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Comment by qsi | Hacker News Reader

Without clever innovations in notation, a lot of math (and physics) would be utterly intractable. For instance, without Einstein's notation hack [1], using and manipulating tensors is extremely painful. Arguably, General Relativity would not have been possible without the clarity of the Einstein notation.

Also long-established notation like integrals and differentials were once new and innovative, and paved the way for new discoveries.

[1] https://en.wikipedia.org/wiki/Einstein_notation

(edit: capitalize GR, grammar)

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>using and manipulating tensors [without summing over repeated indices] is extremely painful

It's a good notation, but writing a few sigmas in front of every equation is at worst a constant-factor overhead. I know of a text that does it that way in a (misguided?) attempt to be more approachable to nonspecialists.

Do you think mathematical innovation is dominantly enabled by clever design?

Oh most definitely. Our conventional algebraic notation is only a few hundred years old. Things that we have children do in math class now were the domain of professionals 500 years ago. Imagine doing basic algebra without our familiar single-letter variables and the notation for addition, subtraction, multiplication and division you learned in grade school.

Leibniz's notation opened up Calculus in ways that Newton's geometric analysis did not (which is why high school students are writing ∫x²dx and not doing a geometric analysis of that expression). This is also why programmers are looking for the bright shiny language that will let them express what they really mean in their code without ambiguity. I think that we might still be looking for the algebraic notation or Leibniz notation for programming that will make everything completely obvious and will make future generations look back at our C, Python, Rust, et al and marvel that we were able to do anything.

Let’s not forget Robert Recorde’s[1] sign for equality, a pair of parallel horizontal line segments. He revolutionized notation. It’s hard to overstate the importance of the equation.

[1] https://en.m.wikipedia.org/wiki/Robert_Recorde

https://blog.plover.com/math/recorde.html

Really cool post, thanks for sharing!

I wonder if mathematics would have been pioneered in Asia if they had arabic numerals instead of abacuses.

They had rod numerals.

APL? That's a very math-like notation if I recall.

Very math-like, yes, but most programming isn’t really math or rather, the requirements for the program aren’t, and that’s what you want to stay close to in the code.

APL was not originally a programming language (pardon the pun) but as a notation for array mathematics.

https://en.wikipedia.org/wiki/APL_(programming_language)

"A mathematical notation for manipulating arrays was developed by Kenneth E. Iverson, starting in 1957 at Harvard University. In 1960, he began work for IBM where he developed this notation with Adin Falkoff and published it in his book A Programming Language in 1962."

It didn't become an actual (usable) programming language until 1963.

I was reading a thread this morning that categorized programming languages into "lisps" and "fortrans" based on how much you were directing an abstract machine vs a real hardware machine and I think this is a similar idea

I wonder where forths sit on that spectrum.

The thread: https://transmom.love/@elilla/108157666333644028

> I think that we might still be looking for the algebraic notation or Leibniz notation for programming that will make everything completely obvious and will make future generations look back at our C, Python, Rust, et al and marvel that we were able to do anything.

I used to think this too, and spent years looking for that notation, but now I think I was wrong. I think any gains now from notational inventions will be dwarfed by further gains from scaling up neural nets. I think we've 'been living in Symbolia but now are in the age of Weightville.

In my opinion, "dominant" is too strong of a characterization.

I often see Einstein notation as APL vs C :)

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