128 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] |
| 639 | Socrates developed definitions as the basis of syllogisms, and also inductive arguments [Socrates, by Aristotle] |
| 9542 | The best known axiomatization of PL is Whitehead/Russell, with four axioms and two rules [Russell/Whitehead, by Hughes/Cresswell] |
| 15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
| 15946 | Cantor developed sets from a progression into infinity by addition, multiplication and exponentiation [Cantor, by Lavine] |
| 9616 | A set is a collection into a whole of distinct objects of our intuition or thought [Cantor] |
| 13444 | Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD] |
| 18098 | Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock] |
| 15505 | If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis] |
| 10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
| 10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
| 17831 | Cantor gives informal versions of ZF axioms as ways of getting from one set to another [Cantor, by Lake] |
| 13016 | The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy] |
| 21720 | Russell saw Reducibility as legitimate for reducing classes to logic [Linsky,B on Russell/Whitehead] |
| 14199 | Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley] |
| 10044 | Russell denies extensional sets, because the null can't be a collection, and the singleton is just its element [Russell/Whitehead, by Shapiro] |
| 18208 | We regard classes as mere symbolic or linguistic conveniences [Russell/Whitehead] |
| 8204 | Lewis's 'strict implication' preserved Russell's confusion of 'if...then' with implication [Quine on Russell/Whitehead] |
| 9359 | Russell's implication means that random sentences imply one another [Lewis,CI on Russell/Whitehead] |
| 21707 | Russell unusually saw logic as 'interpreted' (though very general, and neutral) [Russell/Whitehead, by Linsky,B] |
| 10036 | In 'Principia' a new abstract theory of relations appeared, and was applied [Russell/Whitehead, by Gödel] |
| 10082 | There are infinite sets that are not enumerable [Cantor, by Smith,P] |
| 13483 | Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD] |
| 8710 | The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend] |
| 15910 | Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine] |
| 15905 | Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine] |
| 9983 | Cantor took the ordinal numbers to be primary [Cantor, by Tait] |
| 17798 | Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry] |
| 9971 | Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait] |
| 9892 | Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett] |
| 15911 | Ordinals are generated by endless succession, followed by a limit ordinal [Cantor, by Lavine] |
| 14136 | A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor] |
| 15906 | Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine] |
| 11015 | Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read] |
| 15903 | A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine] |
| 18251 | Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine] |
| 18248 | A real number is the class of rationals less than the number [Russell/Whitehead, by Shapiro] |
| 15902 | Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine] |
| 15908 | It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine] |
| 13464 | Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD] |
| 10112 | The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman] |
| 15896 | Cantor needed Power Set for the reals, but then couldn't count the new collections [Cantor, by Lavine] |
| 17889 | CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner] |
| 13447 | Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD] |
| 10883 | Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten] |
| 8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
| 13528 | Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS] |
| 9555 | Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara] |
| 18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
| 15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
| 18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy] |
| 18152 | Russell takes numbers to be classes, but then reduces the classes to numerical quantifiers [Russell/Whitehead, by Bostock] |
| 9992 | The 'extension of a concept' in general may be quantitatively completely indeterminate [Cantor] |
| 10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro] |
| 18176 | Pure mathematics is pure set theory [Cantor] |
| 8631 | Cantor says that maths originates only by abstraction from objects [Cantor, by Frege] |
| 10025 | Russell and Whitehead took arithmetic to be higher-order logic [Russell/Whitehead, by Hodes] |
| 8683 | Russell and Whitehead were not realists, but embraced nearly all of maths in logic [Russell/Whitehead, by Friend] |
| 10037 | 'Principia' lacks a precise statement of the syntax [Gödel on Russell/Whitehead] |
| 10093 | The ramified theory of types used propositional functions, and covered bound variables [Russell/Whitehead, by George/Velleman] |
| 8691 | The Russell/Whitehead type theory was limited, and was not really logic [Friend on Russell/Whitehead] |
| 10305 | In 'Principia Mathematica', logic is exceeded in the axioms of infinity and reducibility, and in the domains [Bernays on Russell/Whitehead] |
| 8684 | Russell and Whitehead consider the paradoxes to indicate that we create mathematical reality [Russell/Whitehead, by Friend] |
| 8746 | To avoid vicious circularity Russell produced ramified type theory, but Ramsey simplified it [Russell/Whitehead, by Shapiro] |
| 1652 | Socrates did not consider universals or definitions as having separate existence, but Plato made Forms of them [Socrates, by Aristotle] |
| 12033 | An object is identical with itself, and no different indiscernible object can share that [Russell/Whitehead, by Adams,RM] |
| 10040 | Russell showed, through the paradoxes, that our basic logical intuitions are self-contradictory [Russell/Whitehead, by Gödel] |
| 1650 | For Socrates our soul, though hard to define, is our self [Vlastos on Socrates] |
| 23252 | Socrates first proposed that we are run by mind or reason [Socrates, by Frede,M] |
| 21725 | The multiple relations theory says assertions about propositions are about their ingredients [Russell/Whitehead, by Linsky,B] |
| 23474 | A judgement is a complex entity, of mind and various objects [Russell/Whitehead] |
| 23455 | The meaning of 'Socrates is human' is completed by a judgement [Russell/Whitehead] |
| 23480 | The multiple relation theory of judgement couldn't explain the unity of sentences [Morris,M on Russell/Whitehead] |
| 18275 | Only the act of judging completes the meaning of a statement [Russell/Whitehead] |
| 8715 | Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend] |
| 13454 | Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor] |
| 9145 | We form the image of a cardinal number by a double abstraction, from the elements and from their order [Cantor] |
| 23453 | Propositions as objects of judgement don't exist, because we judge several objects, not one [Russell/Whitehead] |
| 195 | No one willingly commits an evil or base act [Socrates] |
| 1653 | Socrates did not accept the tripartite soul (which permits akrasia) [Vlastos on Socrates] |
| 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] |
| 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] |
| 344 | If death is like a night of dreamless sleep, such nights are very pleasant [Socrates] |
| 339 | Men fear death as a great evil when it may be a great blessing [Socrates] |
| 5837 | Things are both good and fine by the same standard [Socrates, by Xenophon] |
| 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] |
| 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] |
| 10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
| 13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |
| 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] |