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It's worth understanding the context Bourbaki arose in.

An entire generation of French mathematicians was turned to bits of blood and gristle in the trenches of World War I, and so French mathematicians in the 1920s and early 1930s faced an acute shortage of teachers who were current with modern mathematics.

The premise of the Bourbaki effort was to write everything down in enough detail that a sufficiently motivated reader could learn it without having to learn it master-apprentice style -- because too many potential masters were dead.

In that case, that means they have entirely failed. The very few people I've met who actually bothered with the books only made fun of how horrible they were to read and understand. I never even did open one myself! I always admired the principle though, because you can reduce everything to pure logic (and even in some cases you can "brute-force" formalism to obtain new results). Which makes me think category theory kind of fills this role in a better way?

Even without the context, there's something to be said for a formalised approach. When I was in undergrad there was a lecture course given by a notoriously aloof and formal lecturer. One of the other - more popular - lecturers decided to give an "understandable" alternative to the course at the same time of the day. Myself and a few others were in the 5% that continued with the official schedule. Those notes were hard as hell to work through, but once you understood something, you REALLY understood it.

One of the exam questions was conveniently targeted at one of the lectures from the more difficult course. I think it was proving that A5 is simple by considering the rotations of a dodecahedron.

I disagree. An overly dry and formal style (definition, definition, lemma, theorem, corollary, definition, lemma, lemma, theorem, ...) does not make students “really understand” the material. It just focuses students on low level details of formal definitions and symbolic manipulation and gives a lot of practice regurgitating/performing those, often at the expense of knowing the purpose or meaning of the subject. Low-level details are certainly essential, but the only way to really understand is to figure out what the formalism is for (what problems does it solve), grapple with the possibility space of definitions and theorems (if we picked this alternate definition, would that also get us where we want?), figure out how topics and structures relate to each-other, spend some time doing personal explorations, and build up mental models of what the definitions mean, not just their formal content.

A too-dry mathematics course/book is like a screenwriting course where you focus on snappy dialog and details of the setting but never talk about the plot or themes of the story.

I don't disagree. What we found was that, starting from the lecture notes and a few examples, getting to the point where we could complete the exercises meant that we had to do most of the above figuring-out for ourselves. And because we did it ourselves rather than have it laid out for us, the learning was more established.

You seem to assume that a dry approach necessarily leads to not understanding the context.

TL;DR of the below: people differ in the way they learn, and anybody who disregards the dry approach simply because it doesn't work for the majority, is doing a disservice to someone it does work well for.

I've personally prepared for my high school and university math tests and exams (those consisting of mostly math "problems") by only focusing on the "dry theory" from mostly "dry" textbooks. I understood the context perfectly well, and I had multiple tests and written exams focused on traditional applied math problems where I came up with "novel" approaches by simply putting my dry knowledge to use (as in, approached a problem from theoretical definitions and theorems but completely not in the way they taught it or expected). I wasn't as fast as I would have been if I learned all the tricks of math problem solving vs just going from the dry theory (iow, I'd be getting B grades from very little preparation or even class attendance, but not for getting anything wrong, but just for not having the time to figure out everything).

But most people, teachers and professors included, seem to disregard people like me who are great at applying and seeing context from abstract theory. I never enjoyed practicing math problems just to be fast at math problems, but I very much enjoyed abstract theory building and application: my motivation was never competitive, at least not after 6th grade. For someone like me, it's not just "symbolic manipulation", but actually abstract concept manipulation.

In a way, I was seriously underserved by mathematics classes focused on different types of students than me. And this goes from primary school all the way to university.

So, if the goal is to find students like me, who are likely to excel at pure mathematics and won't have trouble applying it, we actually need a drier approach. Obviously, that's not the goal of primary education, but the same focus leads to less stellar outcomes even at higher levels of education.

What this article proposes is an even larger move in this direction, and people are arguing for it at all levels of education, without ever recognizing that there are people for whom the dry approach might work just as well, even if they are a minority.

> assume that a dry approach necessarily leads to not understanding the context

If you never hear what the purpose and context is of something, you certainly won’t magically infer the details. You might be able to guess some parts in broad strokes, but that speculation is just as likely to be completely wrong. For instance, someone learning about set theory for the first time, even if they are some kind of genius, isn’t going to immediately guess that it was established after a crisis in analysis sparked by Fourier theory, which was itself invented to solve tricky partial differential equations arising in physics. (In case anyone wants to learn about this, see https://www.youtube.com/watch?v=hBcWRZMP6xs)

Which is not to say that the presentation of mathematical topics should be primarily historical, but only that the context in mathematics is fractally deep everywhere you look.

