Disentangling Metamathematics from Cognitive Science


Many have mused on whether Gödel’s celebrated Incompleteness Theorems have genuine non-metamathematical applications, especially in discussions about human and artificial minds (e.g. Nagel and Newman (1958), Lucas (1961), Penrose (1989)). Typically, the theorems have been used as cornerstones of arguments which purport to show that the mind cannot possibly coincide with any idealized finite machine. Such proposals will be called “negative” Gödelian theses, for they attempt to impose limitations on orthodox computational accounts of the mental. Furthermore, they call for drastic revisions of Cognitive Science. The first half of the paper is devoted to diagnosing, by means of a couple of thought experiments, some of the shortcomings that I maintain all negative proposals suffer from. The same theorems have rather curiously been used to actually illuminate and vindicate computational accounts of the mental, instead of pointing towards inadequacies of such accounts. For instance, Hofstadter (1979, 2007) advances a “positive” Gödelian thesis by taking Gödel’s work to be the key to understanding the relationship between animate beings (such as humans) and their inanimate components (such as neurons). His proposal has not been directly confronted in academic papers. Consequently, in the second half of the paper I will aim to show that Hofstadter’s arguments do not actually hinge on anything that is purely metamathematical, so the incompleteness theorems are not a quintessential part of his project (even though he takes them to be). Actually, the insights that Hofstadter actually needs can be extracted from mathematical areas that are not in the ballpark of Logic. As an extra, I will also briefly comment on some problems concerning his remarks about the aboutness of metamathematical sentences (e.g. the Henkin sentence). By engaging with both positive and negative theses, this paper aims to call for a divorce between metamathematics and neuropsychology, broadly construed.

April 24, 2021
The 24th annual CUNY Graduate Philosophy Conference
New York