green numbers give full details.     |    back to list of philosophers     |     expand these ideas

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

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

 2002 Philosophies of Mathematics
 p.214 10131 If mathematics is not about particulars, observing particulars must be irrelevant
 Intro p.8 10089 Talk of 'abstract entities' is more a label for the problem than a solution to it
 Ch.2 p.19 9946 Logicists say mathematics is applicable because it is totally general
 Ch.2 p.29 9955 Contextual definitions replace a complete sentence containing the expression
 Ch.2 p.35 10031 Impredicative definitions quantify over the thing being defined
 Ch.3 p.46 10092 In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc.
 Ch.3 p.47 10094 The theory of types seems to rule out harmless sets as well as paradoxical ones.
 Ch.3 p.47 10095 Type theory has only finitely many items at each level, which is a problem for mathematics
 Ch.3 p.47 17901 Type theory prohibits (oddly) a set containing an individual and a set of individuals
 Ch.3 p.47 17900 The Axiom of Reducibility made impredicative definitions possible
 Ch.3 p.48 10096 Even the elements of sets in ZFC are sets, resting on the pure empty set
 Ch.3 p.50 10098 The 'power set' of A is all the subsets of A
 Ch.3 p.50 10097 Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = y
 Ch.3 p.52 10100 Axiom of Pairing: for all sets x and y, there is a set z containing just x and y
 Ch.3 p.56 10099 The 'ordered pair' , for two sets a and b, is the set {{a, b},{a}}
 Ch.3 p.56 10101 Cartesian Product A x B: the set of all ordered pairs in which a∈A and b∈B
 Ch.3 p.58 17902 A successor is the union of a set with its singleton
 Ch.3 p.63 10103 Grouping by property is common in mathematics, usually using equivalence
 Ch.3 p.63 10102 The integers are answers to subtraction problems involving natural numbers
 Ch.3 p.64 10104 'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words
 Ch.3 p.69 10106 Rational numbers give answers to division problems with integers
 Ch.3 p.69 10105 Differences between isomorphic structures seem unimportant
 Ch.3 p.70 10107 Real numbers provide answers to square root problems
 Ch.4 p.90 10109 ZFC can prove that there is no set corresponding to the concept 'set'
 Ch.4 p.90 10108 As a reduction of arithmetic, set theory is not fully general, and so not logical
 Ch.4 p.90 10110 Corresponding to every concept there is a class (some of them sets)
 Ch.4 p.91 10111 Asserting Excluded Middle is a hallmark of realism about the natural world
 Ch.6 p.149 10114 Bounded quantification is originally finitary, as conjunctions and disjunctions
 Ch.6 p.162 10119 Consistency is a purely syntactic property, unlike the semantic property of soundness
 Ch.6 p.162 10120 Soundness is a semantic property, unlike the purely syntactic property of consistency
 Ch.6 p.168 10123 The intuitionists are the idealists of mathematics
 Ch.6 p.170 10125 The classical mathematician believes the real numbers form an actual set
 Ch.7 p.177 10127 A 'complete' theory contains either any sentence or its negation
 Ch.7 p.177 10126 A 'consistent' theory cannot contain both a sentence and its negation
 Ch.7 p.182 10128 The Incompleteness proofs use arithmetic to talk about formal arithmetic
 Ch.7 p.193 10129 A 'model' is a meaning-assignment which makes all the axioms true
 Ch.7 p.199 10130 Set theory can prove the Peano Postulates
 Ch.7 n7 p.195 17899 Second-order induction is stronger as it covers all concepts, not just first-order definable ones
 Ch.8 p.168 10124 Gödel's First Theorem suggests there are truths which are independent of proof
 Ch.8 p.219 10134 Much infinite mathematics can still be justified finitely
 Ch.8 n1 p.215 10133 Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle