69 ideas
| 9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
| 10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
| 15879 | The Square of Opposition has two contradictory pairs, one contrary pair, and one sub-contrary pair [Harré] |
| 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 |
| 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] |
| 15891 | Traditional quantifiers combine ordinary language generality and ontology assumptions [Harré] |
| 15878 | Some quantifiers, such as 'any', rule out any notion of order within their range [Harré] |
| 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] |
| 15874 | Scientific properties are not observed qualities, but the dispositions which create them [Harré] |
| 15884 | Laws of nature remain the same through any conditions, if the underlying mechanisms are unchanged [Harré] |
| 20769 | Sphaerus he was not assenting to the presence of pomegranates, but that it was 'reasonable' [Sphaerus, by Diog. Laertius] |
| 15880 | In physical sciences particular observations are ordered, but in biology only the classes are ordered [Harré] |
| 15869 | Reports of experiments eliminate the experimenter, and present results as the behaviour of nature [Harré] |
| 15881 | We can save laws from counter-instances by treating the latter as analytic definitions [Harré] |
| 15882 | Since there are three different dimensions for generalising laws, no one system of logic can cover them [Harré] |
| 15887 | 'Grue' introduces a new causal hypothesis - that emeralds can change colour [Harré] |
| 15888 | The grue problem shows that natural kinds are central to science [Harré] |
| 15889 | It is because ravens are birds that their species and their colour might be connected [Harré] |
| 15890 | Non-black non-ravens just aren't part of the presuppositions of 'all ravens are black' [Harré] |
| 15885 | The necessity of Newton's First Law derives from the nature of material things, not from a mechanism [Harré] |
| 15868 | Idealisation idealises all of a thing's properties, but abstraction leaves some of them out [Harré] |
| 10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
| 15886 | Science rests on the principle that nature is a hierarchy of natural kinds [Harré] |
| 15864 | Classification is just as important as laws in natural science [Harré] |
| 15865 | Newton's First Law cannot be demonstrated experimentally, as that needs absence of external forces [Harré] |
| 15862 | Laws can come from data, from theory, from imagination and concepts, or from procedures [Harré] |
| 15870 | Are laws of nature about events, or types and universals, or dispositions, or all three? [Harré] |
| 15871 | Are laws about what has or might happen, or do they also cover all the possibilities? [Harré] |
| 15876 | Maybe laws of nature are just relations between properties? [Harré] |
| 15860 | We take it that only necessary happenings could be laws [Harré] |
| 15872 | Must laws of nature be universal, or could they be local? [Harré] |
| 15867 | Laws describe abstract idealisations, not the actual mess of nature [Harré] |
| 15892 | Laws of nature state necessary connections of things, events and properties, based on models of mechanisms [Harré] |
| 15875 | In counterfactuals we keep substances constant, and imagine new situations for them [Harré] |