### All the ideas for William Paley, Empedocles and A.George / D.J.Velleman

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68 ideas

###### 2. Reason / D. Definition / 7. Contextual Definition
 9955 Contextual definitions replace a complete sentence containing the expression [George/Velleman]
###### 2. Reason / D. Definition / 8. Impredicative Definition
 10031 Impredicative definitions quantify over the thing being defined [George/Velleman]
###### 4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
 10098 The 'power set' of A is all the subsets of A [George/Velleman]
 10101 Cartesian Product A x B: the set of all ordered pairs in which a∈A and b∈B [George/Velleman]
 10099 The 'ordered pair' , for two sets a and b, is the set {{a, b},{a}} [George/Velleman]
###### 4. Formal Logic / F. Set Theory ST / 3. Types of Set / e. Equivalence classes
 10103 Grouping by property is common in mathematics, usually using equivalence [George/Velleman]
 10104 'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words [George/Velleman]
###### 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
 10096 Even the elements of sets in ZFC are sets, resting on the pure empty set [George/Velleman]
###### 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I
 10097 Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = y [George/Velleman]
###### 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / c. Axiom of Pairing II
 10100 Axiom of Pairing: for all sets x and y, there is a set z containing just x and y [George/Velleman]
###### 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / p. Axiom of Reducibility
 17900 The Axiom of Reducibility made impredicative definitions possible [George/Velleman]
###### 4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / a. Sets as existing
 10109 ZFC can prove that there is no set corresponding to the concept 'set' [George/Velleman]
###### 4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
 10108 As a reduction of arithmetic, set theory is not fully general, and so not logical [George/Velleman]
###### 5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
 10111 Asserting Excluded Middle is a hallmark of realism about the natural world [George/Velleman]
###### 5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
 10129 A 'model' is a meaning-assignment which makes all the axioms true [George/Velleman]
###### 5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
 10105 Differences between isomorphic structures seem unimportant [George/Velleman]
###### 5. Theory of Logic / K. Features of Logics / 2. Consistency
 10119 Consistency is a purely syntactic property, unlike the semantic property of soundness [George/Velleman]
 10126 A 'consistent' theory cannot contain both a sentence and its negation [George/Velleman]
###### 5. Theory of Logic / K. Features of Logics / 3. Soundness
 10120 Soundness is a semantic property, unlike the purely syntactic property of consistency [George/Velleman]
###### 5. Theory of Logic / K. Features of Logics / 4. Completeness
 10127 A 'complete' theory contains either any sentence or its negation [George/Velleman]
###### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
 10106 Rational numbers give answers to division problems with integers [George/Velleman]
 10102 The integers are answers to subtraction problems involving natural numbers [George/Velleman]
###### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
 10107 Real numbers provide answers to square root problems [George/Velleman]
###### 6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
 9946 Logicists say mathematics is applicable because it is totally general [George/Velleman]
###### 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
 10125 The classical mathematician believes the real numbers form an actual set [George/Velleman]
###### 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
 17899 Second-order induction is stronger as it covers all concepts, not just first-order definable ones [George/Velleman]
###### 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
 10128 The Incompleteness proofs use arithmetic to talk about formal arithmetic [George/Velleman]
###### 6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
 17902 A successor is the union of a set with its singleton [George/Velleman]
###### 6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
 10133 Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle [George/Velleman]
###### 6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
 10130 Set theory can prove the Peano Postulates [George/Velleman]
###### 6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
 10089 Talk of 'abstract entities' is more a label for the problem than a solution to it [George/Velleman]
