Physicists measure spin in units of 1/2, but if you double these you get natural numbers.

A particle of spin j and one of spin k can turn into one of spin ℓ only if:

• no one of j, k, ℓ exceeds the sum of the other two;

• j+k+ℓ is even.

See what's going on?

(1/n)

Penrose noticed that these rules are automatic if we think of the numbers j, k, ℓ as numbers of "wires" or "strands".

It's like a particle of spin j is made of j identical things!

He decided that the quantum theory of spin was based on the theory of finite sets.

(2/n)

Penrose wrote some papers and notes on this. They're hard to get, so he let me put some here:

http://math.ucr.edu/home/baez/penrose/

This math led to the theory of "spin networks" in loop quantum gravity. But nobody took much advantage of the connection to finite sets.

(3/n)

Mathematically this connection is called "Schur-Weyl duality":

https://en.wikipedia.org/wiki/Schur%E2%80%93Weyl_duality

The representations of SU(2), which describe spin, come from the representations of Sₙ that permute n identical "bosonic" things!

(4/n, n = 4)