That said, without understanding, Math can't evolve. Comprehension of a proof is very important, but not what Math is fundamentally about.
Computer programs are Math. You can use them without understanding how they work.
Secondly, there is no truly objective truth to the area of a triangle. At bottom, this “truth” is simply “everyone is convinced, and for good reason”.
Without persuading other people of the “truths” that you discover, there is no real mathematics.
Why? If I sat around and studied math by myself and discovered something true yet not yet known but didn't share it, it's still true. Are you saying I didn't "do math" because I didn't share the result? Math exists on another plane and it has 'truths' that we haven't discovered, yet are still 'true', no?
The area of a triangle doesn't have 1 unique formula, it has many, depending on the system you use. A triangle in plane geometry has a different area than a triangle in spherical geometry, and different again in hyperbolic geometry.
When you get to studying the geometry of manifolds, you realize the area of a triangle can be any damn thing you want, depending on how you construct the manifold you embed it in.
A proof can just be "assuming these axioms.....the area of a triangle is X"
Math has been almost purely arbitrary since ~ late 19th/early 20th century. There are uncountably many correct mathematical theorems. Almost all of them can't even be written in symbols. Even if you have a tape with infinite length (which is already longer than the whole physical universe!) filled with theorems, they are still only 0% of all correct theorems. That's how arbitrary math is.
> The area of a triangle
Yes, even this is arbitrary. The rigorous definition of triangle is arbitrary. People just subconsciously choose something that vaguely approximates their physical intuition.
You seem to be vaguely waving in the general direction of a point, without making any concrete claims or bothering to engage with the GP’s argument.
The tastes and interests of humans are absolutely not arbitrary. They are dictated by fate, the sun and the moon gods. Or maybe by the unitary evolution of the universe’s quantum state. Or by the probability distribution of finding ourselves in a particular branch of the universe.
What bearing does this have on whether math is a collaborative endeavor?
And what is unique to math, that your argument wouldn’t apply equally to physics, sociology or financial markets? All, “truth seeking” disciplines.
Nonetheless, the person writes, “ Math has been almost purely arbitrary”.
This is simply a misusage of the word arbitrary, which is a word with a specific meaning you can look up if you are unaware, since mathematics is (obviously) not arbitrary in the sense this person wants to convey, in part for the reasons I state. Humans are not choosing arbitrary logical statements to prove true or false.
What are you talking about? A theorem is a statement that has been proved from some axioms. A statement itself is a finite sequence of symbols satisfying some syntactical rules. The set of symbols for set theory, arithmetic, etc is finite, so the set of statements, and a fortiori the set of theorems, is at most countable. Where do you get the uncountability?
I suspect you are confusing theorems and theories. Assuming the set theory ZF (for instance) is consistent, then a consequence of Gödel's incompleteness theorem is that there are indeed uncountably many inequivalent extensions of ZF that are complete and consistent. Also, not a single one of these extensions can be described in symbols in the sense that there does not exist a computer program that enumerates a possible set of axioms for the extension.
As for your general point about arbitrariness, the late 19th century was a period when mathematicians started being concerned with the rigorous formalization of mathematics. Sure, there are some arbitrariness in the particular choice of formalization in the same way that the particular form of a programming language like C is arbitrary. However, the Gödel stuff has nothing to do with that arbitrariness, it is about the limit of formalization itself. The programming analogue is the undecidability of the halting problem. Saying that mathematics are arbitrary sounds to me a bit like saying that an algorithm like Quicksort is arbitrary because you saw an implementation in C and the particular form of the C language is arbitrary. Obviously, if you don't like the C language, you can implement Quicksort in another language. The same is true for mathematics. If some day, somebody finds a contradiction in ZF, or simply a new formalization that people find more convenient, then most mathematics will simply get translated and very little will change.
Whatever philosophy you prefer, math is about establishing objectively valid logical results, completely independent of the human process used to arrive at them.
Mathematicians even argue which axioms we should have, it isn't objective in the slightest, mathematics is therefore very closely linked to our feelings and intuition. Remove that and you just have formal logic, a very different field.
From my perspective, I feel you restated what I said with the opposite conclusion. You say that axioms "defines" mathematics. If I were Claude, I'd say that the word "define" is doing a lot of work, is load bearing or something like that.
"Define" is where we turn these axioms into consequences - what I call "truth". As opposed to all the other stuff people could say that don't follow from these axioms. These are nonsense and, most certainly, un-mathematical.
it starts as a tool for humans, then evolves into a set of interesting properties of those tools, then grows into an art form, a set of "games" where cooperation is half of the point. the other half is discovering beauty in this weird parallel world of our reasoning and imagination. once tools become autonomous and start making up their own games we can't even play then mathematics loses it's meaning as a discipline. the only retort you can come up with is that "it's going to be useful". how would you know? because your autonomous tool that's too smart for you told you so? they could be as useful as morning orange juice to Claude Shannon was in terms of inventing information theory. I.e. you drinking it won't make you any closer to inventing anything of the sort anytime in your lifetime.
do triangles exist IRL? is the world discrete or continuous? can you prove it? If you have an answer to all of those I know you're wrong.
also in your computer program example just shows you don't understand it at all. those programs ARE NOT understood by you, but someone else who built them did. someone who bothered to read and architect it did. The whole Google codebase might be incomprehensible in its totality if you go bottom up but it is comprehensible by construction by us. Same with math. You don't understand all the bits of it, but someone built every brick and so you know it is "true". once the bricks become black boxes you're screwed.