68 ideas
| 12667 | Metaphysics aims at the simplest explanation, without regard to testability [Ellis] |
| 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] |
| 10101 |
Cartesian Product A x B: the set of all ordered pairs |
| 10099 | The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}} [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] |
| 12666 | We can base logic on acceptability, and abandon the Fregean account by truth-preservation [Ellis] |
| 10111 | Asserting Excluded Middle is a hallmark of realism about the natural world [George/Velleman] |
| 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] |
| 12688 | Mathematics is the formal study of the categorical dimensions of things [Ellis] |
| 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] |
| 12683 | Objects and substances are a subcategory of the natural kinds of processes [Ellis] |
| 12670 | A physical event is any change of distribution of energy [Ellis] |
| 12673 | Physical properties are those relevant to how a physical system might act [Ellis] |
| 12665 | I support categorical properties, although most people only want causal powers [Ellis] |
| 12682 | Essentialism needs categorical properties (spatiotemporal and numerical relations) and dispositions [Ellis] |
| 12684 | Spatial, temporal and numerical relations have causal roles, without being causal [Ellis] |
| 12672 | Properties and relations are discovered, so they can't be mere sets of individuals [Ellis] |
| 12676 | Causal powers can't rest on things which lack causal power [Ellis] |
| 23781 | Categoricals exist to influence powers. Such as structures, orientations and magnitudes [Ellis, by Williams,NE] |
| 12686 | Causal powers are a proper subset of the dispositional properties [Ellis] |
| 12685 | Categorical properties depend only on the structures they represent [Ellis] |
| 12679 | A real essence is a kind's distinctive properties [Ellis] |
| 12668 | Metaphysical necessity holds between things in the world and things they make true [Ellis] |
| 12687 | Metaphysical necessities are those depending on the essential nature of things [Ellis] |
| 12669 | Science aims to explain things, not just describe them [Ellis] |
| 10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
| 13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
| 12681 | There are natural kinds of processes [Ellis] |
| 12680 | Natural kind structures go right down to the bottom level [Ellis] |
| 12675 | Laws of nature are just descriptions of how things are disposed to behave [Ellis] |
| 12671 | I deny forces as entities that intervene in causation, but are not themselves causal [Ellis] |
| 12674 | Energy is the key multi-valued property, vital to scientific realism [Ellis] |
| 12689 | Simultaneity can be temporal equidistance from the Big Bang [Ellis] |
| 20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
| 12690 | The present is the collapse of the light wavefront from the Big Bang [Ellis] |