209 ideas
| 16604 | Philosophy consists of choosing between Plato, Aristotle and Democritus [Pasnau] |
| 16606 | Original philosophers invariably seek inspiration from past thinkers [Pasnau] |
| 16586 | The commentaries of Averroes were the leading guide to Aristotle [Pasnau] |
| 16568 | Modernity begins in the late 12th century, with Averroes's commentaries on Aristotle [Pasnau] |
| 16653 | Once accidents were seen as real, 'Categories' became the major text for ontology [Pasnau] |
| 16704 | In 1347, the Church effectively stopped philosophy for the next 300 years [Pasnau] |
| 16605 | After c.1450 all of Plato was available. Before that, only the first half of 'Timaeus' was known [Pasnau] |
| 16607 | Renaissance Platonism is peripheral [Pasnau] |
| 16715 | Plato only made an impact locally in 15th century Italy [Pasnau] |
| 16610 | Philosophy could easily have died in 17th century, if it weren't for Descartes [Pasnau] |
| 16781 | The 17th century is a metaphysical train wreck [Pasnau] |
| 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] |
| 16677 | Anti-Razor: if you can't account for a truth, keep positing things until you can [Pasnau] |
| 9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
| 10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
| 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] |
| 10098 | The 'power set' of A is all the subsets of A [George/Velleman] |
| 10099 | The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}} [George/Velleman] |
| 10101 |
Cartesian Product A x B: the set of all ordered pairs |
| 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] |
| 16598 | Priority was a major topic of dispute for scholastics [Pasnau] |
| 16727 | In mixtures, the four elements ceased to exist, replaced by a mixed body with a form [Pasnau] |
| 16732 | 17th C qualities are either microphysical, or phenomenal, or powers [Pasnau] |
| 16733 | 17th century authors only recognised categorical properties, never dispositions [Pasnau] |
| 16662 | The biggest question for scholastics is whether properties are real, or modes of substances [Pasnau] |
| 16767 | There is no centralised power, but we still need essence for a metaphysical understanding [Pasnau] |
| 16788 | Instead of adding Aristotelian forms to physical stuff, one could add dispositions [Pasnau] |
| 16738 | Scholastics reject dispositions, because they are not actual, as forms require [Pasnau] |
| 16649 | Scholastics say there is a genuine thing if it is 'separable' [Pasnau] |
| 16785 | If you reject essences, questions of individuation become extremely difficult [Pasnau] |
| 16680 | Scholastics thought Quantity could be the principle of individuation [Pasnau] |
| 16628 | Corpuscularianism promised a decent account of substance [Pasnau] |
| 16617 | Corpuscularian critics of scholasticism say only substances exist [Pasnau] |
| 16741 | Scholastics wanted to treat Aristotelianism as physics, rather than as metaphysics [Pasnau] |
| 16777 | If crowds are things at all, they seem to be Substances, since they bear properties [Pasnau] |
| 16615 | Scholastics use 'substantia' for thick concrete entities, and for thin metaphysical ones [Pasnau] |
| 16775 | For corpuscularians, a substance is just its integral parts [Pasnau] |
| 16769 | If clay survives destruction of the statue, the statue wasn't a substance, but a mere accident [Pasnau] |
| 16602 | Corpuscularianism rejected not only form, but also the dependence of matter on form [Pasnau] |
| 16612 | Hylomorphism may not be a rival to science, but an abstract account of unity and endurance [Pasnau] |
| 16613 | Hylomorphism declined because scholastics made it into a testable physical theory [Pasnau] |
| 16747 | Scholastics made forms substantial, in a way unintended by Aristotle [Pasnau] |
| 16759 | Scholastics began to see substantial form more as Aristotle's 'efficient' cause [Pasnau] |
| 16748 | Aquinas says a substance has one form; Scotists say it has many forms [Pasnau] |
| 16671 | Scholastic Quantity either gives a body parts, or spreads them out in a unified way [Pasnau] |
| 16579 | There may be different types of substrate, or temporary substrates [Pasnau] |
| 16596 | A substratum can't be 'bare', because it has a job to do [Pasnau] |
| 16584 | If a substrate gives causal support for change, quite a lot of the ingredients must endure [Pasnau] |
| 16580 | A substrate may be 'prime matter', which endures through every change [Pasnau] |
| 22525 | The whole is prior to its parts, because parts are defined by their role [Aristotle] |
| 16749 | Aristotelians deny that all necessary properties are essential [Pasnau] |
| 16694 | Typical successive things are time and motion [Pasnau] |
| 16583 | Weak ex nihilo says it all comes from something; strong version says the old must partly endure [Pasnau] |
| 22587 | Understanding is the aim of our nature [Aristotle] |
| 22522 | To grasp something, trace it back to its natural origins [Aristotle] |
| 22524 | The nature of each thing is its mature state [Aristotle] |
| 16783 | Essences must explain, so we can infer them causally from the accidents [Pasnau] |
| 22528 | The nature of all animate things is to have one part which rules it [Aristotle] |
| 10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
| 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] |
| 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] |
| 22569 | The Heraeans replaced election with lot, to thwart campaigning [Aristotle] |
| 22550 | It is dreadful to neither give a share nor receive a share [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] |
| 22527 | Justice is the order in a political community [Aristotle] |
| 22547 | Justice is equality for equals, and inequality for unequals [Aristotle] |
| 2825 | The good is obviously justice, which benefits the whole community, and involves equality in some sense [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] |
| 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] |
| 22554 | It is said that we should not stick strictly to written law, as it is too vague [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] |
| 16609 | Atomists say causation is mechanical collisions, and all true qualities are microscopic [Pasnau] |
| 16603 | In the 17th C matter became body, and was then studied by science [Pasnau] |
| 16592 | Atomism is the commonest version of corpuscularianism, but isn't required by it [Pasnau] |
| 16750 | If there are just arrangements of corpuscles, where are the boundaries between substances? [Pasnau] |
| 16722 | Scholastic causation is by changes in the primary qualities of hot, cold, wet, dry [Pasnau] |
| 16760 | Substantial forms were a step towards scientific essentialism [Pasnau] |
| 16581 | Scholastic authors agree that matter was created by God, out of nothing [Pasnau] |
| 2836 | God is not blessed and happy because of external goods, but because of his own nature [Aristotle] |
| 2802 | Men imagine gods to be of human shape, with a human lifestyle [Aristotle] |
| 16642 | Transubstantion says accidents of bread and wine don't inhere in the substance [Pasnau] |