Combining Philosophers

Ideas for Paul Bernays, Harry G. Frankfurt and Gottfried Leibniz

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15 ideas

6. Mathematics / A. Nature of Mathematics / 2. Geometry
13163 Circles must be bounded, so cannot be infinite [Leibniz]
13008 Geometry, unlike sensation, lets us glimpse eternal truths and their necessity [Leibniz]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units
9147 Number cannot be defined as addition of ones, since that needs the number; it is a single act of abstraction [Fine,K on Leibniz]
12956 Only whole numbers are multitudes of units [Leibniz]
12920 There is no multiplicity without true units [Leibniz]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
19390 Everything is subsumed under number, which is a metaphysical statics of the universe, revealing powers [Leibniz]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
19406 I strongly believe in the actual infinite, which indicates the perfections of its author [Leibniz]
13190 I don't admit infinite numbers, and consider infinitesimals to be useful fictions [Leibniz]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / j. Infinite divisibility
19375 The continuum is not divided like sand, but folded like paper [Leibniz, by Arthur,R]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / k. Infinitesimals
18080 A tangent is a line connecting two points on a curve that are infinitely close together [Leibniz]
18081 Nature uses the infinite everywhere [Leibniz]
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
12937 We shouldn't just accept Euclid's axioms, but try to demonstrate them [Leibniz]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
23026 We know mathematical axioms, such as subtracting equals from equals leaves equals, by a natural light [Leibniz]
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
10303 Restricted Platonism is just an ideal projection of a domain of thought [Bernays]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
10306 Mathematical abstraction just goes in a different direction from logic [Bernays]