数学研究方法初探

Preliminary Study of the Methods of Mathematical Research

为了引导开始从事数学科研的研究生和青年教师尽快进入数学研究之门,根据笔者多年从事数学研究的体会,并结合一些数学大师数学研究的经验,从数学研究的层面和境界,研究课题的选取,数学研究的策略和方法以及数学研究所需要的基本条件等对数学研究的方法与策略进行了总结和阐述。本文主要从如下层面展开论述。对数学研究的步骤,本文强调应首先了解所考虑问题的研究背景,然后基于对问题的剖析发现新的突破口,最后给出解决问题的方法;对于课题选取,本文给出了归纳类比推理、因果关系探讨、逆向思维等逻辑方法;而对于数学研究所需要的条件,给出了严谨的治学态度,扎实的基础和收集、分析资料的能力,周密的思考和和易于闪现的灵感,深层次的学术交流,与导师良好的沟通以及与人密切的合作等基本条件。

Many young people may have some puzzles in the beginning of their mathematical researches. Based on my experience in mathematical research and according to the researching experiences of some mathematicians, a preliminary study of the researching method in mathematics is given in this paper from the following respects ： the level of research, the selection of researching topics, the research strategies and some basic requirements for mathematical research. The purpose of this paper is to give a guidance to the young people in their mathematical research. This paper contains the following issues. For the researching procedure, we suggest that the researchers should first make a deep understanding of the background of the concerned problem to find a breakthrough based on his deep analysis, and then try to construct an efficient solution method to the problem. For the selection of research topics, we list some recommended strategies such as induction and analogy, the cause and effect, retro perspective, etc. To obtain a great achievement in mathematical research, we stress that young researchers should be rigorous in pursuing his studies, have solid foundation thorough thinking and inspiration, academic exchange and a harmonious communication with his supervisors and collaborators.