99 ideas
| 13773 | For the truth you need Prodicus's fifty-drachma course, not his one-drachma course [Socrates] |
| 343 | The unexamined life is not worth living for men [Socrates] |
| 7421 | A philosopher is one who cares about what other people care about [Socrates, by Foucault] |
| 1649 | Socrates opened philosophy to all, but Plato confined moral enquiry to a tiny elite [Vlastos on Socrates] |
| 5842 | Philosophical discussion involves dividing subject-matter into categories [Socrates, by Xenophon] |
| 648 | Socrates began the quest for something universal with his definitions, but he didn't make them separate [Socrates, by Aristotle] |
| 164 | It is legitimate to play the devil's advocate [Socrates] |
| 1647 | In Socratic dialogue you must say what you believe, so unasserted premises are not debated [Vlastos on Socrates] |
| 115 | Socrates was pleased if his mistakes were proved wrong [Socrates] |
| 22099 | The method of Socrates shows the student is discovering the truth within himself [Socrates, by Carlisle] |
| 5844 | Socrates always proceeded in argument by general agreement at each stage [Socrates, by Xenophon] |
| 11389 | Socrates sought essences, which are the basis of formal logic [Socrates, by Aristotle] |
| 9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
| 10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
| 639 | Socrates developed definitions as the basis of syllogisms, and also inductive arguments [Socrates, by Aristotle] |
| 10098 | The 'power set' of A is all the subsets of A [George/Velleman] |
| 10101 |
Cartesian Product A x B: the set of all ordered pairs |
| 10099 | The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}} [George/Velleman] |
| 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] |
| 1652 | Socrates did not consider universals or definitions as having separate existence, but Plato made Forms of them [Socrates, by Aristotle] |
| 1650 | For Socrates our soul, though hard to define, is our self [Vlastos on Socrates] |
| 2170 | Homer does not distinguish between soul and body [Homer, by Williams,B] |
| 23252 | Socrates first proposed that we are run by mind or reason [Socrates, by Frede,M] |
| 10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
| 2171 | The 'will' doesn't exist; there is just conclusion, then action [Homer, by Williams,B] |
| 5843 | People do what they think they should do, and only ever do what they think they should do [Socrates, by Xenophon] |
| 5253 | Socrates was shocked by the idea of akrasia, but observation shows that it happens [Aristotle on Socrates] |
| 199 | The common belief is that people can know the best without acting on it [Socrates] |
| 1653 | Socrates did not accept the tripartite soul (which permits akrasia) [Vlastos on Socrates] |
| 195 | No one willingly commits an evil or base act [Socrates] |
| 5839 | For Socrates, wisdom and prudence were the same thing [Socrates, by Xenophon] |
| 5867 | For Socrates, virtues are forms of knowledge, so knowing justice produces justice [Socrates, by Aristotle] |
| 5069 | Socrates was the first to base ethics upon reason, and use reason to explain it [Taylor,R on Socrates] |
| 5836 | All human virtues are increased by study and practice [Socrates, by Xenophon] |
| 5840 | The wise perform good actions, and people fail to be good without wisdom [Socrates, by Xenophon] |
| 185 | Socrates despised good looks [Socrates, by Plato] |
| 5070 | Socrates conservatively assumed that Athenian conventions were natural and true [Taylor,R on Socrates] |
| 5838 | A well-made dung basket is fine, and a badly-made gold shield is base, because of function [Socrates, by Xenophon] |
| 339 | Men fear death as a great evil when it may be a great blessing [Socrates] |
| 344 | If death is like a night of dreamless sleep, such nights are very pleasant [Socrates] |
| 5837 | Things are both good and fine by the same standard [Socrates, by Xenophon] |
| 21819 | Plato says the Good produces the Intellectual-Principle, which in turn produces the Soul [Homer, by Plotinus] |
| 3017 | The only good is knowledge, and the only evil is ignorance [Socrates, by Diog. Laertius] |
| 1646 | Socrates was the first to put 'eudaimonia' at the centre of ethics [Socrates, by Vlastos] |
| 2 | We should not even harm someone who harms us [Socrates] |
| 1663 | By 'areté' Socrates means just what we mean by moral virtue [Vlastos on Socrates] |
| 345 | A good man cannot be harmed, either in life or in death [Socrates] |
| 4323 | Socrates is torn between intellectual virtue, which is united and teachable, and natural virtue, which isn't [PG on Socrates] |
| 8003 | Socrates agrees that virtue is teachable, but then denies that there are teachers [Socrates, by MacIntyre] |
| 126 | We should ask what sort of people we want to be [Socrates] |
| 4111 | Socrates believed that basically there is only one virtue, the power of right judgement [Socrates, by Williams,B] |
| 7808 | Socrates made the civic values of justice and friendship paramount [Socrates, by Grayling] |
| 346 | One ought not to return a wrong or injury to any person, whatever the provocation [Socrates] |
| 23907 | Courage is scientific knowledge [Socrates, by Aristotle] |
| 341 | Wealth is good if it is accompanied by virtue [Socrates] |
| 7585 | Socrates emphasises that the knower is an existing individual, with existence his main task [Socrates, by Kierkegaard] |
| 11388 | Let there be one ruler [Homer] |
| 5841 | Obedience to the law gives the best life, and success in war [Socrates, by Xenophon] |
| 347 | Will I stand up against the law, simply because I have been unjustly judged? [Socrates] |
| 1661 | Socrates was the first to grasp that a cruelty is not justified by another cruelty [Vlastos on Socrates] |
| 5846 | A lover using force is a villain, but a seducer is much worse, because he corrupts character [Socrates, by Xenophon] |
| 1657 | Socrates holds that right reason entails virtue, and this must also apply to the gods [Vlastos on Socrates] |
| 1662 | A new concept of God as unswerving goodness emerges from Socrates' commitment to virtue [Vlastos on Socrates] |
| 338 | Socrates is accused of denying the gods, saying sun is stone and moon is earth [Socrates, by Plato] |
| 14829 | Homer so enjoys the company of the gods that he must have been deeply irreligious [Homer, by Nietzsche] |