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I was just reading this math.meta question, and it motivated me to ask about my current situation.

I'm an undergraduate student who's reading a textbook (for self-study) that is riddled with errors -- both typographical and mathematical. It is an advanced undergraduate- or graduate- level textbook.

Since the textbook does contain so many errors, it takes me a comparatively long time to progress through the material, since I have to keep stopping whenever I detect a mathematical error. But since I haven't learned the material (that's why I'm studying the textbook in the first place!), I don't have the subject-matter experience to quickly understand precisely what is wrong; instead, my experience with the (prerequisite) mathematics that the textbook subject is built upon, combined with an ability to detect logical errors (from, again, many hours spent studying more-foundational mathematics), has allowed me to do remarkably well in detecting errors when studying the text. The down-side to this is that I have to devote a significant amount of time to research to find out what is precisely wrong.

I've also been in contact with the author on multiple occasions, supplying a list of mathematical and typographical errors after I complete each chapter. So far, the author seems to be thankful for the free editing done on my behalf, although, I am worried that, if the current trend continues, I might end up offending them or upsetting them somehow, since I am only an undergraduate, and I'm literally sending in lists of (significant) mathematical errors to a full-blown professor of mathematics, who has taught the material for many years at a university. Honestly, even I feel deeply uncomfortable before I send these emails, because (1) I don't want to be presumptuous, and (2) in my experience, some mathematicians are very ... sensitive about students finding errors in their work.

Despite this, I have found this, so far, to be a surprisingly beneficial experience:

  1. I'd be lying if I said that it didn't boost my ego/confidence/whatever to find so many mathematical errors in a textbook written by an experienced professor of mathematics. Studying mathematics is a very stressful process, where your self-confidence is continually degraded by the difficulty inherent in learning/understanding such difficult and complex subject-matter, so it's nice to experience moments like this that demonstrate to yourself and others that you are competent.
  2. It is often said that the best way to learn is to teach, since, in order to teach material, you must truly understand it. Obviously, in mathematics of all subjects, what is acknowledged as "true understanding" of some subject-matter is going to be drastically different between individuals based on their experience and the context in question. But what I've found is that, in order to precisely understand what the error is, and then convey it in a coherent and understandable way to the author, I've had to really understand the material in which the error is found. The phenomenon seems to be analogous to when you try to teach or explain concepts to someone else -- you must truly understand what you're talking about in order to transfer that information effectively.

    This necessitates revising theorems and proofs, and even doing research to learn new ones. This level of revision and research would not have been necessary were it not for the fact that the error was present. (And then multiply this learning process by the number of errors.)

  3. When I find an error, I highlight it in the textbook and add a note that describes the error and adds a correction. For some neurological reasons that I am not aware of, this seems to increase familiarity with the textbook and long-term memory retention.

I'm curious to hear from the more advanced members of this website: How beneficial do you think it is to be in this type of situation?

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    $\begingroup$ Purely subjective, but it sounds like a waste of time to me. If a reference is so filled with errors that a beginner in the field has no trouble spotting them...well, it's hard to imagine that it gets the subtle points right. $\endgroup$ – lulu Jul 13 '18 at 16:29
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    $\begingroup$ Re point 3... Spending extra time on some detail and thinking about it i order to understand an error, and marking it, will certainly help you to remember it. This applies to any subject..... Even in the absence of errors it is still best to study the steps in a proof, repeatedly if needed, until the proof seems to be a completed "story" that you understand, after which you will find it hard to forget the result. $\endgroup$ – DanielWainfleet Jul 13 '18 at 18:23
  • $\begingroup$ Errors, especially mathematical ones, are not a good thing. However, this is something everyone needs to become accustomed to, because even books acknowledged as the best to learn a subject from (as measured by coverage of material and good pedagogy) are often full of them. Books by Bourbaki, and probably Dieudonné (such as EGA), are exceptional in how few errors they contain. $\endgroup$ – Dave Jul 13 '18 at 19:21

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