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
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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