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.
From what I understand, most maths are ‘discovered’ backwards. Starting with an answer or problem, then working out how to get there or prove a solution.
The problem here is that you are assuming that mathematics was invented. It’s like gravity, it wasn’t invented, just discovered. With math, there is only one answer, you can’t add 2 and 2 and get 5. It will always be 4. It’s a constant and it is why it is a universal language.
At first, we didn’t know we were doing it right. We made some observations, like that if we knew the lengths of two sides of a triangle and the angle between them, if we made another triangle that had the same side lengths and angle, it would look like an identical twin. That was just something to learn, like “ice is cold.”
It wasn’t until later that mathematicians like Euclid were able to *prove* it, that is, to show that Side-Angle-Side is a *theorem*, not a *postulate.*
someone already said it but ill say it again, math is discovered, not invented. As for the getting the right answers, it’s usually done by proof. If you can prove your answer is right, then it is. You might ask yourself how did they know their proving methods worked? etc. That’s because all the theorems and lemmas are “built” on top of asymptotes aka the stuff that is inherently true, without the need of any proof.
I’ll disagree with some of the others and say mathematics was invented. What they’re describing is *counting*. If I take 2 apples and put them on a rock (tables haven’t been invented yet), then put 2 more apples on the rock, I can count 4 apples.
Mathematics, on the other hand, was invented to be able to write that down in the abstract. 2+2=4 is divorced from whether or not I have 2 or 4 apples in front of me, or no apples anywhere in sight. I can go back to the apples and check my math, but once proven, I’m confident that 2+2=4.
Mathematics was invented to solve not just how many apple there are, but those all-important questions like:. If a man owns a farm, 5 pigs, and 6 fig trees, how much tax does he owe? If each of my graineries holds X bushels of grain, and I have 6 graineries, how much grain did I take from the farmers? We should probably tax the farms more fairly (emphasis on the more). So we invent the concept of area.
Or even more complex: If I have Y bushels of grain stored, and 2,000 townspeople, 20 priests, and the 8 royal family to feed, how many standard bowls of grain do I give out a day to last until harvest? Given that the priests get more, and the royals get even more.
So far math is pretty grounded in the real world. But by the Classical period, we start thinking about more abstract things, like the area and volume of geometric shapes. We see mathematical *proofs*, where we build on already proven concepts to *prove* a higher concept. One of these the *method of exhaustion*, where we divide a shape into smaller and smaller shapes of known area/volume and sum those. They got pretty close to calculus, but that had to wait for Leibniz and Newton, who again used proofs of earlier known concepts to build a new field of mathematics.
Speaking from the angle of abstract math.
Abstract mathematics is about giving a certain set of rules, and then finding consequences of those rules which aren’t immediately obvious from them.
The first key point is that the rules you work with have to have no ambiguity. If anything has even a slight amount of room for interpretation, this is immediately seen as a problem, and mathematicians will demand that the rules be made more precise until they fully accept any deductions from them.
Once you have rules with no ambiguity, every step in your deduction then has to be justified using one of those rules. If you do manage to deduce a fact using nothing but the rules agreed upon, that is then a mathematical result; but that result only applies under the specific rules agreed upon.
Finally, there’s the fact that humans aren’t perfect, and mistakes in following the rules may occur. The safeguard against this is that other mathematicians also know the same rules, and can check your reasoning. Results start getting accepted as true if no-one can find a step where the rules weren’t being followed. Also, in general if a result is important and/or unexpected, it tends to face higher scrutiny.
This process unfortunately isn’t perfect, as new abstract mathematics is often at a level of complexity where mistakes can still sometimes slip through. Yet this approach to mistake checking has worked well enough so far, and mainly means that the further from the widely known “mainstream” you go in the maths you use, the more responsibility there is on you to know the reasonings behind the facts you use.
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