It is hard to answer because of exactly what ndriscoll said[0]
> It's like trying to argue about the distinction between U(1), the complex numbers with magnitude 1, and the unit circle, and getting upset when the mathematicians say "those are 3 names for the same thing". Or saying that writing C is programming but writing in a functional language like Scala or Haskell (or Lean) is not.
As ndriscoll suggests, it is tautological. I mean look at what I said. I really need you to hear it. I said that coding is math. So what I hear is "How programming languages helps with programming languages?" Why are you expecting me to hear anything different?
> What math says about zero based indexes?
Start at 0? Start at 1? Who cares, it is the same thing. The natural numbers, non-negative integers, integers, even integers, who cares? They're the same thing. And who cares about indexing at 0 or 1 in programming? That's always been a silly argument that's inconsequential.
> How do you prevent off by one errors?
By not being off by one? What's the question? Like being confused about if you start at 0 or start at 1 and how to get the right bound? It is a shift from one to the other, but they are isomorphic. We can perfectly map. But I really don't get the question. You can formalize these relationships with equations you know. I know it isn't "cool" but you can grab a pen and paper (or a whiteboard) and write down your program structure if you are often falling for these mistakes. This seems more about the difficulties of keeping track of a lot of things in your head all at once.
> How do you prevent buffer overflows?
By not going over your bounds? I'm so confused. I mean you are asking something like "if f(x) = inf when x > 10, how does math help you prevent the output of the function from being infinite?"
Maybe what will help is seeing what some of the Programming Languages people do and why they like Haskell[1].
Or maybe check out Bartosz Milewski[2,3]. His blog[2], is titled "Bartosz Milewski's Programming Cafe: Category Theory, Haskell, Concurrency, C++". It may look very mathy, and you'd be right(!), but it is all about programming! Go check out his Category Theory Course[3], it is _for programmers_.
Don't trust me, go look at papers published in programming language conferences [4]. You'll find plenty of papers that are VERY mathy as well as plenty that are not. It really depends on the topic and what is the best language for the problems they're solving. But you'll certainly find some of the answers you're looking for.
Seriously, don't trust me, verify these things yourself. Google them. Ask an LLM. I don't know what to tell you because these are verifiable things (i.e. my claims are falsifiable!). The only thing you need to do is look.
Nonmathematical problems are difficult to answer when you try to find mathematical answers. It should be obvious, why it doesn't work.
Mentioning Scala is ironic, it's very light on math spik and to begin with was created to unify object oriented with functional programming, which is mathematically meaningless, because both are Turing complete and thus equivalent, tautological.
>So what I hear is "How programming languages helps with programming languages?"
Oh, right, in mathematics axiom is argument, but in reality it isn't. In programming you should assert what you assume, otherwise your assumptions can be wrong due to divergence from reality, but there no reality in mathematics, only fantasy, so you can't understand this with mathematics alone.
Maybe what will help is seeing what some of the Programming Languages people do and why they like Haskell[1].
Or maybe check out Bartosz Milewski[2,3]. His blog[2], is titled "Bartosz Milewski's Programming Cafe: Category Theory, Haskell, Concurrency, C++". It may look very mathy, and you'd be right(!), but it is all about programming! Go check out his Category Theory Course[3], it is _for programmers_.
Don't trust me, go look at papers published in programming language conferences [4]. You'll find plenty of papers that are VERY mathy as well as plenty that are not. It really depends on the topic and what is the best language for the problems they're solving. But you'll certainly find some of the answers you're looking for.
Seriously, don't trust me, verify these things yourself. Google them. Ask an LLM. I don't know what to tell you because these are verifiable things (i.e. my claims are falsifiable!). The only thing you need to do is look.
[0] https://news.ycombinator.com/item?id=43882197
[1] https://excessivelyadequate.com/posts/isomorphisms.html
[2] https://bartoszmilewski.com/
[3] https://www.youtube.com/watch?v=I8LbkfSSR58&list=PLbgaMIhjbm...
[4] https://www.sigplan.org/Conferences/