Combining Philosophers

Ideas for Hermarchus, E.J. Lowe and Euclid

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18 ideas

6. Mathematics / A. Nature of Mathematics / 1. Mathematics
4240 It might be argued that mathematics does not, or should not, aim at truth [Lowe]
6. Mathematics / A. Nature of Mathematics / 2. Geometry
6297 Euclid's geometry is synthetic, but Descartes produced an analytic version of it [Euclid, by Resnik]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
9603 An assumption that there is a largest prime leads to a contradiction [Euclid, by Brown,JR]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / m. One
9894 A unit is that according to which each existing thing is said to be one [Euclid]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
8738 Postulate 2 says a line can be extended continuously [Euclid, by Shapiro]
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
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]
14157 Modern geometries only accept various parts of the Euclid propositions [Russell on Euclid]
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]
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
8298 Sets are instances of numbers (rather than 'collections'); numbers explain sets, not vice versa [Lowe]
8311 If 2 is a particular, then adding particulars to themselves does nothing, and 2+2=2 [Lowe]
4241 If there are infinite numbers and finite concrete objects, this implies that numbers are abstract objects [Lowe]
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
8310 Does the existence of numbers matter, in the way space, time and persons do? [Lowe]