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  • Print publication year: 2013
  • Online publication date: March 2014

Reply to Mark van Atten: on Husserl-Computable Functions

from In Review

Summary

Abstract: In this note I discuss two specific issues raised by Mark van Atten in his review of my book Logic and Philosophy of Mathematics in the Early Husserl, both concerning my reconstruction of Husserl's elaboration of the notion of “conceivable arithmetical operation” in chapter 13 of his Philosophy of Arithmetic. The first issue is more technical in nature: it concerns the role of a kind of restricted inversion operator in the proof of the equivalence between the class of partial recursive functions and that of the Husserl-computable functions (as I defined them in my book). The second issue has to do with the absence, in Husserl's reflections on computable operations, of a sort of “Husserl thesis” analogous to the Church-Turing thesis. I argue that the theoretical issues and the philosophical context which underlie Husserl's reflections on the totality of “conceivable arithmetical operations” are completely different from those motivating the elaboration of computability theory in the 1930s by Godel, Church, Kleene, Turing and others.

Keywords: Edmund Husserl; Mark van Atten; computable functions, Church-Turing thesis

Mark van Atten's review of my book Logic and Philosophy of Mathematics in the Early Husserl mainly focuses on two very specific issues:

i. the claim (see EH, appendix 1 to chapter 1) that “the class H of functions that Husserl defines in the Philosophy of Arithmetic is extensionally equivalent to the class of partial recursive functions” (MvA, 371); and

ii. the claim (see EH, chapter 3, in particular §3.6) that “the notion of (relative) definiteness of an axiom system is to be understood as its syntactical completeness” (MvA, 371).