数学知识的反个体主义透视

Study on Mathematical Knowledge from a Perspective of Anti-Individualism

数学知识究竟是什么?仅仅诉诸数学家个人的头脑和内在禀赋难以解释其何以能够拥有看似抽象而又客观的数学知识。不管是将数学知识的内涵局限于被证明的命题,还是将数学知识重新诠释为命题间的蕴涵关系、进行数学推理的操作规则和程序以及思考和处理数学问题的技能等,最终均不得不考察数学家与其所处认知环境间的关系才能给出完整解释。可见数学知识无论内容和形式如何,均存在着一个不可或缺的环境维度,认知主体之所以能够拥有数学知识,在构成上依赖于其处于特定的物理环境和社会环境之中,此即"数学知识反个体主义"。

What is mathematical knowledge? Merely invoking the brilliant mind and inner genius of the individual mathematician cannot explain how one can gain seemingly abstract but objective mathematical knowledge. Whether confining the contents of mathematical knowledge to those proved propositions,or reinterpreting mathematical knowledge as the implicational relations between different propositions,the operational principles and procedures in mathematical reasoning or the skills in thinking of and solving mathematical problems,all attempts to fully understand mathematical knowledge have to investigate on the relationship between a mathematician and his environment. Thus there is an indispensable environmental dimension in mathematical knowledge whatever its contents and forms are. Whether an agent has mathematical knowledge is constitutively dependent on the physical and social environment that he is in,which is the core idea of anti-individualism of mathematical knowledge.