In mathematics, proof is the process of deducing some propositions from axioms and theorems according to certain rules or standards in a specific axiom system. Compared with evidence, mathematical proof generally relies on deductive reasoning rather than natural induction and empirical reasoning. The proposition derived in this way is also called theorem in this system.
Mathematical proof is based on logic, but it usually contains natural language, so there may be some ambiguous parts. In fact, if most of the proof is written in mathematical form, it can be regarded as the application of informal logic. In the category of proof theory, only proofs written in pure formal language are considered.
Mathematical proof must be strictly in accordance with unified standards.
The object to be proved by 1. must be a universal concept, and the so-called "proof" of the set concept is not allowed.
2. The proof method must be correct deductive proof (mathematical induction must be under the formula that can unify all the elements of this universal concept, and mathematical induction without a unified formula is invalid).
3. The argument must be correct.
4. Do not use vague concepts, that is, the concept must be the only explanation, and there can be no ambiguity (for example, it is forbidden to use the so-called "almost prime number" and "big enough").
5. All conclusions must be operable, that is, after the conclusion is proved, people can know the result through this conclusion calculation without producing contradictory results.
6. The conclusion must be full name, and the conclusion is invalid.
7. The proof process must be transitive, and the proof without transitivity is invalid. For example, in the process of proving Fermat's last theorem, Fermat's last theorem and Taniyama's conjecture are not transitive, so the proof is invalid.