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简介《聚焦哥德尔定理》

已有 535 次阅读 2021-9-28 00:00 |个人分类:解读哥德尔不完全性定理|系统分类:科研笔记

聚焦哥德尔定理》(Gödel’s Theorem in Focus)是丽贝卡-戈尔茨坦推荐的一本介绍哥德尔定理的丽贝卡-戈尔茨坦的《不完备性:哥德尔的证明和悖论》中的“阅读建议”

- John Dawson也有两篇关于哥德尔的论文,通俗而有趣。"Kurt Godel in Sharper Focus " "The Reception of Godel's Incompleteness Theorems"。这两篇论文都转载于Stuart Shanker编辑的《聚焦哥德尔定理》,以及其他有趣的文章,包括Solomon Feferman "Kurt Godel: 信念与谨慎Kurt Godel: Conviction and Caution»


  • 一,目录

Stuart Shanker 引言 

I John Dawson:透视哥德尔 

II 库尔特-哥德尔:论《数学原理》和相关系统的形式上不可判定的命题(1931)。

III Stephen Kleene:库尔特-哥德尔的工作 

IV John Dawson: 哥德尔不完备性定理的接纳过程

V Solemon Feferman:库尔特-哥德尔:信念与谨慎

VI Michael Resnik:论一致性证明的哲学意义

VII Michael Detlefsen:论哥德尔第二定理的解释

VIII Stuart Shanker: 维特根斯坦对哥德尔定理意义的评论



二,引言的译文


越来越多的哲学分支学科--从科学哲学,语言哲学到心灵哲学和美学-- 要求他们的学生对哥德尔的不完备性定理有一定的了解。因为哥德尔定理提出的问题正是现代人试图振兴形而上学和/或机械论的核心所在。不幸的是,鉴于哥德尔证明的高度技术性,其证明对于那些没有受过数理逻辑训练的人来说仍然是相对难以企及的。本书旨在满足这些需求,对哥德尔证明的机械论和数学意义做了清晰的介绍。我们以John Dawson哥德尔的简短传记开始,然后是Stephen Kleene对哥德尔在数理逻辑方面的工作的概述。有了这一背景,我们接着讨论数学哲学中围绕哥德尔定理的哲学意义所产生的越来越多的争议。


一些人无疑会认为后一种现象反映了科学和数学发现与哲学理解之间不可避免的时间差,也许这甚至会被视为证实了知识社会学中越来越流行的论点,即只有当一门科学消化了某项突破的全部含义之后,它才会成为哲学家们夸大和歪曲的财产。但与标准的数学结果不同,哥德尔定理与所引发的认识论争议有着千丝万缕的联系;事实上,这一点在哥德尔本人的著作中再明显不过了。因此,哥德尔定理给我们带来一系列哲学问题并不奇怪,其中许多问题我们才刚刚开始认识,更不用说解决了。


首先是哥德尔的第一不完备性定理不寻常接受;用John Dawson的话说,逻辑和数学史上最深刻的发现之一是如何被哥德尔的同时代人迅速地、几乎毫无异议地吸收的呢?”正如Solomon Feferman向我们展示的那样,这个问题与更大的问题密切相关,即哥德尔后来在数理逻辑方面的工作以及他在表达柏拉图主义信念时越来越自信,与他先前对第二个不完备性定理的解释和表述有什么关系,这把我们带到了可以讨论最后三篇论文的主要关注点的地方。首先,Michael Resnik概述了哥德尔定理所造成的怀疑问题的严重性。然后,Michael Detlefsen接受挑战,试图将希尔伯特计划从哥德尔显然留下的僵局中拯救出来。在最后一篇论文中,我根据维特根斯坦试图通过化解哥德尔定理所造成的危机来审视这一困境的性质。

像所有的哲学怀疑问题一样,哥德尔定理所提出的问题充满了可能性,也充满了危险。后者中最主要的是一种不可避免的倾向,即远离这些发展的原点和出发点,因为依赖评论家的结论变得越来越有诱惑力,也越来越容易被接受,而这些评论家本身可能是基于早期的总结来进行解读的。当然,通常的做法是接受某一领域的专家所得到的裁决,而不去检查他们的发现,但这种习惯以一致性前提。本书的最终目的是概述迄今为止取得的共识的基础,然后对其提出质疑。我们希望这将激发人们对哥德尔不完备性定理的意义的持续解释的兴趣。


三,原文


Gödel's Theorem in Focus, Croom Helm 1988, S.G. Shanker (ed.).


