Descriptive Complexity, Computational Tractability, and the Logical and Cognitive Foundations of Mathematics

Abstract

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.

On the Learnability of Physical Concepts: Can a Neural Network Understand What's Real?

“…Humans are capable of understanding concepts that require higher-order logic. It is intriguing that there is speculation, and some evidence, that human brains may encode second-order logic [21]. However, it should be clear that data-driven methods to discover laws, rules, causal factors and constraints, including laws of physics, can understand no more than what they can encode, which is at present limited.…”

“…Both classes of architectures can express the language of first-order logic, which cannot directly capture some basic problems ubiquitous in perception such as latent-variable hard subset selection (e.g., robust linear regression with a preponderance of high-cost outliers [9]). It has been argued [21] that human cognition can capture second-order logic, despite it modeling intractable computational problems, and that bounded resources in the brain should not be equated with restriction to P-class complexity problems. We do not touch upon the vast literature in biological perception, cognitive neuroscience, and philosophy, which are well beyond our scope here, but discuss open problems throughout the manuscript, in Section 4, and in the appendix.…”

From Maximal Intersubjectivity to Objectivity: An Argument from the Development of Arithmetical Cognition

“…There are several reasons to believe that there can be unknowable mathematical truths. First, there are mathematical problems that are generally considered to be too complex computationally to be solved for sufficiently large inputs (Arora and Barak 2007;Pantsar 2021b). Second, Gödel (1931) proved that no consistent formal system strong enough to express arithmetic can be complete, i.e., prove all true sentences in the system.…”

On the development of geometric cognition: Beyond nature vs. nurture

“…Leaving aside potential problems with different understandings of the word "intuition", the underlying message appears to be clear. While they certainly seem to downplay the cognitive complexity (Pantsar, 2021c(Pantsar, , 2021d of Euclidean geometry, Spelke and colleagues are not claiming it is "natural geometry" because we have straight-forward, natural cognitive access to it. What they are claiming, to the best of my understanding, is that Euclidean geometry is in an important way based on cognitive abilities that are easily acquired due to their close relation to the cognitive core systems, which are the product of biological (rather than cultural) evolution.…”