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 | 10095 | Type theory has only finitely many items at each level, which is a problem for mathematics |
Ch.3 | p.47 | 10094 | The theory of types seems to rule out harmless sets as well as paradoxical ones. |
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 | 10097 | Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = y |
Ch.3 | p.50 | 10098 | The 'power set' of A is all the subsets of A |
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' <a, b>, 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 | 10102 | The integers are answers to subtraction problems involving natural numbers |
Ch.3 | p.63 | 10103 | Grouping by property is common in mathematics, usually using equivalence |
Ch.3 | p.64 | 10104 | 'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words |
Ch.3 | p.69 | 10105 | Differences between isomorphic structures seem unimportant |
Ch.3 | p.69 | 10106 | Rational numbers give answers to division problems with integers |
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 | 10120 | Soundness is a semantic property, unlike the purely syntactic property of consistency |
Ch.6 | p.162 | 10119 | Consistency is a purely syntactic property, unlike the semantic property of soundness |
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 | 10126 | A 'consistent' theory cannot contain both a sentence and its negation |
Ch.7 | p.177 | 10127 | A 'complete' theory contains either any sentence or 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 |