84 ideas
| 17240 | Definitions are the first step in philosophy [Hobbes] |
| 21900 | Deleuze relies on Spinoza (immanence), Bergson (duration), and difference (Nietzsche) [May] |
| 17237 | Definitions of things that are caused must express their manner of generation [Hobbes] |
| 17239 | Definition is resolution of names into successive genera, and finally the difference [Hobbes] |
| 9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
| 17241 | A defined name should not appear in the definition [Hobbes] |
| 10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
| 17242 | 'Petitio principii' is reusing the idea to be defined, in disguised words [Hobbes] |
| 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] |
| 17245 | A part of a part is a part of a whole [Hobbes] |
| 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] |
| 17258 | If we just say one, one, one, one, we don't know where we have got to [Hobbes] |
| 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] |
| 17253 | Change is nothing but movement [Hobbes] |
| 16670 | Accidents are just modes of thinking about bodies [Hobbes] |
| 16621 | Accidents are not parts of bodies (like blood in a cloth); they have accidents as things have a size [Hobbes] |
| 16734 | The complete power of an event is just the aggregate of the qualities that produced it [Hobbes] |
| 17247 | The only generalities or universals are names or signs [Hobbes] |
| 14960 | Bodies are independent of thought, and coincide with part of space [Hobbes] |
| 17250 | If you separate the two places of one thing, you will also separate the thing [Hobbes] |
| 17249 | If you separated two things in the same place, you would also separate the places [Hobbes] |
| 17248 | If a whole body is moved, its parts must move with it [Hobbes] |
| 16790 | A body is always the same, whether the parts are together or dispersed [Hobbes] |
| 17244 | To make a whole, parts needn't be put together, but can be united in the mind [Hobbes] |
| 17233 | Particulars contain universal things [Hobbes] |
| 17246 | Some accidental features are permanent, unless the object perishes [Hobbes] |
| 17251 | The feature which picks out or names a thing is usually called its 'essence' [Hobbes] |
| 17257 | It is the same river if it has the same source, no matter what flows in it [Hobbes] |
| 12853 | Some individuate the ship by unity of matter, and others by unity of form [Hobbes] |
| 17256 | If a new ship were made of the discarded planks, would two ships be numerically the same? [Hobbes] |
| 16794 | As an infant, Socrates was not the same body, but he was the same human being [Hobbes] |
| 17255 | Two bodies differ when (at some time) you can say something of one you can't say of the other [Hobbes] |
| 16582 | We can imagine a point swelling and contracting - but not how this could be done [Hobbes] |
| 17238 | Science aims to show causes and generation of things [Hobbes] |
| 17260 | Imagination is just weakened sensation [Hobbes] |
| 19373 | A 'conatus' is an initial motion, experienced by us as desire or aversion [Hobbes, by Arthur,R] |
| 2948 | Sensation is merely internal motion of the sentient being [Hobbes] |
| 17261 | Apart from pleasure and pain, the only emotions are appetite and aversion [Hobbes] |
| 17236 | Words are not for communication, but as marks for remembering what we have learned [Hobbes] |
| 10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
| 21898 | For existentialists the present is empty without the pull of the future and weight of the past [May] |
| 21905 | Liberal theory starts from the governed, not from the governor [May] |
| 16600 | Prime matter is body considered with mere size and extension, and potential [Hobbes] |
| 17252 | Acting on a body is either creating or destroying a property in it [Hobbes] |
| 17254 | An effect needs a sufficient and necessary cause [Hobbes] |
| 17235 | A cause is the complete sum of the features which necessitate the effect [Hobbes] |
| 17234 | Motion is losing one place and acquiring another [Hobbes] |
| 17259 | 'Force' is the quantity of movement imposed on something [Hobbes] |
| 17243 | Past times can't exist anywhere, apart from in our memories [Hobbes] |