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3 ideas
| 11044 | One is prior to two, because its existence is implied by two [Aristotle] |
| Full Idea: One is prior to two because if there are two it follows at once that there is one, whereas if there is one there is not necessarily two. | |
| From: Aristotle (Categories [c.331 BCE], 14a29) | |
| A reaction: The axiomatic introduction of a 'successor' to a number does not seem to introduce this notion of priority, based on inclusiveness. Introducing order by '>' also does not seem to indicate any logical priority. |
| 11042 | Parts of a line join at a point, so it is continuous [Aristotle] |
| Full Idea: A line is a continuous quantity. For it is possible to find a common boundary at which its parts join together, a point. | |
| From: Aristotle (Categories [c.331 BCE], 04b33) | |
| A reaction: This appears to be the essential concept of a Dedekind cut. It seems to be an open question whether a cut defines a unique number, but a boundary seems to be intrinsically unique. Aristotle wins again. |
| 12273 | Unit is the starting point of number [Aristotle] |
| Full Idea: They say that the unit [monada] is the starting point of number (and the point the starting-point of a line). | |
| From: Aristotle (Topics [c.331 BCE], 108b30) | |
| A reaction: Yes, despite Frege's objections in the early part of the 'Grundlagen' (1884). I take arithmetic to be rooted in counting, despite all abstract definitions of number by Frege and Dedekind. Identity gives the unit, which is countable. See also Topics 141b9 |