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'Unconscious Cerebral Initiative', 'Naturalism in Mathematics' and 'Introduction to the Philosophy of Mathematics'
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26 ideas
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
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
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 / d. Actual infinite
18190
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Completed infinities resulted from giving foundations to calculus [Maddy]
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18171
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Cantor and Dedekind brought completed infinities into mathematics [Maddy]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
18172
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Infinity has degrees, and large cardinals are the heart of set theory [Maddy]
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18175
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For any cardinal there is always a larger one (so there is no set of all sets) [Maddy]
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18196
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An 'inaccessible' cardinal cannot be reached by union sets or power sets [Maddy]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / j. Infinite divisibility
17941
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Infinitesimals were sometimes zero, and sometimes close to zero [Colyvan]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / l. Limits
18187
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Theorems about limits could only be proved once the real numbers were understood [Maddy]
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6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
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 / 4. Axioms for Number / f. Mathematical induction
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 / 5. Definitions of Number / c. Fregean numbers
18182
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The extension of concepts is not important to me [Maddy]
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18177
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In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets [Maddy]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
18164
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Frege solves the Caesar problem by explicitly defining each number [Maddy]
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
18184
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Making set theory foundational to mathematics leads to very fruitful axioms [Maddy]
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18185
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Unified set theory gives a final court of appeal for mathematics [Maddy]
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18183
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Set theory brings mathematics into one arena, where interrelations become clearer [Maddy]
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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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18186
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Identifying geometric points with real numbers revealed the power of set theory [Maddy]
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18188
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The line of rationals has gaps, but set theory provided an ordered continuum [Maddy]
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18163
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Mathematics rests on the logic of proofs, and on the set theoretic axioms [Maddy]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
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
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 / 4. Mathematical Empiricism / b. Indispensability of mathematics
18204
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Scientists posit as few entities as possible, but set theorist posit as many as possible [Maddy]
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18207
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Maybe applications of continuum mathematics are all idealisations [Maddy]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
18167
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We can get arithmetic directly from HP; Law V was used to get HP from the definition of number [Maddy]
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