82 ideas
10859 | A set is 'well-ordered' if every subset has a first element [Clegg] |
10857 | Set theory made a closer study of infinity possible [Clegg] |
10864 | Any set can always generate a larger set - its powerset, of subsets [Clegg] |
10872 | Extensionality: Two sets are equal if and only if they have the same elements [Clegg] |
10875 | Pairing: For any two sets there exists a set to which they both belong [Clegg] |
10876 | Unions: There is a set of all the elements which belong to at least one set in a collection [Clegg] |
10878 | Infinity: There exists a set of the empty set and the successor of each element [Clegg] |
10877 | Powers: All the subsets of a given set form their own new powerset [Clegg] |
10879 | Choice: For every set a mechanism will choose one member of any non-empty subset [Clegg] |
10871 | Axiom of Existence: there exists at least one set [Clegg] |
10874 | Specification: a condition applied to a set will always produce a new set [Clegg] |
10880 | Mathematics can be 'pure' (unapplied), 'real' (physically grounded); or 'applied' (just applicable) [Clegg] |
10861 | Beyond infinity cardinals and ordinals can come apart [Clegg] |
10860 | An ordinal number is defined by the set that comes before it [Clegg] |
10854 | Transcendental numbers can't be fitted to finite equations [Clegg] |
10858 | By adding an axis of imaginary numbers, we get the useful 'number plane' instead of number line [Clegg] |
10853 | Either lack of zero made early mathematics geometrical, or the geometrical approach made zero meaningless [Clegg] |
10866 | Cantor's account of infinities has the shaky foundation of irrational numbers [Clegg] |
10869 | The Continuum Hypothesis is independent of the axioms of set theory [Clegg] |
10862 | The 'continuum hypothesis' says aleph-one is the cardinality of the reals [Clegg] |
458 | Nothing could come out of nothing, and existence could never completely cease [Empedocles] |
5112 | Empedocles says things are at rest, unless love unites them, or hatred splits them [Empedocles, by Aristotle] |
13209 | There is no coming-to-be of anything, but only mixing and separating [Empedocles, by Aristotle] |
457 | Substance is not created or destroyed in mortals, but there is only mixing and exchange [Empedocles] |
5937 | The goodness of opinions depends on their grounds, and corresponding degrees of conviction [Ross] |
5936 | Knowledge is superior to opinion because it is certain [Ross] |
5927 | I prefer the causal theory to sense data, because sensations are events, not apprehensions [Ross] |
462 | One vision is produced by both eyes [Empedocles] |
5940 | Two goods may be comparable, although they are not commensurable [Ross] |
22765 | Wisdom and thought are shared by all things [Empedocles] |
5924 | Identical objects must have identical value [Ross] |
1524 | For Empedocles thinking is almost identical to perception [Empedocles, by Theophrastus] |
5933 | Aesthetic enjoyment combines pleasure with insight [Ross] |
5928 | Beauty is neither objective nor subjective, but a power of producing certain mental events [Ross] |
5911 | Moral duties are as fundamental to the universe as the axioms of mathematics [Ross] |
5926 | The beauty of a patch of colour might be the most important fact about it [Ross] |
7259 | Ross said moral principles are self-evident from the facts, but not from pure thought [Ross, by Dancy,J] |
5913 | The moral convictions of thoughtful educated people are the raw data of ethics [Ross] |
5920 | Value is held to be either a quality, or a relation (usually between a thing and a mind) [Ross] |
5923 | The arguments for value being an objective or a relation fail, so it appears to be a quality [Ross] |
5918 | The thing is intrinsically good if it would be good when nothing else existed [Ross] |
5930 | All things being equal, we all prefer the virtuous to be happy, not the vicious [Ross] |
5922 | An instrumentally good thing might stay the same, but change its value because of circumstances [Ross] |
552 | Empedocles said good and evil were the basic principles [Empedocles, by Aristotle] |
5921 | We can ask of pleasure or beauty whether they are valuable, but not of goodness [Ross] |
5932 | The four goods are: virtue, pleasure, just allocation of pleasure, and knowledge [Ross] |
5910 | The three intrinsic goods are virtue, knowledge and pleasure [Ross] |
5898 | 'Right' and 'good' differ in meaning, as in a 'right action' and a 'good man' [Ross] |
5899 | If there are two equally good acts, they may both be right, but neither a duty [Ross] |
5904 | In the past 'right' just meant what is conventionally accepted [Ross] |
5919 | Goodness is a wider concept than just correct ethical conduct [Ross] |
5941 | Motives decide whether an action is good, and what is done decides whether it was right [Ross] |
5938 | Virtue is superior to pleasure, as pleasure is never a duty, but goodness is [Ross] |
5931 | All other things being equal, a universe with more understanding is better [Ross] |
5939 | Morality is not entirely social; a good moral character should love truth [Ross] |
5905 | We clearly value good character or understanding, as well as pleasure [Ross] |
5929 | No one thinks it doesn't matter whether pleasure is virtuously or viciously acquired [Ross] |
5906 | Promise-keeping is bound by the past, and is not concerned with consequences [Ross] |
18622 | Promises create a new duty to a particular person; they aren't just a strategy to achieve well-being [Ross] |
5908 | Prima facie duties rest self-evidently on particular circumstance [Ross] |
5917 | People lose their rights if they do not respect the rights of others [Ross] |
5900 | We should do our duty, but not from a sense of duty [Ross] |
5942 | We like people who act from love, but admire more the people who act from duty [Ross] |
5909 | Be faithful, grateful, just, beneficent, non-malevolent, and improve yourself [Ross, by PG] |
5914 | An act may be described in innumerable ways [Ross] |
5912 | We should use money to pay debts before giving to charity [Ross] |
5916 | Rights were originally legal, and broadened to include other things [Ross] |
5915 | Rights can be justly claimed, so animals have no rights, as they cannot claim any [Ross] |
589 | 'Nature' is just a word invented by people [Empedocles] |
21823 | The principle of 'Friendship' in Empedocles is the One, and is bodiless [Empedocles, by Plotinus] |
2680 | Empedocles said that there are four material elements, and two further creative elements [Empedocles, by Aristotle] |
6002 | Empedocles says bone is water, fire and earth in ratio 2:4:2 [Empedocles, by Inwood] |
13207 | Fire, Water, Air and Earth are elements, being simple as well as homoeomerous [Empedocles, by Aristotle] |
459 | All change is unity through love or division through hate [Empedocles] |
13218 | The elements combine in coming-to-be, but how do the elements themselves come-to-be? [Aristotle on Empedocles] |
13225 | Love and Strife only explain movement if their effects are distinctive [Aristotle on Empedocles] |
460 | If the one Being ever diminishes it would no longer exist, and what could ever increase it? [Empedocles] |
5090 | Maybe bodies are designed by accident, and the creatures that don't work are destroyed [Empedocles, by Aristotle] |
461 | God is a pure, solitary, and eternal sphere [Empedocles] |
466 | God is pure mind permeating the universe [Empedocles] |
1719 | In Empedocles' theory God is ignorant because, unlike humans, he doesn't know one of the elements (strife) [Aristotle on Empedocles] |
1522 | It is wretched not to want to think clearly about the gods [Empedocles] |