176 ideas
| 13786 | Wisdom is called 'beautiful', because it performs fine works [Plato] |
| 13780 | Good people are no different from wise ones [Plato] |
| 2845 | Free and great-souled men do not keep asking "what is the use of it?" [Aristotle] |
| 22521 | Our method of inquiry is to examine the smallest parts that make up the whole [Aristotle] |
| 1570 | Human beings, alone of the animals, have logos [Aristotle] |
| 1574 | Reasoning distinguishes what is beneficial, and hence what is right [Aristotle] |
| 2801 | Intelligence which looks ahead is a natural master, while bodily strength is a natural slave [Aristotle] |
| 13778 | A dialectician is someone who knows how to ask and to answer questions [Plato] |
| 22529 | Men are natural leaders (apart from the unnatural ones) [Aristotle] |
| 22571 | 'If each is small, so too are all' is in one way false, for the whole composed of all is not small [Aristotle] |
| 13776 | Truths say of what is that it is, falsehoods say of what is that it is not [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] |
| 13777 | A name is a sort of tool [Plato] |
| 13790 | A name-giver might misname something, then force other names to conform to it [Plato] |
| 13791 | Things must be known before they are named, so it can't be the names that give us knowledge [Plato] |
| 13789 | Anyone who knows a thing's name also knows the thing [Plato] |
| 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] |
| 2063 | How can beauty have identity if it changes? [Plato] |
| 13775 | We only succeed in cutting if we use appropriate tools, not if we approach it randomly [Plato] |
| 13787 | Doesn't each thing have an essence, just as it has other qualities? [Plato] |
| 22525 | The whole is prior to its parts, because parts are defined by their role [Aristotle] |
| 13774 | Things don't have every attribute, and essence isn't private, so each thing has an essence [Plato] |
| 13772 | Is the being or essence of each thing private to each person? [Plato] |
| 13788 | If we made a perfect duplicate of Cratylus, there would be two Cratyluses [Plato] |
| 22587 | Understanding is the aim of our nature [Aristotle] |
| 13792 | There can't be any knowledge if things are constantly changing [Plato] |
| 22522 | To grasp something, trace it back to its natural origins [Aristotle] |
| 22524 | The nature of each thing is its mature state [Aristotle] |
| 13781 | Soul causes the body to live, and gives it power to breathe and to be revitalized [Plato] |
| 22528 | The nature of all animate things is to have one part which rules it [Aristotle] |
| 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] |
| 22570 | Rhetoric now enables good speakers to become popular leaders [Aristotle] |
| 22574 | A community can lack self-control [Aristotle] |
| 2837 | Nothing contrary to nature is beautiful [Aristotle] |
| 2824 | The collective judgement of many people on art is better than that of an individual [Aristotle] |
| 2846 | Music can mould the character to be virtuous (just as gymnastics trains the body) [Aristotle] |
| 2807 | Some say slavery is unnatural and created by convention, and is therefore forced, and unjust [Aristotle] |
| 22582 | Spirit [thumos] is the capacity by which we love [Aristotle] |
| 2810 | Selfishness is wrong not because it is self-love, but because it is excessive [Aristotle] |
| 22555 | The function of good men is to confer benefits [Aristotle] |
| 13785 | 'Arete' signifies lack of complexity and a free-flowing soul [Plato] |
| 22557 | Virtuous people are like the citizens of the best city [Aristotle] |
| 2841 | People become good because of nature, habit and reason [Aristotle] |
| 2829 | The law is the mean [Aristotle] |
| 22590 | Virtue is concerned with correct feelings [Aristotle] |
| 2813 | It is quite possible to live a moderate life and yet be miserable [Aristotle] |
| 22553 | Justice is a virtue of communities [Aristotle] |
| 22561 | The rich are seen as noble, because they don't need to commit crimes [Aristotle] |
| 2808 | Master and slave can have friendship through common interests [Aristotle] |
| 2803 | Man is by nature a political animal [Aristotle] |
| 2820 | People want to live together, even when they don't want mutual help [Aristotle] |
| 22586 | Only humans have reason [Aristotle] |
| 22523 | The community (of villages) becomes a city when it is totally self-sufficient [Aristotle] |
| 2805 | A community must share a common view of good and justice [Aristotle] |
| 22526 | People who are anti-social or wholly self-sufficient are no part of a city [Aristotle] |
| 22535 | Friendship is the best good for cities, because it reduces factions [Aristotle] |
| 22532 | A city can't become entirely one, because its very nature is to be a multitude [Aristotle] |
| 22584 | A community should all share to some extent in something like land or food [Aristotle] |
| 22581 | The size of a city is decided by the maximum self-sufficient community that can be surveyed [Aristotle] |
| 22548 | A city aims at living well [Aristotle] |
| 22577 | What is the best life for everyone, and is that a communal or an individual problem? [Aristotle] |
| 22579 | The same four cardinal virtues which apply to individuals also apply to a city [Aristotle] |
| 2431 | Every state is an association formed for some good purpose [Aristotle] |
| 22578 | The happiest city is the one that acts most nobly [Aristotle] |
| 2832 | The state aims to consist as far as possible of those who are like and equal [Aristotle] |
| 2821 | The six constitutions are monarchy/tyranny, aristocracy/oligarchy, and polity/democracy [Aristotle] |
| 22546 | A city is a community of free people, and the constitution should aim at the common advantage [Aristotle] |
| 2835 | Any constitution can be made to last for a day or two [Aristotle] |
| 22580 | The best constitution enables everyone to live the best life [Aristotle] |
| 22558 | Constitutions specify distribution of offices, the authorities, and the community's aim [Aristotle] |
| 22566 | The greed of the rich is more destructive than the greed of the people [Aristotle] |
| 2973 | We must decide the most desirable human life before designing a constitution [Aristotle] |
