160 ideas
22396 | We take courage, temperance, wisdom and justice as moral, but Aristotle takes wisdom as intellectual [Foot] |
22496 | Wisdom only implies the knowledge achievable in any normal lifetime [Foot] |
22397 | Wisdom is open to all, and not just to the clever or well trained [Foot] |
22462 | We should speak the truth, but also preserve and pursue it [Foot] |
15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
15946 | Cantor developed sets from a progression into infinity by addition, multiplication and exponentiation [Cantor, by Lavine] |
9616 | A set is a collection into a whole of distinct objects of our intuition or thought [Cantor] |
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] |
17831 | Cantor gives informal versions of ZF axioms as ways of getting from one set to another [Cantor, by Lake] |
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] |
1553 | No perceptible object is truly straight or curved [Protagoras] |
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] |
15911 | Ordinals are generated by endless succession, followed by a limit ordinal [Cantor, by Lavine] |
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] |
15896 | Cantor needed Power Set for the reals, but then couldn't count the new collections [Cantor, by Lavine] |
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] |
8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
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] |
15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy] |
9992 | The 'extension of a concept' in general may be quantitatively completely indeterminate [Cantor] |
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] |
1549 | Everything that exists consists in being perceived [Protagoras] |
22449 | When we say 'is red' we don't mean 'seems red to most people' [Foot] |
1545 | Protagoras was the first to claim that there are two contradictory arguments about everything [Protagoras, by Diog. Laertius] |
1547 | Man is the measure of all things - of things that are, and of things that are not [Protagoras] |
3305 | There is no more purely metaphysical doctrine than Protagorean relativism [Benardete,JA on Protagoras] |
3317 | You can only state the problem of the relative warmth of an object by agreeing on the underlying object [Benardete,JA on Protagoras] |
3313 | If my hot wind is your cold wind, then wind is neither hot nor cold, and so not as cold as itself [Benardete,JA on Protagoras] |
247 | God is "the measure of all things", more than any man [Plato on Protagoras] |
606 | Protagoras absurdly thought that the knowing or perceiving man is 'the measure of all things' [Aristotle on Protagoras] |
612 | Relativists think if you poke your eye and see double, there must be two things [Aristotle on Protagoras] |
22371 | Determinism threatens free will if actions can be causally traced to external factors [Foot] |
23438 | Full rationality must include morality [Foot] |
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] |
9145 | We form the image of a cardinal number by a double abstraction, from the elements and from their order [Cantor] |
23437 | Practical reason is goodness in choosing actions [Foot] |
22480 | Possessing the virtue of justice disposes a person to good practical rationality [Foot] |
23694 | All criterions of practical rationality derive from goodness of will [Foot] |
22372 | Not all actions need motives, but it is irrational to perform troublesome actions with no motive [Foot] |
22393 | I don't understand the idea of a reason for acting, but it is probably the agent's interests or desires [Foot] |
23436 | It is an odd Humean view to think a reason to act must always involve caring [Foot] |
22481 | There is no restitution after a dilemma, if it only involved the agent, or just needed an explanation [Foot, by PG] |
22482 | I can't understand how someone can be necessarily wrong whatever he does [Foot] |
22384 | A 'double effect' is a foreseen but not desired side-effect, which may be forgivable [Foot] |
22465 | We see a moral distinction between doing and allowing to happen [Foot] |
22385 | The doctrine of double effect can excuse an outcome because it wasn't directly intended [Foot] |
22386 | Double effect says foreseeing you will kill someone is not the same as intending it [Foot] |
22387 | Without double effect, bad men can make us do evil by threatening something worse [Foot] |
22388 | Double effect seems to rely on a distinction between what we do and what we allow [Foot] |
22466 | We see a moral distinction between our aims and their foreseen consequences [Foot] |
22467 | Acts and omissions only matter if they concern doing something versus allowing it [Foot] |
4692 | It is not true that killing and allowing to die (or acts and omissions) are morally indistinguishable [Foot] |
4694 | Making a runaway tram kill one person instead of five is diverting a fatal sequence, not initiating one [Foot] |
22445 | Morality shows murder is wrong, but not what counts as a murder [Foot] |
22444 | A moral system must deal with the dangers and benefits of life [Foot] |
23683 | Moral norms are objective, connected to facts about human goods [Foot, by Hacker-Wright] |
22392 | Morality is inescapable, in descriptive words such as 'dishonest', 'unjust' and 'uncharitable' [Foot] |
22451 | All people need affection, cooperation, community and help in trouble [Foot] |
22485 | Non-cognitivists give the conditions of use of moral sentences as facts about the speaker [Foot] |
22474 | Unlike aesthetic evaluation, moral evaluation needs a concept of responsibility [Foot] |
23684 | Morality gives everyone reasons to act, irrespective of their desires [Foot, by Hacker-Wright] |
23690 | We all have reason to cultivate the virtues, even when we lack the desire [Foot, by Hacker-Wright] |
23685 | Reason is not a motivator of morality [Foot, by Hacker-Wright] |
23691 | Rejecting moral rules may be villainous, but it isn't inconsistent [Foot] |
23686 | Moral reason is not just neutral, because morality is part of the standard of rationality [Foot, by Hacker-Wright] |
23693 | Practical rationality must weigh both what is morally and what is non-morally required [Foot] |
