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Ideas for 'works', 'Intuitionism: an Introduction' and 'Categories'

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4 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.
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
8754 Logic is dependent on mathematics, not the other way round [Heyting, by Shapiro]
     Full Idea: Heyting (the intuitionist pupil of Brouwer) said that 'logic is dependent on mathematics', not the other way round.
     From: report of Arend Heyting (Intuitionism: an Introduction [1956]) by Stewart Shapiro - Thinking About Mathematics 7.3
     A reaction: To me, this claim makes logicism sound much more plausible, as I don't see how mathematics could get beyond basic counting without a capacity for logical thought. Logic runs much deeper, psychologically and metaphysically.