99 ideas
| 18330 | Judging by the positive forces, the Renaissance was the last great age [Nietzsche] |
| 2900 | I revere Heraclitus [Nietzsche] |
| 2913 | Thucydides was the perfect anti-platonist sophist [Nietzsche] |
| 2909 | Thinking has to be learned in the way dancing has to be learned [Nietzsche] |
| 2892 | Wanting a system in philosophy is a lack of integrity [Nietzsche] |
| 2896 | I want to understand the Socratic idea that 'reason equals virtue equals happiness' [Nietzsche] |
| 2897 | With dialectics the rabble gets on top [Nietzsche] |
| 9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
| 10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
| 2898 | Anything which must first be proved is of little value [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] |
| 18739 | Three stages of philosophical logic: syntactic (1905-55), possible worlds (1963-85), widening (1990-) [Horsten/Pettigrew] |
| 10111 | Asserting Excluded Middle is a hallmark of realism about the natural world [George/Velleman] |
| 18741 | Logical formalization makes concepts precise, and also shows their interrelation [Horsten/Pettigrew] |
| 10129 | A 'model' is a meaning-assignment which makes all the axioms true [George/Velleman] |
| 18744 | Models are sets with functions and relations, and truth built up from the components [Horsten/Pettigrew] |
| 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] |
| 18740 | If 'exist' doesn't express a property, we can hardly ask for its essence [Horsten/Pettigrew] |
| 18317 | The 'real being' of things is a nothingness constructed from contradictions in the actual world [Nietzsche] |
| 18315 | We get the concept of 'being' from the concept of the 'ego' [Nietzsche] |
| 18316 | The grounds for an assertion that the world is only apparent actually establish its reality [Nietzsche] |
| 18314 | In language we treat 'ego' as a substance, and it is thus that we create the concept 'thing' [Nietzsche] |
| 18745 | A Tarskian model can be seen as a possible state of affairs [Horsten/Pettigrew] |
| 18747 | The 'spheres model' was added to possible worlds, to cope with counterfactuals [Horsten/Pettigrew] |
| 18748 | Epistemic logic introduced impossible worlds [Horsten/Pettigrew] |
| 18746 | Possible worlds models contain sets of possible worlds; this is a large metaphysical commitment [Horsten/Pettigrew] |
| 18750 | Using possible worlds for knowledge and morality may be a step too far [Horsten/Pettigrew] |
| 18309 | The evidence of the senses is falsified by reason [Nietzsche] |
| 18323 | Any explanation will be accepted as true if it gives pleasure and a feeling of power [Nietzsche] |
| 18310 | The 'highest' concepts are the most general and empty concepts [Nietzsche] |
| 20368 | There are no 'individual' persons; we are each the sum of humanity up to this moment [Nietzsche] |
| 2899 | The fanatical rationality of Greek philosophy shows that they were in a state of emergency [Nietzsche] |
| 10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
| 18313 | The big error is to think the will is a faculty producing effects; in fact, it is just a word [Nietzsche] |
| 20133 | The 'motive' is superficial, and may even hide the antecedents of a deed [Nietzsche] |
| 18326 | The beautiful never stands alone; it derives from man's pleasure in man [Nietzsche] |
| 20101 | Without music life would be a mistake [Nietzsche] |
| 2902 | Healthy morality is dominated by an instinct for life [Nietzsche] |
| 18311 | Philosophers hate values having an origin, and want values to be self-sufficient [Nietzsche] |
| 18324 | There are no moral facts, and moralists believe in realities which do not exist [Nietzsche] |
| 2904 | The doctrine of free will has been invented essentially in order to blame and punish people [Nietzsche] |
| 2895 | The value of life cannot be estimated [Nietzsche] |
| 18322 | When we establish values, that is life itself establishing them, through us [Nietzsche] |
| 2893 | In every age the wisest people have judged life to be worthless [Nietzsche] |
| 18308 | A philosopher fails in wisdom if he thinks the value of life is a problem [Nietzsche] |
| 2894 | Value judgements about life can never be true [Nietzsche] |
| 18321 | To evaluate life one must know it, but also be situated outside of it [Nietzsche] |
| 18319 | Love is the spiritualisation of sensuality [Nietzsche] |
| 2903 | A good human will be virtuous because they are happy [Nietzsche] |
| 2891 | Only the English actually strive after happiness [Nietzsche] |
| 18327 | A wholly altruistic morality, with no egoism, is a thoroughly bad thing [Nietzsche] |
| 15606 | Military idea: what does not kill me makes me stronger [Nietzsche] |
| 18328 | Invalids are parasites [Nietzsche] |
| 18331 | Democracy is organisational power in decline [Nietzsche] |
| 18332 | The creation of institutions needs a determination which is necessarily anti-liberal [Nietzsche] |
| 2911 | True justice is equality for equals and inequality for unequals [Nietzsche] |
| 18320 | To renounce war is to renounce the grand life [Nietzsche] |
| 2908 | There is a need for educators who are themselves educated [Nietzsche] |
| 18329 | Sometimes it is an error to have been born - but we can rectify it [Nietzsche] |
| 2905 | 'Purpose' is just a human fiction [Nietzsche] |
| 18312 | The supreme general but empty concepts must be compatible, and hence we get 'God' [Nietzsche] |
| 2906 | By denying God we deny human accountability, and thus we redeem the world [Nietzsche] |
| 2901 | How could the Church intelligently fight against passion if it preferred poorness of spirit to intelligence? [Nietzsche] |
| 18325 | Christians believe that only God can know what is good for man [Nietzsche] |
| 18318 | People who disparage actual life avenge themselves by imagining a better one [Nietzsche] |