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I’ve been craving some Physics courses since it’s been about a decade since I was in school. I picked up a Classical Mechanics book to get back into the swing of things and of course it went through some basic linear algebra. It’s been a while since I’ve thought about the dot product of two vectors.

You know what blew me away though? Not one textbook I looked in mentioned “why” the dot product is important; that is it’s useful for determining the similarity of two vectors. They all focused on the mechanical details of computing the dot product, but never spelled out the reason it can be useful. I went through a few other resources before I broke down and had a little chat with ChatGPT to discuss the meaning behind it and it makes perfect sense after that.

In comparison to when I was in college, things are much slower paced so I can take the time I need to ensure I have a full grasp of a concept before moving forward. I guess all of this is to say that as I’ve continued forward through more concepts I keep finding that the books I’m reading offer a mechanical view instead of a holistic view of the material. This feels like the biggest issue with most math books I’ve read and it makes me wonder where books that offer more semantic meaning of concepts instead of recipes exist.

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It's not just the books it is the whole method of teaching. I remember learning the steps to calculate an eigenvector without a single comment on why one would ever want to do that. I think it is done so that the educator can claim "this course teaches all of calculus and linear algebra and quantum mechanics". To actually explain things properly would require more modest course goals.

I was taught linear algebra and multivariate calculus as a business major. They could hardly justify why they were teaching it in that context - they were weeder courses - but I always wished they had at least tried to give us a hint of applications. Nothing, it was all algebra for the sake of algebra. Atrocious.

I think the concepts of linear systems and multivaribale calculus are important for just understanding systems in general. Even without applying them all the time you can think about dynamics with them

Multivariate calculus is also useful in probability, which in my degree was rigorous too, and is broadly useful in business, so perhaps I’m being unfair about all that math not being useful in business management. I’m grateful because later I got a degree in software engineering… But the point about math being taught like shit stands; if calculus and algebra can be useful in thinking about systems they should make an effort to show it.

100% agree that mathematical pedagogy in the USA is in terrible shape. It’s a hard problem, but we can do so much better

This was in Spain, but I have no problem believing it's roughly the same everywhere in the world, with few exceptions.

I still would argue that weeder courses are fine. If you start as a major whatever the field is - you should be interested in the field enough to plow through.

Otherwise at the end you get people dissatisfied they cannot get job but they have a degree.

I think I only learned linear algebra about 3 or 4 years after I graduated. I learned how to do the computations during the course but the teacher was so bad I had no idea what anything was for. Could've been an IQ test course for all it mattered. Here transpose this matrix now. Ok.

Pretty common with lin alg and diffeq unfortunately. Many schools teach it as a toolbox instead of for understanding.

Math like words are tools. When you want application instead of tool operating instructions, go into physics or computer science (or both).

I disagree from a pedagogical standpoint. Math can be used as a tool, but that’s not the most effective way to teach math. I’m assuming the goal is to teach comprehension and understanding.

I still would argue that method of teaching is perfectly fine.

You cannot simply explain to someone complex stuff - best way is to let people grind through to build their own understanding.

Parent poster wrote that "it’s useful for determining the similarity of two vectors" - now I would ask why do I need to determine similarity of two vectors as it does not mean much to me - if I would be grinding through math problems I would most likely find out why, but there is no way I could understand and retain it when someone would just tell me.

I think more intuitive/holistic ways of teaching would be a lot better. But it's hard to do, especially in dead tree format.

To get someone understand something holistically, as in link to their previous knowledge base, requires knowledge of what their knowledge base is. Traditionally this has been done with structuring the teaching with prerequisites etc and hoping it works.

I struggle with this quite a bit when I teach students with heterogeneous background. To be effective, one has to first probe what the students already knows to be able to relate the new stuff to that, and this requires interaction. Hypertext is/would be helpful for self-learning, but it's sadly very underutilized. LLMs may be better. But probably even those can't at least in the current form replace interactive human teaching as they don't really form/retain a model of what the user knows.

> Ask why do I need to determine similarity of two vectors

Simple:

Start with

a) Suppose you are making a video game..

b) Suppose you are determining ballistic trajectory of your missile system based on model rockets

c) Suppose you are running a fighter robot group..

Or any of the stuff children are supposed to *actually* do and then take these classes with determination to do the actual creative things that they wanna do all life.

