Although if you look at most maths textbooks or papers there's a fair bit of English waffle per equation. I guess both have their place.

People definitely could stand to write a lot more comments in their code. And like... yea, textbook style prose, not just re-stating the code in slightly less logical wording.

Welcome to the world of advocating for Literate Programming:

http://literateprogramming.com/

Yes exactly. Or like signposts on a road.

"You came from these few places, you might go to these few places, watch out for these bugbears if you go down that one path."

As somebody that occasionally studies pure math books those can be very, very light on regular English.

That makes them much easier to read though, its so hard to find a specific statement in English compared to math notation since its easier to find a specific symbol than a specific word.

Yes, plain language text to support and translate symbology to concepts facilitates initial comprehension. It's like two ends of a connection negotiating protocols: once agreed upon, communication proceeds using only symbols.

Textbooks aren't just communicating theorems and proofs (which are often just written in formal symbolic language), but also the language required to teach these concepts, why these are important, how these could be used and sometimes even the story behind the discovery of fields.

So this is far from an accurate comparison.

> Textbooks aren't just communicating theorems and proofs

Not even maths papers, which are vehicle for theorem's and proofs, are purely symbolic language and equations. Natural language prose is included when appropriate.

Theorems and proofs are almost never written in formal symbolic language.

My experience in reading computer science papers is almost exactly the opposite of yours: theorems are almost always written in formal symbolic language. Proofs vary more, from brief prose sketching a simple proof to critical components of proofs given symbolically with prose tying it together.

(Uncommonly, some papers - mostly those related to type theory - go so far as to reference hundreds of lines of machine verified symbolic proofs.)

Can you give an example of the type of theorem or proof you're talking about?

Here's one paper covering the derivation of a typed functional LALR(1) parser in which derivations are given explicitly in symbolic language, while proofs are just prose claims that an inductive proof is similar to the derivation:

    https://scholar.google.com/scholar?&q=Hinze%2C%20R.%2C%20Paterson%2C%20R.%3A%20Derivation%20of%20a%20typed%20functional%20LR%20parser%20%282003%29

Here's one for the semantics of the Cedille functional language core in which proofs are given as key components in symbolic language with prose to to tie them together; all theorems, lemmas, etc are given symbolically.

    https://arxiv.org/abs/1806.04709

And here's one introducing dependent intersection types (as used in Cedille) which references formal machine-checked proofs and only provides a sketch of the proof result in prose:

   https://doi.org/10.1109/LICS.2003.1210048

(For the latter, actually finding the machine checked proof might be tricky: I didn't see it overtly cited and I didn't go looking).

Common expressions such as f = O(n) are not formal at all -- the "=" symbol does not represent equality, and the "n" symbol does not represent a number.