I am a very confused person. My high school teachers told me division by 0 is undefined because "it just is. The mathematicians just did it that way." So when I found out for real why division by 0 is undefined, I realized that every basic thing I thought I learned couldn't be trusted.
I understand axioms as "something we make up to see what happens," postulates as "things we suspect are true, but haven't proven," and theorems as "things we've proven based on the axioms we've picked." Very tidy, I can dig it.
But I can't find anywhere in several textbooks up to Calc 1 that describes what a "property" is. Is it a particular kind of axiom? Or is it a simple theorem that's close to the axioms? I've found out about indicator functions, but it looks like they only describe properties, without being the properties themselves. How can I prove that $a \times 1 = a$?
Relations, I get them. Mappings from a set to a set, based on rules of any kind. And functions are just a kind of relation where we say each thing in the domain set only gets to have one mapping to the range set. And operators being functions being relations makes sense too, except, where do the operators come from? I'm used to functions being things we make up to study polynomials. But how do we "make up" addition? I can think of any number of algorithms to perform addition, and any number of plain English descriptions of addition, but I couldn't write the right-hand side of $(a, b) = ?$. All the functions I've ever seen are just compositions of addition and the other operators. And why do operators have to be functions, anyway? Couldn't we define division by 0 just by saying division is a relation but not a function? Then it could just return the set of all numbers for $a/0$.
And then numbers. There's an infinity of them, whatever kind you care to pick. How? I figure you could lay down the existence of 1 as an axiom, but how could you construct 2 from that without using addition? Another axiom? Then there'd have to be an infinity of axioms, and that can't be workable.
Yes. I am a very confused person. I hope I picked the right tags.