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Most of the advanced textbooks (college level textbooks, especially graduate), are written as a list of theorems and definitions, little or none of the exercises have solutions and many proofs and explanations are left to the reader.

My question is: Is there any reason why they do this? What is the purpose of writing a textbook if you make it hard for the reader to learn from it?

If I had to make a textbook I would try to make it as friendly and as intuitive as possible since I know that students buy it to learn something from it, otherwise why would I even write it? I couldn't imagine myself thinking "oh wait, let me give 0 examples, and let me leave this to the students otherwise it's explained too well" it's pretty funny (or depressing).

Is there any advantage of studying from this kind of textbooks? And there's no way that reading the same page for days in a row trying to understand the same line (when maybe one just needed a little example) is productive or it will stick in your mind for longer, if something is explained well and easy to understand you'd be able to learn more stuff (therefore be more productive) and remember it for a long time as well.

(P.s: I understand that for older textbooks, sometimes they were very concise to save paper, but now it's 2020..)

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    I believe that it is much harder to understand a topic when you are not forced to think about it. So missing detail might help the reader to develop his or her own thoughts about the subject. – asdq Sep 05 '20 at 09:15
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    @asdq I don't think so.. "understand would you better wrote ? i if this like" just because you had to sit there and put the sentences in the right order or "you would understand just as much if wrote like this?" –  Sep 05 '20 at 09:18
  • It is not necessarily the textbook which "makes it hard to understand" the topic. The topic itself might not be accessible the easy and lazy way, but requires a lot of exercising by yourself, and not by the book. Perhaps one should have an additional exercise text with solutions. The textbook should focus on the new material, without being more or less an exercise book. – Dietrich Burde Sep 05 '20 at 09:19
  • @Vikise There is a difference between not understanding the words that are written, which you are giving an example of, and not understanding the subject they convey. If all you do as you immerse yourself in a new subject is reading proofs and not writing any of your own, you will possibly have a lackluster understanding in the end, compared to if you were forced to prove a couple of the theorems along the way. If you were handfed examples, rather than trying to come up with your own, you might miss out on deeper appreciation. I do not know that any of this is true, but it is common belief. – Arthur Sep 05 '20 at 09:23
  • @DietrichBurde Well yeah that's what I meant, but "the topics might not be accessible the easy and lazy way" is just as bad as "makes it hard to understand" –  Sep 05 '20 at 09:26
  • @Arthur Yeah i guess that's reason then.. even though it doesn't make sense at all in my opinion.. if you're a teacher you should teach and not teach just a little so that students can teach themselves –  Sep 05 '20 at 09:27
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    Many people use the "difficult"/"dense" books like Rudin's PMA, either as a reference, or as a "consolidation" book for topics that they already have some intuition for. But finding these dense books difficult is normal: reading and writing proofs is a skill that takes time to get good at, and is an entirely different skill than having some intuition about a topic. There are also soft books on how to approach writing proofs, like, "How to prove it, a structured approach" by Daniel J Velleman. Reading this book may help (you don't know unless you try it). – Adam Rubinson Sep 05 '20 at 09:27
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    @AdamRubinson Yeah I agree, many books are used as a second course, but the books that makes me angry are the ones that are super concise and unfriendly and are called "introduction to.." "A first course in.." like really? they should be called "A first not explained course in.." "introduction to who knows .. already" –  Sep 05 '20 at 09:32
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    I think I understand your question more now, and I agree with the sentiment. However, there are more and more "bridging the gap" textbooks, and more "soft" maths textbooks nowadays. I bought a few recently. Also, some of the "classic" textbooks are less dense and give more explanation than others. Have you actually gained access to every textbook in your topic of study? Are you sure there isn't a textbook that would suit you well? An "introduction to..." textbook simply means that the textbook covers 5% of all the knowledge out there of that particular area of maths, not that it's "easy". – Adam Rubinson Sep 05 '20 at 09:41

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The way you learn mathematics when you are an advanced college student, a grad student or even a fresh Ph.D. is very different from the way you do when you are in school. That is mainly because you need to acquire working knowledge about the subject and the potential ability to improve on it. For, you need an active attitude and an habit of independent thinking about the subject.

A well-chosen list of problems ranging from straightforward to difficult (or very difficult) is very useful to train your abilities.

In fact, having to think independently about the subject may help understanding its fine points. I remember, back in my undergrad years in Rome, a professor telling that the best way to approach a grad level textbook is: read first the statements of the theorems, then try to solve the exercises and only after that go back to the proofs of the theorems and try to understand them.

