Why do many math books have so much detail and so little enlightenment? (2010)

There is a huge amount of information encoded in the choice of exactly how to define thing, and which theorems people care about. This reflects a long process of trial-and-error as the field was constructed. For a famous philosophical treatise on this using the Euler characteristic as an example, see "Proofs and Refutations” by Imre Lakatos

https://en.wikipedia.org/wiki/Proofs_and_Refutations

Part 1: https://math.berkeley.edu/~kpmann/Lakatos.pdf

Most of that foundational information is lost when it's not written down somewhere accessible; contrary to the answerer, only a small fraction is reconstructed by students as they learn the subject.

It's a huge problem, it applies to physics textbooks too, and it doesn't have to be this way. Unfortunately, the problem has been known for decades and there's not much reason to expect things to change. (Lakatos wrote the above in 1976.)

http://www.maa.org/external_archive/devlin/devlin_03_08.html

“Sadly, our present system of mathematics education is precisely this kind of nightmare. In fact, if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done— I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.”

[1] https://archive.org/details/ZeldovichMyskisElementsOfApplied...

In physics, there are so many times where we present the same mathematical ideas in a manner that's way more ergonomic for the minds we're talking to-- other physicists.

Inexperienced mathematicians tend to see these presentations and are kind of gobsmacked, "why did no-one tell me it was that simple and concrete?" Experienced mathematicians tend to see these presentations and cringe, "don't they understand it's not that simple and concrete? For instance, look at this pathological example here that contradicts everything they just said."

The slick presentations are great if you're a physicist or anyone just wanting to use the ideas. And they're great if you're an inexperienced mathematician trying to get your bearings as fast as possible. But once you're a research mathematician, trying to build new ideas, too many slick presentations of ideas just creates groupthink. You want as many people to generate their own mental pictures as possible.

That said, few people studying mathematics go on to be research mathematicians. The mathematics culture should be much, much friendlier to people whose endgame is not being a math prof. Apathy toward those whose ambitions lie outside the ivory tower is a problem endemic to most academic subcultures.

Finally, when reading anything generated by academia, remember most of it is crap. Academics are fired if they don't make enough noise. This is truer than you think, even taking into account metrics like "impact". You can make a lot of "impact" just by making enough noise. And you generally can't make "impact" on a short timescale without making enough noise. Once your name shows up enough in a field, you'll get known as "a name", and start being cited in that field. Even when your work is drivel. If your name only shows up a couple of time in a field-- unless your work incredibly well written and you get a good dose of luck-- it'll take years before it gets uncovered and appreciated, if it ever does.

Think of how much more attention you pay to what someone like maxxxxxxxx says here. Sure. He's good. But he's also familiar. It's primarily his familiarity, more than his quality, that makes you feel like he's one of "names" in this little subfield of the internet called hacker news.

He takes the approach of assuming that his readers can range from absolute beginners to mathematicians, and that if you prefer, you can skip the equations and the exercises.

But if you attempt to understand them, then your picture of the Universe will become richer that no other pop science book can promise.

It's an enticing thought, and that and his gentle prose are just about his only tricks to push his readers through his enormous tome.

I haven't finished it yet but it does feel rewarding that much of the maths that I've come across (Alevels and engineering degree) are there, laid out in a new framework of meaning, the Universe.

P.S. There's also an eBook version, and an online resource that gives solutions to the exercises.

This history infects mathematics books.

The other traditional problem with mathematics is terse and obscure notation. There are many implicit assumptions embedded in published mathematical papers and books. This is not helpful. It's like reading code snippets without the declarations.

[1] https://en.wikipedia.org/wiki/Wrangler_(University_of_Cambri...

I say fortunately because during my first round at Calculus II I couldn't fathom the slightest underlying motivation for all the statements and proofs that were regurgitated during class. I had no bearing - I nearly panicked with the Cauchy Problem - and was doubting whether or not I was cut for Engineering at all.

