The Problem With Math Textbooks - Grant Sanderson @3blue1brown

[Dere]

My favourite ever maths module at uni was where the lecturer started the first lecture by showing us a really fascinating theorem and its applications and saying “our aim over the course is to prove this”. Then each lecture we would learn something new that was needed for the proof and he would show us how it brought us closer.

[sfs]

"The most efficient logical order for a subject is usually different from the best psychological order in which to learn it. Much mathematical writing is based too closely on the logical order of deduction a subject, with too many definitions without, or before, the examples which motivate them, and too many answers before, or without, the questions they address." - William Thurston

[gregegan]

Great point, and relevant for most discussions, not just math. People discuss things to far up their individual axiomatic trees. For any meaningful discussion, it needs to be the axioms under inspection first to determine alignment. Of course consideration has to be given for the motivating factors behind the discussion. Euclid set a pretty good standard.

[Martin-uomp]

That’s exactly what was on my mind when I first learned about linear algebra. We spend 2-3 months learning axioms and deducing something from them before they gave us any kind of motivation why these things are interesting. Just merely before the end of the semester they told us about the page rank algorithm, which was cool but that was the only real application of these things outside of pure math. Now, a year later having seen very much applications of LA, I totally understand why the things we learned there are so useful and I wish I had learned them more carefully

[Zutraxi]

Agree textbooks so very often forget to excite people with the problem we are trying to solve.
Just giving conclusions when you need to understand at the core of your being, isn't very satisfying

[FractalMannequin]

Yeah, completely agree and I've been staying the same things for a long time. But the sad truth is that writing a list of theorems and definitions is a lot easier than writing narrative with theorems and definitions as parts of the story.

[NicleT]

That was the thing I found most frustrating in Maths in school, I wanted to know where the axioms were coming from. A bit of history or a simple demonstration to make us understand and accept these axioms as bases, would have been a lot of energy saved with way more initiative to learn.

[Kurtlane]

This is why it is a very good idea to begin by studying Euclidian geometry in the 5th or 6th grade of school. And studying it thoroughly, first the axioms, and then theorems based on the axioms, and then theorems based on those theorems. With proving theorems as part of classwork or homework. Until at least some of the whole building of geometry starts coming into shape.

In Euclidian geometry the number of axioms is at a minimum.

And later (probably beyond high school) you can delete just one axiom from that basic set, and discover the world of non-Euclidian geometry.

[JJ-trrx]

I've always said, getting the idea for the intuition behind an axiom is perhaps the most important step. The rest is just mechanical.

[mtarnowski]

Very well said; like many people, I found group axioms quite arbitrary when I saw them first. It was only through Cayley’s theorem when it started to make sense, because permutations are not as arbitrary. That's why I believe group theory should be dobe after exhausting the topic of function composition, especially iterations. I wrote it once to Grant Sanderson and he probably agreed.

[sfs]

I have great admiration for this man, he is the reason I am alive

[evangelion]

Talking abou books, those three books from Stein he has behind him are pure gold. PURE GOLD!

[Xenogyst]

I had a disappointing moment in college when I was in calc 3 learning some particular multidimensional formula and I raised my hand and asked "so what can we do with this?"
The professor looked at me, almost puzzled by the question, and said something like "well, this is another tool in our tool box for solving problems in this space".
And I had the realization that the only utility he sees with this is that we are learning math... to solve more math. Like math solutioning is just an end into its own sake, isnt it cool that we learn more math things?
I felt this strong disconnect, one which I think is born out when wondering about particular mathematical axioms.
Because, practically, the math we use is almost entirely born from utility. And advanced math was largely born out of the desire to understand physics and the natural world.
When Newton came up with his version of calculus, he wasnt just navel gazing about the abstract glory of math. He was trying to describe the motion of celestial bodies, and came into a need for a set of describable rules for that task.
Part of why we arrived at the mathematics we did was because we have tried to explain the physical world with it, and the set of axioms we picked are not just random, but are chosen to some degree because they work well for that task (and possibly contain some interesting truth of the universe baked into them).

[kennethflorek]

You can't just assume any bunch of things whatever and know they are consistent. To check a hunch, I once tried assuming something plausible was true to see how useful it really would be, and it was tremendously useful. After just a few days I was able to deduce a result not only in one way, but two quite different ways. The trouble came when using one carefully chosen instance you got two different numbers, following the two different methods. OK, so then I knew my assumption was false, however useful it would be. Kinda too bad. It was fun while it lasted. 😂

It is perfectly possible to have inconsistent axioms.

[daviddelaney]

I find it very enjoyable to be able to take time to read and think about the material presented in math texts.
This approach differs from general coursework where there is time pressure to complete assignments. For me the time pressure of most math courses is too much to handle and I will fall behind rather quickly. It is a little like Lucy and the chocolate on the conveyor belt episode if you may be aware.

Plus there are a lot of side bar issues that are ignored during regular coursework which can be dealt with in an independent study approach.

I set a 10 year goal for myself of understanding Maxwell's Equations. Starting with the basics. I am at year 2.5 at this time. Sidebar topics that have arose include study of the Python language, and Ham Radio, and FFTs.

Fun journey anyway.

[johnlacey]

I could never stand those math courses in college that consisted of nothing but theorem/proof; theorem/proof...

[ItsNotAllRainbows_and_Unicorns]

That was my problem when I was completed my Ba in mathematics. I said to myself a number of times, " and where and how does this apply?" I got tired of the theorems rather quickly. I am more of an applied type of person.

[xandercorp]

That's what math history courses are for. Physics history is similarly illuminating.

[PaulPower]

I never got round to it when I was delivering talks to FE students, but a talk I've always wanted to write is "Minus times minus makes a plus, the reason for this we shall *now* discuss" and just start from the axioms of arithmetic and work up to showing why multiplying two negative numbers makes a positive.

Fairly basic topic in the grand scheme of things, but it would definitely be a motivating result for teenagers.

[sereysothe.a]

my measure theory class was basically a semester long build-up to the radon-nikodym theorem, which I feel like they could've just told me from the start