What we call mathematics actually exists as a single point in a space of an infinite number of sets of axioms and inference rules.
We like this particular set of axioms and rules because they can answer questions about physics, but they were stumbled upon almost by accident when we started using the concept of numbers and asking "What does it mean to prove something?".
One could imagine other points in this space though, which define, say, the game of chess, or perhaps something as complex as English grammar. It just so happened that the point we naturally settled on for mathematics is one that was powerful enough to lead to the rules of algebra and geometry etc., and also consistent/deterministic in the same way that most physics seems to be (which is convenient for humans, whose brains are limited by physics).
So, mathematics doesn't "underlie" physics in the sense of being more real than it, but merely is a point we have chosen because of its similarity to physics. I suppose the real question, then, is why should such a useful point exist in this infinite space at all.
>What we call mathematics actually exists as a single point in a space of an infinite number of sets of axioms and inference rules.
No, the word mathematics actually describes this entire space. You can go a level below math down to logic. Logic are the rules that bind all these different mathematical fields together.
In addition to axioms that are assumed to be true, Logic in itself is also assumed to be true and we use a singular set of logical rules for all of mathematics.
>We like this particular set of axioms and rules because they can answer questions about physics, but they were stumbled upon almost by accident when we started using the concept of numbers and asking "What does it mean to prove something?".
Much of abstract mathematics has no real world counterpart.
It seems to me some bits are clearly created and some discovered. Eg. pi is discovered but the symbols we use to represent it's value in decimal form, 1 2 3 . etc are human inventions. There may be some value in trying to figure what's what there? I think it can help to think what alien mathematicians would have the same and what would be different. No doubt they'd find pi and e if they were any good as those are the two most fundamental transcendental constants in math but our decimal representation is one amongst a very large set of possible representations.
I get what you're saying but my point still applies.
Did we discover the decimal form of representation out of all possible forms of representing fractional quantities or did we create that form?
Was the english language discovered out of the set of all possible languages or was it created?
It's good to examine the intuitive differences though. Why does the choice of representation feel more "creative" than say the "choice" of what axioms to use in a certain field in mathematics?
Many times you will discover that the intuition is just a biased flaw in the way we think, but many times you will discover that your intuition is demarcating an actual difference.
I think our intuition is marking an actual difference here. The set of all symbols representing the number 3 is infinite not bounded by any rules. The set of all sets of axioms we can use to formulate a mathematical theory is infinite but it is actually a subset. We only choose axioms in math that are internally consistent. The number Inconsistent sets of axioms are a huge and many are discarded to find a consistent set.
It is possibly this blurry demarcation of restriction and size of the domain that marks our intuitive feelings of what we label as "created" or "discovered." Who knows, I suspect the rules are much more complicated and if we dive in deeper you will find that our intuition is flawed.
For example why does it feel extremely wrong to say that the Americas were created rather than discovered?
But again, realize that we are discussing the intricacies of arbitrary words defined in language. Outside of the domain of language the concept is uninteresting.
