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Neo-Russellian Abstractionism.

Bahram Assadian1

  • 1School of Philosophy, Religion and History of Science, University of Leeds, Woodhouse Lane, Leeds, LS2 9JT UK.

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|March 16, 2026
PubMed
Summary

This paper explores a new variant of abstractionism, proposing that number expressions refer to objects. It addresses challenges to this neo-Russellian approach, focusing on ontology, infinity, and reference to numbers.

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.

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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.