Reprinted from: eaglefantasy.com
Inspired by the article by the legendary Matrix67 (I strongly suggest everyone read it), I would also like to express some of my own views on the writing of textbooks.
1. A Rant on Linear Algebra
(Students who have not studied Linear Algebra may skip the following three paragraphs and continue reading.)
I have always felt that writing Linear Algebra textbooks in a strictly axiomatic language is completely unsuitable for beginners. When I first started learning Linear Algebra, I had absolutely no idea about the intuitive meaning of many concepts. Our textbook was Qiu Weisheng’s Concise Linear Algebra. I will express my disdain for this textbook without reservation: it actually teaches Linear Algebra in the following order: Systems of Linear Equations \rightarrow Determinants \rightarrow Further Discussion of Linear Equations \rightarrow Matrix Operations \rightarrow [A bunch of stuff] \rightarrow Linear Spaces \rightarrow Linear Maps \rightarrow [A bunch of stuff]. This ridiculous order led me to believe for the first half of the semester that Linear Algebra was simply the study of how to solve systems of linear equations. I thought to myself: such a simple problem, anyone can solve specific instances; is it really worth defining such a massive pile of things with such fanfare? It wasn’t until the chapter on Linear Spaces that I finally realized what Linear Algebra was actually about. After learning more, I realized that Linear Algebra is essentially the study of linear spaces and linear maps. Systems of linear equations are not that important. A more reasonable order would be: first teach linear spaces and linear maps, explicitly stating that a matrix is a linear map, then teach determinants, and then present linear equations merely as an example.
And then there is that unreliable concept of the determinant. I don’t understand what kind of consideration leads domestic [Chinese] textbooks to teach determinants before matrices and linear maps. This results in the definition of the determinant being weirdly based on "inversion pairs." How could we possibly understand what a determinant actually is or what it does with such a definition right at the start?! Then there is the concept of a matrix. Our textbook introduced matrices through systems of linear equations, which made me think for the first half of the semester that a matrix was just a bunch of numbers arranged in a square grid. Consequently, I could not for the life of me understand why multiplication could be introduced between matrices, or why the definition of multiplication was so bizarre! It wasn’t until we reached linear maps that our teacher finally explained it clearly: it turns out a matrix is used to represent a linear map. Matrix multiplication represents the sequential application of two linear maps. The reason matrix multiplication is defined that way is that, with such a definition, the resulting new matrix indeed equivalently represents the two sequential linear maps. But as for what a determinant actually is, the textbook never explained it clearly even by the end. It was a single sentence in Matrix67’s article that clarified the essence: “In fact, the true definition of a determinant is just one sentence: the area (or volume) of a unit square after a linear transformation.” Why can’t all textbooks put such a sentence in their pages?!
Many other concepts also lack clear explanations of their intuitive meanings. Many students studying economics or other disciplines might finish an entire course in Linear Algebra without ever understanding what calculating eigenvalues and eigenvectors is for, or why such a weird problem is studied. If our teacher hadn’t mentioned in class that it manifests infinitely important value in quantum mechanics, I certainly wouldn’t have understood what was going on just by looking at the textbook. Eigenvalues are actually the observable physical quantities corresponding to operators in quantum mechanics (this might be hard to understand without studying quantum mechanics). Then there is the concept of the trace. To this day, I haven’t figured out why such a quantity is defined; I have no idea what it represents. Consequently, I can neither remember its various properties nor do I know in what physical problems I would use this weird concept of the trace. If I don’t understand it to this extent, what is the use of learning it? It’s not just my problem; I’ve asked people around me, and they likewise cannot answer what a trace is.
Many students in China likely resonate with this because our domestic textbooks only provide strict mathematical definitions for concepts and basically offer no intuitive explanations. This leads many people to finish Linear Algebra without knowing what they have learned. Understanding the intuitive imagery of mathematical concepts not only helps in remembering their properties but can even help in outlining proof strategies or assist mathematicians in inventing new mathematics. Yet, existing textbooks simply fail to explain the intuitive side!
