Philosophical Studies

, Volume 175, Issue 7, pp 1631–1660 | Cite as

Julius Caesar and the numbers

  • Nathan SalmónEmail author


This article offers an interpretation of a controversial aspect of Frege’s The Foundations of Arithmetic, the so-called Julius Caesar problem. Frege raises the Caesar problem against proposed purely logical definitions for ‘0’, ‘successor’, and ‘number’, and also against a proposed definition for ‘direction’ as applied to lines in geometry. Dummett and other interpreters have seen in Frege’s criticism a demanding requirement on such definitions, often put by saying that such definitions must provide a criterion of identity of a certain kind (for numbers or for linear directions). These interpretations are criticized and an alternative interpretation is defended. The Caesar problem is that the proposed definitions fail to well-define ‘number’ and ‘direction’. That is, the proposed definitions, even when taken together with the extra-definitional facts (such as that Caesar is not a number and that England is not a direction), fail to fix unique semantic extensions for ‘number’ and ‘direction’, and thereby fail to fix unique truth-values for sentences like ‘Caesar is a number’ and ‘England is a direction’. A minor modification of the criticized definitions well-defines ‘0’, ‘successor’ and ‘number’, thereby avoiding the Caesar problem as Frege understands it, but without providing any criterion of number identity in the usual sense.


Abstraction principle Julius Caesar problem Frege Hume’s principle Logicism 



I am indebted to Harry Deutsch, Steven Humphrey, Gary Kemp, Oliver Marshall, Michael Rescorla, Teresa Robertson, and Clark Sexton for discussion and comments, to Jutta Schamp for assistance in translation, and to Yoko Graham Ishii (age 6) for correcting a transcription error.

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Department of PhilosophyUniversity of CaliforniaSanta BarbaraUSA

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