100 ideas
| 19693 | There is practical wisdom (for action), and theoretical wisdom (for deep understanding) [Aristotle, by Whitcomb] |
| 1575 | For Aristotle logos is essentially the ability to talk rationally about questions of value [Roochnik on Aristotle] |
| 19917 | Without reason and human help, human life is misery [Spinoza] |
| 1589 | Aristotle is the supreme optimist about the ability of logos to explain nature [Roochnik on Aristotle] |
| 8200 | Aristotelian definitions aim to give the essential properties of the thing defined [Aristotle, by Quine] |
| 4385 | Aristotelian definition involves first stating the genus, then the differentia of the thing [Aristotle, by Urmson] |
| 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] |
| 13282 | Aristotle relativises the notion of wholeness to different measures [Aristotle, by Koslicki] |
| 4730 | For Aristotle, the subject-predicate structure of Greek reflected a substance-accident structure of reality [Aristotle, by O'Grady] |
| 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] |
| 13276 | The unmoved mover and the soul show Aristotelian form as the ultimate mereological atom [Aristotle, by Koslicki] |
| 13277 | The 'form' is the recipe for building wholes of a particular kind [Aristotle, by Koslicki] |
| 5991 | For Aristotle, knowledge is of causes, and is theoretical, practical or productive [Aristotle, by Code] |
| 11239 | The notion of a priori truth is absent in Aristotle [Aristotle, by Politis] |
| 23312 | Aristotle is a rationalist, but reason is slowly acquired through perception and experience [Aristotle, by Frede,M] |
| 16111 | Aristotle wants to fit common intuitions, and therefore uses language as a guide [Aristotle, by Gill,ML] |
| 16971 | Plato says sciences are unified around Forms; Aristotle says they're unified around substance [Aristotle, by Moravcsik] |
| 11243 | Aristotelian explanations are facts, while modern explanations depend on human conceptions [Aristotle, by Politis] |
| 3320 | Aristotle's standard analysis of species and genus involves specifying things in terms of something more general [Aristotle, by Benardete,JA] |
| 12000 | Aristotle regularly says that essential properties explain other significant properties [Aristotle, by Kung] |
| 19922 | People are only free if they are guided entirely by reason [Spinoza] |
| 23300 | Aristotle and the Stoics denied rationality to animals, while Platonists affirmed it [Aristotle, by Sorabji] |
| 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] |
| 11240 | The notion of analytic truth is absent in Aristotle [Aristotle, by Politis] |
| 6559 | Aristotle never actually says that man is a rational animal [Aristotle, by Fogelin] |
| 19935 | Peoples are created by individuals, not by nature, and only distinguished by language and law [Spinoza] |
| 19914 | In nature everything has an absolute right to do anything it is capable of doing [Spinoza] |
| 19915 | Natural rights are determined by desire and power, not by reason [Spinoza] |
| 7487 | Society exists to extend human awareness [Spinoza, by Watson] |
| 19943 | The state aims to allow personal development, so its main purpose is freedom [Spinoza] |
| 19930 | Sovereignty must include the power to make people submit to it [Spinoza] |
| 19936 | Kings tend to fight wars for glory, rather than for peace and liberty [Spinoza] |
| 19937 | Monarchs are always proud, and can't back down [Spinoza] |
| 19940 | Deposing a monarch is dangerous, because the people are used to royal authority [Spinoza] |
| 19931 | Every state is more frightened of its own citizens than of external enemies [Spinoza] |
| 19920 | Democracy is a legitimate gathering of people who do whatever they can do [Spinoza] |
| 19933 | If religion is law, then piety is justice, impiety is crime, and non-believers must leave [Spinoza] |
| 19938 | Allowing religious ministers any control of the state is bad for both parties [Spinoza] |
| 19923 | Slavery is not just obedience, but acting only in the interests of the master [Spinoza] |
| 19939 | Government is oppressive if opinions can be crimes, because people can't give them up [Spinoza] |
| 19944 | Without liberty of thought there is no trust in the state, and corruption follows [Spinoza] |
| 19942 | Treason may be committed as much by words as by deeds [Spinoza] |
| 19924 | The freest state is a rational one, where people can submit themselves to reason [Spinoza] |
| 7827 | Spinoza wanted democracy based on individual rights, and is thus the first modern political philosopher [Stewart,M on Spinoza] |
| 19926 | The sovereignty has absolute power over citizens [Spinoza] |
| 19918 | Forming a society meant following reason, and giving up dangerous appetites and mutual harm [Spinoza] |
| 19919 | People only give up their rights, and keep promises, if they hope for some greater good [Spinoza] |
| 19921 | Once you have given up your rights, there is no going back [Spinoza] |
| 19925 | In democracy we don't abandon our rights, but transfer them to the majority of us [Spinoza] |
| 19928 | No one, in giving up their power and right, ceases to be a human being [Spinoza] |
| 19929 | Everyone who gives up their rights must fear the recipients of them [Spinoza] |
| 19932 | The early Hebrews, following Moses, gave up their rights to God alone [Spinoza] |
| 19916 | The order of nature does not prohibit anything, and allows whatever appetite produces [Spinoza] |
| 19927 | State and religious law can clash, so the state must make decisions about religion [Spinoza] |
| 11150 | It is the mark of an educated mind to be able to entertain an idea without accepting it [Aristotle] |
| 3037 | Aristotle said the educated were superior to the uneducated as the living are to the dead [Aristotle, by Diog. Laertius] |
| 8660 | There are potential infinities (never running out), but actual infinity is incoherent [Aristotle, by Friend] |
| 12058 | Aristotle's matter can become any other kind of matter [Aristotle, by Wiggins] |
| 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] |
| 22729 | The concepts of gods arose from observing the soul, and the cosmos [Aristotle, by Sext.Empiricus] |
| 19934 | Hebrews were very hostile to other states, who had not given up their rights to God [Spinoza] |
| 4300 | The Bible has nothing in common with reasoning and philosophy [Spinoza] |