| 9838 | Treating 0 as a number avoids antinomies involving treating 'nobody' as a person [Frege, by Dummett] |
| Full Idea: Frege's point was that by treating 0 as a number, we run into none of the antinomies that result from treating 'never' as the name of a time, or 'nobody' as the name of a person. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Michael Dummett - Frege philosophy of mathematics Ch.8 | |
| A reaction: I don't think that is a good enough reason. Daft problems like that are solved by settling the underlying proposition or logical form (of a sentence containing 'nobody') before one begins to reason. Other antinomies arise with zero. |
| 9564 | For Frege 'concept' and 'extension' are primitive, but 'zero' and 'successor' are defined [Frege, by Chihara] |
| Full Idea: In Frege's system 'concept' and 'extension of a concept' are primitive notions; whereas 'zero' and 'successor' are defined. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Charles Chihara - A Structural Account of Mathematics 7.5 | |
| A reaction: This is in contrast to the earlier Peano Postulates for arithmetic, which treat 'zero' and 'successor' as primitive. Interesting, given that Frege is famous for being a platonist. |
| 10551 | If objects exist because they fall under a concept, 0 is the object under which no objects fall [Frege, by Dummett] |
| Full Idea: On Frege's approach (of accepting abstract objects if they fall under a concept) the existence of the number 0, from which the series of numbers starts, is of course guaranteed by the citation of a concept under which nothing falls. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Michael Dummett - Frege Philosophy of Language (2nd ed) Ch.14 | |
| A reaction: Frege cites the set of all non-self-identical objects, but he could have cited the set of circular squares. Given his Russell Paradox problems, this whole claim is thrown in doubt. Actually doesn't Frege's view make 0 impossible? Am I missing something? |
| 8653 | Nought is the number belonging to the concept 'not identical with itself' [Frege] |
| Full Idea: I define nought as the Number which belongs to the concept 'not identical with itself'. ...I choose this definition as it can be proved on purely logical grounds. | |
| From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §74) | |
| A reaction: An important part of Frege's logicist programme, along with his use of Hume's Principle (Idea 8649). He needed a prior definition of 'Number' (in §68). Clever, but intuitively a rather weird idea of zero. It is more of an example than a definition. |
| 9837 | 0 is not a number, as it answers 'how many?' negatively [Husserl, by Dummett] |
| Full Idea: Husserl contends that 0 is not a number, on the grounds that 'nought' is a negative answer to the question 'how many?'. | |
| From: report of Edmund Husserl (Philosophy of Arithmetic [1894], p.144) by Michael Dummett - Frege philosophy of mathematics Ch.8 | |
| A reaction: I seem to be in a tiny minority in thinking that Husserl may have a good point. One apple is different from one orange, but no apples are the same as no oranges. That makes 0 a very peculiar number. See Idea 9838. |
| 10574 | Unless we know whether 0 is identical with the null set, we create confusions [Fine,K] |
| Full Idea: What is the union of the singleton {0}, of zero, and the singleton {φ}, of the null set? Is it the one-element set {0}, or the two-element set {0, φ}? Unless the question of identity between 0 and φ is resolved, we cannot say. | |
| From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 2) |
| 10853 | Either lack of zero made early mathematics geometrical, or the geometrical approach made zero meaningless [Clegg] |
| Full Idea: It is a chicken-and-egg problem, whether the lack of zero forced forced classical mathematicians to rely mostly on a geometric approach to mathematics, or the geometric approach made 0 a meaningless concept, but the two remain strongly tied together. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch. 6) |