70 ideas
| 6123 | Empirical investigation can't discover if holes exist, or if two things share a colour [Merricks] |
| 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 |
| 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] |
| 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] |
| 6143 | Prolonged events don't seem to endure or exist at any particular time [Merricks] |
| 6135 | A crumbling statue can't become vague, because vagueness is incoherent [Merricks] |
| 6145 | Intrinsic properties are those an object still has even if only that object exists [Merricks] |
| 6124 | I say that most of the objects of folk ontology do not exist [Merricks] |
| 6134 | Is swimming pool water an object, composed of its mass or parts? [Merricks] |
| 6125 | We can eliminate objects without a commitment to simples [Merricks] |
| 14229 | Merricks agrees that there are no composite objects, but offers a different semantics [Merricks, by Liggins] |
| 6142 | The 'folk' way of carving up the world is not intrinsically better than quite arbitrary ways [Merricks] |
| 14472 | If atoms 'arranged baseballwise' break a window, that analytically entails that a baseball did it [Merricks, by Thomasson] |
| 14469 | Overdetermination: the atoms do all the causing, so the baseball causes no breakage [Merricks] |
| 6137 | Clay does not 'constitute' a statue, as they have different persistence conditions (flaking, squashing) [Merricks] |
| 6127 | 'Unrestricted composition' says any two things can make up a third thing [Merricks] |
| 6131 | Composition as identity is false, as identity is never between a single thing and many things [Merricks] |
| 6132 | Composition as identity is false, as it implies that things never change their parts [Merricks] |
| 6141 | There is no visible difference between statues, and atoms arranged statuewise [Merricks] |
| 6130 | 'Composition' says things are their parts; 'constitution' says a whole substance is an object [Merricks] |
| 6138 | It seems wrong that constitution entails that two objects are wholly co-located [Merricks] |
| 6128 | Objects decompose (it seems) into non-overlapping parts that fill its whole region [Merricks] |
| 6136 | Eliminativism about objects gives the best understanding of the Sorites paradox [Merricks] |
| 6133 | If my counterpart is happy, that is irrelevant to whether I 'could' have been happy [Merricks] |
| 6150 | The 'warrant' for a belief is what turns a true belief into knowledge [Merricks] |
| 6144 | You hold a child in your arms, so it is not mental substance, or mental state, or software [Merricks] |
| 6140 | Maybe the word 'I' can only refer to persons [Merricks] |
| 6149 | Free will and determinism are incompatible, since determinism destroys human choice [Merricks] |
| 6148 | Human organisms can exercise downward causation [Merricks] |
| 6146 | Before Creation it is assumed that God still had many many mental properties [Merricks] |
| 6147 | The hypothesis of solipsism doesn't seem to be made incoherent by the nature of mental properties [Merricks] |
| 10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |