Combining Philosophers
Ideas for B Hale / C Wright, Giuseppe Peano and Cardinal/Hayward/Jones
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17 ideas
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
3338
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Numbers have been defined in terms of 'successors' to the concept of 'zero' [Peano, by Blackburn]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
13949
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All models of Peano axioms are isomorphic, so the models all seem equally good for natural numbers [Cartwright,R on Peano]
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18113
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PA concerns any entities which satisfy the axioms [Peano, by Bostock]
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17634
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Peano axioms not only support arithmetic, but are also fairly obvious [Peano, by Russell]
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5897
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0 is a non-successor number, all successors are numbers, successors can't duplicate, if P(n) and P(n+1) then P(all-n) [Peano, by Flew]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
15653
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We can add Reflexion Principles to Peano Arithmetic, which assert its consistency or soundness [Halbach on Peano]
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10624
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The incompletability of formal arithmetic reveals that logic also cannot be completely characterized [Hale/Wright]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
8784
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Neo-logicism founds arithmetic on Hume's Principle along with second-order logic [Hale/Wright]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
8787
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The Julius Caesar problem asks for a criterion for the concept of a 'number' [Hale/Wright]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
10629
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If structures are relative, this undermines truth-value and objectivity [Hale/Wright]
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10628
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The structural view of numbers doesn't fit their usage outside arithmetical contexts [Hale/Wright]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
17635
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Arithmetic can have even simpler logical premises than the Peano Axioms [Russell on Peano]
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8788
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Logicism is only noteworthy if logic has a privileged position in our ontology and epistemology [Hale/Wright]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
10622
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The neo-Fregean is more optimistic than Frege about contextual definitions of numbers [Hale/Wright]
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8783
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Logicism might also be revived with a quantificational approach, or an abstraction-free approach [Hale/Wright]
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12225
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Neo-Fregeanism might be better with truth-makers, rather than quantifier commitment [Hale/Wright]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
12224
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Are neo-Fregeans 'maximalists' - that everything which can exist does exist? [Hale/Wright]
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