Combining Texts

Ideas for 'works', 'I.39 On Solitude' and 'Categories'

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2 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
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.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
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.