
Physics Found a Geometric Structure for Math to Play With - MindGods
https://www.quantamagazine.org/how-physics-gifted-math-with-a-new-geometry-20200729/
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Koshkin
_complex numbers. These numbers involve i, the square root of −1, and they
take the form a + bi, where a is the real part and b is the imaginary part._

This is the shortest explanation I've seen of what a complex number is!
(Reminds me of when a piece on some advancement in quantum physics in a pop-
science magazine tries to explain to the reader what an atom is.)

Edit: OK, let me try then to explain what homology is: it is a way of
discovering (and counting) interesting features of the space in consideration
(like holes in a topological space or fixed points of a symplectic
transformation) by looking at a series of algebraic structures, such as
Abelian groups or vector spaces, which one can often conjure up from objects
of the space (points, paths, etc.)

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golergka
It's short, but I don't think that it's very helpful for someone who hears
about it for the first time. When you see

a + bi

your first reaction, as a 12 year old, is to look at the plus sign and wonder
- what's the actual result here is? How do these things add up?

I think that explaining it through geometry and vectors first is far more
natural to begin with, and gives a solid intuition of what happens with
complex numbers when they go through simpler operations.

~~~
Koshkin
This is perhaps true for the complex numbers, but I am not sure if any field
extension can be explained geometrically in a sensible way. (For example,
think about extending the rationals by adding the square root of 2.)

~~~
golergka
That's why you first develop a very good intuition of what complex numbers
really are, understand Euler's formula (and probably asked to come up with a
proof for it), and only then move forward.

In mathematics, it's really easy and dangerous to think that you understand a
concept only because you can memorize it's definitions and properties; any
math education should fight this pseudo-understanding and make sure that you
really grok it.

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Koshkin
What you are saying is true of course, except nobody in their right mind
should think that algebra lacks intuition and that learning it consists of
rote memorization.

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kens
If you're interested in William Ronan Hamilton's life, I recommend this
YouTube video, a clever takeoff on _Hamilton_ :
[https://youtu.be/SZXHoWwBcDc](https://youtu.be/SZXHoWwBcDc)

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wwarner
That was a great read!

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andi999
'Relatively new...', well at least 35 years..

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alexpetralia
I know Eric Weinstein has introduced a (fairly undocumented) "theory of
everything" which pins the fundamental structure of the universe to be
geometric in nature (11 dimensions). Does anyone know if this is at all
related? Or perhaps the geometric approach in physics is fairly typical?

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Koshkin
I do not believe it's related. Symplectic spaces are useful in classical
mechanics. Not sure about "typical", but yes, geometry in general is the most
important part of the conceptual mathematical framework of modern theoretical
physics.

~~~
alexpetralia
Good to know - thanks for clarifying.

