Combining Texts
Ideas for
'Philosophical Explanations', 'Naturalism in Mathematics' and 'Introducing the Philosophy of Mathematics'
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15 ideas
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
8667
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The 'integers' are the positive and negative natural numbers, plus zero [Friend]
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8668
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The 'rational' numbers are those representable as fractions [Friend]
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8670
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A number is 'irrational' if it cannot be represented as a fraction [Friend]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
8661
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The natural numbers are primitive, and the ordinals are up one level of abstraction [Friend]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
8664
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Cardinal numbers answer 'how many?', with the order being irrelevant [Friend]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
8671
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The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps [Friend]
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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 / h. Ordinal infinity
8663
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Raising omega to successive powers of omega reveal an infinity of infinities [Friend]
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8662
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The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
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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18172
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Infinity has degrees, and large cardinals are the heart of set theory [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
8669
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Between any two rational numbers there is an infinite number of rational numbers [Friend]
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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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