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In his early Contributions to a Better-Grounded Presentation of Mathematics (1810) Bernard Bolzano tries to characterize rigorous proofs (strenge Beweise). Rigorous is, prima facie, any proof that indicates the grounds for its conclusion. Bolzano lists a number of methodological constraints all rigorous proofs should comply with, and tests them systematically against a specific collection of elementary inference schemata that, according to him, are evidently of ground-consequence-kind. This paper intends to give a detailed and critical account of the fragmentary logic of the Contributions, and to point out as well some difficulties Bolzano’s attempt runs into, notably as to his methodological ban on ‘kind crossing’.
Abstract: This paper provides answers to some remarks that Mirja Hartimo and Robert Tragesser have made on a number of themes dealt with in my book Logic and Philosophy of Mathematics in the Early Husserl. In the first part (replies to Mirja Hartimo) I focus on the much debated problem of the influences behind Husserl's turn away from his earlier psychologistic positions, and consider in particular the role of Frege and Bolzano in determining Husserl's objectivist turn. In the second part (replies to Robert Tragesser) I discuss Husserl's concept of a universal arithmetic and answer some questions raised by Tragesser on my critical explanation of Husserl's idea that a field of objects can be said to exist, if it is possible to set up an axiomatic existential framework for it. I finally spell out a parallel between Husserl's existential axiomatics and Kit Fine's method of procedural postulationism, trying to bring some evidence for my claims.
Keywords: Edmund Husserl; Robert Tragesser; Mirja Hartimo; philosophy of mathematics; philosophy of logic
“The Intentions behind the Ideas”: On the Question of the Influences behind Husserl's Refutation of Psychologism in the Prolegomena to Pure Logic
“Husserl's idea of pure logic,” according to Mirja Hartimo, “is the cornerstone of his arguments against psychologism (in the Prolegomena to Pure Logic)” Previously she observed that “[m]uch of the secondary literature on Husserl has been devoted to the question of possible influences behind Husserl's rather radical arguments against psychologism in the Prolegomena” and rightly stressed that “one cannot intelligibly address these influences without examining the development of Husserl's conception of logic and mathematics.”
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).
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