Contents

Preface

I John W. Dawson, Jr. : Kurt Godel in Sharper Focus 

II Kurt Godel : On Formally Undecidable Propositions of Principia Mathematica and Related Systems I (1931)

III Stephen C. Kleene : The Work of Kurt Godel 

IV John W. Dawson, Jr.: The Reception of Godel’s Incompleteness Theorems

V Solemon Feferman : Kurt Godel : Conviction and Caution

VI Michael D. Resnik : On the Philosophical Significance of Consistency Proofs

VII Michael Detlefsen : On Interpreting Godel’s Second Theorem

VIII S.G.Shanker : Wittgenstein’s Remarks on the Significance of Godel’s Theorem


Preface 


An ever-growing number of sub-disciplines in philosophy—ranging from the philosophies of science and language to the philosophy of mind and aesthetics—now demand a working acquaintance with Kurt Gödel’s incompleteness theorems from their students. For Gödel’s theorems raise issues which lie at the very heart of modem attempts to revitalise metaphysics and/or the Mechanist Thesis. Unfortunately, given the highly technical nature of Gödel’s proof these debates have remained relatively inaccessible to those not trained in mathematical logic. The present book has been designed to meet these needs by providing a lucid introduction to the mechanics and mathematical import of Gödel’s proof. We begin with a short biographical sketch of Kurt Gödel by John W.Dawson, Jr., followed by Stephen Kleene’s overview of Gödel’s work in mathematical logic. With this background in place we then address the mounting controversy in the philosophy of mathematics surrounding the philosophical significance of Gödel’s theorems. 


Some will no doubt regard the latter phenomenon as a reflection of the inevitable time-lag between scientific and mathematical discoveries versus philosophical comprehension. Perhaps it will even be seen to corroborate the increasingly popular thesis in the sociology of knowledge that it is only once a science has digested the full implications of a breakthrough that it becomes the property of philosophers to exaggerate and distort. But unlike standard mathematical results Gödel’s theorems are inextricably linked to the epistemological disputes which they have sparked off; indeed, nowhere could this be more evident than in the writings of Gödel himself. It is not surprising, therefore, that Gödel’s theorems should present us with a catalogue of philosophical problems, many of which we are only just beginning to recognise, let alone resolve. 


To begin with there is the anomalous reception of Gödel’s first incompleteness theorem; in the words of John W.Dawson, Jr., how was it that ‘one of the most profound discoveries in the history of logic and mathematics was assimilated promptly and almost without objection by Gödel’s contemporaries’? As Solomon Feferman shows us, this issue is intimately connected with the larger question of how Gödel’s subsequent work in mathematical logic and the growing confidence with which he expressed his platonist convictions relate to his earlier interpretation and presentation of the second incompleteness theorem. This brings us to the point where we can address the main concerns of the final three papers. First Michael Resnik outlines the seriousness of the sceptical problem created by Gödel’s theorems. Michael Detlefsen then takes up the challenge of attempting to rescue Hilbert’s Programme from the impasse in which Gödel apparently left it. In the final paper I examine the nature of this dilemma in the light of Wittgenstein’s attempt to resolve by dissolving the crisis created by Gödel’s theorems.


Like all philosophical sceptical problems the issues raised by Gödel’s theorems are pregnant with possibilities and fraught with dangers. Chief amongst the latter is an inevitable tendency to become distanced from the fons et origo of these developments. For it becomes ever more tempting and acceptable to rely on the findings of commentators who might themselves have based their readings on earlier summaries. To be sure, it is common practice to accept the verdict obtained by the experts in a field without inspecting their findings. But such custom presupposes concord. The ultimate aim of this book has been to outline the basis of the consensus which has hitherto obtained in order then to question it. It is our hope that this will stimulate renewed interest in the ongoing interpretation of the significance of Gödel’s incompleteness theorems. 




参考资料:

https://pdfslide.net/reader/f/godels-theorem-in-focus-philosophers-in-focus




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