Neo-Russellian Abstractionism

TL;DR

This paper examines a philosophical approach to numbers that combines ideas from Russell and neo-Fregeanism, addressing challenges to its viability.

Contribution

It proposes a neo-Russellian abstractionism that addresses challenges related to ontological modesty and arithmetical platonism.

Findings

The neo-Russellian approach lacks ontological modesty compared to traditional abstraction principles.

It fails to ground the infinity of natural numbers due to its defused objectual character.

The approach does not support identifying reference to numbers as particular objects.

Abstract

A central thesis of neo-Fregean abstractionism is that numerical expressions of the form ‘the number of Fs’, introduced by Hume’s Principle, should be read as genuine singular terms whose semantic function is to refer to particular objects. This paper explores the prospects of a variant of abstractionism in which such expressions have existential assertoric content, as in Russell’s analysis of definite descriptions. The neo-Russellian abstractionist faces three initial challenges: (i) the Russellian rendering of Hume’s Principle does not retain the ontological modesty that any admissible abstraction principle must respect. (ii) It defuses the objectual character of natural numbers, and thereby fails to ground their infinity. And (iii) it does not involve the means for engaging in identifying reference to numbers as particular objects, thereby constituting a derogation of arithmetical platonism. I shall investigate these challenges and propose solutions to address them.