A fresh look at research strategies in computational cognitive science: The case of enculturated mathematical problem solving

2019

Erkenn

Self Cite

Following Marr's famous three-level distinction between explanations in cognitive science, it is often accepted that focus on modeling cognitive tasks should be on the computational level rather than the algorithmic level. When it comes to mathematical problem solving, this approach suggests that the complexity of the task of solving a problem can be characterized by the computational complexity of that problem. In this paper, I argue that human cognizers use heuristic and didactic tools and thus engage in cognitive processes that make their problem solving algorithms computationally suboptimal, in contrast with the optimal algorithms studied in the computational approach. Therefore, in order to accurately model the human cognitive tasks involved in mathematical problem solving, we need to expand our methodology to also include aspects relevant to the algorithmic level. This allows us to study algorithms that are cognitively optimal for human problem solvers. Since problem solving methods are not universal, I propose that they should be studied in the framework of enculturation, which can explain the expected cultural variance in the humanly optimal algorithms. While mathematical problem solving is used as the case study, the considerations in this paper concern modeling of cognitive tasks in general.

Section: Humanly Optimal Algorithmsmentioning

confidence: 98%

Cognitive and Computational Complexity: Considerations from Mathematical Problem Solving

2019

Erkenn

Self Cite

“…In the case of mathematical problem solving, this implies that mathematical cognitive processes should be computationally modelled by functions whose values can be computed by algorithms for solving decision problems in the complexity class P. 7 With Regina E. Fabry, we have argued that such a direct connection between computational and cognitive complexity is potentially problematic since the computational complexity approach does not transfer in a straight-forward manner into studying the complexity of human cognitive processes (Pantsar 2019b;Fabry & Pantsar 2019). One of the reasons for this is that human mathematical problem solving often includes visual and heuristic reasoning (e.g., the use of diagrams and spatial manipulation of symbols (Fabry & Pantsar 2019)) that is not usually included in the algorithmic approach to problem solving used in theoretical computer science (Pantsar 2019b). Thus it is important to realize that human mathematical problem solving has its own particular characteristics that do not always correspond to the computational complexity approach.…”

Section: Computational Complexity and Tractable Cognitionmentioning

confidence: 99%

“…As I have argued in (Pantsar 2019b), in mathematics human problem solvers use many heuristic and didactic methods (e.g., diagrams) that involve suboptimal problem solving algorithms. In (Fabry & Pantsar 2019), we argue that mathematical problem solving is a culturally shaped ability that is tightly connected to spatial manipulation of symbols and other ways of engaging with cognitive tools in the problem solving process. Thus the computationally optimal algorithms that are standardly studied in the research of computational complexity can be a bad fit with modelling human problem solving algorithms.…”

Section: Tractable Cognition Thesis Reconsideredmentioning

confidence: 99%

“…The reason for this is that both in terms of input sizes and the applied algorithms, human problem solving processes can greatly differ from those studied in computer science, for which the complexity classes and the Cobham-Edmonds tractability thesis are meant to apply. In addition to the humanly optimal algorithms as studied in (Pantsar 2019b) and (Fabry & Pantsar 2019), this should prompt us to consider bounded inputs, limited to humanly relevant input sizes. 24 One such alternative to the P-cognition thesis has been suggested by van Rooij (2008; see Van Rooij et al 2019 for more details), who argues that by introducing suitable parameters to the computational problem, also super-polynomial time computation can be feasible in modelling cognitive tasks.…”

Section: Tractable Cognition Thesis Reconsideredmentioning

confidence: 99%

“…First, as we will see, computational complexity measures are an important part of the discussion on the theory of computational modelling, both among cognitive scientists and philosophers. While there have been important critical assessments of how computational complexity should be applied in cognitive science (e.g., van Rooij 2008;Isaac et al 2014;Szymanik 2016;Szymanik and Verbrugge 2018;Van Rooij et al 2019;Pantsar 2019b;Fabry & Pantsar 2019), limitations like the above P-cognition thesis are still too often misunderstood and given excessive importance. Second, these problems can extend to applications of descriptive complexity through the connections between computational complexity measures and the complexity of logical systems.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

2020

Minds & Machines

Self Cite

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.

Developing Artificial Human-Like Arithmetical Intelligence (and Why)

2023

Minds & Machines

Why would we want to develop artificial human-like arithmetical intelligence, when computers already outperform humans in arithmetical calculations? Aside from arithmetic consisting of much more than mere calculations, one suggested reason is that AI research can help us explain the development of human arithmetical cognition. Here I argue that this question needs to be studied already in the context of basic, non-symbolic, numerical cognition. Analyzing recent machine learning research on artificial neural networks, I show how AI studies could potentially shed light on the development of human numerical abilities, from the proto-arithmetical abilities of subitizing and estimating to counting procedures. Although the current results are far from conclusive and much more work is needed, I argue that AI research should be included in the interdisciplinary toolbox when we try to explain the development and character of numerical cognition and arithmetical intelligence. This makes it relevant also for the epistemology of mathematics.