The following year the other Prof took a digression and for two weeks laid out Inf-dimentional spaces, Banach spaces, functionals, contractions, fixed points and so on. Then one morning, over the course of 2 hours he plugs in differential equations, the Cauchy problem and deals with it with such elegance, clarity and insight that I can still recollect the broad strokes.

I had first hand proof of the blessing and the curse of mathematics...

Volcanoes.

Why are you wasting my precious time with this nonsense before getting to the meat? There are other things I need to read and study. There are problems I need to use these ideas to solve... and not only the ones in the textbook.

But then, I can also recognize that such winding introductions to a subject or an idea might be helpful to people who "get bored" as the author describes. Though I suspect the remedy might be to acquire and deliberately practice study skills. It would be nice if publishers sold versions of their textbooks both with and without what the individual posing the original question would identify as "motivation." I tend to think of it as fatty prose.

I find more deplorable the fact that even basic math topics are often covered in the same dry way, without discussing practical applications or introducing topics through real-world scenarios. Many math books take the attitude "You have to know this, because I say so." Whenever I write about math, I try to start with a concrete example or a useful application of the theoretical result—it's always possible to come up with something for most of first year stuff. Seriously, you'd be surprised how much better reading UX is if you start each chapter/section with a motivating example.

This was never the case for me. When I learned trig ratios, I always understood some basic things that trig ratios could be used for. The teacher always introduced some applications, we always had a lot of word problems, and I could fill in the gaps myself.

Same for calculus. When I learned calculus, I always understood some things that calculus could be used for.

So I understood how those things could be applied to general, everyday sorts of problems. What was missing, though, was that I had nothing to which I could apply those techniques, besides homework.

Learning math (and reading STEM papers) has become easier for me since I now have actual problems to solve. Don't get me wrong: I'm not solving particularly challenging problems or using particularly advanced math. Nothing that tens of thousands of people haven't done before me. But I do need to understand the problems, solutions, and some of the context in order to successfully implement them. This provides a motivation that was always missing before.

I suspect this general narrative is true for a lot of people: that having an actual problem to solve is almost necessary to get a student to really learn the material, instead of just coasting along for a grade.

Today's technology in natural language processing is reaching the point where it will be possible to marry a natural language processing system to an automated theorem prover and have it generate formally verified proofs from math prose proofs.

Once this technology can readily process the textbooks making up the PhD curriculum, I think there will be a culture shift. Quickly there will be a new standard that math results should be formally verified. The hallmark of math, after all, is that it can be proven correct!

But with an increased role for computers will come an increased appreciation for the things that only humans can provide. Motivation and explication will be more valued when the technical aspects of theorem-proving are automated.

https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...

There is a series on calculus as well. It is honestly beautiful.

For example, take something like measure theory. A reasonable motivation for measure theory to me is remembering your introductory probability class how you had to learn a probability mass function for discrete spaces and a probability density function for continuous spaces. Now these two ideas are obviously nearly the same idea. I mean the notation used is a pretty big hint. But you need measure theory to have the right concepts to describe how they are the same.

Where are all those kinds of motivation for things like topology and cohomology?

> When I was a graduate student, we had a wonderful working seminar on Sunday mornings with bagels and cream cheese, where I learned a lot about differential geometry and Lie groups with my classmates.

At the end, the students needed context and enlightment. We cannot underestimate other dimensions of the learning experience.

* We know that teaching mathematics is difficult. But, contrary to what most people think, there is no magic pill to solve it. Motivation and intuition are important, yes, but so are definitions and formality. Also, even when giving them to the students, they might not be useful: sometimes each person needs a different motivation of a concept, sometimes the intuition relies on more advanced concepts... Furthermore, things don't usually click the first time you are exposed to them, but only after you have worked with them for some time. Don't think that just because you are given a motivation and intuition behind the concepts everything is just going to become easy to understand. Even motivations and intuitions need work to be understood.