There is an aspect of jest in the above comment, but it also contains some likely truth. Children love doing stuff, and these are the things that may enable them.

Take a random group of students from the general population and one of those examples (Edit: or any single given example whatsoeve). Turns out 95% are not really interested.

Edit 2: The teacher probably gave some example from biology or something that you didn't care about and therefore forgot about it.

The core skill of the teacher lies in recognizing the interests of the pupil and then working on refining those skills so that the pupil can use those skills for at least their betterment, if not the society.

And that is one of the toughest things to get right. Children are extremely curious, that's how they learn and master absolutely anything including arts, dancing, music, history, skating, catching insects, street smarts etc. It's on us as teachers to not let that curiosity wither into nothingness.

This is exactly how I would teach SO many subjects in school: take what kids are actually interested about and make them see the connection.

The worst thing was to learn something just because the teacher said so. If I hadn't had the motivation not to fail, I would definitely not have gone this far in my studies and in life.

If I'm writing an instruction manual for a lathe, I'm probably not going to run through all the varying things you produce with the lathe. I'm going to talk about correct use of the lathe. It won't help you understand what a lathe is for, but it will help you use the lathe correctly.

I think the same goes for mathematics instruction. The thinking is that you will soon take courses that will make specific application of these tools that have broad application.

You will be correct for the most part. However, there is a question of motivation as well. Kids can make very cool toys with a lathe.

I am not from the west, but I watched Band of Brothers. From most of the ww2 media there is always a term being used. "Why we fight?"

> You cannot simply explain to someone complex stuff

This is absolutely not true. Many times I've experienced the moment of something complex "clicking" after hearing or reading an appropriate explanation for a phenomenon - finally seeing the right visual or appropriate example or comparison. I find it hard to believe you've never experienced this other than through grinding problem sets.

I don't believe something can "click" - without grinding first.

I believe something "clicked" only because you were grinding or already quite familiar with the topic.

There is no way to simply explain complex topic so someone would just get it or someone would "click" on after reading one book on the topic.

Like I said, I've had many of those moments. It is strange you haven't. You telling me I just wasn't aware of my own grinding is a bit strange. I've literally had examples where person A explains X and I don't get it, then 2 hours later person B explains X again and I get it. There's no grinding in the middle. There's good and bad ways to explain complex things, and the success rate will vary and the amount of "grinding" you need to do I think also varies depending on the quality of the explanations you get. Maybe I'm missing the core of your point.

I agree and disagree. I've definitely experienced this, but I've come to the conclusion that the problem is bouncing around in my head for those 2 hours, quietly grinding away in the background. Then, when person B explains it, my brain is more receptive to it. At work as a developer, I'll often encounter a difficult problem, and walk away for a few minutes. Then, if still stuck, I'll go to the gym. Usually when I come back I'll have the answer coded in 5 minutes.

However, I've definitely had the case where person B just explains it better (for me). I still can't completely discount that my brain was primed for it by person A.

The only charitable reading of the comment is that the GP is a purely sequential learner - they've dont have epiphanies and gasps of insight like a partially global learner. The learning style stuff is semi-debunked since usually people dont just fit in one category, but they are classically divided as:

Sequential learners prefer to organize information in a linear, orderly fashion. They learn in logically sequenced steps and work with information in an organized and systematic way.

Global learners prefer to organize information more holistically and in a seemingly random manner without seeing connections. They often appear scattered and disorganised in their thinking yet often arrive at a creative or correct end product.

Just the fact that there are bad explanations for things immediately proves there are ones relatively better, i.e. good.

Exactly, I don't understand how people are bending backwards to not accept this. I could be explained integrals as "it's a banana", 2 hours later someone explains it properly and I get it, and some of these commenters would tell me my brain was working on "it's a banana" and that's why I understand integrals the second time.

Well I've definitely had those moments but I happen to have an alternative explanation.

Person B explains X and I don't get it, then 2 hours later person A explains X again and I get it. There's no grinding in the middle. I just needed both perspectives to make sense of X.

I do not argue that guidance is not needed, it also is, but there is no shortcut that let's people skip own hard work by telling someone incantation of words or sentences in special order that will make things click - without even trying to grasp the subject.

