One example I see frequently around here is the much-maligned monad tutorial. Writers keep trying to connect it by analogy to concepts with which imperative programmers are already familiar. As a math student, I would much prefer to be given a basic statement of the definition of a monad, a list of properties, and maybe a theorem to prove that some structure is a monad instead of having to fall back to the definition. From there, I would just work through a few exercises and start to understand what it is and how it works. Most people don’t work this way, however. People seem to be extremely resistant to learning by experimentation. Instead, they prefer to be given all the answers.

I suppose this isn’t really a problem that’s unique to math. We’ve also discussed how a lot of computer users fail to truly understand how the software works and instead prefer to follow steps by rote in order to carry out one particular task.

Aside: as with monads, group theory has many disparate applications, from solving Rubik’s cubes [1] to particle physics [2] to crystallography [3]. Trying to teach someone group theory using only a Rubik’s cube may not be the best idea, in light of the abstract nature of the structure.

[1] https://en.wikipedia.org/wiki/Rubik%27s_Cube_group

[2] https://en.wikipedia.org/wiki/Particle_physics_and_represent...

[3] https://en.wikipedia.org/wiki/Crystallographic_point_group

Have you looked into the original paper?

https://homepages.inf.ed.ac.uk/wadler/papers/marktoberdorf/b...

Of course not! If that programmer than goes out and tells somebody else that "monads are burritos", they'll have no idea what that vague, nearly nonsensical statement means. You need the context of grappling with the idea over time for an analogy to actually help.

Good textbooks are already extremely optimized for ease of learning; the authors of these books are universally people who started in your position, thinking that everybody would be so much easier to learn if it were just written in sensible language, usually the language they were happening to use when the idea finally clicked for them. And the result ends up hard to understand, because the concept genuinely is hard to understand.

It's a problem with academia in general. Academics don't get rewarded for explaining things in an easy to understand way. Moreover, it is often a huge amount of work to distill results down like this. And if you succeed the response you may get is "well that is obvious".

Anyway, this is my counter argument to the "formalization of mathematics": why don't we incentivize people to bring some clarity to the exposition? If mathematics is suffering from faulty proofs, then I have little sympathy. Make it simpler, organize the concepts. Muscular calculations only go so far.

e.g. look at this advice from the academia stackexchange, to a young aspiring mathematician:

"If you're relatively young and inexperienced and hoping for best results on the rapid publication of your work in strong journals, I would stick pretty mercilessly to the format: (i) strong introduction motivating your work and explaining clearly the value added both in the results themselves and the techniques of proof and (ii) the rest of the paper contains careful proofs of all the results, in a very clear, linear, easy to follow fashion, e.g. "Section A.B: Proof of Lemma C".

Survey articles are explicitly supposed to bring people up on the important results and current state of a subject.

I "only" have a masters degree in Math, but this sounds really wrong to me. Most advanced math concepts are complex and can take months to understand for someone with an unrelated math background (e.g., say, Lebesgue measure theory or Galois theory). I wouldn't even know how to start teaching something like that to someone who doesn't have a formal mathematical background.

I do agree that it's pretty hard to be rigorous without setting up all of this edifice, but I've also found that it's very easy (at least, as an applied mathematician) to take an idea and continue generalizing it until the original application becomes a very small corollary of the new statement—one that appears usually unmotivated and "magical."

In some sense, this is a wonderful thing: you can usually prove a much stronger statement using the same tools you used to prove the weaker one, but in some sense it makes many things unbelievably difficult to understand as a first or even second-time reader.

To give an example, I was reading a book on dissipative PDE-solving as an optimization problem. The book's main chapter starts with a theorem that is so general and so far removed from any one specific application that, even as someone who works on several related fields, I struggled to understand its significance. The book then spends the next several chapters showing how existence/uniqueness of solutions to a bunch of PDEs are all corollaries (with some work) of this theorem.

The problem I have is that the presentation would've been much clearer going the other way. Start with the easy-to-prove existence theorems for the PDEs you care about (which mostly relied on simple tools and could've been proven in a few lines) and build up the generalization to the "main theorem." There, the reader doesn't have to guess the meaning (and/or marvel at the "ingenuity") of the original statement since the motivation is perfectly natural and clear, and they certainly don't have to struggle through a proof without a clear ending. Plus, you usually get the nice side-benefit that, by the time you present the most general statement, the proof is almost always immediate ± some small tricks.

While I do think that math requires a large amount of definitions to work and to put everything on rigorous footing, I also think there's this kind of weird way of writing where definitions and proofs are presented as "magical" and then shown to be right or useful later, rather than the other way around. This is fine for a few things, but I've found that most presentations that work don't follow this template (yet most mathematical papers I've read sadly do).

For every complaint about a textbook, there is an equal and opposite complaint if it had done the opposite thing. At some point we have to accept that the underlying concepts inherently take time and effort to assimilate, that the easiest style to learn from is different for different people, and that the authors really did think about all these issues when they were writing. They clearly care about exposition, otherwise they would not have written a book in the first place, a thankless and financially useless task.

