In computational complexity theory, decision problems are divided into complexity classes based on the amount of computational resources it takes for algorithms to solve them. In theoretical computer science, it is commonly accepted that only functions for solving problems in the complexity class P, solvable by a deterministic Turing machine in polynomial time, are considered to be tractable. In cognitive science and philosophy, this tractability result has been used to argue that only functions in P can feasibly work as computational models of human cognitive capacities. One interesting area of computational complexity theory is descriptive complexity, which connects the expressive strength of systems of logic with the computational complexity classes. In descriptive complexity theory, it is established that only first-order (classical) systems are connected to P, or one of its subclasses. Consequently, second-order systems of logic are considered to be computationally intractable, and may therefore seem to be unfit to model human cognitive capacities. This would be problematic when we think of the role of logic as the foundations of mathematics. In order to express many important mathematical concepts and systematically prove theorems involving them, we need to have a system of logic stronger than classical first-order logic. But if such a system is considered to be intractable, it means that the logical foundation of mathematics can be prohibitively complex for human cognition. In this paper I will argue, however, that this problem is the result of an unjustified direct use of computational complexity classes in cognitive modelling. Placing my account in the recent literature on the topic, I argue that the problem can be solved by considering computational complexity for humanly relevant problem solving algorithms and input sizes.

中文翻译：

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描述的复杂性、计算的可处理性以及数学的逻辑和认知基础

在计算复杂性理论中，决策问题根据算法解决它们所需的计算资源量分为复杂性类别。在理论计算机科学中，人们普遍认为只有用于解决复杂性类 P 中的问题的函数，可在多项式时间内由确定性图灵机解决，才被认为是易处理的。在认知科学和哲学中，这种易处理性结果已被用来论证只有 P 中的函数才能作为人类认知能力的计算模型可行。计算复杂性理论的一个有趣领域是描述性复杂性，它将逻辑系统的表达强度与计算复杂性类别联系起来。在描述复杂性理论中，确定只有一阶（经典）系统连接到 P 或其子类之一。因此，二阶逻辑系统被认为是计算上难以处理的，因此似乎不适合模拟人类的认知能力。当我们将逻辑的作用视为数学的基础时，这将是有问题的。为了表达许多重要的数学概念并系统地证明涉及它们的定理，我们需要一个比经典一阶逻辑更强大的逻辑系统。但如果认为这样的系统难以处理，则意味着数学的逻辑基础对于人类认知来说可能过于复杂。然而，在本文中，我将论证，这个问题是在认知建模中不合理地直接使用计算复杂性类的结果。将我的帐户放在有关该主题的最新文献中，我认为可以通过考虑人类相关问题解决算法和输入大小的计算复杂性来解决该问题。