A crucial question, on which great minds are divided, is whether mathematics is discovered or invented. “God gave us the integers; the rest is the work of Man”—19C mathematician Leopold Kronecker is one point of view. But physicists have long remarked at how well mathematics describes physical reality, and even seemingly abstract pure mathematics sometimes turns out, long after its invention, to be applicable to physics in the real world.
Choosing axioms/postulates is not so easy. Even Euclid’s 3000-year-old parallel postulate (given a line & point not on the line, 1 & only 1 parallel line through the point exists) has been controversial. It can’t be deduced from the others so it must be assumed, in order to prove certain theorems about plane figures, but non-Euclidean geometries (where the postulate is false) successfully describe other real objects.
Euclid’s notion that “things equal to the same thing equal each other” is stated, in modern terms, as the transitive property of the equality relation: If a=b and b=c then a=c. Such logical underpinnings of math seem obvious as universal rules of the game. I don’t think you can remove the very basis of logical inference & still call it mathematics or logic.
If your hope is to settle on a simple system of axioms, from which all mathematical theorems may be deduced, your world was shattered in 1931 by Kurt Godel, who showed that there are always statements which can be proven neither true nor false; they are undecidable. This was a game-changer in terms of how we view logical systems that include numbers, especially the status of unproven mathematical conjectures. Self-consistency and completeness don’t necessarily co-exist. A good explanation is in Godel, Escher Bach by Douglas Hofstadter.