[论文解读] Actor Programming Languages

Carl Hewitt|arXiv (Cornell University)|Jul 20, 2009

Computability, Logic, AI Algorithms被引用 4

一句话总结

本文挑战了人们普遍认为哥德尔证明了数学无法证明自身一致性的观点，主张由于类型语法中缺乏必要的固定点，哥德尔证明中使用的自指句子无法被构造。本文在不牺牲表达力的前提下确立了数学的一致性，将不一致鲁棒性定位为数学发展的进步力量。

ABSTRACT

Inconsistency Robustness is performance of information systems with pervasively inconsistent information. Inconsistency Robustness of the community of professional mathematicians is their performance repeatedly repairing contradictions over the centuries. In the Inconsistency Robustness paradigm, deriving contradictions have been a progressive development and not game stoppers. Contradictions can be helpful instead of being something to be swept under the rug by denying their existence, which has been repeatedly attempted by Establishment Philosophers (beginning with some Pythagoreans). Such denial has delayed mathematical development. This article reports how considerations of Inconsistency Robustness have recently influenced the foundations of mathematics for Computer Science continuing a tradition developing the sociological basis for foundations. The current common understanding is that Godel proved Mathematics cannot prove its own consistency, if it is consistent. However, the consistency of mathematics is proved by a simple argument in this article. Consequently, the current common understanding that Godel proved Mathematics cannot prove its own consistency, if it is consistent is inaccurate. Wittgenstein long ago showed that contradiction in mathematics results from the kind of self-referential sentence that Godel used in his argument that mathematics cannot prove its own consistency. However, using a typed grammar for mathematical sentences, it can be proved that the kind self-referential sentence that Godel used in his argument cannot be constructed because required the fixed point that Godel used to the construct the self-referential sentence does not exist. In this way, consistency of mathematics is preserved without giving up power.

研究动机与目标

• 挑战普遍认为哥德尔证明了数学无法证明自身一致性的解读。
• 证明数学中的矛盾并非致命，而是可以通过不一致鲁棒性系统性地管理。
• 表明哥德尔不完全性定理核心的自指句子在类型数学语法中无法构造。
• 在通过语法约束确保一致性的前提下，保留数学的完整表达力。
• 通过强调矛盾修复在数个世纪以来的作用，重新建立数学基础的社会学与历史基础。

提出的方法

• 使用类型语法形式化分析哥德尔证明中自指句子的句法结构。
• 证明构造哥德尔自指句子所需的固定点在类型语法框架中不存在。
• 应用维特根斯坦对自指的批判，表明此类句子在类型系统中在句法上不一致。
• 通过排除可能形成问题自指句子的可能性，重构数学一致性的证明。
• 主张不一致鲁棒性——数学家历史上实践的处理矛盾的方式——可被形式化为一致数学发展的基础。
• 将哥德尔定理重新解释为并非一致性的障碍，而是无类型系统中句法越界的后果。

实验结果

研究问题

• RQ1 哥德尔不完备性定理证明中使用的自指句子是否能在类型数学语法中构造？
• RQ2 类型语法中固定点的缺失是否使哥德尔关于数学无法证明自身一致性的论点失效？
• RQ3 如果通过语法约束排除自指悖论，数学是否既能保持强大又可保持一致？
• RQ4 数学史上数学家解决矛盾的实践如何为现代数学基础提供启示？
• RQ5 普遍认为哥德尔证明了数学无法证明自身一致性的观点，是否准确反映了他工作的本意？

主要发现

• 由于所需固定点不存在，哥德尔证明核心的自指句子在类型语法系统中无法构造。
• 通过排除句法上无效的自指构造，数学的一致性得以保留，且未牺牲表达力。
• 普遍认为哥德尔证明了数学无法证明自身一致性的解读是不准确的，源于对其定理的误读。
• 数学中的矛盾并非本质上具有破坏性，历史上反而通过修复与完善推动了发展。
• 不一致鲁棒性——即承认并解决矛盾——是一种可行且经历史验证的数学发展基础。
• 维特根斯坦对自指的批判通过形式证明得到验证：此类句子在类型系统中不可构造。