Combining Philosophers
Ideas for Herodotus, E.J. Lowe and David Hilbert
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31 ideas
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
8717
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Hilbert wanted to prove the consistency of all of mathematics (which realists take for granted) [Hilbert, by Friend]
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12456
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I aim to establish certainty for mathematical methods [Hilbert]
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12461
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We believe all mathematical problems are solvable [Hilbert]
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4240
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It might be argued that mathematics does not, or should not, aim at truth [Lowe]
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6. Mathematics / A. Nature of Mathematics / 2. Geometry
13472
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Hilbert aimed to eliminate number from geometry [Hilbert, by Hart,WD]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
9633
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No one shall drive us out of the paradise the Cantor has created for us [Hilbert]
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12460
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We extend finite statements with ideal ones, in order to preserve our logic [Hilbert]
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12462
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Only the finite can bring certainty to the infinite [Hilbert]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
12455
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The idea of an infinite totality is an illusion [Hilbert]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / j. Infinite divisibility
12457
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There is no continuum in reality to realise the infinitely small [Hilbert]
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6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics
17967
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To decide some questions, we must study the essence of mathematical proof itself [Hilbert]
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6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
9546
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Euclid axioms concerns possibilities of construction, but Hilbert's assert the existence of objects [Hilbert, by Chihara]
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18742
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Hilbert's formalisation revealed implicit congruence axioms in Euclid [Hilbert, by Horsten/Pettigrew]
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18217
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Hilbert's geometry is interesting because it captures Euclid without using real numbers [Hilbert, by Field,H]
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17965
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The whole of Euclidean geometry derives from a basic equation and transformations [Hilbert]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
17964
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Number theory just needs calculation laws and rules for integers [Hilbert]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
8297
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Numbers are universals, being sets whose instances are sets of appropriate cardinality [Lowe]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
8266
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Simple counting is more basic than spotting that one-to-one correlation makes sets equinumerous [Lowe]
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8302
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Fs and Gs are identical in number if they one-to-one correlate with one another [Lowe]
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6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
8298
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Sets are instances of numbers (rather than 'collections'); numbers explain sets, not vice versa [Lowe]
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8311
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If 2 is a particular, then adding particulars to themselves does nothing, and 2+2=2 [Lowe]
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4241
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If there are infinite numbers and finite concrete objects, this implies that numbers are abstract objects [Lowe]
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6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
8310
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Does the existence of numbers matter, in the way space, time and persons do? [Lowe]
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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
17697
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The existence of an arbitrarily large number refutes the idea that numbers come from experience [Hilbert]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
17698
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Logic already contains some arithmetic, so the two must be developed together [Hilbert]
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6. Mathematics / C. Sources of Mathematics / 7. Formalism
10113
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The grounding of mathematics is 'in the beginning was the sign' [Hilbert]
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10115
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Hilbert substituted a syntactic for a semantic account of consistency [Hilbert, by George/Velleman]
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22293
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Hilbert said (to block paradoxes) that mathematical existence is entailed by consistency [Hilbert, by Potter]
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12459
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The subject matter of mathematics is immediate and clear concrete symbols [Hilbert]
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6. Mathematics / C. Sources of Mathematics / 8. Finitism
10116
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Hilbert aimed to prove the consistency of mathematics finitely, to show infinities won't produce contradictions [Hilbert, by George/Velleman]
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18112
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Mathematics divides in two: meaningful finitary statements, and empty idealised statements [Hilbert]
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