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Ideas of A.George / D.J.Velleman, by Text

[American, fl. 2002, Two professors at Amherst College.]

2002 Philosophies of Mathematics
p.214 If mathematics is not about particulars, observing particulars must be irrelevant
Intro p.8 Talk of 'abstract entities' is more a label for the problem than a solution to it
Ch.2 p.19 Logicists say mathematics is applicable because it is totally general
Ch.2 p.29 Contextual definitions replace a complete sentence containing the expression
Ch.2 p.35 Impredicative definitions quantify over the thing being defined
Ch.3 p.46 In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc.
Ch.3 p.47 The theory of types seems to rule out harmless sets as well as paradoxical ones.
Ch.3 p.47 Type theory has only finitely many items at each level, which is a problem for mathematics
Ch.3 p.47 Type theory prohibits (oddly) a set containing an individual and a set of individuals
Ch.3 p.47 The Axiom of Reducibility made impredicative definitions possible
Ch.3 p.48 Even the elements of sets in ZFC are sets, resting on the pure empty set
Ch.3 p.50 Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = y
Ch.3 p.50 The 'power set' of A is all the subsets of A
Ch.3 p.52 Axiom of Pairing: for all sets x and y, there is a set z containing just x and y
Ch.3 p.56 Cartesian Product A x B: the set of all ordered pairs in which a∈A and b∈B
Ch.3 p.56 The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}}
Ch.3 p.58 A successor is the union of a set with its singleton
Ch.3 p.63 The integers are answers to subtraction problems involving natural numbers
Ch.3 p.63 Grouping by property is common in mathematics, usually using equivalence
Ch.3 p.64 'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words
Ch.3 p.69 Differences between isomorphic structures seem unimportant
Ch.3 p.69 Rational numbers give answers to division problems with integers
Ch.3 p.70 Real numbers provide answers to square root problems
Ch.4 p.90 ZFC can prove that there is no set corresponding to the concept 'set'
Ch.4 p.90 As a reduction of arithmetic, set theory is not fully general, and so not logical
Ch.4 p.90 Corresponding to every concept there is a class (some of them sets)
Ch.4 p.91 Asserting Excluded Middle is a hallmark of realism about the natural world
Ch.6 p.149 Bounded quantification is originally finitary, as conjunctions and disjunctions
Ch.6 p.162 Consistency is a purely syntactic property, unlike the semantic property of soundness
Ch.6 p.162 Soundness is a semantic property, unlike the purely syntactic property of consistency
Ch.6 p.168 The intuitionists are the idealists of mathematics
Ch.6 p.170 The classical mathematician believes the real numbers form an actual set
Ch.7 p.177 A 'complete' theory contains either any sentence or its negation
Ch.7 p.177 A 'consistent' theory cannot contain both a sentence and its negation
Ch.7 p.182 The Incompleteness proofs use arithmetic to talk about formal arithmetic
Ch.7 p.193 A 'model' is a meaning-assignment which makes all the axioms true
Ch.7 p.199 Set theory can prove the Peano Postulates
Ch.7 n7 p.195 Second-order induction is stronger as it covers all concepts, not just first-order definable ones
Ch.8 p.168 Gödel's First Theorem suggests there are truths which are independent of proof
Ch.8 p.219 Much infinite mathematics can still be justified finitely
Ch.8 n1 p.215 Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle