How in the world was math discovered? Who/what dictates what sort equations work and stuff?
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I think the simplest answer is this: Math is invented when it is practical and needed. Equations are designed to fit reality, so they “work” to the extent they predict what actually happens, and if they make bad predictions, they are bad equations.
Math was likely “invented” in order to keep track of debts and business transactions. That is simple things like addition, subtraction, counting. Then it would move to measuring how heavy things are, how far away places are to travel to.
The next development would be to describe stuff. So engineering, whereas previously it was commerce. Think of things like building the pyramids, trebuchets, etc. math as you’re thinking of didn’t exist then, but they were still crunching the same numbers to design massive buildings or launch rocks at castles. A lot of early math is geometry, trigonometry (people were fascinated with the stars, and orbits are circular), etc.
Now we get to a more advanced age, where lots of progress has been made on number crunching in business and engineering. Jobs are very specialized, global wealth and population are up. People have more free time. Then they can explore these concepts more thoroughly. For example, the euler’s number e and the logarithm are a consequence of infinitely compound interest. Differentiation or integration is formalism for describing, e.g. the path of a thrown rocket. Eventually, the specific motivations can be abstracted away.
High level math today is very abstract, but the people in the field can relate it to real things. Because it is abstract, it could be applied or tested in many different scenarios, and the success of that test determines (to some extent) if the equations “work.” The other idea, is that math is somewhat of an art form. So the techniques used to write or formulate an equation can also have value in themselves, as they can be adapted or applied to other situations.
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