Combining Texts

Ideas for 'fragments/reports', 'Categories' and 'Knowing One's Own Mind'

unexpand these ideas     |    start again     |     choose another area for these texts

display all the ideas for this combination of texts

---

3 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.

6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic

11041	Some quantities are discrete, like number, and others continuous, like lines, time and space [Aristotle]
	Full Idea: Of quantities, some are discrete, others continuous. ...Discrete are number and language; continuous are lines, surfaces, bodies, and also, besides these, time and place.
	From: Aristotle (Categories [c.331 BCE], 04b20)
	A reaction: This distinction seems to me to be extremely illuminating, when comparing natural numbers with real numbers, and it is the foundation of the Greek view of mathematics.