What you're describing is well-known to mathematicians. The idea is that if you struggle hard with something (perhaps till you have to give up), your brain feels there is "unfinished business" and keeps working on the problem even when you're not consciously thinking ("grinding" in your words) about it. The second time you engage with the problem, you're primed to understand it - or understanding may just pop into your head, like an unexpected pizza delivery. A couple of examples:

The great mathematician Henri Poincare was struggling with a problem on fuchsian functions. He made some progress, and then: "Just at this time, I left Caen, where I was living, to go on a geological exursion under the auspices of the Schools of Mines. The incidents of the travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it ..." [1]

"Incubation is the work of the subconscious during the waiting time, which may be several years. Illumination, which can happen in a fraction of a second, is the emergence of the creative idea into the conscious. This almost always occurs when the mind is in a state of relaxation, and engaged lightly with ordinary matters. Helholtz's ideas usually came to him when he was walking in hilly country. ... Incidentally, the relaxed activity of shaving can be a fruitful source of minor idea; I used to postpone it, when possible, till after a period of work." [2]

[1] Jacques Hadamard, "The Psychology of Invention in the Mathematical Field" (p. 12--13)

[2] J. E. Littlewood, "Littlewood's Miscellany" (p. 192) [A wonderful book, by the way!]

No, that is different. I'm aware when my background brain does work because like you say next time I arrive at the subject it's easier. This is not that.

Knowing why and when to use math is equally as important as knowing it. One of the reasons I lost my love for learning it was this missing information.

What I found very annoying with calculus specifically was the previous 15 years I had been memorizing formulas. Formula to get the area of something remember this thing. Formula to get the volume of something remember this formula. Formula to get the angle of something remember this formula. But if I had known the way of calculus and derivatives I could make those formulas. I now have the ability to have a formula factory instead of devoting tons of mental space to keeping those formulas. I feel I wasted 15 years rote memorizing things instead of understanding the N spaces things live in and how to get the formulas.

But you--or at least most of your classmates--probably weren't in a place to just learn calculus before taking high school physics or even simple geometry. And this happens at a lot of different levels with math, physics, chemistry, etc. There are a lot of inter-relationships and often moving forward requires taking some things on faith (for now).

Illustrative point for the “the value isn’t explained” issue: I’ve spent two years on calculus in school plus some more time on my own and don’t know how or why I’d e.g. use it to derive the formula for the area of an oval (I think that’s the kind of thing you’re getting at?)

Actually, I can count the times I’ve applied math from later than 6th or 7th grade on one hand. I’m almost 40 and have been writing code for pay since I was 15. I struggle with this with my own kids and dread their reaching those later classes because I have no compelling answer for “why do I have to learn this boring shit?”

How did you learn to make those formulas?

You're making a false dichotomy. Learning is a combination of guidance and own hard work.

Maybe you prefer to figure out everything yourself, but you have just one lifetime, and having access to guidance while grinding will allow you to learn things faster (and thus more).

This is quite an old book. I wish I had an access to or even knew about this book during my school days 40 years ago. When I discovered this book, I was already a middle-aged engineer and was just looking for books for my kids. The first two pages blew my mind. If only I had this book back then... However, I remember now the several sleepless days and nights in sequence when I tried to make sense of finite fields at university. I could not sleep; I could not rest... And then I understood that you can create your own algebra whenever you want; you just need to follow the rules. This was so mind-blowing, and at the same time, no single teacher even tried to point out this wondrous fact that actually changed my mind in such a significant way. It could literally have saved a couple of years of my life if I had read this book back in high school. I'm not a mathematician; math exists for me only when it is applicable to what I have at hand. But you realize that you missed a HUGE AMOUNT OF TOOLS, too late in your life.

What I argue is that there is no approach to guidance that will substitute for own hard work.

I do not argue that guidance is not needed, it also is, but there is no shortcut that lets people skip own hard work by telling someone incantation of words or sentences in special order that will make things click.

And you'd be dead wrong by all methods of pedagogy that we find useful! People learn best with stories and meaning, its why the ancients were able to reproduce stories of almost inhuman length via memory.

Grinding through to build your own understanding when someone can just give you useful meaning and context to connect to your other parts of learning is a core teaching skill, and anyone avoiding that because its "too hard" is doing a deep disservice to their students.

