Set Theory, Arithmetic, and Foundations of Mathematics: History of constructivism in the 20th century
Set Theory, Arithmetic, and Foundations of Mathematics: History of constructivism in the 20th century
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集合论、算术和数学基础：20 世纪建构主义史
DOI：
10.1017/cbo9780511910616.009
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发表时间：
2011
期刊：
影响因子：
--
通讯作者：
A. Troelstra
中科院分区：
文献类型：
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作者：
A. Troelstra
第1节。导论.在建构主义的历史调查中，1更多的空间致力于早期的发展（直到约。1965年），比后几十年的工作。这不仅是因为大多数概念和一般见解都是在1965年之前出现的，而且也是出于实际原因：自1965年以来的许多工作都具有太多的技术性和复杂性，无法在本文的范围内充分描述。建构主义是一种关于数学方法和对象的观点（或态度），它是规范性的：它不仅根据某些原则解释现有的数学，而且还拒绝不符合这些原则的方法和结果，因为它们是毫无根据的或投机的（拒绝并不总是绝对的，但有时只是一个程度问题：对建设性概念和方法的决定性偏好）。在这个意义上，各种形式的建构主义都是“意识形态”的性质。建构主义作为一种特殊的观点出现在19世纪的最后25年，可以看作是对高度抽象的概念和证明方法在数学中的迅速增加的使用的反应，这一趋势以R. Dedekind和G.康托从今天的观点来看，19世纪最后25年以前的数学主要是建构性的，只有几何学是一个显著的例外，在几何学中，反证法被普遍接受并被广泛使用。
§ 1. Introduction. In this survey of the history of constructivism, 1 more space has been devoted to early developments (up till ca. 1965) than to the work of the later decades. Not only because most of the concepts and general insights have emerged before 1965, but also for practical reasons: much of the work since 1965 is of a too technical and complicated nature to be described adequately within the limits of this article. Constructivism is a point of view (or an attitude) concerning the methods and objects of mathematics which is normative: not only does it interpret existing mathematics according to certain principles, but it also rejects methods and results not conforming to such principles as unfounded or speculative (the rejection is not always absolute, but sometimes only a matter of degree: a decided preference for constructive concepts and methods). In this sense the various forms of constructivism are all ‘ideological’in character. Constructivism as a specific viewpoint emerges in the final quarter of the 19th century, and may be regarded as a reaction to the rapidly increasing use of highly abstract concepts and methods of proof in mathematics, a trend exemplified by the works of R. Dedekind and G. Cantor. The mathematics before the last quarter of the 19th century is, from the viewpoint of today, in the main constructive, with the notable exception of geometry, where proof by contradiction was commonly accepted and widely employed.