* The mathematics people study in high school and first-year engineering courses is very different from the mathematics that mathematicians study. Funnily enough, most mathematicians I know don't like the approach of mathematics as a tool for other disciplines, mainly because it focuses on things that are hardly interesting (here's the 123th trick for how to solve this weird integral that we are teaching you only because you will need it to solve some engineering equation) and avoids proofs that, despite being hard, most of the time helps understanding the concepts.

* Related to the last points, an important number of math books are written not for laypersons but for mathematicians. They go straight to the point, avoid long motivations/intuitions that are not specially useful and assume familiarity with a lot of concepts. Sometimes they are written as reference books for people that already know the ideas. That, of course, makes them hardly approachable by non-mathematicians, but there are already books that do that. Just because a book or a way to teach math is not of your liking does not mean it needs to be changed.

* Not all mathematical concepts need to have a direct application. Sometimes, things are studied just because they seem interesting or beautiful. Searching for motivation needlessly in these cases tends to be a waste of time. It's like reading a novel or solving a puzzle: you don't need to know "why", just enjoy it. If you're lucky applications can appear later, but there's nothing wrong if they don't.

At some point, you graduate from "being taught" to "teaching yourself." By the time you get PhD, you need to teach yourself, because you're studying things no one has ever studied before.

Reminds me of the joke that philosophy students university learn the work of people who didn't go to university.

Would a text written by the discoverer of a field be more interesting?

I've found important papers easier to read than textbooks covering them... OTOH the original paper can be too close to their own motivation, and not treat - or yet be aware - of its general significance.

A motivating problem doesn't have to be practical or relatable... it just must lack obvious solution. A puzzle.

Another example is the Feynman Video Lectures are effectively inaccessible, due to being in an ugly format - they should be on youtube as a planetary resource.

I was recently looking at reviewing University chem so I can have informed discussions with my teenage son.. Atkins has a great book, but at an exorbitant price. College texts seem to be a kind of scam. Some would argue its morally justified to pirate such content.

We seem to be stuck in all these local minima for minor psychological or economic reasons. There should be open texts that cover most of the basics of highschool and freshman science, with upkeep costs shared by universities/schools or covered by a library tax.

We can think of education methods as a technology itself, and the best technology, namely the best explanations, are much much better than the average - how do we propagate the best ideas ?

Having ratings on quality, like those on goodreads and amazon, seems to be a practical benefit, but maybe we need smaller granularity - eg. rate a particular diagram on an open biochem textbook wiki.

Text books are usually written by university professors, often the professors from whom you are taking the class. There are absurdly many entry-level textbooks for a subject whose beginning parts probably haven't changed in over 100 years. They do this at the behest of the university, which will be capturing their royalties under an IP clause in their employment contract. The university intends to sell this book for upwards of $200 to a captured audience of teenagers who don't yet have the life experience to know they can opt out of putting up with bullshit. If the university doesn't have a constant churn of new editions, then the books will very quickly filter into the secondary market where they will be drastically discounted. So universities pressure professors to write textbooks quickly. And university professors aren't exactly the best-paid vocation. Invoke "good, fast, cheap: pick two" and you see it's "good" that is getting dropped on the floor.

With such pervasive usage of TV, internet, cellphones, telephones, material items, shopping, entertainment over the last few generations, we have gotten to the point where we are no longer producing Einsteins, or Thomas Edisons or Babbages. These days, your typical 'scientist' is a guy more concerned with dressing and appearing hip and is more concerned with being a CEO/marketing than with actual trench work in the scientific plains. These are the same types who encourage in turn for younger generations to follow their 'hearts', to be more 'social', to refrain from 'dry and boring studies' that consume one's entire lifespan.

Though, compare it with https://terrytao.wordpress.com/, full of insight.

The most basic example I can think of is "ab" meaning "a x b" but the "times" function has been deleted but still needs to be understood.

I find when reading mathematical books I cannot follow examples and progressions to the next level then the next level etc as at every progression there are significant pieces of information that are missing yet text books rarely if ever mention what is not written, so how can the novice possibly figure it out.

Also, for self-teaching, I've found some math books that are setup to be in the form "Topic A with Applications to Topic B." If I care about B, such a book will typically does a good job motivating A, even if it isn't the purest introduction to the area. I can always read a book that is a more canonical intro to A later.