A long time ago, when I was in high school, we had an introductory course to differentials and integral calculus. When I asked what the purpose of integration was, the teacher shouted that I should save the stupid questions for my parents ... She was a shit teacher for various reasons, but that was the day that I lost my drive for maths.

It wasn't until years later that I found that it was all about "the area under the curve" and why that would be useful. At no point in those high school classes did we ever work a practical example. I was pissed off all over again when I found out how useful that stuff could be, and how much I'd missed out on.

I'm sure most teachers mean well, and I'm sure most of them try. But by god there are some truly awful twats out there who should never set foot in a classroom again.

We had a good math teacher. There was a formula he just told us to memorize, the class asked how it worked but he just said we don't need to know why or how, just like we don't know how a calculator works. What he didn't know was that the class last week in electronics was about how calculators work.

He had to confess he didn't know why or how either of them works, he just uses them :-)

> When I asked what the purpose of integration was, the teacher shouted that I should save the stupid questions for my parents

What an awful person. Chances are she was getting defensive and covering for her own lack of understanding. If I were a parent, I would confront her about that, not least of all her contempt for students and for learning, but toward parents.

Teachers don't know everything, and when they don't know, they should be able to admit that without hesitation or defensiveness. This sets a good example in general, of humility, instead of inculcating the notion that life is about having all the answers, or rather, pretending to have all the answers. All this does is set up people to become imposters. Of course, if you're teaching calculus, you should have at least a basic grasp of the material, and if you don't, you should say so, so that you've not put in a position where you have to teach it.

> I'm sure most teachers mean well, and I'm sure most of them try.

I think it is generally accepted that primary education isn't exactly packed with the best candidates, both from the point of view of pedagogical ability as well as mastery of the material.

I like this view of the integral (and generally regard this site as helpful to build intuitions).

https://betterexplained.com/articles/a-calculus-analogy-inte...

I took both applied math and pure math (+ comp. sci) at uni, and was similarly frustrated at the applied math, which was all triple integrals and partial derivatives etc, and exactly zero examples given of where this stuff would be useful. I just stopped attending the class, since the downside of doing so was minimal. To this day I still don't know where one might apply triple integrals, although no doubt there are uses.

Perhaps a bit oddly I didn't have any problem with the pure math prof. also not offering any use cases, although of course it also does!

In high school trigonometry I am sure I was clear that sine and cosine formed the circle. How could I not?

But that fact’s significance and too obvious simplicity, with all its ramifications, only hit me deeply and profoundly when a year later I realized I could use those functions to draw a circle on a screen.

Before that they were abstractions related to other abstractions that I had to memorize to pass a course.

To this day I am frustrated when reading papers about abstract algebraic relations and other such concepts, without even a sentence or two discussing any intuitive way to think about them. Just their symbolic relations.

I appreciate that in the game of math that view becomes natural. But most of us learn math with additional motivations and are interested in any perspective that highlights potential usefulness or connection to the real world. Many of us mentally organize our knowledge teleologically.

Yet even when usefulness is known to exist, it is often neither mentioned or referenced. Or even considered relevant.

Edit: the same goes for not showing a single concrete example of an abstract concept. A kind of communication that would unlock many mathematical papers to a much larger audience of intelligent and relevant readers.

I’ve been teaching mathematics for over 30 years at the community college level. Most people at the time of taking a course don’t have a sophisticated enough understanding of math to really appreciate “intuitive explanations” because they don’t have intuition.

Take parametric curves. I explain that they generalize the concept of a function. Every function can be parametrized in a trivial way. They don’t really understand this concept. They have a hard time parametrizing a function and do so only becuase of a formula.

The fact is most people need to go through the mechanical process of doin g before they can get to a point of understanding. It takes almost the entire semester for me to convince beginning algebra students that the reason that 2x + 3x is 5x is because of the distributive property. And when they do understand it they don’t understand why that is important.

Later on when things click for someone they will often say things like, “Why didn’t they just tell this when we took the course?” Usually we did. You just didn’t have a sophisticated enough understanding of things to grok it at the time you took the course.

I am sure you are right, and a little convo here isn't going to do the topic justice. So many aspects to how people understand and learn things.

And I know picking on your example isn't in the league of a general solution.