| 22563 | The middle classes are neither ambitious nor anarchic, which is good [Aristotle] |
| 2818 | The virtues of a good citizen are relative to a particular constitution [Aristotle] |
| 22545 | A person can be an excellent citizen without being an excellent man [Aristotle] |
| 22544 | A citizen is someone who is allowed to hold official posts in a city [Aristotle] |
| 22541 | Kings should be selected according to character [Aristotle] |
| 2819 | The only virtue special to a ruler is practical wisdom [Aristotle] |
| 22542 | People who buy public office will probably expect to profit from it [Aristotle] |
| 22552 | The rich can claim to rule, because of land ownership, and being more trustworthy [Aristotle] |
| 22583 | The guardians should not be harsh to strangers, as no one should behave like that [Aristotle] |
| 22543 | In large communities it is better if more people participate in the offices [Aristotle] |
| 22539 | Election of officials by the elected is dangerous, because factions can control it [Aristotle] |
| 22572 | Officers should like the constitution, be capable, and have appropriate virtues and justice [Aristotle] |
| 2826 | Like water, large numbers of people are harder to corrupt than a few [Aristotle] |
| 22567 | Democracy arises when people who are given equal freedom assume unconditional equality [Aristotle] |
| 22560 | Popular leaders only arise in democracies that are not in accord with the law [Aristotle] |
| 22562 | Choosing officials by lot is democratic [Aristotle] |
| 2823 | The many may add up to something good, even if they are inferior as individuals [Aristotle] |
| 22533 | If the people are equal in nature, then they should all share in ruling [Aristotle] |
| 2817 | It is wrong that a worthy officer of state should seek the office [Aristotle] |
| 22576 | No office is permanent in a democracy [Aristotle] |
| 22549 | In many cases, the claim that the majority is superior would apply equally to wild beasts [Aristotle] |
| 22575 | Ultimate democracy is tyranny [Aristotle] |
| 22531 | We aim to understand the best possible community for free people [Aristotle] |
| 22564 | Community is based on friends, who are equal and similar, and share things [Aristotle] |
| 22585 | Look at all of the citizens before judging a city to be happy [Aristotle] |
| 22565 | The best communities rely on a large and strong middle class [Aristotle] |
| 22589 | Citizens do not just own themselves, but are also parts of the city [Aristotle] |
| 22534 | People care less about what is communal, and more about what is their own [Aristotle] |
| 22538 | Owning and sharing property communally increases disagreements [Aristotle] |
| 22536 | There could be private land and public crops, or public land and private crops, or both public [Aristotle] |
| 22530 | Both women and children should be educated, as this contributes to a city's excellence [Aristotle] |
| 21358 | Natural slaves are those naturally belonging to another, or who can manage no more than labouring [Aristotle] |
| 20092 | One principle of liberty is to take turns ruling and being ruled [Aristotle] |
| 2833 | Equality is obviously there to help people who do not get priority in the constitution [Aristotle] |
| 2834 | It is always the weak who want justice and equality, not the strong [Aristotle] |
| 2830 | We can claim an equal right to aristocratic virtue, as well as to wealth or freedom [Aristotle] |
| 22550 | It is dreadful to neither give a share nor receive a share [Aristotle] |
| 22569 | The Heraeans replaced election with lot, to thwart campaigning [Aristotle] |
| 22568 | Faction is for inferiors to be equal, and equals to become superior [Aristotle] |
| 2814 | Phaleas proposed equality of property, provided there is equality of education [Aristotle] |
| 22540 | Wealth could be quickly leveled by only the rich giving marriage dowries [Aristotle] |
| 2828 | Law is intelligence without appetite [Aristotle] |
| 22537 | Property should be owned privately, but used communally [Aristotle] |
| 22573 | The virtue of justice may be relative to a particular constitution [Aristotle] |
| 2825 | The good is obviously justice, which benefits the whole community, and involves equality in some sense [Aristotle] |
| 22527 | Justice is the order in a political community [Aristotle] |
| 22547 | Justice is equality for equals, and inequality for unequals [Aristotle] |
| 2816 | If it is easy to change the laws, that makes them weaker [Aristotle] |
| 2806 | Man is the worst of all animals when divorced from law and justice [Aristotle] |
| 22556 | Laws that match people's habits are more effective than mere written rules [Aristotle] |
| 22554 | It is said that we should not stick strictly to written law, as it is too vague [Aristotle] |
| 2827 | It is preferable that law should rule rather than any single citizen [Aristotle] |
| 22551 | Correct law should be in control, with rulers only deciding uncertain issues [Aristotle] |
| 2840 | The whole state should pay for the worship of the gods [Aristotle] |
| 2811 | A state is plural, and needs education to make it a community [Aristotle] |
| 22588 | A city has a single end, so education must focus on that, and be communal, not private [Aristotle] |
| 2847 | The aim of serious childhood play is the amusement of the complete adult [Aristotle] |
| 2842 | Men learn partly by habit, and partly by listening [Aristotle] |
| 2844 | Abortions should be procured before the embryo has acquired life and sensation [Aristotle] |
| 2809 | If nature makes everything for a purpose, then plants and animals must have been made for man [Aristotle] |
| 2800 | The best instruments have one purpose, not many [Aristotle] |
| 10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
| 13779 | The natural offspring of a lion is called a 'lion' (but what about the offspring of a king?) [Plato] |
| 2836 | God is not blessed and happy because of external goods, but because of his own nature [Aristotle] |
| 13783 | Even the gods love play [Plato] |
| 13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |
| 2802 | Men imagine gods to be of human shape, with a human lifestyle [Aristotle] |