I suppose you could argue that it's the student's responsibility but maybe that would require insight and inspiration on a par with that of the person who came up with the answers in the first place and I think it's fair to say that that's an unrealistic and unreasonable expectation.

The bottom line seems to be that the "noise" level keeps increasing to the point where the core fundamental concepts are lost. Which is to bad, because teaching "tricks" which don't clarify the problems, or make them simpler to solve is IMHO just a waste of time.

BTW; For abstract algebra there are a number of "applications" books which teach the core concepts necessarily to understand error correction or whatever. Use those books first, and then read the more traditional text books. The textbooks should be considered more of a reference book than something that should be read from cover to cover. (although that itself is a problem because jumping into the middle isn't easy).

Personally, I prefer the dry stuff (e.g., Herstein) because that's the abstract-est abstraction. Rotations and matrices are groups? Don't care. I just wanna see the strange hidden properties of abstract structures reveal themselves.

Throwing on you dull definition of matrix multiplication is completely useless unless you know where this definition comes from.

As of the teaching side, I went through my entire math bachelor without understanding what were the applications of what I was learning. Now that I do cryptography I finally understand some of these. There was also this course by Gilbert Strang on matrices that taught me that they were used in compression. None of my teachers could have told me that...

One way that seems to have begun common in Math is to make it magical. The entry burden is hard as hell, and most reasonable things in math are expressed in a way that as best sounds mystical to common people.

This gives math some kind of credibility. Nobody wants to be the one to say how little they actually understand about it, so they continue to support the crazy magicians instead of losing face. And in the end humanity is better off for it.

PS: You can see that it is not necessary to be so cryptic but still math continues to do so, when you compare it to programming. For instance math continues to use a greek letter ∑ (sigma) to express it. In programming it's mostly a "for loop" or a "sum() function". Most people can understand "sum([1,2,3])" or "for <something> do <something> done", but they don't know the greek alphabet. So it's possible to encode math more readable, but math continues not to do so, and as argued before, probably also has good reason to do so.

Another book i recommend that helped me was "the architecture of math" by pierre basieux.

These books really assume nothing beyond the ability to read and do a great job!

- e; The story of a number (Eli Moar) - An imaginary tale; The story of -1 (Paul Nahin) - The Poincare Conjecture (Daniel O'Shea) - The man who loved only numbers (Paul Hoffman) - Prime Obsession (John Derbyshire)

They're part biography, part history, and give a little colour to the subject that isn't available from your typical college textbook.

https://news.ycombinator.com/item?id=11745486

Reference and theory are very useful, but context is how they are tied together in implementation.

If you can no longer "see" a thing, you can not usefully explain it.

I used limits of sequences.

In our university, we encounter limits in either a handwavy way in Calc I or an epsilon-delta way in Advanced Calculus. Sequences and series are introduced in Calculus II.

Continuity is then described in the same terms. Or perhaps in terms of open sets if there is some topological topic.

I think this is pretty terrible for intuition. Not just that, but it's not even general enough - it requires normed spaces, so then you have to generalize again.

Much simpler to talk about ancient Greeks and Zeno's paradox. And then rigorously define limits of sequences, and define limits of real valued functions in terms of "for any sequence x_n that approaches x, y_n = f(x_n) approaches y". Simple and right away lets me show counterexamples beyond "left and right limits", like cos(1/x), and show students how to produce two sequences that converge to different numbers.

There are similar ah-ha moments when discussing fundamental concepts of linear algebra, number theory, complex numbers etc.

The two best books I have ever found on teaching Complex Numbers are:

1) Bak and Newman

2) Schaum's Outlines

They actually give you the understanding and ceeling behind WHY analytic functions are the way they are, and derive holomorphic functions from that. Imho a more terrible approach is that of Serge Lang and proving everything the other way, with Taylor power series.

Bottom line - make a directed graph of how you will teach your subject and then figure out the best entry point an direction for the greatest cohesion, as you would when telling a story.

