How in the world was math discovered? Who/what dictates what sort equations work and stuff?
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No one really dictates it – math exists in nature. All we did was set up a language that describes it.
For example: 1 + 1 = 2. That just exists. If you have “a thing” and add “a thing” to it, you get “a thing and a thing”. All we did was define that “a thing” equals 1 and “a thing and a thing” equals 2. We then defined that “putting things together” is a +, and the result is =. The math already existed – we just defined the terms to describe it.
Every single piece of math after that simple statement can be logically derived without anyone “defining” anything. We give additional functions – addition, multiplication, division, etc. – specific symbols, but we don’t _define them_ really – we just prove how they work given the rules of logic. That is what the famous text the _Principa Mathmatica_ does – it defines the _language_ of math, and then proves basically every high-level function, starting from 1 + 1 = 2 and building on each proof.
I’d highly recommend the book “mathematics for non mathematicians” it walks through basically the whole history of how and why… What problems people had that they kept inventing new math to solve, or new observations that required new math to describe…
I think it basically starts with commerce… Tracking who bought/sold how much of what to whom and who paid or still owes how much
If you start generalizing and want some math to define the relationship between thing you get algebra pretty quickly which is good for figuring out weights, volumes, or dimensions of things given partial information… Which is super practical for everyday life… Cooking, shipping, storage…
Later on when Newton started figuring out gravity, he started running into situations where values affect each other over time… Two things get closer, which increases the pull, which brings them closer faster, which increases the pull faster… And I think previously people addressed these sort of problems by dividing up the time into steps… Crunch the numbers for each step and you get pretty close like tracing someone’s path on gps… if it reported every hour you’d get a pretty rough, angular trail. Vs if it reported every minute, that’d be better, vs every second you’d get pretty close… But Newton figured out how to use an infinite number of steps to get the true answer, and this is the jist of calculus… It let’s you get true answers for systems where the values that are changing affect each other over time (among many other uses)
There’s a bunch more, like, it seems obvious to me that fractions and decimals make sense, being able to describe partial things… And that negative numbers make sense… Being able to talk about a lack of something, or debt? But it took people a long time to figure out they needed a 0 in their number system, let alone negative numbers… Then there a whole story about why imaginary number exist and the journey to proove they’re useful for real-world applications and not just a quirk that looks like you broke math…
But there’s also always some math that mathematicians have discovered but seems like nonsense until we find a way to apply it to describe real world phenomena… But historically, we tend to find one eventually
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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