How to play mathematics
aeon.co
aeon.co
Indeed, teaching mathematics without thinking of it as an embodied practice is how you end up with terrible math courses.
You need the intuition.
You need to know the motivation (usually applications in physics/chemistry/CS/business in lower-level mathematics, but even highly abstract mathematics without any known immediate applications usually has a motivation.)
And of course, you also need the abstraction and formality and rigor. The certitude of proof is, after all, the raison d'etre of modern mathematics!
> By thinking about mathematics as performance, we liberate it from the straightjacket of abstraction into which it has been too narrowly confined.
We can also think about this the other way around. By thinking about mathematical objects formally, we free ourselves from the straight-jacket of uncertainty into which the exclusively performative approach inevitably binds us. The advent of formal systems and the development of mathematical theories that grew too hairy for purely intuitive reasoning coincided for a reason.
The most significant and beautiful pieces of mathematics -- and many of the examples in the article -- were discovered and fully understood precisely because we hone our intuition with the rigor of mathematical abstraction.
The blog post linked below comes to mind.
https://terrytao.wordpress.com/career-advice/there%E2%80%99s...
I wish someone had told me this before college. I wasn't prepared psychologically to deal with the realization that I knew absolutely nothing about mathematics. It truly felt as if I was drowning.
It's very similar to how most anyone can catch a ball, but it takes way more to catch the ball on paper.
Math education could be so much better, when you compare school to self-directed learning with knowledgeable peers. I went to a good high school by U.S. standards, and tested into the advanced freshman math class at Caltech, and still had to ask if the roots of a polynomial with real coefficients came in complex-conjugate pairs -- I wasn't sure. (Maybe nowadays with math circles and the web, the frosh are a lot better prepared?)
I think there's also a problem with the culture of math writing undervaluing things like examples and motivating background.
http://worrydream.com/KillMath/ talks about going beyond the limits of paper for media for doing math.
Then when it's time to do math you can use your techniques so avoid letting small details trip you up and let you focus on the bigger picture.
This is a hypothesis of yours. What evidence is there for this? I could argue that there is plenty of evidence that the universe is doing mathematics. That the universe is a game.
I'm interested to hear what you think.
Mathematics is a language to describe the universe, both observable and theoretical.
The universe is more akin to the specification, whereas mathematics is our implementation.
A way to expound, introspect and reach understanding.
I wouldn't say that the universe performs mathematics. We could equally understand it through a different method, perhaps if we had moved towards magic rather than method in the days of alchemy, we might have something as pure as mathematics, but looking vastly differently.
Maybe they mean that the surface of the sea slug is some sort of approximation of hyperbolic space?
On a tangent, but commenting on the article:
Even something as simple as the surface of a sphere has an intrinsic geometry which is non-Euclidean. So it's frankly a bit ridiculous for Wertheim to affect wonderment at the apparent "intelligence" implied by the sea slug's production of a surface with a non-Euclidean intrinsic geometry. It's disingenuous at best. Take the example of a falling rock: it doesn't know anything about elegant 19th C formulations of mechanics. It doesn't need to, but humans nevertheless make leaps of understanding of unchanging phenomena.
I don't remember learning plane geometry in 5th grade. Maybe times have changed