Reddit reviews Synthetic Philosophy of Contemporary Mathematics

Well, Penelope Maddy has a highly influential paper related to justifying the axioms of ZFC. She does offer a good review of why people pushed for certain set-theoretical axioms historically. Caveat: you need to know some set theory to appreciate it.

What could justification for the rules^1 of logic even look like? While not directly tackling your question, there's lots of relevant information scattered in Fernando Zalamea's book. Caveat: it won't make much sense unless you already know a wide range of higher level mathematics.

One should look at proof theory for a possible justification for the rules of logic. As hinted at by Zalamea, the most important work in that area was done by Jean-Yves Girard. His book, The Blind Spot, disguised as a textbook of proof theory, provides an introduction into how "geometric" considerations could ultimately illuminate the rules of logic. Caveat: to read it, you absolutely have to know proof theory, at least at the level of a beginning graduate student working in the area.

The truly profound questions (e.g. why do classical logic proofs seem to work?) are very elusive. Explanations are very limited, and provided in the form of techincal results.

You'll notice that every paragraph has the same caveat: it requires technical grounding. OP, with all due respect, judging by your recent contributions, you don't have that techincal grounding yet. You'll have to get it by studying mathematics first.

^1 axiom means something else in that context

> It's not useful in this context to group the mathematics underpinning computer design under philosophy.

It's absolutely useful in the context of a discussion about Stephen Hawking declaring "philosophy is dead". If mathematics is a branch of philosophy than it's impossible for that statement to be true.

>I know my math better than my philosophy, but I do know both well enough to know there really isn't that much. But I am willing to be proven wrong. What overlap do you see?

I'd suggest maybe taking a look at some of the case studies in Synthetic Philosophy of Contemporary Mathematics starting at page 133 or skipping over to part three on page 269 and referring back when necessary. The book leans towards the philosophical side of things so it may be insightful for you.

In reality depending on how you define mathematics, philosophy, logic, and other terms we're discussing here you'll find yourself under the assumption that it's inherently part of philosophy, overlaps it in places, or has nothing to do with it. I'm certainly not under the impression I can prove to you that they overlap heavily or not but that is my stance on the matter.

EDIT: For more (or different) reading you can check out the philosophy section over at ncatlab.org.