Logicism, Interpretability, and Knowledge of Arithmetic

Abstract

A crucial part of the contemporary interest in logicism in the philosophy of mathematics resides in its idea that arithmetical knowledge may be based on logical knowledge. Here, an implementation of this idea is considered that holds that knowledge of arithmetical principles may be based on two things: (i) knowledge of logical principles and (ii) knowledge that the arithmetical principles are representable in the logical principles. The notions of representation considered here are related to theory-based and structure-based notions of representation from contemporary mathematical logic. It is argued that the theory-based versions of such logicism are either too liberal (the plethora problem) or are committed to intuitively incorrect closure conditions (the consistency problem). Structure-based versions must on the other hand respond to a charge of begging the question (the circularity problem) or explain how one may have a knowledge of structure in advance of a knowledge of axioms (the signature problem). This discussion is significant because it gives us a better idea of what a notion of representation must look like if it is to aid in realizing some of the traditional epistemic aims of logicism in the philosophy of mathematics.

The Strength of Abstraction With Predicative Comprehension

“…While the Russell paradox shows that Basic Law V is inconsistent with the full comprehension schema (1.2) (cf. [29] p. 1682), nevertheless Basic Law V is consistent with predicative restrictions, as was shown by Parsons ([25]), Heck ([19]), and Ferreira-Wehmeier ( [9]). This thus suggests the project of understanding whether there is a version of Frege's theorem centered around the consistent predicative fragments of the Grundgesetze.…”

“…The traditional proof of Frege's Theorem uses instances of this comprehension schema in which some of the formulas in question contain higher-order quantifiers (cf. [29] p. 1690 equations (44)-(45)). However, there is a long tradition of predicative mathematics, in which one attempts to ascertain how much one can accomplish without directly appealing to such instances of the comprehension schema.…”

“…Roughly, Robinson's Q is the fragment of first-order Peano arithmetic obtained by removing all the induction axioms. (For a precise definition of Robinson's Q, see [17] p. 28, [26] p. 4, [29] Here Q m is the expansion of Robinson's Q by finitely many primitive recursive function symbols and their defining equations along with induction for bounded formulas ([3] pp. 60-63), so that Burgess records the prediction that predicative Fregean theories will be interpretable in weak arithmetics.…”

“…This paper is part of a series of three papers, the other two being [31] and [32]. These papers collectively constitute a sequel to our paper [29], particularly as it concerns the methods and components related to Basic Law V. In that earlier paper, we showed that Hume's Principle (1.1) with predicative comprehension did not interpret second-order Peano arithmetic with predicative comprehension (cf. [29] p. 1704).…”

On the Consistency of the Arithmetic System

“…In consequence, one can obtain another arithmetic system (cf. [1], [2], [4], [6], [7], [10], [13], [14], [16], [17], [18], [21], [23], [25], [28], [30], [31], [32], [34], [36], [37], [40], [47], [48], [50], [51], [52], [53], [54], [58], [59], [62], [71], [72], [77], [78]), namely,…”