Comment by oefrha | Hacker News Reader

Comment by oefrha | Hacker News Reader

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> I can't remember once being shown the physical problems that drove progress in math until I reached calculus.

Progress in math often (or usually) isn’t driven by physical problems, so there’s that. Just like progress in physics often (or usually) isn’t driven by “real world” problems. Discoveries are often made before applications are found.

As a concrete example, number theory is an incredibly profound and fruitful branch of mathematics. It is also largely “useless” to this day (outside of cryptography, which uses only a small subset of number theory, and isn’t really “physical” to begin with).

Another well-known example is non-Euclidean geometry, which came into being due to the purely intellectual pursuit of the Fifth Postulate, and almost magically found its place in the theory of gravitation long afterwards.

It takes a certain kind of personality to enjoy mathematics as is, even if it seems “useless”.

(Theoretical physicist who double majored in math and physics in college here. Definitely possess the “certain kind of personality”.)

> Progress in math often (or usually) isn’t driven by physical problems

This is true at the largest scale, but the math classes that most people take are directly applicable to (and often were developed partly to solve) physical problems. I, too, found the high school presentation of math dry and boring until I took a good physics class, and I was amazed by how much of the math I learned was directly useful for that subject.

Unfortunately, the dry treatment math gets in high schools also usually fails to highlight the beauty and subtlety of pure math. I didn't realize how much I loved pure math until I took my first proof-based analysis class after junior year of college.

I think we're seriously failing both the applied and pure camps of students with the way we currently present math. The fact that it took me till nearly the end of college to really like it, despite my proclivity towards the abstract and technical, really drove this point home for me.

Practical does not mean interesting. A lot of math is based around accounting and taxes.

How big is this field and how do I split it evenly into N pieces. That’s more or less why Geometry was considered foundational for so long. Carpenters are often used as examples, but they generally avoid anything mathematically complicated instead using simple rules.

> failing both the applied and pure camps

Yes. Neither its use nor truth. Just a bunch of stuff to learn.

Much less could be covered if driven by applications/problems or from axioms, and the rest of the math ecosystem is built around it. So, hard to change.

The pure mathematician is a relatively recent beast though, and most non-graduate students of math (i.e. almost all of them) study mathematics that almost exclusively was developed to solve a problem in the physical world.

Euclid was a pure mathematician.

Euclid used his geometry to solve problems... its practical utility is what made it last for so long.

Almost nothing is known about who Euclid was. Might not have even been a single person.

You might be other things as well, but if you write a textbook on pure mathematics, you are a pure mathematician.

>Progress in math often (or usually) isn’t driven by physical problems, so there’s that. Just like progress in physics often (or usually) isn’t driven by “real world” problems. Discoveries are often made before applications are found.

The use of math however is driven by physical problems for most people; teaching people how to use math (or physics, or chemistry) ought to be the point of most non-grad level courses? Anecdotally, I got more out of my math classes that were through the lens of either a solver e.g. matlab/mathmatica/R or a 'practical' subject like quantum chemistry.

That doesn't work for everyone though. I'm studying pure math and I find most applied math problems very boring. I also like solving puzzles, though. If a problem is little more than knowing which formula to use (with some manipulation) and then punching the values into my calculator then it just feels like I'm doing a computer's job.

So I think if we gear our curriculum too heavily to the applied side of things, we risk alienating some very gifted students who might otherwise go on to become brilliant mathematicians.

I would say 'plug and chug' applied problems don't work for anyone, past the first couple. At risk of anecdote: I like solving puzzles as much as anyone[1] but the purer side wasn't a puzzle to me because there is no purpose, no picture to be pieced together[2].

[1] Back to the original topic, chemistry interestingly inverts its approach with organic chem - a class in puzzle solving that often frustrated premeds trying to memorize their way through.

[2] I don't mean that the problem was open-ended, e.g. research but rather that it didn't exist. Most of my math education hasn't been trying to prove P=NP (or other easier puzzles) but demonstrating specific transformations (e.g. here are the rules for integration, after a trivial treatment of the proof). imo something like an intro to algorithms in highschool or early college would have do more for both the uninterested and the gifted than "integration, but with 1 more dimension!"

That’s disappointing. My math education so far has included the usual calculus and the basics of analysis up to and including multi-variable calculus as well as linear algebra on abstract vector spaces, inner product spaces, and complex vector spaces. Additionally, I’ve delved into combinatorics with enumeration, combinatorial proofs, and generating functions as well as graph theory, abstract algebra including group theory and ring theory, and mathematical logic.

I plan to take Euclidean geometry, real and complex analysis, fields and Galois theory, and eventually measure theory as well as differential geometry and maybe commutative algebra and representation theory.

The problems I’ve studied in all of these courses have been varied, interesting, and highly challenging. Far more entertaining than any problem I encountered in high school. But not all of the material was too complex for a high school student to understand. I could definitely teach a high school class in graph theory and I bet the students would have a lot of fun with it. Despite it being a beautiful and sometimes very abstract field, it can be applied to computer science as well as games and puzzles of all sorts. Yeah, it’s not useful to any of the hard engineering disciplines but it’s far more entertaining than grinding integrals all day.

Gerry Sussman has a really good talk touching on [2]. It was surprising to me that his real interests are in classical mechanics but he uses computation as a tool to convey the ideas.

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