You may argue that long lists of problems like those in Hartshorne's Algebraic Geometry GTM book or Lang's (in)famous approach to homological algebra ("Take any textbook in homological algebra, read the statements and prove them") may not considered friendly to the student, but why learning in depth a technical subject should be regarded as needing a friendly approach?

Talking about exercises, I think there is a very important reason why answers should not be provided. A math problem may have different ways to be solved, sometimes using different ideas. If the author gives an answer, the student may be led to think that that way of tackling the problem is the standard one, or the canon. But this would discourage independent thinking which is--or should be--a main goal in teaching mathematics (or teaching anything except maybe religious dogma, for that matter).

Andrea Mori
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    Why not just write in the preface of the book "A math problem may have different ways to be solved, sometimes using different ideas, therefore try to solve the exercises in different ways" and give one of the possibilities? Everyone would be happy. And how do you "acquire working knowledge about the subject and the potential ability to improve on it" if you can't even start to learn the subject? And "read first the statements of the theorems, then try to solve the exercises" how can you solve the exercises if you can't even know if your solution is correct? it just doesn't makes sense.. –  Sep 05 '20 at 10:16
  • @Vikise: μὴ εἰ̃ναι βασιλικὴν ἀτραπὸν ἐπὶ γεωμετρίαν – Andrea Mori Sep 05 '20 at 10:31
  • This is a quote saying that there is no royal way to learn geometry. – Konstantinos Gaitanas Sep 05 '20 at 10:59
  • @KonstantinosGaitanas: yes. According to tradition is Euclid's answer to the ruler Ptolemy when the latter asked him if there were any simpler way to learn geometry than reading his Elements – Andrea Mori Sep 05 '20 at 11:05
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By the time that a student is at a graduate level of mathematics, a number of things should have happened:

  1. Their academic maturity should be sufficiently well-developed that the need for a more exercise- or computational-oriented text is significantly less.
  2. The direction of their mathematical development ought to be toward asking questions about various mathematical statements or claims, and thinking about how those statements could be proven. In a sense, brevity is a means by which some authors evoke this type of thinking in the reader--it is a feature, not a bug.

Additionally:

