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All the ideas for 'Thinking About Mathematics', 'Philosophical Fragments' and 'Things and Their Parts'

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25 ideas

4. Formal Logic / G. Formal Mereology / 1. Mereology
13331 Part and whole contribute asymmetrically to one another, so must differ [Fine,K]
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]
9. Objects / B. Unity of Objects / 1. Unifying an Object / c. Unity as conceptual
13332 Hierarchical set membership models objects better than the subset or aggregate relations do [Fine,K]
9. Objects / C. Structure of Objects / 3. Matter of an Object
13333 The matter is a relatively unstructured version of the object, like a set without membership structure [Fine,K]
9. Objects / C. Structure of Objects / 8. Parts of Objects / a. Parts of objects
13326 A 'temporary' part is a part at one time, but may not be at another, like a carburetor [Fine,K]
13327 A 'timeless' part just is a part, not a part at some time; some atoms are timeless parts of a water molecule [Fine,K]
9. Objects / C. Structure of Objects / 8. Parts of Objects / b. Sums of parts
13329 An 'aggregative' sum is spread in time, and exists whenever a component exists [Fine,K]
13330 An 'compound' sum is not spread in time, and only exists when all the components exists [Fine,K]
9. Objects / C. Structure of Objects / 8. Parts of Objects / c. Wholes from parts
13328 Two sorts of whole have 'rigid embodiment' (timeless parts) or 'variable embodiment' (temporary parts) [Fine,K]
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]