Erkenntnis 86 (4):961-997 (2021)

Authors
Markus Pantsar
University of Helsinki
Abstract
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.
Keywords computational complexity  mathematical problem solving  computational modelling  cognitive complexity  mathematical cognition  philosophy of mathematics  philosophy of cognitive science
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DOI 10.1007/s10670-019-00140-3
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References found in this work BETA

Aspects of the Theory of Syntax.Noam Chomsky - 1965 - Cambridge, MA, USA: MIT Press.
On Computable Numbers, with an Application to the N Tscheidungsproblem.Alan Turing - 1936 - Proceedings of the London Mathematical Society 42 (1):230-265.
Physical Symbol Systems.Allen Newell - 1980 - Cognitive Science 4 (2):135-83.
The Knowledge Level.Allen Newell - 1982 - Artificial Intelligence 18 (1):81-132.

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