2. A Rant on Theoretical Mechanics
However, I suddenly realized that I have also always felt that Theoretical Mechanics textbooks merely use Newtonian mechanics as an example to derive more general principles, which makes me feel like I cannot grasp its logic. The drawback of such textbooks is exactly the opposite of Linear Algebra. (Students who have not studied Theoretical Mechanics can skip the next four paragraphs and read on.)
At least all the Theoretical Mechanics textbooks I have seen (the one printed by Peking University, and Goldstein’s Classical Mechanics) follow roughly this order: use Newton’s laws to derive the Lagrange equations, then use the Lagrange equations to derive the canonical equations (i.e., Hamilton’s equations), then derive the Principle of Least Action (or Hamilton’s Principle) from the Lagrange equations, and then derive others like the Hamilton-Jacobi equation and Noether’s theorem. /* Note: Landau’s Mechanics does indeed start from the Principle of Least Action. I might not have fully understood it at the time. But I still feel that even Landau’s Mechanics does not clearly explain the definition of L. */
If the status of these equations were equivalent, this teaching order would be fine. But the most critical issue is that the canonical equations and the Principle of Least Action are universally applicable to any physical system (some even believe they apply to any system, such as finance), whereas it is well known that Newtonian mechanics has significant limitations. In other words, we are actually deriving universally applicable principles from a special case. Thus, we are merely using Newtonian mechanics as an example to briefly explain how these two universal principles are equivalent to Newtonian mechanics under macroscopic, low-speed conditions. However, Theoretical Mechanics fails to point out how the canonical equations and the Principle of Least Action remain valid in cases where Newtonian mechanics does not apply. What I find particularly unbearable is that the textbooks mention nothing about how the Lagrangian L and the Hamiltonian H are defined once Newtonian mechanics no longer holds. Consequently, I cannot grasp the logical foundation of the entire subject of Theoretical Mechanics.
I remember a few weeks ago, a teacher in our group wrote the canonical equations on the blackboard for an arbitrary system (not limited to physical systems) and stated that the canonical equations are universally applicable to any system! Initially, I didn’t quite follow his logic; I just thought the conclusion was incredibly miraculous and was excited for a long time. Later, upon careful reflection, it felt more and more off. What premises did his derivation use? To write the canonical equations, one must first have a Hamiltonian H. So how is the H of any system given? How do you define H in any system? I discussed this with classmates and asked that teacher, and finally understood the logic of Theoretical Mechanics:
First, we have a belief: that the Principle of Least Action always holds. That is, in any system (at least any physical system), we can always find a Lagrangian L such that the Principle of Least Action is satisfied. This is the defining expression for L. Once we have L, H can be easily defined. With the definition of H, one can derive the canonical equations. Thus, that teacher’s derivation wasn’t miraculous at all; it was essentially deriving the canonical equations from the Principle of Least Action, which is something certainly practiced thoroughly in Theoretical Mechanics class. However, taking the Principle of Least Action as the most fundamental principle presents a problem: it does not provide a method for how to find such an L. Therefore, finding the appropriate L in a specific physical system becomes a very difficult and core challenge.
In short, which principles hold axiomatic status and how those physical quantities are defined in general cases are never mentioned in existing Theoretical Mechanics textbooks. Thus, what is missing in Theoretical Mechanics textbooks is exactly what frustrated me in Linear Algebra textbooks: a strict axiomatic system and rigorous logic.
3. Conclusion
So, thinking about it now, a truly good textbook should actually write the content twice: The first time, explain it as intuitively as possible, using minimal mathematics to let everyone understand what the object of study actually is, so they have an intuitive physical image in their minds; the order of teaching can follow the historical development of these concepts. The second time, present the axiomatic system, strictly listing all definitions and axioms of the theory, and then deduce all necessary theorems according to logical priority.
I have a plan: when I am able to teach and educate others, I will definitely write a Linear Algebra textbook and a Theoretical Mechanics textbook myself to promote this philosophy, so that others can avoid these detours when they learn in the future.
Yes, that’s it.
When reposting, please include the original address of this article: https://kexue.fm/archives/1328
For more detailed information regarding reposting, please refer to: Scientific Space FAQ