If you are curious, now I teach a course on "Thinking Matematically" and here are this semester's results of that approach:

Numbers and Algebra: https://qbix.com/docs/mathematically1.pdf (https://vimeo.com/215335666)

Sets and Infinity: https://qbix.com/docs/mathematically2.pdf (https://vimeo.com/210500111)

Boolean Algebra: https://qbix.com/docs/mathematically3.pdf

Logic and Probability (Coming up)

The above videos were made with help from video exditors on upwork.com

I always found mathematics self-explanatory. From all the repetitions the understanding came naturally, and I think there is definitely a subset of the population that has what you might call 'aptitude' for maths. It makes sense, because philosophically it is self-explanatory by definition. LHS = RHS - the trick is to prove it or fill in the blanks.

But I dare say the majority of people aren't 'apt' for maths in the way in which it is taught, but at the same time it's totally unfair to rule them out as maths-stupid. Many people who have no mathematical knowledge whatsoever still prove rather deft at deducing when their partners are lying or when an argument in a debate is self-contradictory. People who failed their mathematical education go on thinking they suck at math, but when they play video games and figure out optimal strategies and formulae for success in the game, it's not all that different. Sometimes the mathematical understanding comes later to people as a direct result of being forced to reason about mathematical problems in an applied setting and discover that they could have been good at maths all along if they had only understood the 'why' of the exercises and formulae they were doing.

I think there are two big stumbling blocks that stop people from taking a bigger interest and investing more concentration and care into maths: firstly people mix up arithmetic with maths, and assume from the fact that they take a long time to divide a couple of numbers (or get the wrong answer when they do so) that they are doomed to be crap at maths.

But that's utter BS. I'm hopeless at mental arithmetic, but maths isn't about adding numeric values, its about deduction and reasoning, and I'm certainly not alone in being a terrible arithmetician and a good mathematician.

The other issue that people struggle with is that the notation itself isn't clearly explained or appears daunting, especially if they didn't go beyond compulsory education. It can prove quite distracting because a person might have questions as to why limit notation looks the way it does or what the hell the integral sign means, when really it's no different to choosing commas and brackets to denote semantic delineation in written language.

Take limit notation or summation. The placement of the various numbers is arbitrary on a fundamental level, it's just the standard everybody agreed to use. If someone good enough at math sat down and tried to build up math from first principles, with no knowledge of modern mathematical standards and notation, he/she could come up with the same formulae and concepts but an entirely different way of representing them.

That's exactly what happened really. you had Leibniz and Newton just organizing information in a way that made sense to them, and it stuck. An explanation of how arbitrary the notation is on a fundamental level would probably make a board full of symbols and algebraic letters considerably less daunting for those who could-but-daren't understand it.

I think there's a big problem in education as well. I'm from Great Britain and growing up I think the biggest mistake in our education system is the dire failure to explain mathematics in a way that is friendly for people who aren't tip-top at abstract thinking or imagining.

Take the humble function, for example. teacher says "a function takes an input and produces an output. so f of x equals y minus 5x..." - and half the class stares blankly, daydreams for 15 minutes, chatters through the "work your way through the textbook" phase and does one of two things before the teacher goes through the solutions:

    A. copy answers from the back of the textbook
    B. copies an adjacent boffin.

I was endlessly helping classmates in school and I often think about how the education system could have been better and made math more accessible. The conspiracy theorist might say that society cannot afford for everybody to grasp mathematics and get a /good/ job, but hopefully that's not possible.

Geometry in education is a big fail as well. when I was growing up, trigonometry wasn't explained at all, it was just repetition of applying trig tables. Nothing visually relating right-angled triangles and circles like Thales was even touched upon until college.

Secondary education in UK tries to cover to many things. Not that there's any way of chopping math down to a single textbook or subject area, but there's often little in the way of structure and it's about cramming as many formulae down the student's throats and then examining them on their ability to apply what they have repeated. but you can't apply what you repeat, you apply what you learn and grok.

