Induction, Constructivity, and Grounding

IF 0.6 3区 数学 Q2 LOGIC
T. McCarthy
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引用次数: 0

Abstract

This paper is divided into two parts, the first being a point of departure for the second. I will begin by discussing a well-known negative argument due to Mark Lange concerning the explanatory role of mathematical induction. In the first part of the paper, I offer yet another response to Lange’s argument and attempt to characterize the sort of explanatory role played by inductive proofs. That account depends on two structural principles about explanatory proof that look like a fragment of a constructive semantics for that concept. The remainder of the paper fills out this semantics and explores its consequences. It will be clear that this framework does not constitute a fully general characterization of the concept of mathematical proof; the question will be whether there is a natural class of proofs that it does characterize. My answer will be that it nicely describes what I shall call grounding explanatory proofs. A proof of this sort explains the sentence proved in terms of the grounds of the fact that it describes. I will conclude by briefly exploring the connections between grounding proofs and the notion of purity.
归纳,构造和接地
本文分为两部分,第一部分是第二部分的出发点。我将首先讨论马克·兰格关于数学归纳法解释作用的一个著名的否定论点。在本文的第一部分,我对兰格的论点提出了另一种回应,并试图描述归纳证明所起的解释作用。这种解释依赖于解释性证明的两个结构原则,这两个原则看起来像是该概念的建构语义的片段。本文的其余部分将补充这种语义并探讨其后果。很明显,这个框架并没有构成对数学证明概念的完全一般的描述;问题将是,是否存在一类自然的证明,它确实具有特征。我的回答是,它很好地描述了我称之为基础的解释性证明。这种证明是根据它所描述的事实的根据来解释被证明的句子的。最后,我将简要地探讨基础证明和纯洁性概念之间的联系。
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来源期刊
CiteScore
1.00
自引率
14.30%
发文量
14
审稿时长
>12 weeks
期刊介绍: The Notre Dame Journal of Formal Logic, founded in 1960, aims to publish high quality and original research papers in philosophical logic, mathematical logic, and related areas, including papers of compelling historical interest. The Journal is also willing to selectively publish expository articles on important current topics of interest as well as book reviews.
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