90 ideas
| 15447 | We shouldn't always follow where the argument leads! [Lewis on Plato] |
| 243 | It is foolish to quarrel with the mind's own reasoning processes [Plato] |
| 241 | We ought to follow where the argument leads us [Plato] |
| 21264 | Mortals are incapable of being fully rational [Plato] |
| 251 | Truth has the supreme value, for both gods and men [Plato] |
| 15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
| 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] |
| 13016 | The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy] |
| 14199 | Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley] |
| 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] |
| 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] |
| 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] |
| 8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
| 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] |
| 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] |
| 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] |
| 21259 | To grasp a thing we need its name, its definition, and what it really is [Plato] |
| 9216 | Each area of enquiry, and its source, has its own distinctive type of necessity [Fine,K] |
| 9214 | Unsupported testimony may still be believable [Fine,K] |
| 21260 | Soul is what is defined by 'self-generating motion' [Plato] |
| 276 | My individuality is my soul, which carries my body around [Plato] |
| 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] |
| 249 | People who value beauty above virtue insult the soul by placing the body above it [Plato] |
| 265 | An action is only just if it is performed by someone with a just character and outlook [Plato] |
| 269 | Attempted murder is like real murder, but we should respect the luck which avoided total ruin [Plato] |
| 240 | It would be strange if the gods rewarded those who experienced the most pleasure in life [Plato] |
| 264 | The conquest of pleasure is the noblest victory of all [Plato] |
| 4332 | Virtue is a concord of reason and emotion, with pleasure and pain trained to correct ends [Plato] |
| 248 | A serious desire for moral excellence is very rare indeed [Plato] |
| 253 | Every crime is the result of excessive self-love [Plato] |
| 263 | The only worthwhile life is one devoted to physical and moral perfection [Plato] |
| 235 | Virtue is the aim of all laws [Plato] |
| 277 | The Guardians must aim to discover the common element in the four cardinal virtues [Plato] |
| 254 | Excessive laughter and tears must be avoided [Plato] |
| 266 | Injustice is the mastery of the soul by bad feelings, even if they do not lead to harm [Plato] |
| 242 | The best people are produced where there is no excess of wealth or poverty [Plato] |
| 256 | Virtue and great wealth are incompatible [Plato] |
| 245 | Totalitarian states destroy friendships and community spirit [Plato] |
| 239 | Education in virtue produces citizens who are active but obedient [Plato] |
| 1402 | Friendship is impossible between master and slave, even if they are made equal [Plato] |
| 262 | Men and women should qualify equally for honours on merit [Plato] |
| 236 | Sound laws achieve the happiness of those who observe them [Plato] |
| 259 | Justice is granting the equality which unequals deserve [Plato] |
| 238 | Children's games should channel their pleasures into adult activity [Plato] |
| 260 | Control of education is the key office of state, and should go to the best citizen [Plato] |
| 257 | Mathematics has the widest application of any subject on the curriculum [Plato] |
| 4331 | Education is channelling a child's feelings into the right course before it understands why [Plato] |
| 250 | The best way to educate the young is not to rebuke them, but to set a good example [Plato] |
| 275 | Creation is not for you; you exist for the sake of creation [Plato] |
| 9215 | Causation is easier to disrupt than logic, so metaphysics is part of nature, not vice versa [Fine,K] |
| 10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
| 273 | Movement is transmitted through everything, and it must have started with self-generated motion [Plato] |
| 13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |
| 8004 | In 'The Laws', to obey the law is to be obey god [Plato, by MacIntyre] |
| 21261 | Self-moving soul has to be the oldest thing there is [Plato] |
| 21258 | The only possible beginning for the endless motions of reality is something self-generated [Plato] |
| 21257 | Self-generating motion is clearly superior to all other kinds of motion [Plato] |
| 274 | Soul must be the cause of all the opposites, such as good and evil or beauty and ugliness [Plato] |
| 21263 | If all the motions of nature reflect calculations of reason, then the best kind of soul must direct it [Plato] |
| 278 | If astronomical movements are seen as necessary instead of by divine will, this leads to atheism [Plato] |
| 21265 | The heavens must be full of gods, controlling nature either externally or from within [Plato] |
| 21262 | There must be at least two souls controlling the cosmos, one doing good, the other the opposite [Plato] |