> I understood the context perfectly well,

This is not plausible for anyone’s first course. Even life-long experts don’t understand the context “perfectly well”. We are talking about a subject with infinite depth and interrelation, and centuries of history.

Sure, but with mathematics, nobody is ever starting from scratch. And even the "driest" of educational books has some context in it. Or maybe I was lucky, and I haven't seen really dry books.

Initial topics on numbers and geometry have students already have some intuition on them, usually brought from home (eg. kids can count, add smaller numbers, understand differences between straight lines and circles...).

Similarly, coming into more advanced courses, I already had a bunch of context already at my disposal.

Like everything, it can go to an extreme in either direction, but I am mostly saying that I am comfortable with books that tend to be more on the "dry" end of the spectrum, and I can sometimes figure out many of those "dry theory" applications myself which brings motivation and joy (but certainly not all; lots of math has, as you point out, taken centuries for brilliant people to find solutions and tricks that work, and I don't kid myself that I can figure out all of that in a few months or years, let alone few hours — not without learning specifically of their tricks and solutions that do work). And it's definitely not the most efficient way to advance the science of mathematics — but we are discussing teaching and learning mathematics here: keeping students motivated is at the core of any successful learning experience.

Historical context is wonderful in mathematics because it allows one to really see what the original motivation for building up an abstract system was, and that's the best context.

> even the "driest" of educational books has some context in it

Have you ever tried reading Bourbaki?

Are you claiming that no one can learn by reading Bourbaki,no matter how intelligent and motivated they are?

How did you turn «doesn’t try to include context» into «completely useless»?

Bourbaki has no history, no diagrams, limited motivating discussion. It strives to be entirely axiomatic/formal, and to be organized in a strictly “logical” fashion. Bourbaki claims that intuition and analogy are dangerous/faulty and should be avoided.

That doesn’t mean you can’t learn anything from it.

I haven't, so I'll look it up.

I looked up Bourbaki, my first go was at Algebra (chapters 1 to 3). Seems pretty decent, a bunch of things are obvious (has an introductory section which describes some of the motivation and historical context.

Then starts by defining a law of composition using a function from E x E -> E: all pretty obvious. It even uses the common operators + and . (or no sign) to indicate addition and multiplication, all of which are intuitively clear and easy to make a parallel with what is already familiar. It even explicitely brings up a law of composition not everywhere defined on E for anyone who has not caught that composition needs to be a function on E x E -> E, or rather that it works for all values from E x E. For instance, subtraction on natural numbers is not a law of composition according to this definition.

And straight up on the first page, there are examples of what compositions are available on the set of natural numbers and on subsets of sets.

I am sure it gets hairer as you go along, but this is roughly the type of books I've used decades ago while studying mathematics, and roughly the type of books I enjoyed when properly interested and motivated.

It's only obvious that one needs to go through the Set Theory first as they rely on the terminology introduced there for precise handling of whatever comes here (I still remember most of the terminology, but I don't trust my memory to get all the specifics right).

It is not completely devoid of context and historical perspective, though it presents it in a slightly backwards way (introduction is clear to highlight how parts between asterisks are not necessary for purely logical reading of the text).

Again, and I've said this before, formal mathematics is hard mostly because you have to memorize so much of the new language, and you can't really grasp the context without having grasped the context for what you are building on (eg. if you don't have understanding of functions/mappings, tough luck getting to the grips of algebraic structures).

Switching to Theory of Sets, this is the type of writing that brings me joy. And it certainly concerns itself with context in the pretty longish introduction, attempts to recognize the limits of the language, covers metamathematics and use of simple arithmetic before the foundation for it has been formally laid...

I enjoy that it starts off with defining symbols of a theory, and then assemblies, which is the first I hear of the term, but I can already feel what assemblies will amount to before the example given of an assembly in Theory of Sets — even if the text warns that "the meaning of this expression will become clear as the chapter progresses."

