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What never ceases to amaze me when reading ancient mathematics is how much harder everything is without proper notation.

For example, you don't have rational numbers, so you are forced to represent everything as awkward sums. Even computing, say, three fifths of a number is a challenge under that system!

You don't have \pi to represent "ratio of circumference to diameter", so you have to use ad-hoc approximations.

You don't have notation for polynomials, so even basic equations are ad-hoc problems.

Another example: (almost) the entire work of Diophantus, the man diophantine equations are named after, is basically obsoleted by \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

Mind you, it's not that these people were dumb. That they were solving those problems with limited tools is proof enough that they were very smart.

It's like trying to write a program in 80s Basic, at a certain point the incidental complexity is eating you alive.

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> at a certain point the incidental complexity is eating you alive.

I had a similar deep insight when studying both theoretical physics and computer language / compiler design: for some areas of human endeavour, the only limit is the human brain. The machines ceased being the limiting factor a long time ago, but the human brain is essentially unchanged over the timescale of millennia.

In areas like this, incidental complexity is the key limit holding us back.

Once you've reached 100% of the brain's capacity, the only way to "get more out of it" is to waste less of its capacity on non-useful things.

You mentioned pi as a significant advancement. Yes, it is. But tau would be better. Every time I (or anyone) mentions this, there's dozens of people coming out of the woodwork saying that it doesn't matter, that it's just notation, that there's an established standard, blah, blah, blah.

Meanwhile, basically nothing in trigonometry really clicked for me until I saw Vihart's rant about why we should use tau. None of it. Not really, because the pi notation is nonsense, an accident of history, nothing more. It filled my brain with incidental complexity: that factor of two everywhere that made no sense to me.

> …for some areas of human endeavour, the only limit is the human brain. … But tau would be better. … [people dismiss this because “it’s just notation”…]

The case for me of this is current flowing from - to +. Franklin didn’t know better and had a 50/50 chance of getting it right, but happened to get it “backwards”.

A small thing but one that always makes me skip a beat while following a circuit.

> In areas like this, incidental complexity is the key limit holding us back.

> Once you've reached 100% of the brain's capacity, the only way to "get more out of it" is to waste less of its capacity on non-useful things.

Queue modern day social media... It has its uses, but I feel like the brainrot and lessening of attention span it induces (and not to mention everything else like polarization, cult-like behaviours etc) is basically not worth the small amount of benefit it actually provides.

Tau is better for some equations (most of them?) and worse for some others. It would probably be a net positive to adopt it if you discount the huge chore of transitioning during decades. But what this fails to account is that most mathematicians regard 2pi as an important quantity on its own already; it's not a single symbol but it's close enough. So 2pi is expected to appear in many equations and any other value, like 3pi, looks fishy.

Really, the only problem with 2pi is that if you multiply it by a constant, it disappears. so people might write 2*2pi as 4pi. Sometimes you may not want to do that.

Anyway you don't need to fully transition to tau in order to benefit from it. Just like the planck constant has a reduced planck constant (which is probably more useful), and both are used, tau can be already used when it makes sense, without ever fully replacing pi.

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…

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.

Diophantus is credited with inventing some new algebraic notation.

The quadratic formula wasn't interesting to Diophantus because he didn't believe in negative or irrational or imaginary numbers; he considered those "impossible".

> (almost) the entire work of Diophantus, the man diophantine equations are named after, is basically obsoleted by \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

This is nowhere close to an accurate statement, even if we only consider works that have survived to the present (most of his work has been lost). Have you tried to read the work of Diophantus or any modern summary?

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