How do you know that various mathematical statements are true?

Answer 1

You start with a set of definitions and "self evident" axioms. These cannot be proven or disproved.

Using the rules of mathematical logic you then deduce other statements theorems). If the axioms are true, then these theorems must also be true. You can then use the axioms and the theorems to derive more true statements and so on. Once proven, you can assume that they are true without having to go back to the axioms every time.

Euclid formalised geometry in this fashion and all was well until his parallel postulate (an axiom) was questioned. The original was phrased differently (and in Egyptian, I guess), but it can be paraphrased as follows:

"Given a straight line and a point outside the line, there is exactly one line that goes through the point and is parallel to the original line."

Mathematicians in the 19th century found that they could develop axiomatic geometries replacing this postulate with its two alternatives: no parallel lines or many parallel lines, along with the other Euclidean axioms. They found that these geometries were wholly consistent.

So, you could have a perfectly good axiomatic geometry with Euclid's parallel postulate as well as with its negations!

Bertrand Russell tried to do the same for mathematics but failed. Then, in 1931 Kurt Godel showed that Russell's project was doomed from the start. Godel's incompleteness theorem proved that any (non-trivial) axiomatic system that was capable of arithmetic had to have statements which could be true and false - both versions were valid within the system.

However, you would need to be working with mathematics at a very high level before you need to deal with the issue of incomleteness.