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'Demonstratives', 'Introducing the Philosophy of Mathematics' and 'The Art of Rhetoric'
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22 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 / 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 / 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 / B. Foundations for Mathematics / 1. Foundations for Mathematics
8676
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Is mathematics based on sets, types, categories, models or topology? [Friend]
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
8678
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Most mathematical theories can be translated into the language of set theory [Friend]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
8701
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The number 8 in isolation from the other numbers is of no interest [Friend]
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8702
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In structuralism the number 8 is not quite the same in different structures, only equivalent [Friend]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
8699
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Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? [Friend]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism
8696
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Structuralist says maths concerns concepts about base objects, not base objects themselves [Friend]
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8695
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Structuralism focuses on relations, predicates and functions, with objects being inessential [Friend]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
8700
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'In re' structuralism says that the process of abstraction is pattern-spotting [Friend]
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6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
8681
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The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? [Friend]
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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
8712
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Mathematics should be treated as true whenever it is indispensable to our best physical theory [Friend]
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6. Mathematics / C. Sources of Mathematics / 7. Formalism
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Formalism is unconstrained, so cannot indicate importance, or directions for research [Friend]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
8706
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Constructivism rejects too much mathematics [Friend]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
8707
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Intuitionists typically retain bivalence but reject the law of excluded middle [Friend]
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