  1. Side explanations and commentary are often not wanted by more experienced mathematicians who studied the material in the past, but need to go back and revisit it/refresh their memory.
  2. Papers written by mathematicians are often very succinct and only provide the minimum necessary details to establish results, with the expectation that the reader (who is also a mathematician, and quite often one who specializes in the same field as the author) is familiar enough with the context to do any calculations or minor points of reasoning for themselves. Therefore, this is a skill that must be fostered in preparation for a career in mathematics research.
heropup
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    1: Just because you're a graduate student, a new topic is still a new topic, and examples and explanations are always useful, even if you got 20 master degrees. 2: i guess.. 3: then they shouldn't call the book "introduction to.." 4: I'm talking about textbooks.. i guess for papers it's different ‍♂️ –  Sep 05 '20 at 09:24
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    @Vikise Well, you don't give any specific examples of texts that you think are not matched in terms of treatment and content. For example, I would not expect a textbook in homological algebra--even at the most introductory level possible--to be written in a style similar to a high school calculus text. You seem to be under the misapprehension that a textbook needs to be a self-contained treatment, rather than one of many tools through which a student expands their knowledge. – heropup Sep 05 '20 at 09:29
  • @Vikise However, to be fair, I agree that there exist texts that are less suitable than others for a given subject. But this is to a certain extent a matter of personal taste. For instance, I disliked Spivak's Calculus on Manifolds. If I were already familiar with the topic, it would be fine. But as a learning tool it was not for me. I used several different texts instead. – heropup Sep 05 '20 at 09:32
  • Yeah I think it's also a matter of personal taste but i think people who like to learn from books that don't try to teach so much are some sort of masochists lol. I didn't give any example because just pick a random graduate textbook and you will unfortunately have it –  Sep 05 '20 at 09:39
  • @Vikise Sometimes when I crack open a promising book, I find out that it really is more like a collection of lecture notes. This absolutely does happen. There is a financial incentive for some authors to turn the courses they teach into a text, and sometimes what you get is maybe just a step away from PowerPoint slides (okay, maybe that's an exaggeration). But there are good texts out there, and there are good teachers. And keep in mind there are students who just pick things up like they were born knowing the secrets of the universe. They become professors who write short textbooks. – heropup Sep 05 '20 at 09:47
  • @Vikise Why do you always presuppose that these books, of which you haven't mentioned a single one, don't try to teach or are hard on purpose. Specialized material usually assumes tons of background knowledge and a certain maturity from the reader. Otherwise the book would be impossible to write and to read. – Jonathan Schilhan Sep 05 '20 at 09:57
  • @Jonathan It's not that i presuppose it is like that, an author has to sit there and think "ok I will give 0 examples" "I will leave this to the reader when i could just easily write it here" that's why. For examples, again, https://www.springer.com/new+&+forthcoming+titles+%28default%29?SGWID=5-40356-404-173621337-136 pick a random book from here and I bet we will have an example –  Sep 05 '20 at 10:10
  • @Vikise I see now from your other questions on this site, that you seem to have a misconception about how university mathematics works. Fields build up on each. Just because a book is called "Introduction to ..." it does not mean that it is on an undergraduate level or even more that a person with no formal background in uni math can read it. The first book that I see at the springer link is an introduction to ergodic dynamics. But this presupposes that you know at least measure theory, analysis, topology, some functional analysis, some group theory, ergodic theory itself,... cont'd – Jonathan Schilhan Sep 05 '20 at 10:24
  • ... the list is endless. To the examples part: the truth is sometimes there are no simple examples. When a definition is given in a book there is often no immediate intuition that would help understand anything. Or there is an intuition but it is impossible to verbalize. Sometimes an example would be much more complex then the definition itself. Mathematics is abstract and things build up on each other. At the end of the day you have a web of definitions, theorems, lemmas and there might be zero intuition so far. That only can build after that and that needs maturity. – Jonathan Schilhan Sep 05 '20 at 10:36
  • @Vikise By the way, the book that I mentioned on the springer site, says in its description: "Motivated by examples throughout, the author offers readers an approachable entry-point ...", "Starting with several simple examples of dynamical systems, the book begins by..." – Jonathan Schilhan Sep 05 '20 at 10:39
  • @Jonathan Yeah exactly, as you said "fields build up on each", I've read a couple of logic books, i've taken a linear algebra, abstract algebra, combinatorics, analysis course in college (computer engineering), i started self studying set theory, of which i meet the fields it's built up on, and the books still sucks, they give a theorem, and 0 examples (or sometimes examples with no solutions), should I name some books? Set Theory and the Continuum Hypothesis, Classic Set Theory: For Guided Independent Study, Set Theory and Its Logic etc. –  Sep 06 '20 at 09:03
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    @Jonathan oh and for the springer website "Motivated by examples throughout, the author offers readers an approachable entry-point ..." lol in the back cover of one of the books i mentioned above it says "designed for students studying on their own, without access to lecturers and other reading. There are thus a large number of exercises within the main body of the text designed to help students engage with the subject, many of which have full teaching solutions." and 80% of the proofs/examples are left to the reader (definitions included!!) –  Sep 06 '20 at 09:07
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    @Vikise For set theory I recommend Kunen's book "Set Theory". It has a lot of explanations (in my opinion). But again, set theory is a very specilized field (i just happen to work in this field). It is not part of the standard curriculum in most places and if it is then it is at the master level (graduate) or advanced bachelor (undergraduate). But hey, if you have a concrete theorem that you do not understand, then you can ask on this website. After all we are here to help precisely for this. – Jonathan Schilhan Sep 06 '20 at 09:22
  • @Jonathan ok thank you! i'll check it out –  Sep 07 '20 at 10:40
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I think that you are correct to point these issues out. When I started out studying mathematics I thought the same. However, many people who went through an academic degree forget that they were not isolated with the textbooks but they were taught by people. So most people actually had the concepts explained to them by someone. Be it through problem solving sessions, lectures, and so on. So by the time you have a bachelors or masters degree or a PhD you usually have the necessary skills to read such books and "teach yourself".

I always felt that textbooks in mathematics were more of a collection of notes (for instance Rudin's books; however, the learning happens in his exercises which are very good!) and I don't think that the logical presentation of a mathematical theory is necessarily the best pedagocical one.

I agree with heropup's answer. But this applies only for people who actually studied at university or something equivalent. For self-studying purposes, i.e. alone without a teacher/professor, most books in mathematics are ill-suited if you don't have the necessary (intellectual) maturity.

  • I guess you're right, most of the textbooks are meant for schools and not for self study, teachers are WAY less rigorous than textbooks (they comment, rephrase theorems etc) –  Sep 06 '20 at 09:08