91 ideas
| 20262 | Don't use wisdom in order to become clever! [Nietzsche] |
| 20255 | Early 19th century German philosophers enjoyed concepts, rather than scientific explanations [Nietzsche] |
| 20260 | Carlyle spent his life vainly trying to make reason appear romantic [Nietzsche] |
| 20256 | What we think is totally dictated by the language available to express it [Nietzsche] |
| 20265 | The desire for a complete system requires making the weak parts look equal to the rest [Nietzsche] |
| 9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
| 10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
| 20380 | Why should truth be omnipotent? It is enough that it is very powerful [Nietzsche] |
| 20235 | Like animals, we seek truth because we want safety [Nietzsche] |
| 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] |
| 20258 | Most people treat knowledge as a private possession [Nietzsche] |
| 20250 | We may be unable to remember, but we may never actually forget [Nietzsche] |
| 20270 | There is no one scientific method; we must try many approaches, and many emotions [Nietzsche] |
| 20131 | We can cultivate our drives, of anger, pity, curiosity, vanity, like a gardener, with good or bad taste [Nietzsche] |
| 20242 | Things are the boundaries of humanity, so all things must be known, for self-knowledge [Nietzsche] |
| 20249 | Our knowledge of the many drives that constitute us is hopelessly incomplete [Nietzsche] |
| 20231 | People used to think that outcomes were from God, rather than consequences of acts [Nietzsche] |
| 10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
| 20266 | It is essential that wise people learn to express their wisdom, possibly even as foolishness [Nietzsche] |
| 20251 | Actions done for a purpose are least understood, because we complacently think it's obvious [Nietzsche] |
| 20271 | Beauty in art is the imitation of happiness [Nietzsche] |
| 20230 | The very idea of a critique of morality is regarded as immoral! [Nietzsche] |
| 20234 | Morality prevents us from developing better customs [Nietzsche] |
| 20237 | Moral feelings are entirely different from the moral concepts used to judge actions [Nietzsche] |
| 20238 | Treating morality as feelings is just obeying your ancestors [Nietzsche] |
| 20243 | Human beings are not majestic, either through divine origins, or through grand aims [Nietzsche] |
| 20268 | Most dying people have probably lost more important things than what they are about to lose [Nietzsche] |
| 20252 | Marriage is too serious to be permitted for people in love! [Nietzsche] |
| 20236 | Marriage upholds the idea that love, though a passion, can endure [Nietzsche] |
| 20263 | Fear reveals the natures of other people much more clearly than love does [Nietzsche] |
| 20233 | Punishment has distorted the pure innocence of the contingency of outcomes [Nietzsche] |
| 20248 | People do nothing for their real ego, but only for a phantom ego created by other people [Nietzsche] |
| 20246 | If you feel to others as they feel to themselves, you must hate a self-hater [Nietzsche] |
| 20272 | Honesty is a new young virtue, and we can promote it, or not [Nietzsche] |
| 20240 | The Jews treated great anger as holy, and were in awe of those who expressed it [Nietzsche] |
| 20244 | Christianity replaces rational philosophical virtues with great passions focused on God [Nietzsche] |
| 20274 | The cardinal virtues want us to be honest, brave, magnanimous and polite [Nietzsche] |
| 20257 | Cool courage and feverish bravery have one name, but are two very different virtues [Nietzsche] |
| 20259 | Teach youth to respect people who differ with them, not people who agree with them [Nietzsche] |
| 20267 | Seeing duty as a burden makes it a bit cruel, and it can thus never become a habit [Nietzsche] |
| 20275 | Most people think they are already complete, but we can cultivate ourselves [Nietzsche] |
| 20229 | No authority ever willingly accepts criticism [Nietzsche] |
| 20254 | People govern for the pleasure of it, or just to avoid being governed [Nietzsche] |
| 20273 | The French Revolution gave trusting Europe the false delusion of instant recovery [Nietzsche] |
| 20232 | Get rid of the idea of punishment! It is a noxious weed! [Nietzsche] |
| 20253 | Modern wars arise from the study of history [Nietzsche] |
| 13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
| 20261 | History does not concern what really happened, but supposed events, which have all the influence [Nietzsche] |
| 20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
| 20241 | Enquirers think finding our origin is salvation, but it turns out to be dull [Nietzsche] |
| 20245 | Christianity hoped for a short cut to perfection, that skipped the hard labour of morality [Nietzsche] |
| 20247 | Christianity was successful because of its heathen rituals [Nietzsche] |
| 20269 | 'I believe because it is absurd' - but how about 'I believe because I am absurd' [Nietzsche] |
| 20264 | The easy and graceful aspects of a person are called 'soul', and inner awkwardness is called 'soulless' [Nietzsche] |