数学哲学(mathematics, philosophy of)

哲学的一支，关注於数学的认识论和本体论。在20世纪早期，兴起了三个主要的思想学派－－称为逻辑主义、形式主义（formalism）和直觉主义－－来解释、并解决在数学的基本原则中所产生的危机。逻辑主义主张，所有数学上的概念，都可化约至抽象（pure）思考的法则，或逻辑原则；在此主张下的另一个不同看法，以数学柏拉图主义为名，则认为数学概念是超越经验的完美典型，或称为形式，是独立於人类的意识之外。形式主义则认为，数学是依据规定好的规则，只是运用有限的符号配置所组成；只是一场独立於对任何符号进行物质（physical）性解释的「游戏」。直觉主义的特徵是，对任何对於真理持超越经验之概念的知识，或着证据，都加以排斥。因此，只有在数目有限的步骤中能够加以建构（参阅Constructivism）的对象（object）才获得承认，至於实际上的无限，以及排除中间项（middle，见思维法则）的法则也不被接受。这三个思想学派各自主要由罗素、希耳伯特和丹麦的数学家布罗韦（Luitzen Egbertus Jan Brouwer, 1881～1966）所率领。

English version：

mathematics, philosophy of

Branch of philosophy concerned with the epistemology and ontology of mathematics. Early in the 20th century, three main schools of thought－called logicism, formalism, and intuitionism－arose to account for and resolve the crisis in the foundations of mathematics. Logicism argues that all mathematical notions are reducible to laws of pure thought, or logical principles; a variant known as mathematical Platonism holds that mathematical notions are transcendent Ideals, or Forms, independent of human consciousness. Formalism holds that mathematics consists simply of the manipulation of finite configurations of symbols according to prescribed rules; a "game" independent of any physical interpretation of the symbols. Intuitionism is characterized by its rejection of any knowledge- or evidence-transcendent notion of truth. Hence, only objects that can be constructed (see constructivism) in a finite number of steps are admitted, while actual infinities and the law of the excluded middle (see laws of thought) are rejected. These three schools of thought were principally led, respectively, by Bertrand Russell, David Hilbert, and the Dutch mathematician Luitzen Egbertus Jan Brouwer (1881-1966).