system
Given the following post, respond as if you're commenting on Reddit:
human
What is a mathematical proof? How is something "proven"?

Can someone also give me an example of a mathematical proof that I can, as an arts major, understand?

**Edit**: I *am* going to read all of this! Not tonight, though. Thanks for the answers, and for any more answers to come.

Also, I thought I'd put this question up here since it generated a lot of discussion:

**Follow up question**: How are the basic laws of mathematics proven?
gpt
Here's an explanation for the layman. In practice a proof is just an explanation of the truth of a statement that is sufficiently convincing to other mathematicians because giving all the details is usually prohibitively time consuming and unnecessary.

More precisely, a mathematical proof is a series of logical deductions and inferences that reduce a proposition to a logical combination of axioms or previously proven propositions, or to a contradiction of such, thereby establishing that the statement is as true as the assumptions and axioms. Technically, a valid proof is reducible to a series of statements in set theory (or some other foundational theory of mathematics and logic) that could in principle be verified directly or by an automated proof checking program.

Axioms -- basic laws of math -- are never proven; they are assumed and generally simple enough that most mathematicians accept them without issue. Some people study the axioms and whether they are consistent, and there are many deep results (Godel's theorem's) and controversies (e.g. the axiom of choice). There are many difficult issues that arise that are not initially obvious, such as the fact that some statements can not be proven true or false in a given system of axioms (they are "undecidable"), though they may be true or false given another set of axioms. Just deciding if a set of axioms is consistent turns out to be extremely difficult.