80 ideas
9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
10098 | The 'power set' of A is all the subsets of A [George/Velleman] |
10099 | The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}} [George/Velleman] |
10101 | Cartesian Product A x B: the set of all ordered pairs in which a∈A and b∈B [George/Velleman] |
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] |
10096 | Even the elements of sets in ZFC are sets, resting on the pure empty set [George/Velleman] |
10097 | Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = y [George/Velleman] |
10100 | Axiom of Pairing: for all sets x and y, there is a set z containing just x and y [George/Velleman] |
17900 | The Axiom of Reducibility made impredicative definitions possible [George/Velleman] |
10109 | ZFC can prove that there is no set corresponding to the concept 'set' [George/Velleman] |
10108 | As a reduction of arithmetic, set theory is not fully general, and so not logical [George/Velleman] |
10111 | Asserting Excluded Middle is a hallmark of realism about the natural world [George/Velleman] |
3005 | 'Jocasta' needs to be distinguished from 'Oedipus's mother' because they are connected by different properties [Fodor] |
10129 | A 'model' is a meaning-assignment which makes all the axioms true [George/Velleman] |
10105 | Differences between isomorphic structures seem unimportant [George/Velleman] |
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] |
10120 | Soundness is a semantic property, unlike the purely syntactic property of consistency [George/Velleman] |
10127 | A 'complete' theory contains either any sentence or its negation [George/Velleman] |
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] |
10107 | Real numbers provide answers to square root problems [George/Velleman] |
9946 | Logicists say mathematics is applicable because it is totally general [George/Velleman] |
10125 | The classical mathematician believes the real numbers form an actual set [George/Velleman] |
17899 | Second-order induction is stronger as it covers all concepts, not just first-order definable ones [George/Velleman] |
10128 | The Incompleteness proofs use arithmetic to talk about formal arithmetic [George/Velleman] |
17902 | A successor is the union of a set with its singleton [George/Velleman] |
10133 | Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle [George/Velleman] |
10130 | Set theory can prove the Peano Postulates [George/Velleman] |
10089 | Talk of 'abstract entities' is more a label for the problem than a solution to it [George/Velleman] |
10131 | If mathematics is not about particulars, observing particulars must be irrelevant [George/Velleman] |
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] |
10114 | Bounded quantification is originally finitary, as conjunctions and disjunctions [George/Velleman] |
10134 | Much infinite mathematics can still be justified finitely [George/Velleman] |
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] |
13076 | Scholastics treat relations as two separate predicates of the relata [Cover/O'Leary-Hawthorne] |
7014 | A particle and a coin heads-or-tails pick out to perfectly well-defined predicates and properties [Fodor] |
13102 | If you individuate things by their origin, you still have to individuate the origins themselves [Cover/O'Leary-Hawthorne] |
13103 | Numerical difference is a symmetrical notion, unlike proper individuation [Cover/O'Leary-Hawthorne] |
13104 | Haecceity as property, or as colourless thisness, or as singleton set [Cover/O'Leary-Hawthorne] |
13100 | Maybe 'substance' is more of a mass-noun than a count-noun [Cover/O'Leary-Hawthorne] |
13068 | We can ask for the nature of substance, about type of substance, and about individual substances [Cover/O'Leary-Hawthorne] |
13069 | The general assumption is that substances cannot possibly be non-substances [Cover/O'Leary-Hawthorne] |
13072 | Modern essences are sets of essential predicate-functions [Cover/O'Leary-Hawthorne] |
17080 | Modern essentialists express essence as functions from worlds to extensions for predicates [Cover/O'Leary-Hawthorne] |
13101 | Necessity-of-origin won't distinguish ex nihilo creations, or things sharing an origin [Cover/O'Leary-Hawthorne] |
13081 | Even extreme modal realists might allow transworld identity for abstract objects [Cover/O'Leary-Hawthorne] |
3008 | Evolution suggests that innate knowledge of human psychology would be beneficial [Fodor] |
2990 | Contrary to commonsense, most of what is in the mind seems to be unlearned [Fodor] |
3009 | Sticklebacks have an innate idea that red things are rivals [Fodor] |
13071 | We can go beyond mere causal explanations if we believe in an 'order of being' [Cover/O'Leary-Hawthorne] |
2994 | In CRTT thought may be represented, content must be [Fodor] |
15494 | We can't use propositions to explain intentional attitudes, because they would need explaining [Fodor] |
7326 | Intentionality doesn't go deep enough to appear on the physicists' ultimate list of things [Fodor] |
3001 | Behaviourism has no theory of mental causation [Fodor] |
2993 | Any piece of software can always be hard-wired [Fodor] |
3011 | Causal powers must be a crucial feature of mental states [Fodor] |
5498 | Mind is a set of hierarchical 'homunculi', which are made up in turn from subcomponents [Fodor, by Lycan] |
2995 | Supervenience gives good support for mental causation [Fodor] |
2991 | Hume's associationism offers no explanation at all of rational thought [Fodor] |
3002 | If mind is just physical, how can it follow the rules required for intelligent thought? [Fodor] |
2992 | We may be able to explain rationality mechanically [Fodor] |
2988 | Folk psychology is the only explanation of behaviour we have [Fodor] |
3010 | Belief and desire are structured states, which need mentalese [Fodor] |
2999 | Obsession with narrow content leads to various sorts of hopeless anti-realism [Fodor] |
3012 | Do identical thoughts have identical causal roles? [Fodor] |
10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
2998 | Grice thinks meaning is inherited from the propositional attitudes which sentences express [Fodor] |
3006 | Whatever in the mind delivers falsehood is parasitic on what delivers truth [Fodor] |
3007 | Many different verification procedures can reach 'star', but it only has one semantic value [Fodor] |
3004 | The meaning of a sentence derives from its use in expressing an attitude [Fodor] |
3000 | Meaning holism is a crazy doctrine [Fodor] |
3003 | Very different mental states can share their contents, so content doesn't seem to be constructed from functional role [Fodor] |
2996 | Mental states may have the same content but different extensions [Fodor] |