Hacker News Reader

Comment by dboreham | Hacker News Reader

Something I would add :

Find out where the mathematics you're learning comes from : who first developed it? what problems where they trying to solve? why were they trying to solve those problems? what problems does it solve for us today?

In my mathematical education I noticed that I had a more pleasant time and felt more motivated to learn the material when the teacher gave us this kind of background story. Since most teachers don't do so today, the student typically needs to get on the Internet to do their own research.

View on HN

This happened almost through all my formal education. The knowledge was just presented, as something that was obvious. There was very little discussion of motivation, or what problems was person/people trying to solve. I understand that there is so much knowledge to cover during high school and college, but IMHO, this causes students to develop mindset, that you either see solution instantly, or you are just not smart enough to crack the problem.

I think Bourbaki are partly to blame for this state of affairs. Vladimir Arnold had a lot to say on the subject: he was a big fan of keeping the intuition and motivation for development in the subject clear.

At times he went a bit overboard, but he makes some valid points in this lecture: https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html

Got to love a piece that begins "Mathematics is a part of physics" !

i realize the entire speech is a troll, but it successfully made me very angry. i'm not a physicist, and have no interest in physics, yet I do need mathematics.

so it's always been irritating to me how much introductory or "applied" texts bias themselves towards physics and engineering. like how a lot of multivariate calculus/introductory real analysis texts pretty much limit themselves exclusively to three dimensions and teach notation to match.

it even plagues more advanced books; i'm struggling through Villani's text on optimal transport and nearly every example is from physics even though the theory was developed for economics and has extremely broad applications across many disciplines. half of the "motivating examples" i can't even make sense of what they're talking about.

as much as I struggle with the hyper-abstract bourbaki-style stuff, at least I have a hope of understanding it without needing to dip into an irrelevant discipline where i have no background or historical context.

As a former physicist it also made me angry that they used geometry, when teaching us about multi-dimensional algebra. Professor would tell us that vectors are orthogonal to each other, but not telling us what it actually means is, that changing one of them doesn't affect other. I mean, yes, in hindsight, this information was there, however, when you are just starting to learn something, you have a problem filtering, what is really important and what not.

And for example, a real revelation with regard to infinity came, when I read (or heard somewhere, it was long time ago) that infinity can sometimes be few millimeters or even micrometers. Up until then I always imagined some really large number, but at that point I realized, you need to put problem before you need to put problem before you into perspective.

I don't know if I can agree with that. I bought every book Serge Lang produced. He was a member of Bourbaki. His Linear Algebra book is far better at providing intuition and motivations than the rest. Yet it is also rigorous. Lang is #1 for Mathematics books in my world.

Fair - I don't doubt that there are some great educators in the Bourbaki school. But as Arnold mentions in his lecture I do think mathematics lost something when it embraced formalism so fully.

For me, the definition "A group is a set of transformations on an object such that..." is so much more enlightening than "A group is a set G together with a binary operation * such that..."

Only yesterday I was trying to learn about differential forms. Most of the notes I found online introduce the wedge product in a deeply unhelpful way, by listing some axioms that it satisfies and deducing results from the axioms. It takes hours of work to understand why those specific axioms were chosen. For me - and maybe I'm wrong here - that's Bourbaki's formalist approach in a nutshell.

It was only when I found Terry Tao's notes [0] and Dan Piponi's notes [1] that I could actually see the use of differential forms. It's an unfortunate state of affairs for the discipline that in order to learn about X, you have to google "X intuition", since it's not given to you as a matter of course.

[0] https://terrytao.wordpress.com/2007/12/25/pcm-article-differ... [1] https://github.com/dpiponi/forms/raw/master/forms.pdf

I totally agree. A lot of times you have to search for intuition rather than it being provided. Rote memorization is useless because mathematical research requires the intuition to give your mind the right direction to go in.

I'm not sure it really takes much time or effort to convey enough of the background: I still remember my high school math teacher telling the class, in passing, that Galois was killed in a duel at age 20.

