什麼是數學基礎

對數學理論和數學方法的範圍提出科學的質問。開始於歐幾里德的《幾何原本》，對數學的邏輯和哲學基礎提出質問。其基本點是，任何系統的公理（例如歐幾里德幾何或微積分）是否可以保證它的完整性和一致性。在近代，經過一段時間的爭論，分成了三派思想。邏輯主義認為抽象的數學物件全部可以從基本的幾組想法以及合理的或邏輯的思想發展出來，稱為數學的柏拉圖主義的一個變型把這些物件看作是觀察者之外的、獨立的存在；形式主義相信數學是按照預先規定好的規則來操縱一些符號的配置，是與這些符號的任何物理解釋無關的一種「遊戲」；直覺主義否認某些邏輯概念，公理方法的註釋已經足夠解釋數學的全部，而不把數學看作是處理與語言和任何外部現實無關的思想構造的一種智力活動。在20世紀，哥德爾定理終止了尋找數學公理基礎的任何希望，因為數學本身既是完整的，也是沒有矛盾的。

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.