Keep in mind it is also true the other way around. Something can be mathematically clear to someone and totally a mystery in code form. Each one has his/her strengths and weaknesses.
For some concepts that can be expressed in both code and math, I prefer the code notation because I can run it, and also make small tweaks and see what happens. For example, I got a better understanding of Löb's theorem [1] by translating the proof into Haskell [2].
If it can be coded, I prefer having both, or implementing the code. It helps in understanding the algorithm behind. But maths is much larger than what can be coded, or is useful in code, so the only thing left is playing with toy examples ("coding" when working with really weird stuff.)
I'd love to see more of APL (and a "larger" set of APL functions, actually) in use. The idea of a notation we could run directly is/was awesome.
Probably true, and I guess if you're a mathematician, you quickly get used the symbols. And I'm not arguing against having those symbols in the first place, its just that some of them have an 19th century feel to them, and do not seem intuitive.
The art of typography and signage really only matured in the 20th century, and I'm certain some of the symbols would look very different if they were designed today. Anything that helps with teaching math and making it appear friendlier is a plus, imho.
I'm not sure what symbols are you hinting at. First I thought it was to Fraktur kind of letters, but obviously this shouldn't be the case, as you point "teaching" as a plus of redesigning them, and Fraktur symbols are used "traditionally" in relatively high level algebra (for some reason some symbols are used more in some realms, for me Fraktur started appearing when talking about complex stuff about ideals). Once you get used to them, it's like a second language, and that's it. I remember reading Feynman used his own symbols for sin, cos and other basic functions (turning them to one-stroke symbols) but he had to give up once he had to talk with other people.
Math symbols are more or less a universal language. Once you know how the symbol appeared, or get used to "reading it right" they are totally natural. I don't see ∂ as a "weird d," I read this as "partial." It wasn't natural at first, but I got used to it, just like I got used to English.
It's like three-letter names in assembly. It's good when you're doing it, but step away from it for a while and you can't remember what the signs mean anymore.