###### 6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
 10131 If mathematics is not about particulars, observing particulars must be irrelevant [George/Velleman]
###### 6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
 10092 In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc. [George/Velleman]
 10094 The theory of types seems to rule out harmless sets as well as paradoxical ones. [George/Velleman]
 10095 Type theory has only finitely many items at each level, which is a problem for mathematics [George/Velleman]
 17901 Type theory prohibits (oddly) a set containing an individual and a set of individuals [George/Velleman]
###### 6. Mathematics / C. Sources of Mathematics / 8. Finitism
 10134 Much infinite mathematics can still be justified finitely [George/Velleman]
 10114 Bounded quantification is originally finitary, as conjunctions and disjunctions [George/Velleman]
###### 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
 10123 The intuitionists are the idealists of mathematics [George/Velleman]
 10124 Gödel's First Theorem suggests there are truths which are independent of proof [George/Velleman]
###### 7. Existence / A. Nature of Existence / 5. Reason for Existence
 458 Nothing could come out of nothing, and existence could never completely cease [Empedocles]
###### 7. Existence / B. Change in Existence / 1. Nature of Change
 5112 Empedocles says things are at rest, unless love unites them, or hatred splits them [Empedocles, by Aristotle]
###### 9. Objects / A. Existence of Objects / 6. Nihilism about Objects
 13209 There is no coming-to-be of anything, but only mixing and separating [Empedocles, by Aristotle]
###### 9. Objects / E. Objects over Time / 10. Beginning of an Object
 457 Substance is not created or destroyed in mortals, but there is only mixing and exchange [Empedocles]
###### 13. Knowledge Criteria / E. Relativism / 3. Subjectivism
 462 One vision is produced by both eyes [Empedocles]
###### 17. Mind and Body / A. Mind-Body Dualism / 3. Panpsychism
 22765 Wisdom and thought are shared by all things [Empedocles]
###### 18. Thought / A. Modes of Thought / 1. Thought
 1524 For Empedocles thinking is almost identical to perception [Empedocles, by Theophrastus]
###### 18. Thought / D. Concepts / 1. Concepts / a. Nature of concepts
 10110 Corresponding to every concept there is a class (some of them sets) [George/Velleman]
###### 26. Natural Theory / A. Speculations on Nature / 1. Nature
 589 'Nature' is just a word invented by people [Empedocles]
###### 26. Natural Theory / A. Speculations on Nature / 6. Early Matter Theories / e. The One
 21823 The principle of 'Friendship' in Empedocles is the One, and is bodiless [Empedocles, by Plotinus]
###### 26. Natural Theory / A. Speculations on Nature / 6. Early Matter Theories / f. Ancient elements
 2680 Empedocles said that there are four material elements, and two further creative elements [Empedocles, by Aristotle]
 6002 Empedocles says bone is water, fire and earth in ratio 2:4:2 [Empedocles, by Inwood]
 13207 Fire, Water, Air and Earth are elements, being simple as well as homoeomerous [Empedocles, by Aristotle]
 13218 The elements combine in coming-to-be, but how do the elements themselves come-to-be? [Aristotle on Empedocles]
 459 All change is unity through love or division through hate [Empedocles]
 13225 Love and Strife only explain movement if their effects are distinctive [Aristotle on Empedocles]
 460 If the one Being ever diminishes it would no longer exist, and what could ever increase it? [Empedocles]
###### 27. Natural Reality / G. Biology / 3. Evolution
 5090 Maybe bodies are designed by accident, and the creatures that don't work are destroyed [Empedocles, by Aristotle]
###### 28. God / A. Divine Nature / 2. Divine Nature
 466 God is pure mind permeating the universe [Empedocles]
 461 God is a pure, solitary, and eternal sphere [Empedocles]
###### 28. God / A. Divine Nature / 4. Divine Contradictions
 1719 In Empedocles' theory God is ignorant because, unlike humans, he doesn't know one of the elements (strife) [Aristotle on Empedocles]
###### 28. God / B. Proving God / 3. Proofs of Evidence / b. Teleological Proof
 21277 Even an imperfect machine can exhibit obvious design [Paley]
 21275 Unlike a stone, the parts of a watch are obviously assembled in order to show the time [Paley]
 21276 From the obvious purpose and structure of a watch we must infer that it was designed [Paley]
 21357 No organ shows purpose more obviously than the eyelid [Paley]
 21278 All the signs of design found in a watch are also found in nature [Paley]
###### 29. Religion / A. Polytheistic Religion / 2. Greek Polytheism
 1522 It is wretched not to want to think clearly about the gods [Empedocles]
###### 29. Religion / A. Polytheistic Religion / 4. Dualist Religion
 552 Empedocles said good and evil were the basic principles [Empedocles, by Aristotle]