Combining Philosophers
Ideas for B Hale / C Wright, Mark Colyvan and Alfred Tarski
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20 ideas
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
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17928
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Ordinal numbers represent order relations [Colyvan]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
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17923
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Intuitionists only accept a few safe infinities [Colyvan]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / j. Infinite divisibility
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17941
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Infinitesimals were sometimes zero, and sometimes close to zero [Colyvan]
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6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
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17922
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Reducing real numbers to rationals suggested arithmetic as the foundation of maths [Colyvan]
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6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
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10157
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Tarski improved Hilbert's geometry axioms, and without set-theory [Tarski, by Feferman/Feferman]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
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17936
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Transfinite induction moves from all cases, up to the limit ordinal [Colyvan]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
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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
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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
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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 / 6. Mathematics as Set Theory / a. Mathematics is set theory
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17940
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Most mathematical proofs are using set theory, but without saying so [Colyvan]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
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17931
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Structuralism say only 'up to isomorphism' matters because that is all there is to it [Colyvan]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
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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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17932
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If 'in re' structures relies on the world, does the world contain rich enough structures? [Colyvan]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
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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
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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
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12224
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Are neo-Fregeans 'maximalists' - that everything which can exist does exist? [Hale/Wright]
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6. Mathematics / C. Sources of Mathematics / 7. Formalism
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10154
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Tarski's theory of truth shifted the approach away from syntax, to set theory and semantics [Feferman/Feferman on Tarski]
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