解决PvNP需要范式转变吗？
Does resolving PvNP require a paradigm shift?
摘要：
我认为解决PvNP问题需要在可证明性域和可计算性域之间建立一个当否桥。前者涉及人类智慧如何决定数论关系的真理，并由一阶皮亚诺算术PA在戴德金的皮亚诺公设公理化之后正式化。后者涉及人类智能如何计算数论函数的值，并通过遵循图灵对可计算函数的分析的图灵机的操作来形式化。我将表明，这样的桥梁需要一个客观的定义都PA的“算法”的解释，和PA的“实例化”的解释。我将表明，这两种解释都隐含在对PA的主观定义的“标准”解释的定义中。然而，这两种客观可定义的解释的存在和区别-以及前者是合理的而后者不是-被根据PA的“标准”解释的外来假设所掩盖，即亚里士多德的具体化必须适用于自然数的结构N。我将认为，认识到这种信念的虚假性，等待一个范式转变，在我们的看法的应用塔斯基的分析（真理的概念，在语言的演绎科学）的“标准”解释PA。然后，我将证明一个算术公式[F]是PA可证明的，当且仅当[F]在PA的算法解释下解释为真。最后，我将说明如何从Goedel构造的一个形式上“不可判定的”算术命题得出，存在一个Halting型PA公式，根据塔斯基的定义，该公式在PA的合理解释下，在算法上可验证为真，但在算法上不可计算为真。
英文摘要：
I shall argue that a resolution of the PvNP problem requires building an iff bridge between the domain of provability and that of computability. The former concerns how a human intelligence decides the truth of number-theoretic relations, and is formalised by the first-order Peano Arithmetic PA following Dededekind's axiomatisation of Peano's Postulates. The latter concerns how a human intelligence computes the values of number-theoretic functions, and is formalised by the operations of a Turing Machine following Turing's analysis of computable functions. I shall show that such a bridge requires objective definitions of both an `algorithmic' interpretation of PA, and an `instantiational' interpretation of PA. I shall show that both interpretations are implicit in the definition of the subjectively defined `standard' interpretation of PA. However the existence of, and distinction between, the two objectively definable interpretations---and the fact that the former is sound whilst the latter is not---is obscured by the extraneous presumption under the `standard' interpretation of PA that Aristotle's particularisation must hold over the structure N of the natural numbers. I shall argue that recognising the falseness of this belief awaits a paradigm shift in our perception of the application of Tarski's analysis (of the concept of truth in the languages of the deductive sciences) to the `standard' interpretation of PA. I shall then show that an arithmetical formula [F] is PA-provable if, and only if, [F] interprets as true under an algorithmic interpretation of PA. I shall finally show how it then follows from Goedel's construction of a formally `undecidable' arithmetical proposition that there is a Halting-type PA formula which---by Tarski's definitions---is algorithmically verifiable as true, but not algorithmically computable as true, under a sound interpretation of PA.
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