7 ms·

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 o

by neel_k 4y ago

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.

woolion 4y ago
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?
n4r9 4y ago
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.
- jacobolus 4y ago
  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.
  - n4r9 4y ago
    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.
  - necovek 4y ago
    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.
bsedlm 4y ago
> 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)

by neel_k 4y ago

woolion 4y ago

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?

n4r9 4y ago

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.

jacobolus 4y ago
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.
- n4r9 4y ago
  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.
- necovek 4y ago
  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.

bsedlm 4y ago

> 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)