从非修正主义的维特根斯坦主义看康托尔的天堂

Q4 Arts and Humanities
J. Gomułka
{"title":"从非修正主义的维特根斯坦主义看康托尔的天堂","authors":"J. Gomułka","doi":"10.24917/20841043.10.2.5","DOIUrl":null,"url":null,"abstract":"\n \n \nCantor’s paradise from the perspective of non‐revisionist Wittgensteinianism: Ludwig Wittgenstein is known for his criticism of transfinite set theory. He forwards the claim that we tend to conceptualise infinity as an object due to the systematic confusion of extension with in‐ tension. There can be no mathematical symbol that directly refers to infinity: a rule is the only form by which the latter can appear in our symbolic operations. In consequence, Wittgenstein rejects such ideas as infinite cardinals, the Cantorian understanding of non‐denumerability, and the view of real numbers as a continuous sequence of points on a number line. Moreover, as he understands mathematics to be an anthropological phenomenon, he rejects set theory due to its lack of application. As I argue here, it is possible to defend Georg Cantor’s theory by taking a standpoint I call quietistic conventionalism. The standpoint broadly resembles Wittgenstein’s formalist middle period and allows us to view transfinite set theory as a result of a series of definitions established by arbitrary decisions that have no ontological consequences. I point to the fact that we are inclined to accept such definitions because of certain psycho‐ logical mechanisms such as the hypothetical Basic Metaphor of Infinity proposed by George Lakoff and Rafael E. Núñez. Regarding Wittgenstein’s criterion of applicability, I argue that it presupposes a static view of science. Therefore, we should not rely on it because we are unable to foresee what will turn out to be useful in the future. \n \n \n","PeriodicalId":30403,"journal":{"name":"Argument Biannual Philosophical Journal","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2021-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Cantor’s paradise from the perspective of non‐revisionist Wittgensteinianism\",\"authors\":\"J. Gomułka\",\"doi\":\"10.24917/20841043.10.2.5\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"\\n \\n \\nCantor’s paradise from the perspective of non‐revisionist Wittgensteinianism: Ludwig Wittgenstein is known for his criticism of transfinite set theory. He forwards the claim that we tend to conceptualise infinity as an object due to the systematic confusion of extension with in‐ tension. There can be no mathematical symbol that directly refers to infinity: a rule is the only form by which the latter can appear in our symbolic operations. In consequence, Wittgenstein rejects such ideas as infinite cardinals, the Cantorian understanding of non‐denumerability, and the view of real numbers as a continuous sequence of points on a number line. Moreover, as he understands mathematics to be an anthropological phenomenon, he rejects set theory due to its lack of application. As I argue here, it is possible to defend Georg Cantor’s theory by taking a standpoint I call quietistic conventionalism. The standpoint broadly resembles Wittgenstein’s formalist middle period and allows us to view transfinite set theory as a result of a series of definitions established by arbitrary decisions that have no ontological consequences. I point to the fact that we are inclined to accept such definitions because of certain psycho‐ logical mechanisms such as the hypothetical Basic Metaphor of Infinity proposed by George Lakoff and Rafael E. Núñez. Regarding Wittgenstein’s criterion of applicability, I argue that it presupposes a static view of science. Therefore, we should not rely on it because we are unable to foresee what will turn out to be useful in the future. \\n \\n \\n\",\"PeriodicalId\":30403,\"journal\":{\"name\":\"Argument Biannual Philosophical Journal\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-03-31\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Argument Biannual Philosophical Journal\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.24917/20841043.10.2.5\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"Arts and Humanities\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Argument Biannual Philosophical Journal","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.24917/20841043.10.2.5","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"Arts and Humanities","Score":null,"Total":0}
引用次数: 0

摘要

从非修正主义的维特根斯坦主义看康托尔的天堂:路德维希·维特根斯坦以其对超有限集合论的批判而闻名。他提出我们倾向于把无限概念化为一个对象是由于系统地混淆了延伸和张力。不可能有任何数学符号直接指向无穷大:规则是无穷大在我们的符号运算中出现的唯一形式。因此,维特根斯坦拒绝了无限基数、康托利亚对不可数性的理解以及实数作为数轴上点的连续序列的观点。此外,由于他认为数学是一种人类学现象,他拒绝集合论,因为它缺乏应用。正如我在这里所说的,我们可以通过采取一种我称之为静寂传统主义的立场来捍卫乔治·康托尔的理论。这种观点与维特根斯坦的形式主义中期观点大体相似,它允许我们将超有限集合论视为一系列定义的结果,这些定义是由没有本体论结果的任意决定建立的。我指出的事实是,我们倾向于接受这样的定义,因为某些心理机制,如乔治·拉科夫和拉斐尔·e·Núñez提出的假设的基本隐喻的无限。关于维特根斯坦的适用性标准,我认为它以一种静态的科学观为前提。因此,我们不应该依赖它,因为我们无法预见什么在未来会变得有用。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Cantor’s paradise from the perspective of non‐revisionist Wittgensteinianism
Cantor’s paradise from the perspective of non‐revisionist Wittgensteinianism: Ludwig Wittgenstein is known for his criticism of transfinite set theory. He forwards the claim that we tend to conceptualise infinity as an object due to the systematic confusion of extension with in‐ tension. There can be no mathematical symbol that directly refers to infinity: a rule is the only form by which the latter can appear in our symbolic operations. In consequence, Wittgenstein rejects such ideas as infinite cardinals, the Cantorian understanding of non‐denumerability, and the view of real numbers as a continuous sequence of points on a number line. Moreover, as he understands mathematics to be an anthropological phenomenon, he rejects set theory due to its lack of application. As I argue here, it is possible to defend Georg Cantor’s theory by taking a standpoint I call quietistic conventionalism. The standpoint broadly resembles Wittgenstein’s formalist middle period and allows us to view transfinite set theory as a result of a series of definitions established by arbitrary decisions that have no ontological consequences. I point to the fact that we are inclined to accept such definitions because of certain psycho‐ logical mechanisms such as the hypothetical Basic Metaphor of Infinity proposed by George Lakoff and Rafael E. Núñez. Regarding Wittgenstein’s criterion of applicability, I argue that it presupposes a static view of science. Therefore, we should not rely on it because we are unable to foresee what will turn out to be useful in the future.
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
Argument  Biannual Philosophical Journal
Argument Biannual Philosophical Journal Arts and Humanities-Religious Studies
自引率
0.00%
发文量
0
审稿时长
16 weeks
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信