And that's exactly my point. I can enjoy learning from a text like this. I know I am in the minority (I was in the minority in Mathematical Gymnasium and university studies who enjoyed it; I wasn't even remotely the best at solving math problems, because I did not enjoy them), but I am calling for people not to discount this approach for everybody.

> a sufficiently motivated reader could learn it without having to learn it master-apprentice style

if that's the case, I would say they failed.

however, what they accomplished would certainly help jog the memory of somebody who knew the material once upon a time.

maybe it's a bit like looking at a zip file directly and uncompressing the contents on the fly in your head? (something about 'understanding' as a compression scheme)

Yeah, I must say I prefer the math teaching approach of:

1. Explain some problem that is difficult to solve using other techniques. Ideally the problem should be interesting, even if potentially somewhat abstract.

2. Propose a technique to solve it, without requiring full rigor, but that allows for intuitively feeling that the technique is probably valid.

3. Show that the technique also works on other different problems.

4. Show show some flaws (or limits of generality) of the technique. Propose fixes for the flaw, and/or better outline where the technique viable.

5. Now either prove the technique's validity more rigorously (need not be a perfect proof, especially if a full proof requires far more complicated mathematics) or extend this technique to be able to solve more (but eventually coming back to proving the validity).

This sort of approach is has been used in educational videos like some of threeblueonebrown's on youtube, and I've also seen this used as a faux historical development of algebra and high school calculus in "Algebra: the easy way", and "Calculus: the easy way", which did a great job of motivating the development of most of the ideas in those courses. I'm not familiar with similar texts for higher level mathematics (which would have a different tone, since they would be targeting adults, not teens/children), but surely some must exist, right?

This is infinitely better than the all too common higher level math textbook approach of:

1. Here is some unfamiliar theorem, that you might not even really understand, and certainly have no clue of the relevance.

2. Now here is how to prove the theorem. (Which you might be able to follow, but you probably don't really care about right now).

3. Now we finally explain what the theorem is, and hint at (but may fail to show why) it might be useful.

Unfortunately what we typically end up with when learning math is:

1. Explain some problem that is difficult to solve using other techniques. The problem is painfully abstract and completely absurd.

2. Propose a technique to solve it. Never mind how we got to this technique or why it makes sense, just apply this equation and you'll get the answer. Memorize the numbers and symbols, that's all that matters.

3. Show that the technique also works on different (but really the same) problems.

4. Exam time! Here's a problem that looks nothing like the ones you've seen so far, but use the magic technique you memorized! You did memorize it right? Even though you didn't understand it at all?

5. This class is over so you will never use that technique again--even though it's generally useful, but since you never learned how to actually apply it or why it's useful, you'll forget it the day the exam is finished.

6. Math is great! Why don't people like math?!

Fully agreed!

I'm also puzzled why so little math courses seems to involve the actual history of an idea. Often, an idea is presented in a vacuum or - even worse - confusing a formal proof with an explanation of purpose.

In my opinion, that's about as effective for teaching mathematical concepts as teaching vi by giving you a copy of the source code.

I think explaining the historical context of an idea and the problems the authors back then were trying to solve with it would be much more valuable for understanding than just the bare proof.

I found a math book aimed at math grads going into official sub-college teaching positions, and was shocked about the historical context provided. Every idea had 2 or 3 variants that differed in origin. To me it would have made math 2x more fun for a lot of pupils. But none of it made it into our classrooms (lack of time ? a dusty pedagogical approach maybe ..). It's probably vital for most people that don't have their neurons naturally properly aligned to a subject to have various informations and point of views about the problems and goals.

Which book?

It's a french book. "mathematiques au concours de professeur des ecoles" (Hachette editions 2006)

For instance the numbering system chapter mentions non roman numeral symbols, babylonian, egyptian up to indian of course. Then various ways to multiply numbers. As a kid everything was pruned and we only saw one thing.

If you're interested in ring theory, for example, Kummer's 1844 manuscript on "ideal complex numbers" clarified a bunch of undergraduate algebra for me. Overviews:

https://projecteuclid.org/journalArticle/Download?urlid=bams...

https://arxiv.org/pdf/1108.6066.pdf

IME, you will get that information if you get a math degree. You are less likely to get it, unless you read the textbook (that is, not from the teacher) from any K-12 or non-major math classes because that's not their focus (for better or worse). But often the motivational cases for the math are present in high school and non-major math course textbooks. Just not part of the class because that's not what you're going to be evaluated on.