Combining Philosophers

Ideas for Hermarchus, E.J. Lowe and Euclid

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9 ideas

6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
14157 Modern geometries only accept various parts of the Euclid propositions [Russell on Euclid]
22278 Euclid relied on obvious properties in diagrams, as well as on his axioms [Potter on Euclid]
8673 Euclid's parallel postulate defines unique non-intersecting parallel lines [Euclid, by Friend]
10250 Euclid needs a principle of continuity, saying some lines must intersect [Shapiro on Euclid]
10302 Euclid says we can 'join' two points, but Hilbert says the straight line 'exists' [Euclid, by Bernays]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic
1600 Euclid's common notions or axioms are what we must have if we are to learn anything at all [Euclid, by Roochnik]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
8297 Numbers are universals, being sets whose instances are sets of appropriate cardinality [Lowe]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
8266 Simple counting is more basic than spotting that one-to-one correlation makes sets equinumerous [Lowe]
8302 Fs and Gs are identical in number if they one-to-one correlate with one another [Lowe]