Comment by sethhochberg | Hacker News Reader

Comment by sethhochberg | Hacker News Reader

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As a person who briefly pursued a degree in a traditional engineering discipline before finding my way back to the arts (music) and then eventually into a software career, I think the difference - at least for me - is that with the arts I've never gotten the impression that the fundamentals stop being interesting if they're still applied well.

To phrase it another way: practicing scales can be boring. But playing a walking bassline is a whole lot more similar to scales than it is anything else, and one of the most fun things to do with an instrument is to jam with other people using the fundamentals you all share.

I never felt the same way with math, because I never felt like math allowed me to put anything unique and of my own into the mix. Learning the same proof from a book that a million other kids taking geometry learned was much less interesting to me than transcribing a solo some musician played so that I could try to riff on the style they put into the world.

Sure, yeah, there's rules in music theory too. You're expected to learn them, and get graded on them in tests if you study music academically. But I don't know of any famous mathematician who ever said something like the various Duke Ellington quotes around "if it sounds good, it IS good". In the art world, similar riffs on an existing idea are everything. In math, they're just... wrong. Or at least thats the understanding I had as a student which led me to only care about the parts of math that seemed to be directly applicable to me, or really intuitive.

As a mathematician, I would claim that math at its best is exactly about riffing and jamming on shared fundamentals and adding your own insights. But it is often misunderstood what these fundamentals are: not the proofs or the formulas, but the elements of logical reasoning that make it possible to say with absolute certainty that given some conditions, some statement must be absolutely true.

A riff on a proof in geometry isn't changing a few of the letters around to see if still works - it might instead be about changing the assumptions. In plane geometry, the angles of a triangle sum to 180 degrees. But what if we are not in the plane, but on the surface of a sphere, such as the earth? Does a triangle connecting the north pole to two points on the equator still have the sum 180 degrees? If not, can we prove something else about it? In the context of the original article, it might be something like "Can we use any parts of our proof about pentagons (5-cycles) for some other shapes? What about hexagons (6-cycles)? Or is there even any insight we can generalize so that it becomes a statement about cycles of any length?"

Sadly, this is way too seldom the way math is taught. I didn't enjoy the endless calculations of long division in grade school, or the memorization of different tricks for solving trigonometric integrals in college, either.

For a famous mathematician quote, how about Paul Erdös concept of "a proof from the Book" - said about math proofs that are so perfect and clear that they must be in God's celestial collection. Often, there is more than one way to prove something true - checking every example by brute force would be the most extreme - but sometimes you can discover an elegant argument that just convinces everyone who reads it that it simply must be true. That could also be thought of as riffing on a proof: Ok, you convinced me that this is true, after many boring pages of calculations - can I find a simpler way to convince myself of the same thing, and improve both our understandings?" Quanta magazine had a nice article about this as well: https://www.quantamagazine.org/gunter-ziegler-and-martin-aig...

Mathematics is indeed not taught in a proof-based manner, but I would also argue that it takes a lot of training to teach proof-based maths to primary grade students. It is far harder than teaching maths to undergrads or high schoolers. Which is the real reason it doesn't happen in schools.

> I didn't enjoy the endless calculations of long division in grade school...

Sure, everyone has their own cup of coffee. But drummers, for instance, practice the same beats over and over again for dozens of hours to perfect them. Not very different from doing long division over and over again.

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