But if 2x (which is x + x) is two apples in a box, and 3x (which is x+x+x) is three apples in another box, then you put those two boxes in a bigger box (another +), people already intuitively can see the distributed property of scalar multiplication vs. addition of some unit, they just didn't have a name for it.

Likewise, a 3x4 square of paper next to a 7x4 piece of paper can be easily seen to be a 10x4 piece of paper. Multiplication of numbers over added numbers.

So one way to introduce distribution is to start by showing examples of several places where people already understand the concept of multiplication distributing, and use it every day, but just didn't know it was one concept with a name.

Once people can recognize distribution as an already familiar relationship in everyday life, then the symbols can be visited as the way we write down the already known and useful concept so we can be very clear and general about it.

Anyway, that's just a reaction to one example, which may not mean much.

Yes, almost everyone gets that. Then you need to explain that x is really 1 x. Then you need to explain that -x is really (-1)x. Everything is great. We all understand. Now simplify (1/2) x + 3x. You’ll lose most people at this step. Then explain (1/2) x - (2/3)x. More confusion. Now explain that ax+x is (a+1)x. You lost a lot of people at this step. Now explain that xy+ x^2y is (x+x^2)y and that this is just the distributive property “in reverse”.

Sometime later a person will really grok all this and then say, “why didn’t they just tell us this is all just the distributive property?”

Ah, I get it.

A different approach I have thought about, which would really tear the textbooks apart, is introducing every concept in its simplest form as early as possible.

Then when it is eventually expanded on, its familiarity will aid in taking further steps more quickly and intuitively.

For instance, something as simple as adding up the area of a fence of varying heights, or the area of multi-height wall to be painted, being referred to as integrating the area, in early grade arithmetic, creates a conceptual link for down the road.

Systematically going over K-12 materials, just making similar small adjustments to terminology and concepts to be highlighted, would be interesting.

As I see it the issue with your integrating example is that area is the correct word for finding “area”. Integrating is not finding the area. The indefinite integral is not about area. The definite integral in dimension 1 has to do with signed area. I don’t think having people ingrained to think finding area is “integration” would be a good thing. Especially since most people don’t take calculus.

To your point, people do constantly try to tweak things to make subjects easier to understand and more intuitive.

That's how Riemann integrals are defined in class. Vertical slices.

Tossing the word "integral" at younger children won't make that easier or harder.

> To this day I am frustrated when reading papers about abstract algebraic relations and other such concepts, without even a sentence or two discussing any intuitive way to think about them. Just their symbolic relations.

If you’re talking about research papers, that’s just because they’re written for domain experts and aren’t really for giving you intuition. They’re written in a deliberately terse (one might say elegant) style to convey the research findings in formal mathematical language and nothing much else. If you want to gain an intuitive grasp of things, read a proper textbook in detail or play around with the ideas on paper. Or both!

I guess the reason is that once you’ve acquired the intuition, having the literature cluttered up with the same explanations again and again becomes clunky and increases the volume of material to be sifted through when you’re just looking for a result you need in your research and don’t need all the extra chatter. It’s just cleaner that way. But to an outsider it does look more opaque. It’s a trade off.

> having the literature cluttered up with the same explanations again and again becomes clunky

I think that really is the best reason for not being more accessible. Along with less work - given a good paper already can take a lot of work to write clearly even for the immediate audience.

But there is tremendous value in reaching a wider audience, for readers, writers, and the very real serendipity of cross pollinating ideas. So an easily skipped concise titled section, that gave a little context or example for the non-inside crowd, would be a nice tradition. Even an appendix - although that might strike the established culture as too quirky.

Some papers manage to do something like that, a colorful example or perspective adding levity as well as clarity. So it is not breaking any barriers. Just not standard or prescribed.

Or maybe it wouldn't have much impact. I tend to find reasons to dive into many different new topics, so it is a prevalent need for one!

> an easily skipped concise titled section, that gave a little context or example for the non-inside crowd, would be a nice tradition.

I completely agree — especially in the modern era where extra pages cost nothing.