I am not, of course, saying that all textbooks (or papers) will be better served by this structure, but I think a surprisingly large amount are. This may also be a consequence of the field I'm in, as well, where most statements I've made have several natural generalizations which are all relatively straightforward to prove, but presenting them all under one large umbrella with a very general construction makes the end result much harder to understand.

Either way, both ways have their merits, but I've (along with several of my peers and editors) found one much harder to work with and read than the other. Though, I will again mention that this may be due to the specific fields I work in.

--

[1] I want to be clear that I'm also not advocating for the "big reveal finish" which many papers (and some terrible textbooks) fail miserably at. Neither of these mediums is a novel. But I am saying that some approach motivating the construction and building the picture in a natural way (instead of a here's the theorem, let's explain why everything is a trivial corollary of it) lends itself to more clear expositions.

Perhaps either presentation can be made clear, but I've found it's very, very difficult to make the latter work in a way that readers can easily understand, while the former falls almost naturally from most research notes I've written and keeps people more engaged.

Some people will, but they probably aren't the target audience for the book, and most people won't.

You don't teach kids who are learning to count about 0 by explaining group theory and identity elements.

Maybe it's obvious that sketching only a little formalism is the best. But then some of your statements end up vague, and students stumble on practice problems where they really need to understand the details of the formalism.

Maybe it's obvious that stating things as simply as possible is the best, even at the cost of generality. But then your statements won't be powerful enough when people try to go beyond the few simple examples you give, and people will feel betrayed.

Maybe it's obvious that a historical approach is the best, for motivation. But historical papers are extremely confusing to read. They're full of redundant checks of things we know are obviously true today, and archaic notation and jargon. People will get annoyed at slogging through ten pages of text to confirm something that is obvious in a modern formalism.

Maybe it's obvious that a modern, clean formulation is best. But then the definitions are unmotivated, seem to come out of nowhere, there's not enough connection to the historical development, and your book won't be compatible with older books.

Maybe it's obvious that no formalism at all is best; you just show how to solve problems concretely. But this turns what could have been a few simple principles into an enormous bag of ad hoc tricks people need to memorize. Textbooks that do this are derisively referred to as "cookbooks".

It is not obvious how to write a book.

I agree, and I also agree that readers will differ in which books they find easiest to read. Different people learn in different ways.

However, in some respects, it is obvious how not to write a book. Mathematics is not magically exempt from effective technical writing skills, nor all we have discovered about cognitive psychology and how people learn, nor anything else that goes into creating a generally good presentation.

Certainly some advanced math concepts require a lot of foundations and therefore take time to build up to them, but usually the individual steps along the way should be quite accessible if well presented.

To follow up on one of your own examples, suppose we have someone who understands the basic ideas of abstract algebra, say someone who had done a first course in topics like groups and rings as part of their undergraduate studies. (I note in passing that even a keen school student considering studying maths at university might be able pick up these concepts with good presentation in a few days.) Then suppose that person are curious about why there is a quadratic formula but people claim that not only do we not know of any corresponding quintic formula, in fact we can prove that no such formula is possible.

Explaining the concepts and key results from Galois theory needed to get to that proof shouldn't take months. There are lots of technical terms like solvability by radicals and field extensions and quotient rings and so on, but if these concepts are properly explained, none of them would in itself be more than a typical undergrad student could understand after a few minutes. Building up the argument would of course take longer than understanding any given step, but it's not prohibitively difficult or complicated.

Unfortunately, in many university lectures and textbooks, those concepts are introduced using only a mixing pot of other technical terms and notation, without any sort of motivation or examples to put them in context. There's no intuition behind them. There's no frame of reference. So then of course the final proof of the unsolvability by radicals of a general quintic just feels like word and symbol soup that you have to learn by rote, because how could anyone possibly have derived it in any sort of deliberate or systematic way? And worse, not only does this result become "magic", the student lacks the ability to use the mathematical tools that support the result for any other purpose either.

A toddler could understand the difference between a smooth curve and a spiky curve, but to actually get a handle on them computationally, the best tool that we have is Calculus -- and even after decades of people iterating on how to teach it, Calculus is still a mystery to most undergrads who muddle through it.

Similarly I could give you some intuition about manifolds by telling you that they're kinda like surfaces that can have higher dimensionality, but until you're able to actually understand and work with the definition of "locally homeomorphic to R^n", you're not going to be able to do anything with them.

I think it's bollocks, this idea that mathematicians are obfuscating easy concepts with complex language. The language doesn't exist to get research grants and gatekeep, it's there because you need some succinct way to describe objects and concepts without running through a full definition every time.

Obviously you can always improve communication, and from my experience most mathematicians try pretty hard to do that, especially in the classroom, but it's not easy.