# Proof

Mathematical proof is logical reasoning that beyond any doubt shows given statement to be true. Such a proof is the best and most objective guarantee of truthfulness we can ever hope for -- this is not to say there can't ever exist any doubt about mathematical proof, but as long as we agree on the axioms, believe logic itself holds and that there is no error in the proof, the proven statement has to hold. Mathematical proof is a **deductive** reasoning (despite the fact that, a bit confusingly, one type of mathematical proof is called *induction*) that absolutely eliminates any chance of being wrong, i.e. it is yet a stronger argument than inductive proofs provided by traditional sciences, which typically just have high level of statistical confidence that however never reaches 100%. This is to say mathematical proof must never leave a slightest doubt about the conclusion, and so while presenting thousands or millions of samples supporting a hypothesis may in traditional science be good enough to call the hypothesis confirmed, it is not enough for a mathematical proof. Mathematical proofs are therefore hard and still amount to art requiring human brain (although some proofs can nowadays be assisted or even carried out completely autonomously by computers), and so in practical life we often require mathematical proofs only in very critical areas -- for example software used in planes or medical devices has to be mathematically PROVEN to be correct (see formal verification), most other software is debugged just by testing, which is just a good enough evidence it's mostly free of bugs.

TODO

**Why do we need mathematical proof if something is obvious?** Well, mathematicians need to be most precise and proof enables them to discover absolute truths without any shadow of a doubt (a luxury most other scientists don't have), so they set it as a standard because many things that seem obvious aren't in fact so -- for example numbers 31, 331, 3331, 33331, 333331, 3333331 and 33333331 are all primes so you might think by this pattern also 333333331 will be a prime, but that's not the case because 333333331 = 19607843 * 17. Sometimes patterns deceive us, mathematicians only take proof for the ultimate solution. But indeed e.g. the industry sometimes accepts even unproven but highly likely conjectures to hold, e.g. that P doesn't equal NP, simply for economic reasons (the chance of being wrong is very low and profitability of being right is high).


**There are mathematical problems that cannot be proven** or disproved (under given set of axioms). See decidability.