I've been trying to work through this in the context of programming a CNC using a recent trilogy of books:

_Make: Geometry: Learn by coding, 3D printing and building_ https://www.goodreads.com/book/show/58059196-make

_Make: Trigonometry: Build your way from triangles to analytic geometry_ https://www.goodreads.com/book/show/123127774-make

_Make: Calculus: Build models to learn, visualize, and explore_ https://www.goodreads.com/book/show/61739368-make

(oddly the Calculus book was published second, so I guess I'll need to re-read it after I finish the trigonometry book)

Hopefully, this will provide me with a sufficient grounding in conic sections that I can solve my next CNC project with a reasonably efficient set of calculations (trying to do it using my rudimentary understanding of triangles from trigonometry had me 4 or 5 triangles deep, barely half-way to the final point I needed, and OpenSCAD badly bogged down performance-wise).

Thanks for that list, I just ordered them! I hadn't heard of that series before

Glad to!

It's quite recent, but seems very good to me, but my math education is about non-existent, so anything would be an improvement.

Not teaching the Why is such a sin! I didn't understand calculus properly at all until I read Steven Strogatz' brilliant book Inifinte Powers, which not only explained the why but the history of why. 10/10 book for me.

https://www.stevenstrogatz.com/books/infinite-powers

Modern education is grounded in a different worldview than the classical liberal arts[0]. The classical liberal arts are so-called because they are freed from the burden of having to be practical or economic in nature (which is not to say they could not or did not incidentally have practical application), intended to produce a free man. Here, too, by "free" we mean free to be good, that is, more fully human, not what we mean by freedom today as doing whatever you happen to feel like doing, a recipe for enslavement, misery, and despair, and therefore directly opposed to the good and to becoming more human.

Opposed to the liberal arts were the illiberal or servile arts. These are necessary and good, of course, but necessarily inferior to the liberal arts because their end is not truth or formation; they are instead practical, concerned with effecting some kind of economic end. The point here is not to disparage, but to understand how all of these are related and ranked according to a "for the sake of" relation. A human being doesn't exist to eat, he eats to exist, for instance.

Modern education is very much oriented toward the servile arts, and what passes for the liberal arts today is anything but the classical notion.

The point is that modern education is less interested in leading to understanding, realizing virtuous habits, and leading to freedom, and more interested in churning out workers. Workers don't ask "why" (though we can agree that those who do can, guided by prudence, contribute more economically). Indeed, that is perhaps the key difference between classical science and modern science: the emphasis of the former is truth, while that of the latter is control of nature. Of course, it isn't that you must choose absolutely between understanding and effectiveness, and the classical tradition does not claim either that study precludes work. Study often requires work, for sake of preparing the way for truth. Rather, it is that the end of the modern educational tradition is different from that of classical education, and this end determines the form of the pedagogical methodology. It is a difference in anthropology, of the vision of man.

All men work, but what do they work for? Do they work for work's sake, or perhaps to make money to satiate their base appetites (modern view)? Or do they work in order to be free to pursue higher ends[1]?

[0] https://www.newadvent.org/cathen/01760a.htm

[1] https://a.co/d/hE5830i

That seems pretty surprising to me. The lower level/physics books I've seen introduce the dot product with both a geometric and algebraic definition, and show they're equivalent in 2-3 dimensions. The "how" is the algebraic definition and the "why" is the geometric definition.

It's not really that it measures similarity. Physics isn't interested in that. It's that it tells you lengths and angles, which you need in all sorts of calculations. In more advanced settings, a dot product is generally taken as the definition of lengths and angles in more abstract spaces (e.g. the angle between two functions).

In machine learning applications, you want a definition of similarity, and one that you could use is that the angle between them is small, so that's where that notion comes in. A more traditional measure of similarity would be the length of the difference (i.e. the distance), which is also calculated using a dot product.

Angle is a measure of similarity (well, distance/nearness).

In physics, the dot product is used to losslessly project a vector onto an orthonormal basis, and the angle measures how much of the vector's magnitude is distributed to each bases vector.

The angle can be defined in terms of the dot product, because you don't need the angle (as in a uniform measure of rotation) in order to compute important physical results.

Angle is one way to measure similarity. A more natural one in most settings is distance. In any case, physics isn't really concerned with similarity.

The angle can be defined in terms of dot product because |a|=sqrt(a•a) can be shown to be a norm, and because a•b/|a||b| can be shown to always be between -1 and 1, and because those things agree with length and cosine of the angle for Euclidean spaces. It's not that you don't need the angle. It's that the dot product gives a good definition of angle in settings where it's otherwise not clear what it would be (e.g. what's the angle between two polynomials `x` and `x^2`)

In programming terms, there are interfaces for things like length and angle (properties that those things should satisfy). If you implement the dot product interface, you get implementations of those other ones automatically. The "autogenerated" implementations agree with the ones we'd normally use in Euclidean geometry.

