Neo-Russellian Abstractionism

Area of Science:

• Philosophy of Mathematics
• Logic and Analytic Philosophy

Background:

• Neo-Fregean abstractionism posits that numerical expressions like 'the number of Fs' are singular terms referring to objects.
• Hume's Principle is central to this view, linking numerical terms to properties of sets.

Purpose of the Study:

• To explore a variant of abstractionism where numerical expressions have existential assertoric content, akin to Russell's analysis of descriptions.
• To investigate and propose solutions for three key challenges facing this neo-Russellian abstractionist approach.

Main Methods:

• Analysis of the semantic and ontological implications of applying Russell's theory of descriptions to Hume's Principle.
• Examination of challenges related to ontological modesty, grounding the infinity of natural numbers, and enabling identifying reference to numbers.

Main Results:

• The Russellian rendering of Hume's Principle faces challenges in maintaining ontological modesty.
• This approach may undermine the objectual character of natural numbers and their infinity.
• It potentially hinders the ability to refer to numbers as particular objects, impacting Platonism.

Conclusions:

• The paper investigates the viability of a neo-Russellian abstractionism.
• It proposes solutions to address the identified challenges concerning ontology, number infinity, and reference.
• The study aims to refine abstractionist theories in the philosophy of mathematics.