On Mei's Mathematical Proof

Mathematical proof is the main approach to form and develop new conceptions. Renew knowledge depends on proof level, thus developing mathematical is the key to increase knowledge. Proof construction depends on cultural background and culture influences mathematical developments by mathematical proof. Being the conflict between Chinese and Western culture, the relationship of mathematics and culture influence mutually is quite special during Ming-Qing Dynasty. Mei Wending's mathematical proof manifested the particularity. There are lots of researches on Mei Wending's mathematical achievements. Although much researches interfere in his mathematical proof, which has not been regarded as special research object. The paper attempts to make a special research of Mei's proof, explain the conceptions and changes and illustrate what attributes he made in some sense. It will help to understand the development of mathematical proof thought during Ming-Qing Dynasty, acquaint with the difficulties and characters of Chinese mathematics integrating with internationality, comprehend why the Chinese mathematics has been replaced eventually.The paper involves four aspects. Summarize as follows:1. The motivation of Mei's proof. Practices indicated Western methods possessed practical values, however, the conceptions existed in its base were opposite to the Confucianism. Therefore, neither accepting nor denying Western knowledge complied with the country's benefits. Mei Wending provided reasonable grounds to introduce Western methods by developing traditional mathematical proof. What he has done was the only scheme could be chosen then. As a result, the increasing demands guided the proof to develop.2. The structure of Mei's proof. Mei distinguished "sky" between mathematics and philosophy, which eliminated the probability of pure formal deduce. The rationalities of mathematical propositions depended on visual testimonies. The ways and proceeds to form mathematical conceptions haven't changed, which benefited to introduce Western methods. Pure formal definitions were not compatible with the ancient traditions. So the ancient traditions decide the proof structure.3. The merits of Mei's proof. Mei's mathematical proof manifested the prominent changes of Chinese mathematical objects, which arose the improvement of proof methods and resulted in the developments of Chinese mathematical conceptions. For instance, to define similar figures didn't depend on area transform because of quoting"angle", "verticality", "parallelism" and so on. So area transform wasn't indispensable in proportional theory. Other examples embody analogous changes.4. The influences of Mei's proof. The later theories' increase oriented to connect with Mei's works. Increasing speed connected with the ancient traditions. Mei's target was to introduce Western methods, which made him have to change people's opinions on western knowledge and not to touch people's opinions on mathematics. Under his leading, later Chinese mathematicians introduced much more Western methods but the main mathematical research target wasn't the general relationship.The paper analyses deeply the changes of Chinese mathematical proof during the special history periods. Explanations on the characters and meanings of Mei's work are much clearer than before. The analyses of Mei's proof are comprehensive. There involved individual examples in Mei's proof on Elements in former papers and the paper offers analyses comprehensively. Concerning the relationship between developing proof and maintaining traditions, the paper gives much more concrete and definite analyses than before.