# “Opinionated” textbooks

It's been said that no one explains anything so well as when they are trying to persuade you of something.

One of my favourite textbooks is E.T. Jayne's "Probability Theory: The Logic of Science". The entire book is designed to convince the reader of the superiority of Bayesian methods in probability, and the writing is particularly clear and understandable. I suspect that the quality of the writing is partially a consequence of the book's opinionated nature.

Another example of this is "Linear Algebra Done Right" by Sheldon Axler. The book is written with the unusual approach of banishing determinants all the way to the final chapter. In trying to prove that this is the right way to go about things Axler has produced an excellent textbook.

So I suspect that other "opinionated" books might also be very well written. What other books or textbooks (at any level and in any topic) are written in a way that tries to push a particular point of view?

Not an "opinionated" book per se, but Tom Apostol's Calculus follows up the chronological order of the concepts of the calculus.

Hence, it starts with an example of Archimedes exhaustion method and after defines the integrals before limits as:

Let $f$ be a function defined and bounded on $[a, b]$. Let $s$ and $t$ denote arbitrary step functions defined on [a, b] such that

$s(x) \leq f(x) \leq t(x)\hspace{3cm}(1)$

for every $x$ in $[a, b]$. If there is one and only one number $I$ such that:

$\int_{a}^{b}s(x)dx \leq I \leq \int_{a}^{b}t(x)dx$

for every pair of step functions $s$ and $t$ satisfying (1), then this number $I$ is called the integral of $f$ from $a$ to $b$, and is denoted by the symbol $\int_a^bf(x)dx$ or $\int_a^bf$.

Later, after the integral is defined and proved solely based on the integration of step functions, the $\epsilon - \delta$ is introduced.

It is an interesting approach, but Apostol never tries to convince us that it is a superior one, so it is not really an "opinionated" book. However, if we consider that writing a two volume textbook is not easy, it does contain the indirect opinion that, at least, Apostol liked this approach better. Also, in his words:

In this book the subject is introduced in an informal way, and ample use is made of geometric intuition whenever it is convenient to so so.

Which are now commonplace in many calculus textbooks.

Permit me to recycle a portion of my answer to an MSE question:

Tristan Needham, Visual Complex Analysis, Oxford Univ. Press. 1997.

This book "brings a beautiful and powerful subject to life by consistently using geometry (not calculation) as the means of explanation." [quoting the jacket]

Here's one figure from the book, p.135, that gives a flavor of his approach:

You can almost guess the theorem from the figure: The two spheres $S_1$ and $S_2$ are orthgonal iff the two circles $C_1$ and $C_2$ are orthogonal.

• Can you comment on how it is opinionated? – user89 Mar 31 '15 at 1:26
• @user89: It is opinionated in believing that the best way to convey complex analysis is through visual representations. – Joseph O'Rourke Mar 31 '15 at 1:33

Maybe I could title my post as "formal, instead of informal", to constrast it with the suggestions presented it so far. Although, I am not as opinionated to suggest one method in exclusion to the others. I think like all good things in life, mathematical method has to be mixed and matched.

I am not sure if letters or papers will count, but E. W. Dijkstra was a prolific writer of letters to communicate with a scientific audience. For those not familiar with him, he is probably most well known today for "Dijkstra's algorithm" to find the shortest path between two nodes on a graph (which he writes, he came up with while out for a coffee with his wife, and thus one of his most trivial achievements). More significantly, he came up with the conceptual underpinnings of modern garbage collection routines now common for memory management on computers and ways to reason about parallel programming algorithms (semaphores were invented by Dijkstra -- terminology and concept). He won the Turing Award in computing science for:

For fundamental contributions to programming as a high, intellectual challenge; for eloquent insistence and practical demonstration that programs should be composed correctly, not just debugged into correctness; for illuminating perception of problems at the foundations of program design.

Dijkstra partially received the Turing award for the eloquence of his opinionated arguments.

It is worth noting that Dijkstra strongly thought of himself as a mathematician, (he was afterall, the chairman of Mathematics Inc.) and that thus, he was involved in computing science as it was an application of mathematics of the highest order. However, Dijkstra was very opinionated about discipline in thinking, which computing simply demanded, and that he believed that most mathematicians at the time lacked.

Here's a superbly convincing (and polemical!) paper written by him: On the cruelty of really teaching computing science

Leslie Lamport, one of the people who collaborated with Dijkstra and was strongly influenced with him recently wrote a similarly opinionated piece (consider the title alone!) regarding how proofs in mathematics should be structured and presented: How to write a 21st century proof.

I don't think you'll think too highly of Spivak's Calculus after you're done with Lamport and Dijkstra!

Lamport in particular uses specific examples where Dijkstra did not, but the general idea of both of these men is that proof construction as taught by mathematicians of old is awfully disorganized and clumsy, and most importantly: horrendous for education.

I must end with a textbook: Algorithmic Problem Solving by Roland Backhouse (another one of Dijkstra's disciples, if you will) presents a whole textbook (aimed for the beginner) on the type of elegant and precise proofs that Dijkstra and Lamport pushed for. He is just as opinionated as Lamport and Dijkstra, even if his language does not match the flourish of Dijkstra's. Perhaps he can be forgiven for that, given that he is writing a textbook, after all, not a piece of rhetoric.

You'll find it difficult to write proofs "the old way", after you are done, and you'll likely hate your high school math teachers even more than before.

Judea Pearl's Causality is both the most opinionated textbook and one of the best textbooks I have ever read. The entire book is designed to convince the reader that it is causation, not correlation, that science should study, and that causation can be detected using statistical tools. It received the 2001 Lakatos Award.

Royden's "Real Analysis" has a Prologue to the student which is somewhat opinated, containing the following:

"A third method of proof is proof by contradiction or reductio ad absurdum: We begin with A and -iB and derive a contradiction. All students are enjoined in the strongest possible terms to eschew proofs by contradiction! There are two reasons for this prohibition: First, such proofs are very often fallacious, the contradiction on the final page arising from an
erroneous deduction on an earlier page, rather than from the
incompatibility of A with —i B. Second, even when correct, such a proof gives little insight into the connection between A and B, whereas both the direct proof and the proof by contraposition construct a chain of argument connecting A with B. One reason that mistakes are so much more likely in proofs by contradiction than in direct proofs or proofs by contraposition is that in a direct proof (assuming the hypothesis is not always false) all deductions from the hypothesis are true in those cases where the hypothesis holds, and similarly for proofs by contraposition (if the conclusion is not always true) the deductions from the negation of the conclusion are true in those cases where the conclusion is false. Either way, one is dealing with true statements, and one's intuition and knowledge about what is true help to keep one from making erroneous statements. In proofs by contradiction, however, you are (assuming the theorem true) in the unreal world where any statement can be derived, and so the falsity of a statement is no indication of an erroneous deduction. "