> so are we basically compiling a dictionary of proofs all stemming from a handful of self-evident truths

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.

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Waterluvian 10 days ago | parent [-]

Thanks for this. So I have half a thought I’m trying to flesh out from what you’ve shared. Bear with me, whoever reads this:

Are there essentially two flavours:

- the maths based on axioms that are fundamentally, cosmically true such as x = x. And in doing so we’re formalizing on paper the universal truths and all subsequent rules we know to be true given these

- the maths that incorporate those plus additional axioms that aren’t necessarily fundamentally true (or maybe just not provable), but work very well at laying a foundation for further rules that build a practically useful, but not necessarily “correct” toolbox

With the latter, is it kind of a “they won’t hold up under normal conditions but if you accept these axioms, there’s interesting things you can explore and achieve?”

	▲	Bjartr 10 days ago \| parent \| next [-]
		A thing to realize here is that there is no "fundamentally, cosmically true" in math. While math can be used to model reality, it is not bound by reality. The only thing that matters is what you choose to take as granted for a particular question. It's like how you can draw a map of a place that doesn't exist. Or like coming up with rules for a game and then trying it to see how it plays. Or it's like a material used for construction. You can build a house out of it, but there's no inherent "houseness" to it, despite how common such a use is.
	▲	daxfohl 10 days ago \| parent \| prev \| next [-]
		The first set is the rules of logic, which applies in non math stuff too. From there you can add new axioms for boolean logic, for integers, for sets, for geometry, whatever. Even given that separation (which itself is fuzzy), even the first group can't be construed as "universally true". For example there's a range of opinion around the law of the excluded middle (whether "P is not false" implies "P is true"). Most other propositional logic axioms like modus ponens are less controversial though. As far as real world math, while ZF set theory axioms are generally viewed as the "foundation", that's due to convention more than any real primacy of ZF. Other set theories and types of "foundations" exist that seem to be just as suitable to be called a "foundation", and most math is "foundation" agnostic. Like, if all you're doing is something with prime numbers, then it doesn't matter what "foundation" you use; so long as it lets you define prime numbers, that's all that matters. "Foundations" only come into play when you're doing really subtle things with different orders of infinities. And then, yes, the answer can be different depending on what foundation you choose. And that's fine. It doesn't mean that either foundation is wrong. They're just different. Which is why "foundation of mathematics" is a bit of a misnomer / fool's errand. Different foundations have different results on certain edge cases, and those are interesting to investigate, not something to be upset about. And like I said, most "ordinary" math is pretty foundation agnostic.
	▲	Tainnor 10 days ago \| parent \| prev [-]
		I would maybe rephrase it as: there are certain axioms that absolutely nobody reasonable takes issue with and then there are others that are more controversial, although it's still important to note that the vast majority of mathematicians accept the ZFC axioms + classical logic.