Munich Center for Mathematical Philosophy (MCMP)

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Frege's Original Conception of Number

Time: Fr 18. Jan. - Sa 19 Jan. 2013

One of Frege's main contributions to the philosophy of mathematics was his novel analysis of the concept of number. His original proposal for how to reconstruct the natural numbers as equivalence classes was, of course, undermined by Russell's antinomy. Consequently, alternative proposals, by Russell, Zermelo, von Neumann, etc., took center stage.

Since the 1980s, there has been a general revival of interest in Frege's philosophy of mathematics, based on the use of neo-Fregean abstraction principles. While exploring the latter, new light has been shed on Frege's original conception of number as well, e.g., on the fact that it is not undermined by Russell's antinomy directly and completely, but only if combined with additional set-theoretic principles. The time seems ripe to pursue such issues more systematically. In this workshop, the goal is to investigate the following aspects further:

  1. The historical background and philosophical motivation for Frege's original conception of number
  2. Relevant developments within Frege's own writings, e.g., from Foundations to Basic Laws
  3. The precise relationship of Frege's construction to Russell's and similar antinomies
  4. Connections to issues about non-well-foundedness in axiomatic set theory
  5. Comparisons to its Russellian, set-theoretic, and neo-logicist alternatives


  • Patricia Blanchette (University of Notre Dame, USA)
  • Roy Cook (University of Minnesota at Minneapolis, USA)
  • Philip Ebert (University of Stirling, UK)
  • Thomas Forster (University of Cambridge, UK)
  • Øystein Linnebo (Birkbeck College, University of London, UK)
  • Erich Reck (University of California at Riverside, USA)


Friday, January 18:
09:00 - 09:15   Welcome and Introduction
09:15 - 11:00 Erich Reck Motivating Frege’s Conception of Number
11:15 - 13:00 Øystein Linnebo Logical Objects and Frege's Context Principle
14:15 - 16:00 Patricia Blanchette The Breadth of the Paradox
Additional talk (beyond the workshop, but not unrelated):
16:15 - 18:00 Thomas Forster Some Background to Holmes’ Recent Proof of the Consistency of NF
Saturday, January 19:
09:15 - 11:00 Philip Ebert Frege on Basic Law V and Hume’s Principle
11:15 - 13:00 Roy Cook Frege Cardinals and Neo-Fregeanism
14:15 - 16:00 Thomas Forster Implementing Cardinals and Other Mathematical Objects as Sets in NF


  • Erich Reck (UC Riverside & MCMP)
  • Roy Cook (University of Minnesota at Minneapolis)