Nope, math is invented, without people there is no math. Math is a logic system invented by humans. You don't have to use math to describe relations between things. Math is a language, it describes real world and is not real world itself. So math is invented.
Nope, math is discovered, without people also there is math. Math is how the universe works. When you describe relations between things, that is Math. Math is notated using many languages, but the real world itself cares not for which notation you use. So math is discovered.
----
Have a look at how various cultures around the world did maths before meeting Europeans. You will quickly stop thinking "Math is a language".
Hell, even European maths wasn't entirely European. The most popular number system in use to this day, arrived in Europe via Arab traders and itself originated in ancient India. A culture that developed its own entirely different set of ways to explain some the logic of the universe.
While the ancient Indian system of arithmetic would look very different to anyone with a standard school education today, both systems describe the exact same things: addition, multiplication, subtraction, and division of things.
If we were to meet an alien civilization, who'd undoubtedly have their own language(s) and culture(s), the fastest way for us to learn how to communicate with them would be to look at how they do maths. Because, while their language and notation of maths may be different, what they describe is going to be same fundamental laws of this Universe.
Nobody's making the claim that Euclidean geometry is all of maths. But the part of the universe that Euclidean geometry represents has always, still does, and will continue to work even when the last traces of Euclidean geometry vanish from recorded knowledge and memory.
> ... link to Gödel's incompleteness theorems
That's a proof of some limits of formal systems — particularly those that want to formalise everything under one unified set of axioms — not limits of mathematics. Mathematics / the universe does care one iota if you use this particular set of axioms or another. Or even any. It continues to work without a care for your need to have a grand unified theory. That you cannot discover all of its secrets because you restricted yourself is not its concern.
Maths is how the universe works, whether you understand it or not.
----
But thank you for linking to Gödel's theorems. Your link directly answers the topic being discussed. You'll notice the text never says "invented" when talking about these or related theorems; it says "discovered".
> Euclidean geometry does not describe the universe, even if it's useful.
A statement that doesn't disprove the thesis in the slightest.
eg: There's non-Euclidean geometry which some say is handy in a post Newton Einstein universe.
If that fails, I feel there'll be something else again that conforms better to the universe as we understand it to be.
> the fact that math is not how the universe works was proven with math
Another statement that fails to prove the thesis; the universe itself is sufficiently complex that there can indeed be things out there that we will never 'prove' to our satisfaction.
You need to do some lifting here (perhaps a little more than 'some') to prove that Godel|Church|Turing results demonstrate beyond doubt that maths cannot underpin the workings of a universe.
Your comment reminds me a little of Gödel's ontological "proof" .. full of sound and fury but not really landing.
Mathematical notation is a language, but actual math isn't. For example, the concept of there being one of something is an inherent feature of our reality, but drawing short vertical lines for them is a thing we do. Similarly, we didn't invent 3.14..., that's just how circles work. We only invented the shapes I just used.
