Combining Texts
Ideas for
'Leibniz: Guide for the Perplexed', 'Nature and Meaning of Numbers' and 'Universals'
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15 ideas
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
9823
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Numbers are free creations of the human mind, to understand differences [Dedekind]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
10090
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Dedekind defined the integers, rationals and reals in terms of just the natural numbers [Dedekind, by George/Velleman]
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7524
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Order, not quantity, is central to defining numbers [Dedekind, by Monk]
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17452
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Ordinals can define cardinals, as the smallest ordinal that maps the set [Dedekind, by Heck]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
14131
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Dedekind's ordinals are just members of any progression whatever [Dedekind, by Russell]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
14437
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Dedekind's axiom that his Cut must be filled has the advantages of theft over honest toil [Dedekind, by Russell]
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18094
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Dedekind says each cut matches a real; logicists say the cuts are the reals [Dedekind, by Bostock]
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
9824
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In counting we see the human ability to relate, correspond and represent [Dedekind]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / b. Mark of the infinite
9826
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A system S is said to be infinite when it is similar to a proper part of itself [Dedekind]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
13508
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Dedekind gives a base number which isn't a successor, then adds successors and induction [Dedekind, by Hart,WD]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
18096
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Zero is a member, and all successors; numbers are the intersection of sets satisfying this [Dedekind, by Bostock]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
18841
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Categoricity implies that Dedekind has characterised the numbers, because it has one domain [Rumfitt on Dedekind]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
14130
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Induction is proved in Dedekind, an axiom in Peano; the latter seems simpler and clearer [Dedekind, by Russell]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
8924
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Dedekind originated the structuralist conception of mathematics [Dedekind, by MacBride]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
9153
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Dedekindian abstraction talks of 'positions', where Cantorian abstraction talks of similar objects [Dedekind, by Fine,K]
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