Logic In Mathematical Proofs

Mathematical truths is established by proofs.Since Euclid,people do mathematical proofs more than two thousand years.However,few research has conducted to the conception of mathematical proofs until 1930 s.Therefore,“what is”is different from “how to do”.This study focus on a simple and basic question: what is the position of logic in mathematical proofs? I notice that,by giving a common mathematical proof,it was very difficult to find out which is logic and which is mathematical.In a common mathematical proof,logic always looks like hidden in the proof.Therefore,deeply examines are needed so that to find it.This paper has conducted to the case of integer and point out which is the logic in mathematical proof.I find out that a sequent calculus may answer the research quesiton.Every sequent calculus has a premise and a conclusion,by using the transform rules make a sequent become another sequent.This article has five parts :The first part has put forward the research quesiton.Logic as if hidden in a common mathematical proofs so that we don't know the reason of conclusion of mathematical proof and so on.In order to do a further explanation of the problem in the first chapter,the second chapter provides an example about integers.A series of integer axioms are put forward.the first section describes nine algebraic axioms of integer and use those integer axioms to prove a basic example.The second section describes axioms of order.I tried to find out reasons for each step in these proofs.I found that there is a type of sequent calculus format which may answer that question :which is the logic part in mathematical proofs? Therefore,I will give a detailed introduction of the sequent calculus and introduce four kinds of inference rules: Structural rules,Connective rules,Quantifier rules and Equality rules.The fourth part is formal proofs.Using the sequent calculus rules in the third chapter to formalize those proofs in the second chapter.It gives us a clearer and more intuitive understandding about those examples through the formalization.The fifth chapter is the conclusion.