# Structure of math textbooks

Math textbooks for undergraduate and graduate students are almost always structured in the same way. Each chapter/section/etc. has it's definitions, theorems, propositions, etc. with proofs following the statements directly.

If we now take a look at, for example, a physics text book, we will probably find a completely different approach in the structure. Instead of building blocks (theorems, etc.) we will find a continuous text, where the problem is usually introduced in the beginning, some calculations and reasoning done in the middle part with the result at the end of the chapter/section/etc. Instead of physics we could have taken almost any other field, I at least don't recall seeing such a strict way of dividing a book in building block as it is done in mathematics in any other field, and found the same result.

Of course there are some exceptions, at least in physics one can find books that are almost math-like in structure (often called Some-field-in-physics for mathematicians), but generally speaking this separation in theorems/etc., or more generally in such building blocks, seems to be something almost exclusive to mathematics.

And now to the question: Why do we have such a strict separation in math textbooks? What are advantages and what are disadvantages? Do you know some examples of textbooks outside of mathematics that have attempted a similar structure?

Honestly, I always preferred the approach that math textbooks had over that of other disciplines (mostly physics, not really familiar with the rest). There are obviously some problems when it comes to the math-approach, like f.e. steps that are not motivated enough, etc. but the fact that I am able to see the thing that we are going to prove on the next few pages in form of a concrete statement (theorem, etc.) helped me to not lose sight of what we are actually trying to do. On the other hand in physics you often start with something and have no clue where exactly you want to go/what you want to show (well, the lecturer/author/etc. knows but the student/reader has usually no idea)... So I was wondering is there a reason why such an approach is usually avoided in physics?

Notes: This question was originally asked on PhysicsSE (see here) but put on-hold for not being on topic. I therefore re-framed it a bit in the hope to make it on-topic for this site. I hope the structure of math textbooks is also part of math education... If not, please feel free to suggest a better site to ask this question.

• I would guess that the structure of the math text is a result of the deductive logic inherent in the discipline together with the long history of doing it this way, e.g. Euclid's Elements. Since few (no?) other fields rely solely on a deductive approach, I'm not surprised this format isn't followed elsewhere. – Aeryk Dec 7 '18 at 21:46
• @Aeryk I see your point, but even if the deductive logic is more pronounced in math than in other subjects, it's still present. Let's use physics as example again. There are things that we just can motivate, but assuming the axioms of quantum mechanics for example, one can deduce quite a bit from them using only logic. – Sito Dec 7 '18 at 22:19
• @user683 I'm having a hard time understanding what you mean... Could you maybe elaborate a bit on The Elements may be a model but not often copied in mathematical texts especially in modern textbooks. – Sito Dec 8 '18 at 21:10
• Math books are better because they are about math. Books from other disciplines are often less disciplined because said discipline is less disciplined. Fortunately, most Mathematicians are most humble though, so, there's that. – James S. Cook Jan 6 at 23:18

## 2 Answers

(Comment)

I'm not really sure that your predicate is as you've described. I took down a copy of my Calc book (Thomas Finney 1980) and my college chemistry book (MSS fifth edition) and both are really rather similar in explication, having examples, derivations, chapters, subsections, etc. Yes, there are some differences: subsections in TF a little more standalone than in MSS; also TF has problems for each subsection, each lesson while MSS has end of chapter problems. But I still don't think it is radically different and TF definitely doesn't come off as a theorem-proof abstraction (despite doing a decent job at derivations and understanding of why methods work). Looking at my bookshelf and about two feet of stuff and none of it is theorem-proof monstrosities. OK, maybe one or two tiny little books I won in science fairs and never got any use out of.

Certainly the mass of textbooks from algebra through engineering math (post ODE), were not in the proof mode. Yes, there is Euclid's Elements and Rudin, but I really think that this is a minority of "math books". As with many Q&As here, I think there is a blind spot, not realizing the (patently obvious) fact that math majors are a tiny minority of UG math teaching/courses. And zero in high school.

So really before answering the question, which may be based on a false predicate, it would be worthwhile to do some development of the predicate itself. Heck, even if it is correct, I think this exercise helps you to answer some of your question (by exposure to the materials and looking in detail into how they really are organized).

• First of all, thank you very much for the comment. I can of course only speak from my own experience, so it is entirely possible that I'm having a wrong impression here. During my university time I had to take several classes together with mathematics students (linear algebra 1&2, real analysis 1&2, complex analysis, PDE's, etc.) and in all of them the textbooks where proof-based, meaning that you have the typical structure of definition-theorem-proof-proposition-proof,etc. with little to no examples. And since I had to read a couple of math-papers in the last few weeks I again encountered... – Sito Dec 8 '18 at 13:04
• [cont'd] the same structure, which led me to believe that this seems to be the norm for math textbooks. – Sito Dec 8 '18 at 13:04

I think the short answer, particularly for graduate texts but to some extent for undergraduate texts in the style you describe, is that the textbooks are written that way because that is the way research papers are written. Part of the point of university, certainly at graduate level, is to learn how the academic research discipline operates.