
How to play mathematics - Hooke
https://aeon.co/essays/theres-more-maths-in-slugs-and-corals-than-we-can-think-of
======
throwaway729
_> 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.

The blog post linked below comes to mind.

[https://terrytao.wordpress.com/career-
advice/there%E2%80%99s...](https://terrytao.wordpress.com/career-
advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/)

~~~
nur0n
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!

~~~
smhost
> 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.

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aloupi
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?

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Vexs
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.

~~~
abecedarius
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.

[http://worrydream.com/KillMath/](http://worrydream.com/KillMath/) talks about
going beyond the limits of paper for media for doing math.

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valgor
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.

~~~
mathgenius
> The universe is not "doing" mathematics

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.

~~~
shakna
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.

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soVeryTired
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?

[1]
[https://www.math.utah.edu/~treiberg/Hilbert/Hilber.pdf](https://www.math.utah.edu/~treiberg/Hilbert/Hilber.pdf)

~~~
theoh
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.

~~~
soVeryTired
That's a fair point. I guess you can reasonably describe a wall as 'planar'
without requiring that it extends out to infinity too.

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jraines
Reminds me of Tegmark's Mathematical Universe Hypothesis, which would say (I
think) that entities can _do_ math or _perform_ math because they _are_ math.
[https://en.wikipedia.org/wiki/Mathematical_universe_hypothes...](https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis)

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paulpauper
_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

~~~
mathgenius
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.

