I am an undergraduate student in mathematics and I met a student from another university recently. He told me that it is useless to study those textbooks that are rather simple compared to others. (For example, Bartle's Analysis compared to Rudin's Analysis.) I feel extremely sad for what he said since I literally studied almost all subjects of pure mathematics for which he mentioned as simple textbooks. I studied Functional Analysis from Kreysig, I studied Real Analysis from Wilcox, I studied Linear Algebra from Anton and Modern Algebra from Fraleigh, etc...

Is it really that bad if I didn't study those advanced texts like Rudin's, Artin, Herstein, etc? Is the choice of books matter that much?

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    $\begingroup$ I don't think it matters all that much, as long as the text is rigorous and contains proofs of the relevant theorems. To test your knowledge, why don't you check out some of the "advanced" texts and see if you understand them? $\endgroup$ – Math1000 Nov 23 at 3:28
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    $\begingroup$ It's mostly a matter of coverage. Do your simple textbooks cover the same material as the advanced ones? If not, you'll need to catch up on what you missed if you want to seriously go into those subjects. (The missing material is often needed in followup courses, or when you try to read research papers. For example, many abstract algebra classes assume a higher level of linear algebra than most linear algebra classes teach. Thus books with titles like "the linear algebra a graduate student should know".) $\endgroup$ – darij grinberg Nov 23 at 3:28
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    $\begingroup$ @Math1000, I can understand the advanced text after I have read the simpler ones. But he said it would be better for me to go for advanced (standard) text used by the top university directly to train mathematical maturity. " You can understand Rudin after you have read Bartle, but you understand because of foundation you had, not because you have reached the mathematical maturity required. If next time, you need to read an advanced text without going through the simpler ones, you wouldn't be able to do so." This is what he adviced me. I don't know what he said is true or not though. $\endgroup$ – Ling Min Hao Nov 23 at 3:36
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    $\begingroup$ Personally, I think Rudin is a terrible book to learn from. In my experience, only people who know the material well think it is a good text to teach from. $\endgroup$ – Elliot G Nov 23 at 3:39
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    $\begingroup$ For private study, I would say that the choices of books are very important, but more advanced books aren't necessarily better. The most important thing is whether you can gain a deeper understanding of the subject by reading the book. A book that suits your needs may not suit the others'. It's a matching problem. $\endgroup$ – user1551 Nov 23 at 4:07

This mostly started as an extended comment, but became far too long to be one.

If you want any context as to my perspective, I recently came out of my undergrad in mathematics, and will be entering a doctoral program this coming autumn. My school isn't particularly prestigious or anything so I've probably not had the most rigorous education in math (though I've done some self-studying and took nearly every math course at my school).

I mostly say that because there's likely a lot of naivete in what follows and it likely gives off a whole "wet behind the ears" vibe to those who know better.

Every text has its merits and its drawbacks; no one textbook can work well with every type of learner, nor can a learner easily adapt to any given textbook. If this student is so confident in the superiority of Rudin's work or these other works, I have to wonder if he can cite the strengths and failings of these works -- or perhaps he is just going off their reputation? I know that's part of why I've wanted to go through some of these works that are heralded as the best in math, or at least very commonly used - Rudin's analysis texts, Euclid's Elements, Spivak's Calculus, Jacobson's Basic Algebra, etc. etc. etc.

In the case of Rudin, from what I've heard (I've yet to read it myself sadly), I think I've heard its main merits described as being concise but challenging, leaving a lot of work to the reader in terms of filling in the blanks and understanding the motivations, with a variety of sometimes difficult exercises. That if nothing else means it has a steep learning curve. When I hear that, I don't look at it as a good book for beginners and students in their first analysis class unless they really want to test themselves, or they're really bright, or they have an amazing professor.

Rather, I see that as a sort of way to test your knowledge and your ability after having grasped the basics from simpler texts. It certainly does seem a good book in that light, as a sort of "big step" for those loving analysis; the conciseness also likely means it's a good text to keep around for reference, for instance.

I feel that the student in question might have a point, however, in mathematical maturity. You can't always have everything proverbially spoon-fed to you -- or, more clearly, if you need to study some advanced, narrow subject for your field (research or whatever), it would take too much time to find an easy text, study that, and work your way up. At some point in your mathematical career, I feel there is the belief that you've grasped the overall "meta mathematics," as it were -- how to learn, how to explain, how to communicate, how to fill in the blanks, all stuff you've learned through experience, in a proper rigorous and mathematical way -- and that you at that point don't need every little detail told explicitly, that you can fill them in on your own.

I'm not sure at what point that comes for a person. It's probably different for every mathematician. Everyone is a different learner -- they learn at different speeds and in different ways. I think that's the main thing you should take away from this discussion.

Do I think it will drastically affect your mathematical career if you learned from someone else as opposed to Rudin? I doubt it.

Do I think reading and working through Rudin might be a nice way to cement your knowledge, and if nothing else help ease some of your worries on the matter? It might.

  • $\begingroup$ I'm undecided on whether this question is a good fit for Math.SE, but you raised IMO good points. $\endgroup$ – Jyrki Lahtonen Nov 23 at 17:46

Personally, much of learning mathematics/physics is about going back over things many times. After you have the undergraduate or masters degree, going back to an introductory text can make you see it the way it was intended, which is difficult if all the material is new. Doing all the exercises is now more possible, for example. So perhaps this second approach is the time to learn the subject from Bourbaki.

An example is learning electromagnetism. After a postgraduate degree I went back to the original text and it seemed so much clearer what the foundations were. I don't think I ever really had a chance of learning it properly the first time. There were too many difficult differential equation ideas which you need to really study clearly in order to understand the various phenomenon. And it gets more difficult quickly. Having this all down clearly after a PhD makes this much easier. Now you pace your way through the foundations properly. I imagine the same is true in quantum mechanics. For pure mathematics, this is no doubt even more true, consider e.g. vector spaces and linear algebra. No way people get the depth of the subject in first year undergraduate.

Rudin/Bourbaki probably isn't a very common way to learn anything for the first time. Not unless you spend a very long time (+1 year) on a single book, which you don't really have time for when you have exams.

Probably the course you take at university, if done properly, is ideal. Then ramp up the independent learning.

  • $\begingroup$ By the time Bourbaki has written the definitive text on electromagnetism, George R. R. Martin may even have finished writing The Winds of Winter! :) $\endgroup$ – Calum Gilhooley Nov 27 at 0:56

Choice of the textbook might matter to some extent, but there are several angles to it.

  • For learning the material, find whichever book suits you the best. It might be useful to go to a library and surf through some books, or ask for some book recommendations from stack-exchange. It might also be fine to go for a combination of different textbooks. For this step, the priority is to see that new concepts make sense to you.
  • Now that you have learnt some new ideas, what's next? Find exercises to solve. Again, you can find them from a combination of multiple books. Furthermore, if you can successfully solve problems from the "unfriendly" books as well, then your understanding is just as good.
  • Try skimming through the unfriendly books anyway, at the end. See if there are things that are things explained in a differrent light. Also, one of the skills that'd be necessary for a mathematician would be to understand new things from hard-to-read sources. I suspect that as a professional mathematician, you would encounter papers which are hard to understand for various reasons. This means, it is useful to be able to understand material which is written in slightly dry ways.
  • $\begingroup$ You mean, study beginner book for the first exposure. Then study the more advanced book whenever I have extra time, right? $\endgroup$ – Ling Min Hao Nov 23 at 15:54

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