74 ideas
| 22496 | Wisdom only implies the knowledge achievable in any normal lifetime [Foot] |
| 16943 | Philosophy is continuous with science, and has no external vantage point [Quine] |
| 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] |
| 16949 | Klein summarised geometry as grouped together by transformations [Quine] |
| 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] |
| 16939 | Mass terms just concern spread, but other terms involve both spread and individuation [Quine] |
| 16948 | Once we know the mechanism of a disposition, we can eliminate 'similarity' [Quine] |
| 16945 | We judge things to be soluble if they are the same kind as, or similar to, things that do dissolve [Quine] |
| 16944 | Science is common sense, with a sophisticated method [Quine] |
| 16940 | Induction is just more of the same: animal expectations [Quine] |
| 16941 | Induction relies on similar effects following from each cause [Quine] |
| 16933 | Grue is a puzzle because the notions of similarity and kind are dubious in science [Quine] |
| 16934 | General terms depend on similarities among things [Quine] |
| 16938 | To learn yellow by observation, must we be told to look at the colour? [Quine] |
| 8486 | Standards of similarity are innate, and the spacing of qualities such as colours can be mapped [Quine] |
| 16947 | Similarity is just interchangeability in the cosmic machine [Quine] |
| 10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
| 16932 | Projectible predicates can be universalised about the kind to which they refer [Quine] |
| 23694 | All criterions of practical rationality derive from goodness of will [Foot] |
| 23686 | Moral reason is not just neutral, because morality is part of the standard of rationality [Foot, by Hacker-Wright] |
| 23693 | Practical rationality must weigh both what is morally and what is non-morally required [Foot] |
| 23687 | Moral virtues arise from human nature, as part of what makes us good human beings [Foot, by Hacker-Wright] |
| 22493 | Sterility is a human defect, but the choice to be childless is not [Foot] |
| 22492 | Virtues are as necessary to humans as stings are to bees [Foot] |
| 22491 | Moral evaluations are not separate from facts, but concern particular facts about functioning [Foot] |
| 22497 | Deep happiness usually comes from the basic things in life [Foot] |
| 22498 | Happiness is enjoying the pursuit and attainment of right ends [Foot] |
| 23695 | Good actions can never be justified by the good they brings to their agent [Foot] |
| 22499 | We all know that just pretending to be someone's friend is not the good life [Foot] |
| 22495 | Someone is a good person because of their rational will, not their body or memory [Foot] |
| 22502 | Refraining from murder is not made good by authenticity or self-fulfilment [Foot] |
| 7375 | Quine probably regrets natural kinds now being treated as essences [Quine, by Dennett] |
| 16935 | If similarity has no degrees, kinds cannot be contained within one another [Quine] |
| 16936 | Comparative similarity allows the kind 'colored' to contain the kind 'red' [Quine] |
| 16937 | You can't base kinds just on resemblance, because chains of resemblance are a muddle [Quine] |
| 16942 | It is hard to see how regularities could be explained [Quine] |