| 10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro] |
| Full Idea: Cantor's theorem entails that there are more property extensions than objects. So there are not enough objects in any domain to serve as extensions for that domain. So Frege's view that numbers are objects led to the Caesar problem. | |
| From: report of George Cantor (works [1880]) by Stewart Shapiro - Philosophy of Mathematics 4.6 | |
| A reaction: So the possibility that Caesar might have to be a number arises because otherwise we are threatening to run out of numbers? Is that really the problem? |
| 10030 | 'Julius Caesar' isn't a number because numbers inherit properties of 0 and successor [Frege, by George/Velleman] |
| Full Idea: 'Julius Caesar' is not a natural number in Frege's account because he does not fall under every concept under which 0 falls and which is hereditary with respect to successor. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.2 | |
| A reaction: Significant for structuralist views. One might say that any object can occupy the structural place of '17', but if you derive your numbers from 0, successor and induction, then the 17-object must also inherit the properties of zero and successors. |
| 8690 | From within logic, how can we tell whether an arbitrary object like Julius Caesar is a number? [Frege, by Friend] |
| Full Idea: The 'Julius Caesar problem' in Frege's theory is that from within logic we cannot tell if an arbitrary objects such as Julius Caesar is a number or not. Logic itself cannot tell us enough to distinguish numbers from other sorts of objects. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Michèle Friend - Introducing the Philosophy of Mathematics 3.4 | |
| A reaction: What a delightful problem (raised by Frege himself). A theory can look beautiful till you ask a question like this. Only a logician would, I suspect, get into this mess. Numbers can be used to count or order things! "I've got Caesar pencils"? |
| 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] |
| Full Idea: Frege proposed that the number 2 is a certain extension, the collection of all pairs. Thus, 2 is not Julius Caesar because, presumably, persons are not extensions. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Stewart Shapiro - Philosophy of Mathematics 3.2 | |
| A reaction: Unfortunately, as Shapiro notes, Frege's account of extension went horribly wrong. Nevertheless, this seems to show why the Julius Caesar problem does not matter for Frege, though it might matter for the neo-logicists. |
| 13889 | Fregean numbers are numbers, and not 'Caesar', because they correlate 1-1 [Frege, by Wright,C] |
| Full Idea: We cannot reasonably suppose that any numerical singular term has the same reference as 'Caesar', because Frege's numbers (unlike persons) are to be identified and distinguished by appeal to facts about 1-1 correlation among concepts. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Crispin Wright - Frege's Concept of Numbers as Objects 3.xiv |
| 18142 | One-one correlations imply normal arithmetic, but don't explain our concept of a number [Frege, by Bostock] |
| Full Idea: Frege inferred from the Julius Caesar problem that even though Hume's Principle sufficed as a single axiom for deducing the arithmetic of the finite cardinal numbers, still it does not explain our ordinary understanding of those numbers. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by David Bostock - Philosophy of Mathematics 9.A.2 |
| 11030 | The words 'There are exactly Julius Caesar moons of Mars' are gibberish [Rumfitt on Frege] |
| Full Idea: The word 'Julius Caesar is prime' may well involve some kind of category error, but the still compose a grammatical sentence. The words 'There are exactly Julius Caesar moons of Mars', by contrast, are gibberish. | |
| From: comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Ian Rumfitt - Concepts and Counting p.48 |
| 9046 | Our definition will not tell us whether or not Julius Caesar is a number [Frege] |
| 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. | |
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §56) | |
| 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? |
| 18143 | Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set! [Bostock] |
| Full Idea: The Julius Caesar problem was one reason that led Frege to give an explicit definition of numbers as special sets. He does not appear to notice that the same problem affects his Axiom V for introducing sets (whether Caesar is or is not a set). | |
| From: David Bostock (Philosophy of Mathematics [2009], 9.A.2) | |
| A reaction: The Julius Caesar problem is a sceptical acid that eats into everything in philosophy of mathematics. You give all sorts of wonderful accounts of numbers, but at what point do you know that you now have a number, and not something else? |
| 13888 | If numbers are extensions, Frege must first solve the Caesar problem for extensions [Wright,C] |
| Full Idea: Identifying numbers with extensions will not solve the Caesar problem for numbers unless we have already solved the Caesar problem for extensions. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 3.xiv) |
| 8787 | The Julius Caesar problem asks for a criterion for the concept of a 'number' [Hale/Wright] |
| Full Idea: The Julius Caesar problem is the problem of supplying a criterion of application for 'number', and thereby setting it up as the concept of a genuine sort of object. (Why is Julius Caesar not a number?) | |
| From: B Hale / C Wright (Logicism in the 21st Century [2007], 3) | |
| A reaction: One response would be to deny that numbers are objects. Another would be to derive numbers from their application in counting objects, rather than the other way round. I suspect that the problem only real bothers platonists. Serves them right. |
| 18164 | Frege solves the Caesar problem by explicitly defining each number [Maddy] |
| Full Idea: To solve the Julius Caesar problem, Frege requires explicit definitions of the numbers, and he proposes his well-known solution: the number of Fs = the extension of the concept 'equinumerous with F' (based on one-one correspondence). | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.1) | |
| A reaction: Why do there have to be Fs before there can be the corresponding number? If there were no F for 523, would that mean that '523' didn't exist (even if 522 and 524 did exist)? |
| 17997 | Some suggest that the Julius Caesar problem involves category mistakes [Magidor] |
| Full Idea: Various authors have argued that identity statements arising in the context of the 'Julius Caesar' problem in philosophy of mathematics constitute category mistakes. | |
| From: Ofra Magidor (Category Mistakes [2013], 1.1 n1) | |
| A reaction: [She cites Benacerraf 1965 and Shapiro 1997:79] |