> Mathematics need not be taught as an abstraction, it can be approached as an embodied practice, like learning a musical instrument.
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.
Thanks for the link, I try to study math on my own, but it easy to lose sight of the big picture without a mentor. I found high quality advice in the blog. Cheers!
> The transition from the first stage to the second is well known to be rather traumatic.
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.
Forgive me if i'm wrong but i think that the essence of this article goes back to (let's wear our sicp hats) the difference of functions and procedures. Procedures are performed intrinsically a lot of times unaware of their functional view. Both can produce the same results.
As for the article itself i find it quite annoying cause it just tries to shove so many things to the reader that it obscures the essence. Is it really needed to go from sea slugs to neurons to hyperbolic geometry to atomic theory and the solution of the Schroedinger equations to holograms and Fourier transforms to make your point?
I've seen many articles like this, and they're always wonderful to read, as they bring up so many interesting interactions between the world and mathematics- however, I remember back to struggling through calculus as kind of an antithesis of sorts- I understood how calculus worked at a more "artistic" level, the interaction of graphs, shapes, and how they all collected together, however, to be able to actually do the operations required of me involved a long series of rather troublesome bits of mat that still bother me to this day!
It's very similar to how most anyone can catch a ball, but it takes way more to catch the ball on paper.
I read somewhere that people catch a fly ball in the same way as that bird in the article caught its prey -- by feedback control, not by calculating the trajectory even subconsciously. (Citation needed, yeah.)
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.
I don't think your analogy is good. The troubles of actually doing the operations but no trouble of what's happening at the artistic level is coming from your missing a few key steps at level of nuts and bolts... it is more similar to how you can picture in your head a beautiful dog... But when you try to convert that image in your head to pencil you get stuck... Knowing some artistic techniques may help (perspectives, shading) Practicing mathematics exercises would be like practicing your technique ... It takes some time to develop the expertise.
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.
The universe is not mathematics. Mathematics is a game. When we talk about the universe (physics, chemistry, etc.) we use mathematics as a language. The universe is not "doing" mathematics, but when we observe the universe we use mathematics to describe what we are seeing to each other.
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 think along what I think might be similar lines to the parent.
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.
I'm not a geometer but I thought I remembered a theorem that said they hyperbolic plane can't be embedded in euclidean 3-space [1] (I can't really remember the difference between an embedding and an immersion, so I could be wrong).
Maybe they mean that the surface of the sea slug is some sort of approximation of hyperbolic space?
The surface has negative curvature, which is all it takes to create a hyperbolic geometry. It doesn't have to extend to infinity and include the whole of hyperbolic space to exemplify the geometry. So it's hyperbolic, but not the whole 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.
Ask any fifth-grader what the angles of a triangle add up to, and she’ll say: ‘180 degrees’. That isn’t true on a hyperbolic surface. Ask our fifth-grader what’s the circumference of a circle and she’ll say: ‘2π times the radius’.
I don't remember learning plane geometry in 5th grade. Maybe times have changed
I remember my sixth grade teacher getting all the students to make triangles and measure the total of the inside angles. I was so annoyed by my classmates all getting near to 180 degrees, they just seemed like a bunch of boring conformists. I went out of my way to construct a triangle with 150 degrees internal angle, which I was quite proud of at the time. And thus began my career as a mathematical weirdo.
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...