The Algorithmicity of Mathematical Cognition

Abstract

This article purports to establish the philosophical inappropriateness of using established theorems in mathematical logic, such as Gödel's (1931) first incompleteness theorem, in order to conclude that human minds have a non-algorithmic nature. First, I will argue that the ongoing debate in the philosophy of mathematics concerning absolute provability is fully independent of the question whether our brains are biologically instantiated computers or not. Second, through a combination of evolutionary considerations and the phenomenon of vagueness, I will demonstrate the fragility of the ill-defined notion of an idealized mathematical mind, which plays a key role in the anticomputationalist arguments. Third, I will make the case that all these arguments adopt a psychologically implausible conception of the nature of the high-level algorithms behind our mathematical abilities. Lastly, I look at the recent advances in artificial intelligence in order to illustrate the philosophical sleight of hand on which the anticomputationalist argument operates.