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Ideas for 'Thinking About Mathematics', 'Lectura' and 'Foundations without Foundationalism'

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22 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
13641 Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals [Shapiro]
8763 The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / d. Natural numbers
13676 Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are [Shapiro]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
13677 Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals [Shapiro]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy
18249 Cauchy gave a formal definition of a converging sequence. [Shapiro]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
13652 The 'continuum' is the cardinality of the powerset of a denumerably infinite set [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
8764 Categories are the best foundation for mathematics [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
13657 First-order arithmetic can't even represent basic number theory [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / f. Zermelo numbers
8762 Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
13656 Some sets of natural numbers are definable in set-theory but not in arithmetic [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
8760 Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro]
8761 A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
13664 Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions [Shapiro]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
13625 Mathematics and logic have no border, and logic must involve mathematics and its ontology [Shapiro]
8744 Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro]
6. Mathematics / C. Sources of Mathematics / 7. Formalism
8749 Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro]
8750 Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro]
8752 Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
8753 Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
8731 Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
13663 Some reject formal properties if they are not defined, or defined impredicatively [Shapiro]
8730 'Impredicative' definitions refer to the thing being described [Shapiro]