I'm not sure why there is this enormous failure in education. It could be because the people who write the books and lesson plans already grok mathematics and forget to give people aids to intuition or intrinsically understanding what they're doing. Or maybe it's because a mathematical education excludes the philosophy of math and set theory which is a mistake. Set theory is perfect to start off with, it gives insight into functions, probability, everything really. Can always discredit the /project/ of founding mathematics on set theory later.

Of course, maybe I went to a bad school, and it was a few years ago now so perhaps things have changed - but I don't think so. most of my younger sister's friends have a worse understanding than the disinterested kids when I was growing up, but that could be down to the whole fast-food, instasnaptwit culture and a hundred other things that seem to be distracting people from anything academic.

This isn't necessarily a problem. It's a good thing to have experts working together to codify the current state of affairs, but it isn't necessarily the best teaching device.

Even when you have people writing books that are intentionally targeted at people new to a topic, it's easy to forget how much you know and how much beginners don't know. I'm working on a Music Theory book with an audience of total beginners, and it's extremely difficult for me to put myself in that mindset. In doing research for that book, I've been going over many of my college music theory textbooks, and re-reading some of the treatises about the subject going back to Pythagorus and Plato.

Things that were infuriatingly opaque to me as an undergrad are now--after decades of study--just plainly obvious and practically self-evident. You forget how little you knew after a while.

If there were one thing I could go back in time and tell myself when I was younger, it would be to make it clear that textbooks are not there to explain it to me like I'm five. Textbooks are scholarly works intended to be as precise as possible about the current consensus of experts in the topic area. Approach a textbook the same way that you would approach a philosophical treatise. You aren't going to get it all the first time around. And I do tell that to my violin students who are now starting to go to college.

Textbooks are designed to be paired with an instructor who can interpret the book and guide the students. They are not actually designed to be self-sufficient courses of study that you can simply read and learn from.

The reason for this is that this is the model of education that's been pretty much generally accepted. The university model really hasn't changed a whole lot since medieval times and arguably going back to greco-roman times. At least in the West. I don't know enough about Asian culture or history to comment about that. You have a master and students and some texts laying out the way of things. And in order to become the master you have to jump through a certain set of hoops.

It's an almost religious sort of process, in some ways. There's a tradition of a certain type of study with anointed masters, there are holy texts that you don't understand but must adhere to. And if you put in enough time and survive certain types of hazing rituals, you get the privilege of becoming anointed yourself, making very little money, and possibly contributing to a holy text yourself.

Out of the many things that get cited as disruptive with respect to the technology revolution, one thing that doesn't get talked about a lot is the way that the internet has disrupted the education market.

An "explain it like I'm 5" post on reddit is far more practically useful than any textbook on any topic, if your goal is to help a person learn about a topic. I spent the vast majority of my life studying violin, and music theory. And I was a professional performer for 20 years. When I got burned out on performing and traveling and never making enough money, I got online and started learning, practically, how to code. At first just enough to get a junior job. And I kept reading, and I got better. It's been 11 years now, and I haven't looked back. I can write code and play the violin when I feel like it, and work on a book about music theory. All that as a college dropout. It's a life I'm deeply grateful for.

A generation ago, changing careers would've meant going back to school, probably taking on some debt, finishing up a degree that I never really wanted anyway, and then going to start something else. Which is a pretty big pause in your life when you're in your late 20s.

I didn't have to do any of that.

At the same time, rigorous textbooks certainly have a place in the world. The same way that Plato's Timmeus or Descartes Compendium musicae have a place. The biggest problem with textbooks is that people misunderstand their intended purpose and audience.

They are tools created by academics to help create more academics. Nothing more; nothing less. If you want to participate in the academic system, you have to play by its rules. That means that you deal with these textbooks and the professors who guide you through them until you yourself are a professor, and you can in turn, guide others.

For those of us who don't want to do that particular type of work and would rather be educated practitioners, there's the internet.

I'm glad we have both options available now.

