Another way of looking at it, would be to imagine what would happen if we started writing proofs in formal logic. Quite quickly, people would notice patterns of inferences that appeared again and again, and save space and time by encoding the pattern in a single statement which anyone could use, not actually an additional axiom since it follows from the others, but working like one. If you do this a lot quite soon you would be almost entirely using short-cuts like this, and begin to notice higher-level patterns in the short-cuts you are using. Keep extending this short-cut-generating process until mathematical papers are actually of a remotely readable length, and you get something that looks quite a lot like modern mathematics.
I’m not claiming this is what actually happened, what actually happened is a few hundred years ago mathematics really was as wishy-washy as you say it is, and people gradually realised what they should actually be doing, then skipped the stage of writing 10000-page proofs in formal logic because mathematicians aren’t stupid and could see where the chain ended from there.
Another way of looking at it, would be to imagine what would happen if we started writing proofs in formal logic. Quite quickly, people would notice patterns of inferences that appeared again and again, and save space and time by encoding the pattern in a single statement which anyone could use, not actually an additional axiom since it follows from the others, but working like one. If you do this a lot quite soon you would be almost entirely using short-cuts like this, and begin to notice higher-level patterns in the short-cuts you are using. Keep extending this short-cut-generating process until mathematical papers are actually of a remotely readable length, and you get something that looks quite a lot like modern mathematics.
I’m not claiming this is what actually happened, what actually happened is a few hundred years ago mathematics really was as wishy-washy as you say it is, and people gradually realised what they should actually be doing, then skipped the stage of writing 10000-page proofs in formal logic because mathematicians aren’t stupid and could see where the chain ended from there.