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This may not be the type of discussion that takes place on this site. If not, I do apologize, I am new here, but I am genuinely interested in people's reaction to this.

I've noticed that almost every math book I read begins with the words "Introduction to" or "Elementary." Are all of these authors seriously downplaying the amount of information one has to absorb to understand these books? Or am I an idiot? I'm in my first year of a PhD program, second semester. I did well the first semester, doing well in this one too. In fact I've always done well in math, in high school, through the BA and through the MA, straight A's. But after all this work it seems like I will never get to the point where I can read a book that's maybe "Intermediate." Let alone "Advanced." I almost feel like I must be faking it because there's so much I don't understand. And furthermore, it seems when you do get to that point, you're actually just reading papers that only maybe 35 other people in the world could read. Okay, I'm exaggerating.

Here's an example. One of the books I'm currently studying is John Lee's Riemannian Geometry: An Introduction to Curvature. It will often take me 3 full days to do the problems at the end of one of his chapters. Sometimes I get so stuck I feel like I'm wasting my time even working on it, but if I look for another reference, I am screwed, because this is the "introduction!"

I have a difficult time ascertaining exactly how much confusion your average mathematician experiences along the course of understanding something. Most of my peers have seen me as one of the more talented students. I expect they have no idea how many hours I spend weekly feeling like I'm banging my head against a wall.

I'd very much like to hear the opinion of a more experienced mathematician than myself. Is there ever a point where you feel so confused that you decide a topic must not be for you? To what extent do you think it's healthy to struggle with a concept? Does it get easier? When?

Thank you to whoever read this for your time.

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    $\begingroup$ When you grieve over taking "three full days to do the problems," you have over-looked the fact that you were able to do them. Try to be patient, and remember that "introduction" is not a synonym for "easy." $\endgroup$
    – David Steinberg
    May 2, 2011 at 6:19
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    $\begingroup$ You should read the FAQ to find out what questions are appropriate here. I am afraid your question is not, and I will vote to close. A brief answer is that your experience is pretty universal, and in some respects it does get "easier" over time. I wouldn't take the word "introduction" so much to heart; it's not as if such books are for dummies; the expectation is that someone undertaking a new subject should be prepared to work hard and struggle through some confusion. $\endgroup$
    – Todd Trimble ♦
    May 2, 2011 at 6:22
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    $\begingroup$ The reason for "introduction to ....", "Elementary ..." has nothing to do with the demands on the reader, but the demands on the author. If you don't say this everyone who you didn't cite will assume that you think their work is not important, and will stop speaking to you. $\endgroup$
    – Ron Maimon
    May 2, 2011 at 6:26
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    $\begingroup$ Although the meaning "simple" of the world "elementary" (like e.g. in "elementary school", or in Sherlock Holmes' famous exclamation) goes back at least to 1600, it is a popular misconception that a maths book's title like Elementary X or Elements of X should promise an "Easy/simple/friendly introduction to X". Rather, it refers to the original meaning of the word, hence, a complete foundation built on the first principles, in the tradition of Euclid's Elements. Quite a demanding title, both for the author and the reader, and nothing to do with user's guides for dummies. $\endgroup$
    – Pietro Majer
    May 2, 2011 at 9:06
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    $\begingroup$ My first attempt at algebraic geometry consisted in me, as a 1st year undergrad, going into the library, browsing index cards for "algebraic geometry", and settling for a cozy-sounded book called Elements of Algebraic geometry, vol. 1. I asked for it, put it in my backback, went home and, once there, sat down with the book and opened it... $\endgroup$
    – Mariano Suárez-Álvarez
    May 2, 2011 at 21:38

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