In a lecture you try and emphasis how some things are important - while other things you can hand wave away. A textbook can't do that.

It's only the gist of it.

For an answer, there have been various influences:

(1) Whatever math was before 1900 or so, by the time of the Russell paradox and its fix with axiomatic set theory, the style of the fix was to be close to Russell-Whitehead (if I have that right) idea that proofs could be checked essentially mechanically by just symbol substitution and manipulation. E.g., in those days there was a book on the natural numbers, that is, 1, 2, 3, ... that apologized for numbering the pages before the natural numbers had been carefully defined!

There was even a name given to this style of writing, telegraph style.

(2) Of course, the books written on axiomatic set theory easily fell into the telegraph style. Even there the writers were getting into trouble: They gave names for the various axioms; they didn't explain why the names they gave were appropriate, and I never could discover why. But I was eager to get out of the sub-sub-basement of axiomatic set theory ASAP so just did a f'get about it.

(3) When books were written on abstract algebra, e.g., basic set theory, construction of the main number systems -- the naturals, integers, rationals, reals, complex -- and then went on to the main algebraic systems defined with axioms -- groups, rings, fields, vector spaces -- it was easy to stay with the telegraph style. E.g., it was tough to find a book on abstract algebra that also discussed group representations and its applications to quantum mechanics and molecular spectroscopy.

(4) Long calculus was often done with a lot of intuition and nothing like some carefully done definitions, theorems, and proofs as in, say, W. Rudin, Principles of Mathematical Analysis. And physics and engineering kept drawing diagrams with "little interval dx" etc. So, when abstract algebra was proving theorems, the calculus authors also wanted to be careful at least about delta-epsilon arguments. Then including a lot of physics, mechanical engineering, touching on the heat equation or fluid flow, was considered off-topic. Bummer.

(5) The series of astoundingly carefully written books, close to telegraph style, by the team Bourbaki was influential.

(6) During the Cold War and the Space Race, US math was awash in grant money and essentially turned its back on the physical science motivations and applications. Some of the funding people started to get angry about that, and we got the Tom Lehrer song and joke about abstract math being about "the analytic algebraic topology of locally Euclidean metrization of infinitely differentiable Riemannian manifolds" or some such.

But, sure, especially in analysis, for a good proof, there's often a good picture and if can see the picture then can construct the proof easily. E.g., for positive integer n, the set of real numbers R, and convex f: R^n --> R, that f is continuous has a really cute picture. Same for Jensen's inequality. In linear algebra, the polar decomposition says that each square matrix is just (A) a rigid motion, rotation and/or reflections followed by (B) moving a sphere into an ellipsoid by stretching and/or shrinking on mutually orthogonal axes. One or more of the axes goes to zero if and only if the matrix is singular. Etc. Nice picture.

Currently, then, there is an opportunity for math authors to include motivations for their subject, definitions, and theorems, intuitive descriptions and helpful pictures, applications, issues, open questions, etc. Uh, when reading a proof, for each of the assumptions, check off where it was used in the proof! The definitions, theorems, and proofs can still be fully precise and solid.

Some students of, say, analysis have long tried to find and draw pictures that would clarify what was going on in some definitions, theorems, and proofs. I would advise new students to do that also.

Great math teachers understand the need to teach intuition. He wasn't a math teacher, but I think Richard Feynman is the pinnacle of this. See [1] to see how he expresses intuition about physics, and his Red Books[2] for how he teaches mathematical physics with all the qualities I believe makes a great maths text for students.

Also, there's a linear algebra MOOC which also teaches great intuition before delving into proofs and heavy detail [3]. I mention these examples because they are exemplars of this idea of teaching intuition.

[1] https://www.youtube.com/watch?v=4zZbX_9ru9U

[2] https://www.amazon.com/Feynman-Lectures-Physics-Vol-Mechanic...

[3]https://www.edx.org/course/linear-algebra-foundations-fronti...

It's almost as if they partly lack theory of mind, and cannot imagine that somebody else does not know what they know, and consequently give explanations that are only suitable for someone who already knows the material.