[hyh1048576]

If you want to see how some very simple notations greatly simplifies some math, check out J. H. Conway's proof of Morley's theorem.

Background: Morley's theorem is a non-trivial theorem in planar euclidean geometry stated in 1899 (first proof appeared 15 years later). The proofs are not easy. One can use complicated trignometry identities to prove it. Even the "simple" proofs are sometimes quite involved.

Conway introduced some notation and almost trivialized it. The notation he introduced was just a* := a + 60 where a is the degree of an angle. No one would believe this notation can do anything good, but with them (and some other insight) Conway can explain the proof in just a few sentences! (One might think anyone who understand that the interior angles of a triangle will always have a sum of 180° can come up with this simple proof, but that just didn't happen for 100 years until Conway revealed it.)

See page 3-6 here: http://thewe.net/math/conway.pdf

[AlexErrant]

Michael Nielsen (with Andy Matuschak) also wrote about how Hindu-Arabic numerals enabled more powerful kinds of thinking (compared with Roman numerals.)

https://numinous.productions/ttft/#how-to-invent-hindu-arabi...

> ...the Hindu-Arabic numerals aren’t just an extraordinary piece of design. They’re also an extraordinary mathematical insight. They involve many non-obvious ideas, if all you know is Roman numerals. Perhaps most remarkably, the meaning of a numeral actually changes, depending on its position within a number. Also remarkable, consider that when we add the numbers 72 and 83 we at some point will likely use 2+3=5; similarly, when we add 27 and 38 we will also use 2+3=5, despite the fact that the meaning of 2 and 3 in the second sum is completely different than in the first sum. In modern user interface terms, the numerals have the same affordances, despite their meaning being very different in the two cases. We take this for granted, but this similarity in behavior is a consequence of deep facts about the number system: commutativity, associativity, and distributivity

[hyh1048576]

Yep. Imagine doing multiplication in Roman numerals :)

[isaacfung]

I think that is partly why LLMs are bad at math and often fail at counting subsequences. Play with the tokenizer and you see long numbers are split into groups of 2 or 3 numbers.

https://huggingface.co/spaces/Xenova/the-tokenizer-playgroun...

[mnky9800n]

Imagine designing and building an aqueduct with Roman numerals.

[alberto_ol]

I think Romans used the abacus to calculate.

https://cacm.acm.org/blogs/blog-cacm/234881-how-did-the-roma...

https://en.wikipedia.org/wiki/Roman_abacus

[mnky9800n]

yes but they didn't use it to write down their plans.

[mthtrw]

You use a counting board rather than paper: https://www.youtube.com/watch?v=UVB27kKAcRs&t=111

The main advantage of Arabic numerals on paper have is that operations are non destructive and you can restarts a calculation if you lose your place. The main disadvantage is memorising the times table and the amount of scratch paper you need.

[vjerancrnjak]

There's also many assumptions in the initial m^2 != 2n^2 proof that many would have a hard time instantly grasping. Problem is stated in three different ways, takes quite a bit of knowledge to instantly see they are the same problem.

The proof stops at 2n-m and m-n, subtly, because that is another smaller case of can two squares fit in a big square. If you do the expansion of 2(m-n)^2 and (2n-m)^2 to see if there's a fit, you see there isn't but the proof does not need to deal with that and to me the reason is very subtle, wordy.

Graph theory proofs feel very similar.

[ChrisKnott]

It's a wonderful proof, thanks for sharing, but the notation is really just syntactic sugar, no? It would have only been slightly clumsier to have "+60" everywhere. The real insight seems more to have done the construction "backwards", starting with the equilateral.

[raincole]

Syntactic sugar is powerful.

There was a time the China accepted calculus, but rejected Leibniz notations.

It looks this like: https://i.imgur.com/foUem6w.png

Edit: Technically this is still Leibniz notation.

[heresie-dabord]

The phrase "syntactic sugar" implies ease, but it's much more than mere "syntactic sugar" when the notation unlocks further information gain and cognitive advantage.

Cheers!

[bmacho]

I like to keep all of my proofs very very very simple. 3 big steps, where each big steps should be intuitively true. Sure sometimes you should internalize the properties of your objects before you go on (big and useful theorems are just properties of the objects), but that's it.

I believe every natural problem has such a simple proof, that's why I use BFS when I try to solve a problem instead of DFS.

[Turneyboy]

A similar thing also happens in the modern formulation of the Stoke's theorem. Once you set up the machinery of differential forms it becomes almost a trviality which is just amazing.