数学基础(mathematics, foundations of)
对数学理论和数学方法的范围提出科学的质问。开始於欧几里德的《几何原本》，对数学的逻辑和哲学基础提出质问。其基本点是，任何系统的公理（例如欧几里德几何或微积分）是否可以保证它的完整性和一致性。在近代，经过一段时间的争论，分成了三派思想。逻辑主义认为抽象的数学物件全部可以从基本的几组想法以及合理的或逻辑的思想发展出来，称为数学的柏拉图主义的一个变型把这些物件看作是观察者之外的、独立的存在；形式主义相信数学是按照预先规定好的规则来操纵一些符号的配置，是与这些符号的任何物理解释无关的一种「游戏」；直觉主义否认某些逻辑概念，公理方法的注释已经足够解释数学的全部，而不把数学看作是处理与语言和任何外部现实无关的思想构造的一种智力活动。在20世纪，哥德尔定理终止了寻找数学公理基础的任何希望，因为数学本身既是完整的，也是没有矛盾的。
English version：
mathematics, foundations of
Scientific inquiry into the nature of mathematical theories and the scope of mathematical methods. It began with Euclid's Elements as an inquiry into the logical and philosophical basis of mathematics—in essence, whether the axioms of any system (be it Euclidean geometry or calculus) can ensure its completeness and consistency. In the modern era, this debate for a time divided into three schools of thought. Logicists supposed that abstract mathematical objects can be entirely developed starting from basic ideas of sets and rational, or logical, thought—a variant known as mathematical Platonism views these objects as existing external to and independent of an observer; Formalists believed mathematics to be the manipulation of configurations of symbols according to prescribed rules, a “game” independent of any physical interpretation of the symbols; and Intuitionists rejected certain concepts of logic and the notion that the axiomatic method would suffice to explain all of mathematics, instead seeing mathematics as an intellectual activity dealing with mental constructions (see constructivism) independent of language and any external reality. In the 20th century, G?del's theorem ended any hope of finding an axiomatic basis of mathematics that was both complete and free from contradictions.