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Full Idea
We can never decide by means of our definitions whether any concept has the number JULIUS CAESAR belonging to it, or whether that same familiar conqueror of Gaul is a number or not.
Gist of Idea
Our definition will not tell us whether or not Julius Caesar is a number
Source
Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §56)
Book Ref
Frege,Gottlob: 'The Foundations of Arithmetic (Austin)', ed/tr. Austin,J.L. [Blackwell 1980], p.68
A Reaction
This has become a famous modern problem. The point is that the definition of a number must explain why this is a number, and not something else. Must you mention that you could use it to count? Count you count using emperors?
| 10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro] |
| 10030 | 'Julius Caesar' isn't a number because numbers inherit properties of 0 and successor [Frege, by George/Velleman] |
| 8690 | From within logic, how can we tell whether an arbitrary object like Julius Caesar is a number? [Frege, by Friend] |
| 10219 | Frege said 2 is the extension of all pairs (so Julius Caesar isn't 2, because he's not an extension) [Frege, by Shapiro] |
| 13889 | Fregean numbers are numbers, and not 'Caesar', because they correlate 1-1 [Frege, by Wright,C] |
| 18142 | One-one correlations imply normal arithmetic, but don't explain our concept of a number [Frege, by Bostock] |
| 11030 | The words 'There are exactly Julius Caesar moons of Mars' are gibberish [Rumfitt on Frege] |
| 9046 | Our definition will not tell us whether or not Julius Caesar is a number [Frege] |
| 18143 | Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set! [Bostock] |
| 13888 | If numbers are extensions, Frege must first solve the Caesar problem for extensions [Wright,C] |
| 8787 | The Julius Caesar problem asks for a criterion for the concept of a 'number' [Hale/Wright] |
| 18164 | Frege solves the Caesar problem by explicitly defining each number [Maddy] |
| 17997 | Some suggest that the Julius Caesar problem involves category mistakes [Magidor] |