This is why 1b3b is so popular. Instead of teaching the mechanics he teaches the intuition.

With that being said, I do remember my math and physics teachers in high school spend lots of time talking about the why and intuitions and let the books state the how.

> Not one textbook I looked in mentioned “why” the dot product is important; that is it’s useful for determining the similarity of two vectors.

Most textbooks motivate it by the angle between the vectors or as projections (e.g., for hyperplanes). Numerics-focused ones will further emphasize how great it it is that you can compute this information so efficiently, parallelizable etc.. Later on it will be about Hilbert space theory or Riemannian geometry and how having a scalar product available gives you lots of structure.

> This feels like the biggest issue with most math books I’ve read and it makes me wonder where books that offer more semantic meaning of concepts instead of recipes exist.

All of the good ones do both. They first give the motivation and intuition and then make matters precise (because intuition can be wrong).

The sheer amount of material a student needs to digest in order to become conversant as even a pseudo-professional is enormous, which I think excuses, to some degree, the strange style of text books. I personally find that education is a process of emanations: first one digests the jargon and the mechanical activity of some subject (taking a dot product, in this case) and then one revisits the concepts with the distracting unfamiliarity of the technical accoutrements diminished by previous exposure. Thus able to digest the concepts better, the student can revisit the technical material again with a deeper appreciation of what is happening. The process repeats ad-infinitum until you ask yourself "what even IS quantum field theory?"

Is there someone now that can explain the intuition behind QFT?

The intuition behind QFT isn't the problem. I'd argue its quite intuitive: write a classical field, assume some plausible commutation relations, turn the crank. To add interactions pretend that you observe the results at infinity or whatever and take some terms of a power series representing the amplitudes, adding a cut off which you calibrate with an experiment. All fine and dandy. Just sucks that the machinery doesn't quite pass a combination of mathematical rigor and philosophical substance.

Colleges often have multiple classes on the same math subject, one made for physics and ME/EE people, one made for psych people, and one for CS. Some people don't realize that they accidentally picked up a textbook meant for a specific college pathway they don't care about.

Understandably college courses & textbooks meant for CS people will be more focused on computation, while a math major who is taking Linear Algebra will get a more theoretically motivated course. Gilbert Strang is an example of an engineering-focused text while Sheldon Axler or Katznelson & Katznelson is an example of what a math major would experience.

There are two ways to see every operation: a mathematical way and a physical way. The mathematical view of the dot product is an operation on vectors that adds their multiplied components, a·b = a_x b_x + a_y b_y + a_z b_z. The physical view of the dot product is what you said, comparing two vectors for similarity, or, in alternatively, multiplying their parallel components like scalars. The difference between these perspectives is in what is regarded as the defining property of the operation, which affects what you keep "fixed" as you vary aspects of the theory you're working in. For instance, when switching to spherical coordinates, the mathematical version of the dot product could still look the same, but the physical version has to change to preserve the underlying concept, which means its form becomes quite messy: (a_r, a_θ, a_φ)·(b_r, b_θ, b_φ) = a_r b_r (sin (a_θ) sin (b_θ) cos (a_φ - b_φ) + cos a_θ cos b_θ.

The difference in pedagogy seems to be which of these perspectives is treated as fundamental. Math education tends to treat the mathematical operations as fundamental. Physics treats the concept as fundamental and regards the operation as an implementation detail. It is very similar to how in software development you (for the most part) treat an API's interface as more fundamental than its implementation.

Unfortunately even physics books don't go over the intuition for underlying math very well, to their detriment. They seem to just assume everyone already perfectly understands multivariable calculus and linear algebra. I think it's because by the time you've gotten through a physics PhD you have to be completely fluent in those and the authors forget what it was like to find them confusing.

You might really enjoy working through the Art of Problem solving series. They’re early math -> calc books for kids that are getting into math competitions, and they explain so much in so much detail and really get to the root of why while also developing intuition. Get the e-book version. The explainers are incredible.

