How did mathematicians know their contribution to the subject was correct? How much testing did Al-Kwarizmi do before concluding that Algebra is solid and true?

Question

Shoutout to all lost geniuses who used Algebra long before Al-Kwarizmi.

In: Mathematics

Answer 1

In math you usually can prove something will work for every possible number.

If something just works for all numbers you tried but cannot prove it works for all of them then you have a conjecture (wich is usually deemed much less usefull than an actual law)

Lots of effort is spend by mathematicians to actually prove that things work in general. For that you basically put some letter variables into your equations and then try to derive the general form of your law with that. If that works then it must work with every number. 

Another way is disproving the opposite. You assume your law is wrong, and show how that leads inevitably to something that must be wrong (reductio ad absurdum).