I remember learning this in my algorithms class many years ago. We used the Comen et al Intro to Algorithms book, which goes a long way to providing a proof for every algorithm discussed: https://mitpress.mit.edu/books/introduction-algorithms-third...

Wikipedia does mention it though:

  In mathematics and computer science, an algorithm is *a finite sequence* of well-defined, computer-implementable instructions, typically to solve a class of specific problems or to perform a computation.

"Finite sequence", meaning it terminates. The wiki citation (The Definitive Glossary of Higher Mathematical Jargon) provides a little more: "A finite series of well-defined, computer-implementable instructions to solve a specific set of computable problems. It takes a finite amount of initial input(s), processes them unambiguously at each operation, before returning its outputs within a finite amount of time." https://mathvault.ca/math-glossary/#algo

So they stipulate finite input and output and finite runtime. This is in contrast to something like a webserver or OS, which has a potentially unbounded number of inputs, and is expected to run effectively forever.

I mean think about it, what does an OS kernel look like in its most distilled form? It's essentially just an entry point to an infinite loop. Same as games, where they're called "event loops" in that domain.

The way I think about it is this: if computers and programming languages and software didn't exist, algorithms still would. I don't think you can say the same about e.q. Quake. Quake isn't a mathematical truth even though it uses them to work, kind of like how engineers use physics to build a bridge, but the bridge itself isn't physics.