I would say, "from a handful of axioms".
It's certainly true that when Euclid started this whole enterprise, it was thought thax axioms should be self-evident. But then, many centuries later, people discovered that there are other interesting geometries that don't satisfy the same axioms.
And when you get to reasoning about infinities, it's very unclear that anything about them can be considered self-evident (a few mathematicians even refuse to work with infinities, although it's definitely a very niche subcommunity).
Some of today's common axioms are indeed self-evident (such as "you should be able to substitute equal subterms"), but things like the axiom of choice have (at least historically) been much more controversial. I would probably say that such axioms can be considered "plausible" and that they generally allow us to be able to prove what we want to prove. But you'll definitely find mathematicians championing different axioms.
> Aside: I recall some famous mathematician had made a list of base proofs that you just hold to be true. Can someone remind me who, and/or what that list is called? I’m guessing they’re considered axioms.
That would be the ZFC axioms. It was originally the ZF axioms (named so after the mathematicians Zermelo and Fraenkel who worked in the early 20th century), and then later the Axiom of Choice (C) was added. It's generally considered to be the "standard" set of axioms for maths, although very few mathematicians actually work directly from these axioms. But in theory, you can take almost every mathematical proof (unless it's explicitly set in some other foundation) and recast it entirely in applications of the ZFC axioms.