly harder to read--but that is because it treats the material in a rigorous, correct way, which is inherently more difficult than heuristics.

Your argument is that Rudin is hard for incidental reasons: he skips steps in his proofs. I personally did not feel this way when I read the book. But I am open to the idea that Rudin is imperfect. I personally admire Rudin's concision, but that may just be my personal aesthetic preference.

My point, and perhaps I did not convey it well, is simply that some subjects (in particular mathematics!) are inherently difficult. And if you want to learn them well, you have to face that difficult head on: you have to read hard books. Euclid's old saying comes to mind: "there is no royal road to mathematics!"

[yaketysax]

I just told you: most analysis textbooks treat the topic in a nearly identical manner (maybe not in full generality). Do you want to dispute this? Why are you pretending like we're arguing separate issues?

"Your argument is that Rudin is hard for incidental reasons: he skips steps in his proofs. I personally did not feel this way when I read the book. But I am open to the idea that Rudin is imperfect. I personally admire Rudin's concision, but that may just be my personal aesthetic preference."

Good god.