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

[basic principles of arithmetic according to the early Greeks]

10 ideas
Some quantities are discrete, like number, and others continuous, like lines, time and space [Aristotle]
The idea of 'one' is the foundation of number [Aristotle]
Each many is just ones, and is measured by the one [Aristotle]
Number is plurality measured by unity [Aristotle]
Euclid's common notions or axioms are what we must have if we are to learn anything at all [Euclid, by Roochnik]
A number is not a multitude, but a unified ratio between quantities [Newton]
Arithmetic is based on definitions, and Sums of equals are equal, and Differences of equals are equal [Mill]
Numbers were once defined on the basis of 1, but neglected infinities and + [Russell]
Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic) [Shapiro]
Greek mathematics is wholly sensory, where ours is wholly inferential [Macbeth]