信念在数学革命与知识范式转换中作用的案例研究

Case Studies on the Role of Beliefs in Mathematical Revolution and Transformation of Knowledge Paradigm

在数学发展的历史上,信念作为一种基本的数学观,在数学范式的形成、演化与转换中一直扮演着极为重要的角色。通过对数学史上毕达哥拉斯主义“万物皆数”的本体论信念、欧几里得主义的公理化认识论信念、笛卡尔主义的“万能方法”信念、非欧几何知识创造中破除康德"先验综合判断”的自然观信念以及从4维空间到高维空间的知识创造中信念的价值等典型历史案例的考察,可以看到,数学革命的发生常常伴随着传统信念的破裂与旁落,以及新的信念的凝聚与主导。数学信念具有促进数学思想变革、加速知识创新并生成新的数学范式的关键作用。

In the history of mathematics,as a fundamental mathematical view,belief has played a very important role in the process of formation,evolution and transformation of mathematical paradigm. By means of exploring the historical cases such as the ontological belief in Pythagoreanism which insisted that "all is number",the epistemological belief in Euclideanism which obeyed to axiomatization,the universal methodological belief in Cartesianism,the view of nature in Non-Euclidean which rejected Kant’s synthetic judgements a priori and belief in knowledge creations from four dimensional space to higher dimensional space,we can see that the mathematical revolution often accompanied with the decline of traditional beliefs and the rise of new beliefs. Mathematical beliefs play a key role in promoting the evolution of mathematical thoughts,accelerating the knowledge innovations and generating the new paradigm of mathematics.