Mathematics is about abstraction, it's literally the study of objects. Different foundations of math define the fundamental objects of the universe in different ways. You have set theory, where everything must be constructed from the empty set, and operations of union, intersection, complement, and inclusion/comprehension.
https://en.wikipedia.org/wiki/Set-theoretic_definition_of_na...
And then you keep on using these complex objects to build even more complex ones. So you have a consistent system. And you derive truths about the system, these get called proofs. Arithmetic is child's play - it has nothing to do with consequence or truth. Math is all about starting with agreeable rules and deriving profound consequences that are a kind of umbrella that those rules cover, and sprout from. Canonically, these get called axioms. And we like to choose ones that agree with our common perception of the world -- ie. the Peano axioms for how objects physically work when you combine them, split them, try to group them into rows and columns (prime factorization), etc..
Numbers are one of the simplest and fascinating abstractions because they take the idea of object and unify everything under it -- so two cats and two dogs get typecast to two objects and two objects, then you can add them and get 4 objects. Numbers are basically a reflection of the most primitive type cast possible in our universe of thought. Such, number theory is called the Queen of Mathematics. Because number theory seems to relate much closer to our actual physicality than other axiomatic systems.
You might say, how the hell do I get an ordered pair (x,y) from sets? Well, this was solved in a myriad of ways. https://en.m.wikipedia.org/wiki/Ordered_pair#Defining_the_or...
You also have category theory where everything is a graph and you start with the global object: 1 -> N then build all numbers recursively. https://en.wikipedia.org/wiki/Natural_numbers_object
And then you keep on using these complex objects to build even more complex ones. So you have a consistent system. And you derive truths about the system, these get called proofs. Arithmetic is child's play - it has nothing to do with consequence or truth. Math is all about starting with agreeable rules and deriving profound consequences that are a kind of umbrella that those rules cover, and sprout from. Canonically, these get called axioms. And we like to choose ones that agree with our common perception of the world -- ie. the Peano axioms for how objects physically work when you combine them, split them, try to group them into rows and columns (prime factorization), etc..
Numbers are one of the simplest and fascinating abstractions because they take the idea of object and unify everything under it -- so two cats and two dogs get typecast to two objects and two objects, then you can add them and get 4 objects. Numbers are basically a reflection of the most primitive type cast possible in our universe of thought. Such, number theory is called the Queen of Mathematics. Because number theory seems to relate much closer to our actual physicality than other axiomatic systems.