Combining Texts

All the ideas for 'Infinity: Quest to Think the Unthinkable', 'The Principles of Art' and 'Platonistic Theories of Universals'

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27 ideas

2. Reason / B. Laws of Thought / 6. Ockham's Razor
8964 Entities can be multiplied either by excessive categories, or excessive entities within a category [Hoffman/Rosenkrantz]
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
10859 A set is 'well-ordered' if every subset has a first element [Clegg]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
10857 Set theory made a closer study of infinity possible [Clegg]
10864 Any set can always generate a larger set - its powerset, of subsets [Clegg]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I
10872 Extensionality: Two sets are equal if and only if they have the same elements [Clegg]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / c. Axiom of Pairing II
10875 Pairing: For any two sets there exists a set to which they both belong [Clegg]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / d. Axiom of Unions III
10876 Unions: There is a set of all the elements which belong to at least one set in a collection [Clegg]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
10878 Infinity: There exists a set of the empty set and the successor of each element [Clegg]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI
10877 Powers: All the subsets of a given set form their own new powerset [Clegg]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
10879 Choice: For every set a mechanism will choose one member of any non-empty subset [Clegg]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / k. Axiom of Existence
10871 Axiom of Existence: there exists at least one set [Clegg]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / l. Axiom of Specification
10874 Specification: a condition applied to a set will always produce a new set [Clegg]
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
10880 Mathematics can be 'pure' (unapplied), 'real' (physically grounded); or 'applied' (just applicable) [Clegg]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
10860 An ordinal number is defined by the set that comes before it [Clegg]
10861 Beyond infinity cardinals and ordinals can come apart [Clegg]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
10854 Transcendental numbers can't be fitted to finite equations [Clegg]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / k. Imaginary numbers
10858 By adding an axis of imaginary numbers, we get the useful 'number plane' instead of number line [Clegg]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / l. Zero
10853 Either lack of zero made early mathematics geometrical, or the geometrical approach made zero meaningless [Clegg]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
10866 Cantor's account of infinities has the shaky foundation of irrational numbers [Clegg]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
10869 The Continuum Hypothesis is independent of the axioms of set theory [Clegg]
10862 The 'continuum hypothesis' says aleph-one is the cardinality of the reals [Clegg]
8. Modes of Existence / D. Universals / 4. Uninstantiated Universals
8962 'There are shapes which are never exemplified' is the toughest example for nominalists [Hoffman/Rosenkrantz]
8. Modes of Existence / E. Nominalism / 1. Nominalism / a. Nominalism
8961 Nominalists are motivated by Ockham's Razor and a distrust of unobservables [Hoffman/Rosenkrantz]
10. Modality / E. Possible worlds / 2. Nature of Possible Worlds / a. Nature of possible worlds
8963 Four theories of possible worlds: conceptualist, combinatorial, abstract, or concrete [Hoffman/Rosenkrantz]
21. Aesthetics / B. Nature of Art / 4. Art as Expression
20420 The emotion expressed is non-conscious, but feels oppressive until expression relieves it [Collingwood]
21. Aesthetics / B. Nature of Art / 7. Ontology of Art
20421 Art exists ideally, purely as experiences in the mind of the perceiver [Collingwood, by Kemp]
21. Aesthetics / C. Artistic Issues / 6. Value of Art
20406 Art clarifies the artist's mind and feelings, thus leading to self-knowledge [Collingwood, by Davies,S]