Note the difference in what I said. I didn’t say math is not part of nature. But I did say math is not natural, that it does not appear naturally in nature. You won’t find an equation roaming around in the fields, waiting for you to capture it with your giant butterfly net. Math is as much part of nature as a game of chess or a computer program is. Anyway, that’s more of a semantic argument that’s not really important right now.
The important thing is Axioms. Axiom is something that we take to be true without need for proof. So, for example, Euclid, when codifying the rules of geometry (geometry at the time being primarily a set of rules developed by farmers, builders and similar people concerned with measurements (some say that most of the early work of creating the rules of geometry happened in Egypt, because of the need to know where everyone’s farm plots were after the Nile floods receded)) said that “In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.” (he actually did not say exactly that, but it’s close enough). But he didn’t say that because he found a line and a point somewhere and tried to draw a parallel line. Because lines do not exist. They are a pattern. A construct of the human brain. Someone, somewhere decided what lines are. He created the idea of lines, the rule of lines, because that idea, that pattern was a useful tool, a useful rule. Any other rule you create from that rule is true and it’s logical with no need for experimentation. You can just prove that it’s 100% true without a speck of doubt. It’s pure logic. There’s nothing to believe in. Because it’s something that isn’t real. It’s an idea, something created in your mind, a set of rules, a logic. The moment you accept the axioms as 100% true then you accept anything deduced from those axioms as being 100% true.
Which is why I mentioned non-eucledian geometries. If you change that little rule Euclid created, you come up with alternative geometries, geometries where, for example, you can have more than one parallel lines crossing a point. They were able to change that rule because they could. Because it wasn’t something true. It was just a rule. You can create any set of rules you want, any set of axioms. And you can use those axioms to to find more rules, more tools, more patterns that emerge from those axioms. But whether or not you create those things is irrelevant for the physical world. They are just rules, just ideas. They may be useful ideas of course, useful rules that you can apply to patterns in the real world and come up with useful results. Or they may be ideas that are completely useless. True, 100% precent true beyond doubt (at least in that system of axioms), but useless.
Or, for another example, consider the rules of a game of chess. We know that the bishop only moves diagonally and stops when it encounters another piece. We accept that when we play chess as something that is true. There’s no need to prove that the bishop only moves diagonally because we have created the rule that says that the bishop only moves diagonally. And we can always create another game, different from chess, that will have a bishop moving in a different manner, with a different set of rules. And for that game, those rules will be equally true and without need for proof. For the same reason you don’t need to prove that 1+1=2. Because it’s just a rule we accept.
What you are talking about with the Fibonacci example is patterns. The human mind is good at spotting patterns. And math is a tool that is perfect for describing and discovering patterns. To say that math exist in nature because of the Fibbonaci example is to say that stocks exist in nature because there is a stock market or that metres exist in nature because there are things that are a metre long. Or to put it differently: There are people who believe in numerology or that the number 23 is mysteriously linked to everything. They see patterns in nature because they look for patterns and they believe that because math has rules for some of those patterns that means that nature is governed by math. Which is not true. Math is just a useful tool, sometimes used for observing and making deduction about the patterns found in nature. In fact, it was created for exactly that purpose. But on its own it is a construct completely irrelevant to nature. It just is.
tl;dr: There’s nothing to prove there because that’s just something that we consider to be true in math. That’s what axiom means, something that doesn’t need proof, the base for the creation of a system.
I hope I made my point. It’s hard to discuss such abstract topics, especially when English is not my first language. I’ll stop now, to prevent the thread from going further off topic. Feel free to continue in another topic if you find the subject interesting.