Combining Philosophers

Ideas for Hilbert,D/Ackermann,W, Agathon and Michle Friend

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22 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number

8667	The 'integers' are the positive and negative natural numbers, plus zero [Friend]

8668	The 'rational' numbers are those representable as fractions [Friend]

8670	A number is 'irrational' if it cannot be represented as a fraction [Friend]

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers

8661	The natural numbers are primitive, and the ordinals are up one level of abstraction [Friend]

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers

8664	Cardinal numbers answer 'how many?', with the order being irrelevant [Friend]

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers

8671	The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps [Friend]

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity

8663	Raising omega to successive powers of omega reveal an infinity of infinities [Friend]

8662	The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend]

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / j. Infinite divisibility

8669	Between any two rational numbers there is an infinite number of rational numbers [Friend]

6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics

8676	Is mathematics based on sets, types, categories, models or topology? [Friend]

6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory

8678	Most mathematical theories can be translated into the language of set theory [Friend]

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism

8701	The number 8 in isolation from the other numbers is of no interest [Friend]

8702	In structuralism the number 8 is not quite the same in different structures, only equivalent [Friend]

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism

8699	Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? [Friend]

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism

8696	Structuralist says maths concerns concepts about base objects, not base objects themselves [Friend]

8695	Structuralism focuses on relations, predicates and functions, with objects being inessential [Friend]

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism

8700	'In re' structuralism says that the process of abstraction is pattern-spotting [Friend]

6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism

8681	The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? [Friend]

6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics

8712	Mathematics should be treated as true whenever it is indispensable to our best physical theory [Friend]

6. Mathematics / C. Sources of Mathematics / 7. Formalism

8716	Formalism is unconstrained, so cannot indicate importance, or directions for research [Friend]

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism

8706	Constructivism rejects too much mathematics [Friend]

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism

8707	Intuitionists typically retain bivalence but reject the law of excluded middle [Friend]