数学的抽象

On Mathematic Abstraction

数学在本质上研究的是抽象的东西,数学的发展所依赖的最重要的基本思想也就是抽象,只有通过抽象才能得到抽象的东西。在柏拉图的理念论之下,数学的概念就不应当是经验意义上的存在,而应当是一种永恒的存在;亚里士多德不赞成柏拉图的想法,认为一般概念是人们在日常生活的经验中,通过对于许多具体存在事物的共同性质抽象而得到的,所以一般概念不可能是真正的存在。亚里士多德的观点仍然需要补充,数学研究所涉及的基本概念并不一定都是直接从现实的具体的存在中抽象出来的,也可以借助符号与类比得到更高层次的抽象,这里既包括感性具体也包括理性具体。我们通过与培根、笛卡尔、洛克、休谟、康德、叔本华等思想家的攀谈,获得正反两方面的思想资源,在现代科学知识的语境下洞察到数学抽象的真实过程和本质。在讨论知识获得的问题时,我们不能忘记一个根本,即人的本性,也就是人之所以被称为人的本质,抽象了的东西是人抽象出来的,这就不能不依赖人的本性。事实上,就某一个抽象了的东西而言,还应当依赖于某一个具体的人或者某一群具体的人,但在这里我们只讨论一般意义上的抽象。人获得知识所凭借的,是先天的同时又依赖于经验的"直观能力",数学抽象能力与这种直观能力是同构的,也是一种依赖于经验的先天抽象能力,现在的数学体系就是人们通过日常生活和生产实践的具体事物,运用数学抽象的能力逐渐构建起来的。讨论这种抽象至少可以给数学教育提供两个重要启迪:一个是受教育者应当在适当的时机给予适当的教育;另一个是在传授知识的同时也应当注重培养受教育者的直观能力。

Mathematics in essence studies abstract things. The most important basic idea that mathematics hinges on in its development is abstraction as well, for abstract things can be obtained only through abstraction. Within Plato＇s theory of ideas, the concept of mathematics should not be an existence in an empirical sense but an eternal existence. Aristotle does not agree with Plato＇s idea but maintains that general concepts are what people obtain from many particular existing things in their experiences of daily life by way of homogeneous abstraction. Thus, general concepts cannot be a real existence. But Aristotle＇s idea still needs supplement due to the fact that the fundamental concepts concerned with mathematic study may not be abstracted directly from particular beings in reality and they might be highorder abstractions as well by way of signs and analogy. If it goes like this, the abstraction must be from both perceptual concretes and rational concretes. Through dialogues with, among others, Francis Bacon, Rene Descartes, John Locke, David Hume, Immanuel Kant, Arthur Schopenhauer, we have acquired resources of thought from positive and negative sides and gained an insight into the true process and essence of mathematic abstraction. In discussion of how knowledge is acquired, we cannot ignore a fundamental thing, namely, the human nature, by which man is able to be so named. Abstracted things are those abstracted by man, so the human nature will have to be reliable. As a matter of fact, a particular abstracted thing still relies on a particular person or a particular group of people, but in this article we are concerned only with abstraction in a general sense. What helps man to acquire knowledge is something innate on the one hand and the ＂directly perceived capability＂ drawing on his experience on the other hand. The capability of mathematic abstraction, homogeneously structured with the directly perceived capability, is also an innate abstraction competence drawing on one＇s experience. The system of mathematics available now was gradually built up by taking advantage of the capability of mathematic abstraction from particular things in everyday life and production practice. The discussion of this kind of abstraction, it can at least provide two significant implications for mathematics education： first, the educated should be given proper instruction at proper times; second, attention should be drawn to developing the directly perceived capability of the educated while giving in- struction on knowledge.