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All the ideas for 'Thinking About Mathematics', 'Philosophical Fragments' and 'Essays on Intellectual Powers 5: Abstraction'

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23 ideas

5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
8729 Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
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 / h. Reals from Cauchy
18249 Cauchy gave a formal definition of a converging sequence. [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 / 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 / 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 / d. Logicism critique
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
8730 'Impredicative' definitions refer to the thing being described [Shapiro]
7. Existence / A. Nature of Existence / 5. Reason for Existence
16007 I assume existence, rather than reasoning towards it [Kierkegaard]
8. Modes of Existence / D. Universals / 5. Universals as Concepts
23651 Universals are not objects of sense and cannot be imagined - but can be conceived [Reid]
8. Modes of Existence / E. Nominalism / 1. Nominalism / b. Nominalism about universals
23650 Only individuals exist [Reid]
8. Modes of Existence / E. Nominalism / 2. Resemblance Nominalism
23649 No one thinks two sheets possess a single whiteness, but all agree they are both white [Reid]
9. Objects / F. Identity among Objects / 1. Concept of Identity
11874 Real identity admits of no degrees [Reid]
10. Modality / A. Necessity / 2. Nature of Necessity
16013 Nothing necessary can come into existence, since it already 'is' [Kierkegaard]
12. Knowledge Sources / C. Rationalism / 1. Rationalism
8725 Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro]
18. Thought / A. Modes of Thought / 1. Thought
23652 We must first conceive things before we can consider them [Reid]
18. Thought / E. Abstraction / 1. Abstract Thought
23648 First we notice and name attributes ('abstracting'); then we notice that subjects share them ('generalising') [Reid]