like we know now how to get certain answers through different processes and all that fun stuff. but when they were first coming up with it, how did they know they were doing it right?
In: Mathematics
We can roughly divide math into two groups: abstract and applied.
Applied mathematics is mathematics as applied to issues of the real world. This was how and why math came to be a thing in the first place: to solve real-world problems. And we know we get the right answers by those answers actually solving those real-world problems. I know 2 + 3 = 5 because when I have two sheep and I buy three more sheep, I end up with 5 sheep.
As we developed math to solve real-world problems, people began to think about math as a thing in its own right, developing math for the sake of developing math. This is abstract math, math that might not have anything to do with any real world situations. In the case of abstract maths, where we can’t “check” it by comparing it to real world answers, we check it by comparing it with itself. That is, does this answer, if true, contradict anything else that we know to be true. If there are no contradictions, the new answer is accepted as true.
In this sense you can think of math as a game whose rules we get to invent. We define what those rules are, then play the game to see all the different states the game can be in. Every rule is well defined and so we can trace all the steps taken to play the game, from its initial state to any current state, and check to see if the rules were followed. If all the rules were followed, then it’s a legal game state.
Latest Answers