The articles in this issue can be divided into three groups. Krajewski’s article, Yong Cheng’s contribution, and a short note by Rudy Rucker, provide detailed mathematical analysis of Lucas-Penrose type arguments. In the second group, with articles by Arnon Avron, Stepan Holub, Panu Raaikiainen, and Albert Visser, the authors discuss the status and various methodological and technical problems of the anti-mechanist arguments. In essence: what does the problem of “minds vs. machines” really mean, and how can it, and how should it, be formulated? Moreover: How to evaluate the merit of arguments that mix formal mathematics and philosophical considerations? The third group consists of the articles that, while including issues from the other two groups, concentrate of more specific themes: an analysis of Georg Kreisel’s observation that it does not logically follow from the fact that a formal system is subject to the second Gödel incompleteness theorems that there are absolutely no means available to prove its consistency (Jeff Buechner); Per Martin-Löf’s proof that there are no absolute unknowables in constructive mathematics (V. Alexis Peluce); diagonal arguments and Chomsky’s approach to linguistic competence as contrasted with arithmetic competence (David Kashtan); and the role in the anti-mechanist arguments of difficulties in capturing the nature of natural numbers in formal systems (Paula Quinon).
Avigad, J. (2018). The Mechanization of Mathematics. Notices of the American Mathematical Society, 65(6), 681–690.
Koellner, P. (2016). Gödel’s Disjunction. In: L. Horsten, P. Welch (Eds.), Gödel’s Disjunction: The Scope and Limits of Mathematical Knowledge (pp. 148–188). Oxford University Press.
Koellner, P. (2018a). On the Question of Whether the Mind Can Be Mechanized. Part 1: From Gödel to Penrose. Journal of Philosophy, 115(7), 337–360.
Koellner, P. (2018b). On the Question of Whether the Mind Can Be Mechanized. Part 2: Penrose’s New Argument. Journal of Philosophy, 115(9), 453–484.
Smoryński, C. (1988). Hilbert’s Programme. CWI Quarterly, 1(4), 3–59.
Wolfram, S. (2018). Logic, Explainability and the Future of Understanding. Retrieved from: https://writings.stephenwolfram.com/2018/11/logic-explainability-and-the-future-of-understanding/