从数学的对象谈恩格斯关于数学的论断

On Engels’ Mathematics Thesis from Mathematics Objects

数学的研究对象问题，是一个古老而常新、困难而有趣的问题，也是数学哲学界、数学史界、数学界、数学教育界共同关注的问题．数学研究的对象是一个动态的概念体系，对数学的研究对象在不同时期有不同的认识，尤其是在同一时期由于认识角度的不同而不同，可以是“数与形”、“量的关系和空间形式”，也可以是更加抽象的“形式和关系”、“模式”、“结构”；可以是对“量”的研究，也可以是对“质的研究”．

Engels pointed out that pure a mathematics object refers to the relation between the spatial form of real-world and a quantity. However, with the further development of mathematics, mathematics research objects had broken through what Engels referred to. The research objects of modem mathematics had turned from ＂the given quantitative relations and spatial forms＂ to ＂possible quantitative relations and spatial forms＂. According to some new explanations, mathematics research objects should be ＂the object-world-and-logically -possible quantitative relations and structural relations, not only on the prescriptive of quantity, but also on that of quality＂. All objects that reject concrete contents and were described in Mathematics makeup language were the objects researched by mathematics. Research object researched by mathematics was a dynamic conceptual system, which varies with different mathematics problems with different methods, and contains richer and deep connotations. It was impossible to get a once-and-for-all answer at any stages of mathematics developments.