30+ years later that remark is still having an effect in that when I took my kids to the top of the tour montparnasse I made them look down at the cemetery to point out that Galois is down there (somewhere).

> I still remember my high school math teacher telling the class, in passing, that Galois was killed in a duel at age 20.

I think that's actually an example of the problem. Galois's politics really had nothing to do with the problem he was trying to solve. Historical features are thrown in as random info with no point. The point of making mathematicians real people is so that students can say "Oh! That's how they came up with that problem! That's why they solved it this way! I could have done that!"

Rather instead, we further mythologize the individuals. You don't need to join radical political organizations and get yourself killed in your 20s to be a great mathematician.

...and in Austria the education was almost the exact other way round - the historic context and the motivations and problems causing the search for a solution was given great importance.

This is a really good point I think! Up until the last few months, I've avoided doing any math with my brain that I'm not required to do (so, no brain math since college). Just recently I've been picking up some books about the history and philosophy of math like Fermat's Enigma and Everything and More (I'm a sucker for DFW, so this was a no-brainer when math was back on my radar).

Now every evening I find myself saying out loud, "why the hell didn't the math teachers explain it like this." For example, we all learn about the Pythagorean Theorem, but why didn't we learn about the Pythagorean Brotherhood[1] a little bit? And feel just a little bit of the excitement that Pythaogoras did when he found an exposed wire of the universe and figured out how to work it.

Fun fact: Pythagoras had to pay his first pupil to listen to him. No one gave a shit, probably not too different than how kids don't give a shit today. But the kids we're teaching don't even get paid, they're told their true reward is something intangible or some vague math-fu. All this while they have a black hole for their mind in their pocket where at least the fall is comfy.

[1]: https://en.wikipedia.org/wiki/Pythagoreanism

I am surprised you would say that about David Foster Wallace. I went to school with him from kindergarten until 6th grade and 10th through 12th grade, in Urbana, Illinois. Math was never his strong point. However he wrote manic funny stories that would make you laugh so hard you would start to cry. That is why you have to limit yourself to 20 Pages a day at most.

Well I think the reason I like his math book is because it's his typical style on a different topic, so it's a more relaxed read and not something like a hard math book. The first words of the book are something like "Unfortunately this is an introduction you actually have to read" which if you've read much DFW you just kind of settle in right away.

He does mention in the book that the only math classes he had any success with were AP classes. I think it's dedicated to his teacher. But really it makes no attempt to disguise that it's mostly the math angle of his uneasiness with the world and yep I agree with you on the 20 pages a day.

> Find out where the mathematics you're learning comes from : who first developed it? what problems where they trying to solve? why were they trying to solve those problems? what problems does it solve for us today?

For some of us, this is incredibly useful advice re: math study habits. I also figured out that once I had the context and history, it was much easier for me to learn and apply.

The 'Why' lends to the 'How'. I agree. I usually like to know why even in things that are not math before attempting to apply or understand them.

I had such a problem with this. At my college, the professor I had for my entire calc sequence once said, in response to a question about applications, "My degree is in pure math. I don't care how it's applied."

Thankfully(?) I was taking physics at the same time, so I did get to see how double and triple integrals and differential equations could be applied. But it frustrated me to no end that the person who was driving my math education cared so little about helping us understand it.

That's a poor professor of calculus. They should have been teaching real analysis instead. (In pure math, "Real" means "divorced from reality") Maybe a fine mathematician.

I think the real means real numbers

Annoyingly they've also laid claim to 'proper' and 'normal'.

Yes, exactly. Real numbers aren't real.

Im not sure that real in the context of math means anything other than the real numbers or (at a stretch) existence. I agree with you. And no one talks about this but a priori i dont see why even the natural numbers have anything to do with reality either. "Counting" some objects seemingly would require a notion of the object and some way to determine whether each part of reality was or was not the object which already seems fairly abstract and removed from the world. I think certain parts of math just seem more intuitive to certain people

Do you know any books or websites that provide this information? I also feel more inspired and motivated when I know what problem they were trying to solve with that particular piece of mathematics.

Reply on news.ycombinator.com