重思数学哲学中的柏拉图主义源头--兼论亚里士多德对柏拉图相数理论的批评

A Second Thinking of the Source of Platonism in the Philosophy of Mathematics: Aristotle's Critique of Eidetic Numbers

柏拉图认为,数学事物(如数、运算符号等等)是相,因而分离于感性事物;相数是以相为单元的数,指涉诸相之间呈现出数学结构的参与关系,这种本原意义上的参与被众物对一个智性之物的统一体的参与所分有,因此后一种参与也呈现出数学结构。亚里士多德则质疑相的"存在状态"与它所涵盖的一群个体的分离,认为相特有的存在方式就是始终作用于感性事物上。对亚里士多德而言,数的存在方式依赖于感性事物,是感性事物的"累积";数的智性特征不是产生于与感性事物的分离,而是产生于从感性事物中的"提升"、"抽离"或"抽象"。实际上,亚里士多德没有看到,柏拉图的相数理论真正处理的问题是,要说明计数所使用的每个基本数所特有的差异化的统一性。

Plato regards mathematical things( for example,numbers,operational signs) as eidê,which are separated from sensible things. Eidetic numbers are numbers consisting of eidê as units,and they refer to the mathematical structural participational relationship among eidê. In the dispute with the Platonic account of the eidê,Aristotle takes issue with its 'separation'of the 'beinghood'( ousia) belonging to a genos from the multitude of single things that are encompassed by it,and holds that the proper mode of being of an eidos is'being-at-work( energeia) 'on a thing that is. For Aristotle,the mode of being of a number relies on sensible thing and is a'heap'of sensible things. Intelligible characteristics of numbers arise not from their separation of sensible things,but from abstraction,from being'lifted off','drawn off'or,in other words,from being'abstracted'from sensible beings. Aristotle did not see the problem that Platonic concept of eidetic numbers actually deals with,namely,how to account for the differentiated unity proper to each of the basic numbers employed by counting.