23431 | Human defects are just like plant or animal defects [Foot] |
23687 | Moral virtues arise from human nature, as part of what makes us good human beings [Foot, by Hacker-Wright] |
22477 | Calling a knife or farmer or speech or root good does not involve attitudes or feelings [Foot] |
22486 | The mistake is to think good grounds aren't enough for moral judgement, which also needs feelings [Foot] |
6016 | Early sophists thought convention improved nature; later they said nature was diminished by it [Protagoras, by Miller,FD] |
23432 | Concepts such as function, welfare, flourishing and interests only apply to living things [Foot] |
22375 | Moral judgements need more than the relevant facts, if the same facts lead to 'x is good' and 'x is bad' [Foot] |
22493 | Sterility is a human defect, but the choice to be childless is not [Foot] |
22492 | Virtues are as necessary to humans as stings are to bees [Foot] |
23433 | Humans need courage like a plant needs roots [Foot] |
22378 | We can't affirm a duty without saying why it matters if it is not performed [Foot] |
22487 | Moral arguments are grounded in human facts [Foot] |
22377 | Whether someone is rude is judged by agreed criteria, so the facts dictate the value [Foot] |
22376 | Facts and values are connected if we cannot choose what counts as evidence of rightness [Foot] |
22491 | Moral evaluations are not separate from facts, but concern particular facts about functioning [Foot] |
23434 | There is no fact-value gap in 'owls should see in the dark' [Foot] |
22447 | Saying something 'just is' right or wrong creates an illusion of fact and objectivity [Foot] |
23439 | Principles are not ultimate, but arise from the necessities of human life [Foot] |
22452 | Do we have a concept of value, other than wanting something, or making an effort to get it? [Foot] |
23435 | If you demonstrate the reason to act, there is no further question of 'why should I?' [Foot] |
22381 | Being a good father seems to depend on intentions, rather than actual abilities [Foot] |
22379 | The meaning of 'good' and other evaluations must include the object to which they attach [Foot] |
22458 | Consequentialists can hurt the innocent in order to prevent further wickedness [Foot] |
22460 | Why might we think that a state of affairs can be morally good or bad? [Foot] |
22461 | Good outcomes are not external guides to morality, but a part of virtuous actions [Foot] |
22464 | The idea of a good state of affairs has no role in the thought of Aristotle, Rawls or Scanlon [Foot] |
22497 | Deep happiness usually comes from the basic things in life [Foot] |
22498 | Happiness is enjoying the pursuit and attainment of right ends [Foot] |
23695 | Good actions can never be justified by the good they brings to their agent [Foot] |
1580 | For Protagoras the only bad behaviour is that which interferes with social harmony [Protagoras, by Roochnik] |
22470 | A good moral system benefits its participants, and so demands reciprocity [Foot] |
22499 | We all know that just pretending to be someone's friend is not the good life [Foot] |
22402 | Most people think virtues can be displayed in bad actions [Foot] |
23145 | Virtues are intended to correct design flaws in human beings [Foot, by Driver] |
22401 | Actions can be in accordance with virtue, but without actually being virtuous [Foot] |
22398 | Virtues are corrective, to resist temptation or strengthen motivation [Foot] |
22478 | The essential thing is the 'needs' of plants and animals, and their operative parts [Foot] |
23692 | Good and bad are a matter of actions, not of internal dispositions [Foot] |
22468 | Virtues can have aims, but good states of affairs are not among them [Foot] |
205 | Protagoras contradicts himself by saying virtue is teachable, but then that it is not knowledge [Plato on Protagoras] |
22495 | Someone is a good person because of their rational will, not their body or memory [Foot] |
22373 | People can act out of vanity without being vain, or even vain about this kind of thing [Foot] |
22456 | Maybe virtues conflict with each other, if some virtue needs a vice for its achievement [Foot] |
22469 | Some virtues imply rules, and others concern attachment [Foot] |
22403 | Temperance is not a virtue if it results from timidity or excessive puritanism [Foot] |
22472 | The practice of justice may well need a recognition of human equality [Foot] |
22479 | Observing justice is necessary to humans, like hunting to wolves or dancing to bees [Foot] |
22400 | Courage overcomes the fears which should be overcome, and doesn't overvalue personal safety [Foot] |
22391 | Saying we 'ought to be moral' makes no sense, unless it relates to some other system [Foot] |
22389 | Morality no more consists of categorical imperatives than etiquette does [Foot] |
22395 | Moral judgements are hypothetical, because they depend on interests and desires [Foot] |
22448 | We sometimes just use the word 'should' to impose a rule of conduct on someone [Foot] |
22463 | Morality is seen as tacit legislation by the community [Foot] |
22459 | For consequentialism, it is irrational to follow a rule which in this instance ends badly [Foot] |
22502 | Refraining from murder is not made good by authenticity or self-fulfilment [Foot] |
4693 | The right of non-interference (with a 'negative duty'), and the right to goods/services ('positive') [Foot] |
1659 | Protagoras seems to have made the huge move of separating punishment from revenge [Protagoras, by Vlastos] |
532 | Successful education must go deep into the soul [Protagoras] |
1552 | He spent public money on education, as it benefits the individual and the state [Protagoras, by Diodorus of Sicily] |
22383 | Abortion is puzzling because we do and don't want the unborn child to have rights [Foot] |
22446 | In the case of something lacking independence, calling it a human being is a matter of choice [Foot] |
22380 | Some words, such as 'knife', have a meaning which involves its function [Foot] |
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] |
1551 | He said he didn't know whether there are gods - but this is the same as atheism [Diogenes of Oen. on Protagoras] |