Maybe the books assume that the geometrical interpretations of the dot product are already known by the reader? I think they (both the projection interpretation and relation to angle between vectors) were taught in high school at the latest. There's also a lot of interpretations and uses for the dot product, some of which aren't necessarily that useful for classical mechanics.

But in general, literature using and/or teaching mathematics does tend to be too algebraic/mechanistic. Languge models can be a very good aide here!

I find the math portions in physics books are just basic refreshers.

I guess if you want to learn math, only a math textbook will actually care.

I remember being shocked in the first year of college that introductory physics and introductory derivatives and integrations were not taught together. The calculus class never explains why these methods are useful, and the physics class expects rote memorization of the final algebraic equations.

It might be because you weren't in a Physics or an Engineering program.

Colleges tend to have two tracks for physics, one that's closer to high school physics, which is as you described. A collection of algebraic equations that you have to either remember or, if your professor was kind, given a crib sheet of.

The other is the "Engineering" or "Calculus" based physics track where, as you can imagine, you're taking Calc 1 and Physics 1 at the same time.

I have seen some, kinder, programs where you take Calc 1 in your first semester and start the Physics classes in your second semester.

I've seen a lot of comments, in this thread and others, to the effect of: "I didn't get math until I looked at it in a different way, with a lengthy span of time in between." Maybe just the two different looks and the time span by themselves are beneficial.

Try 3blue1brown. You'll love it.

So don't leave us hanging ...

I think you hit the nailed on why most the textbooks are lousy at best on providing the 'why', apparently they are focusing more on the mechanical aspects and repeating exercises for scoring the exams, as a filler to the 1000 pages at US$100 textbook.

One of the best books on Electronics according to HN crowd is The Art of Electronics, and it is filled with pages over pages of how-to of designing circuits of more than 1000 pages. But if you want to know why a Colpitts oscillator is the best for your design, all the best for that.

Even the textbooks produced by professors from the best engineering schools (e.g MIT, Stanford, etc) are not spared of this issue. One of my former lecturers (not MIT) for linear algebra and numerical analysis courses claimed that he worked and consulted for NASA, but how I wished that he had cover some of the motivations of doing a dot product. For the ChatGPT responses of the reasons of doing dot product for two vectors see ChatGPT 4 prompt below. I think once ChatGPT/Gemini/etc can operate properly on math equations at least as good they can for NLP, I think we are going to have a wonderful treat even for the understanding the higly complex concepts for examples thermodynamics, electromagnetics and general relativity.

Another classic example is try to find inside your favorite DSP book on why analytic signal in complex number generated by Hilbert Transform is necessary for communication and there is a very high probability that that's not a single explanation for that but there are many equations and graphs related to analytic signal. If we understand why or the raison d'etre, perhaps we can further ask why stop at analytic signal in complex number form, can we use quaternion number and what is the benefit of having more elaborate quaternion number form rather than complex form. If you are trying to do that and check this fact with most of DSP textbooks on this particular important issue then you will probably end up dissapointed even if you have bookshelf that contains majority of DSP books that are published since the invention of FFT.

I have also included the ChatGPT prompt for the dot product, and the responses are delightful to say the least.

>User prompt: Why you want to perform dot product of two vectors

>ChatGPT 4:

The dot product of two vectors, also known as the scalar product, is a fundamental operation in mathematics and physics with various applications, including:

Computing angles between vectors: The dot product can be used to find the cosine of the angle between two vectors. This is particularly useful in geometry and 3D modeling to determine orientations and angles.

Projecting vectors: The dot product is used in the projection of one vector onto another. This is crucial in physics for resolving a vector into components parallel and perpendicular to another vector, and in computer graphics for rendering and shading calculations.

Checking orthogonality: Two vectors are orthogonal (i.e., the angle between them is 90 degrees) if their dot product is zero. This property is vital in many areas including computer graphics, signal processing, and linear algebra.

Work calculation in physics: In physics, the work done by a force (a vector) moving an object through a displacement (another vector) is calculated using the dot product. This is important for understanding how much energy is being used or transferred in a system.

Machine learning and data analysis: The dot product is used in algorithms such as those for finding similarities between data points, calculating vector norms, or in the training processes like those of neural networks where weights and input signals are combined.

Each of these applications demonstrates why the dot product is a key tool in both theoretical and applied disciplines, linking geometry, algebra, and practical computation.

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