Chapter I - An Introduction to Proof Theory
Chapter I - An Introduction to Proof Theory
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第一章-证明论简介
DOI：
10.1016/s0049-237x(98)80016-5
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发表时间：
1998
期刊：
影响因子：
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通讯作者：
S. Buss
中科院分区：
文献类型：
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作者：
S. Buss
证明论是数学领域，研究数学证明和数学可证明性的概念。由于“证明”的概念在数学中起着核心作用，作为建立数学命题的真或假的手段;证明论至少在原则上是对所有数学基础的研究。当然，把证明论作为数学的基础是不可避免的，因为证明论本身就是数学的一个分支，关于数学证明有两种不同的观点。第一种观点认为，证明是数学家相互说服定理的真实性的社会惯例。也就是说，一个证明是用自然语言加上可能的符号和图形来表达的，并且足以让专家相信定理的正确性。社会证明的例子包括在对话中或在文章中发表的证明。当然，在这个社会意义上，不可能精确地定义什么是有效的证据;而且，有效证据的标准可能会随着受众和时间的推移而变化。第二种观点的证明范围更窄：在这种观点中，证明由一串符号组成，这些符号满足某个精确规定的规则集，并证明一个定理，该定理本身也必须表示为一串符号。根据这种观点，数学可以被看作是一种“游戏”，根据一些精确定义的规则，用一串符号进行游戏。后一种证明被称为“形式”证明，以区别于“社会”证明。
Proof Theory is the area of mathematics which studies the concepts of mathematical proof and mathematical provability. Since the notion of “proof” plays a central role in mathematics as the means by which the truth or falsity of mathematical propositions is established; Proof Theory is, in principle at least, the study of the foundations of all of mathematics. Of course, the use of Proof Theory as a foundation for mathematics is of necessity somewhat circular, since Proof Theory is itself a subfield of mathematics.There are two distinct viewpoints of what a mathematical proof is. The first view is that proofs are social conventions by which mathematicians convince one another of the truth of theorems. That is to say, a proof is expressed in natural language plus possibly symbols and figures, and is sufficient to convince an expert of the correctness of a theorem. Examples of social proofs include the kinds of proofs that are presented in conversations or published in articles. Of course, it is impossible to precisely define what constitutes a valid proof in this social sense; and, the standards for valid proofs may vary with the audience and over time. The second view of proofs is more narrow in scope: in this view, a proof consists of a string of symbols which satisfy some precisely stated set of rules and which prove a theorem, which itself must also be expressed as a string of symbols. According to this view, mathematics can be regarded as a ‘game’played with strings of symbols according to some precisely defined rules. Proofs of the latter kind are called “formal” proofs to distinguish them from “social” proofs.