>>22
Ahhh the axiomization of mathematics. You make it sound so simple, but this was a philisophical problem for mathematics only settled in the 19th century. All was going well until Gödel proved in the 20th century that there are things in mathematics that are true and can't be proved with the axiom system. There are still many Platonic mathematicians who hold that numbers are real objects, and not just a language we use axioms to play games with.
Logic is also philosophy. Deductive reasoning, the basis of all formal proofs from axioms, relies on Aristotlean logic. However, most proofs are not stated so formally and are at the mercy of human language and its imperfections and ambiguities. This makes most written mathematical proofs fallible. If written proofs were written axiomatically they could simply be entered into a computer and checked using a simple program, without professors having to spend weeks in some cases agonizing over them just to be hopefully "convinced" by the arguments presented.