Combining Philosophers
Ideas for Cleanthes, Bertrand Russell and Stuart Glennan
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15 ideas
6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics
17627
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It seems absurd to prove 2+2=4, where the conclusion is more certain than premises [Russell]
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6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
10052
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Geometry is united by the intuitive axioms of projective geometry [Russell, by Musgrave]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
14431
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The definition of order needs a transitive relation, to leap over infinite intermediate terms [Russell]
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14124
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Axiom of Archimedes: a finite multiple of a lesser magnitude can always exceed a greater [Russell]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
7530
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Russell tried to replace Peano's Postulates with the simple idea of 'class' [Russell, by Monk]
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18246
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Dedekind failed to distinguish the numbers from other progressions [Shapiro on Russell]
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14422
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Any founded, non-repeating series all reachable in steps will satisfy Peano's axioms [Russell]
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14423
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'0', 'number' and 'successor' cannot be defined by Peano's axioms [Russell]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
14147
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Denying mathematical induction gave us the transfinite [Russell]
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14125
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Finite numbers, unlike infinite numbers, obey mathematical induction [Russell]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic
14116
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Numbers were once defined on the basis of 1, but neglected infinities and + [Russell]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
14117
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Numbers are properties of classes [Russell]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
14425
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A number is something which characterises collections of the same size [Russell]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
14434
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What matters is the logical interrelation of mathematical terms, not their intrinsic nature [Russell]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
9977
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Ordinals can't be defined just by progression; they have intrinsic qualities [Russell]
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