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Comment by jameshart | Hacker News Reader

The question this should inspire: what do we find hard that would be trivial if we developed better notation?

Einstein tensors are a great example of notation as an enabling technology - it’s a mistake to think we have thought of all the notation tricks we might need.

I wonder if, for example, in attempting to reason about neural networks in terms of ‘gradients’ and ‘matrix operations’ we might look, to future mathematicians, like poor ancients struggling to handle arithmetic without a clean notation for rational fractions…

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I think quantum computing is at the start of a notation transition, from using mainly quantum logic circuits to using mainly ZX graphs. It makes things that used to look different obviously the same, so you can easily translate results.

A few examples:

https://arxiv.org/abs/2303.08829 (ZX shows equivalence between approaches that used to be thought of as different)

https://arxiv.org/abs/1704.08670 (ZX shows how to find new ways to implement fault tolerant operations)

https://arxiv.org/abs/2206.12780 (using ZX to build a 4-body parity measurement out of 5 2-body parity measurements instead of 6, and to show the previous constructions are easily related)

The hard part of course is that genuinely new notation (i.e. not just a more convenient way to write the same thing as before) requires genuinely new mathematics to go along with it.

People had been doing derivatives (at least sort of) before Newton and Leibniz, Cardano (sort of) used complex numbers before Euler, algebra before Galois, categories before Mac Lane.

But these people "connected the dots", as it were. It's not merely a trick -- it's a fundamental contribution.

Can’t we view it as new notation creates new mathematics? A placeholder symbol vs an actual number doing the same thing and more (discovery of zero as a number).

The new mathematics at least could be initially tiny, and then the notation provides the leg work for huge forthcoming discoveries.

Can new notation also create new physics, not just mathematics? Thinking about Feynman diagrams. I am neither physicist nor mathematician, so I don't know if this makes sense. Wondering if Feynman diagrams were not just a crisp articulation of existing concepts, but did they also pave the way for new ideas in physics?

Heavyside's vector notation compressed Maxwell's twenty-some component equations into four. Thats elegant for teaching and teeshirts. However when I want to compute them on a computer grid, I use something closer to Maxwell's original equations.

And if you use a formalism accepting general multivectors, you get 1 equation.

The fully expanded coordinate version is based on an arbitrary choice of spacetime coordinates.

Ideally you'd use a computer language/environment with a rich enough basic formalism to let you write the 1 formula in a human-legible way and have your compiler expand out all of the coordinate version as appropriate for the particular problem for the sake of doing floating point arithmetic.

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