Be ruthless

A guide for finding a good math book

TLDR

Don’t waste time on bad textbooks

A good textbook has:

- As many examples as definitions.

- Exercises with solutions.

A list of good math books®

Here is a fact:

There are many shitty textbooks.^[1]

They pose a great danger for you, beware. Engaging with a bad textbook is not only a waste of spacetime. It’s frustrating, demotivating, and demoralizing. Especially when you are at the beginning of the journey.

Being able to judge textbooks is critical to keeping your morale high and learning with a smile. Below I list what makes textbooks good. I wrote it for myself. I wrote it with the hope that the next time I pick a book, skim it, and think “Oh, I really would like to learn bout it. Maybe that’s okay that there are no examples…”, I will go here and this list will help me to make a hard decision to throw that book.

But before we start, I think it’s important to keep in mind: Don’t look for a perfect textbook on some topic. You will probably need a few textbooks to learn some subjects well: one book will help you more about intuitions and motivations, and the other will help you systematize knowledge.

A good textbook…

… has exercises with solutions

This is the single and most important thing.

In Vector calculus, linear algebra, and differential forms: a unified approach^[2] after each concept, the reader has a quick exercise. The solution is in the footnote, on the same page so there is no traction. Perfect! ε>

… explains why and gives context

Why a particular field is important or interesting. Why such and such assumptions are being made, etc.

Analysis I^[3] begins with “Why do analysis?”.
Similarly, in Linear Algebra, Pillar I, the chapter about determinants begins with “Why determinants?”
In Precalculus^[4] the reader finds remarks on why or when certain skills will be important.

… provide many examples, non-examples, and counterexamples

Seven Sketches on Compositionality^[5] has roughly the same number of definitions and the number of examples. Fire!

… gives glimpses of advanced topics & communicates its limitations

In Precalculus^[6] authors mention a naive set theory and its fundamental flaw. The authors provide further readings for the curious high schoolers.

… care about terms

A reader needs a signal when the notation is overused. Or when a phrase is just a remnant of somebody’s mistake.

From some uncareful biology textbooks you will learn that “breaking bonds releases energy”. But chemists, correctly, will say the exact opposite!
In Analysis I^[7] Tao carefully defines a limit of a function at x. His notation goes a bit against the grain, but is more precise and, for me, was less confusing. Here is MSE question about it.

… is honest with a reader

Thurston in On proof and progress in mathematics^[8] shares his observation, that the formal form of papers and textbooks are obstacles to understanding. Is the author pompous? Do they mention context or informal reasons for doing such and such moves in the proof?

Status may prevent authors from being honest. For example, an author might avoid mentioning, that some of their chapters are boring but necessary. Why?! Knowing that is valuable information for a reader who is bored and thinks something is wrong with them.

I love when an author doesn’t hide their enthusiasm. Knowing that some particular piece of text is the author's special interest brings a more engaging, “dialogue-like” energy to the textbook.

I also like to see a few jokes or easter eggs. I think it’s a sign that an author enjoyed writing and wasn’t treated like a robot by the publishers.

In the Introduction to abstract algebra^[9] we read “If you get bored reading it, you have my sympathy. I was bored typing it.”
In Linear algebra done right page 21 is written as “≈7π".
In The integrals of Lebesgue, Denjoy Perron, and Henstock^[10] you have a joke combined with advice (look at “cc” with cursive):

Another sentence (from the same book): “This lemma, known as the Vitali Covering Lemma, will probably seem strange at first and it may require several readings to make sense of it. It is necessary to see the lemma in action several times before appreciating it.”

… mentions metacognitive aspects

How to think about the subject? What metaphors will help? What are the dangers of a particular metaphor? Will your current knowledge interfere with what you are about to learn?

… has neat graphics & layout

Neat graphics not only mean pretty pictures. The reader should be informed what are limitations of a particular picture.

I am a fan of hand drawings in textbooks. They feel more direct.

Some biology textbooks care about scale, etc. This is bad. For a neat example of the opposite see Cell Biology by the Numbers.^[11]
In Road to Reality^[12] there are plenty of author’s drawings. You learn how the author feels about the topic.

… uses simple language

Consider this:

Newton’s laws as stated in...
Dynamics and Relativity^[13]	Fundamentals of Physics^[14]
Left alone, a particle moves with constant velocity.	In the absence of external forces, an object at rest remains at rest and an object in motion continues in motion with a constant velocity (that is, with a constant speed in a straight line).
The acceleration (or, more precisely, the rate of change of momentum) of a particle is proportional to the force acting upon it.	The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
Every action has an equal and opposite reaction.	If two objects interact, the force F12 exerted by object 1 on object 2 is equal in magnitude to and opposite in direction to the force F21 exerted by object 2 on object 1: F12 = -F21

… provides summaries & big picture for each chapter

In Linear Algebra Done Right^[15] every chapter is summarized providing a big picture & motivations.
In each chapter of Online notes for MAT237: Multivariable Calculus, 2018-9 Robert Jerrard writes a “Basic Skills” section.