113 ideas
3722 | Metaphysics goes beyond the empirical, so doesn't need examples [Kant] |
3738 | The hallmark of rationality is setting itself an end [Kant] |
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] |
10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
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] |
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] |
17889 | CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner] |
8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
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] |
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] |
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] |
3726 | The categorical imperative is a practical synthetic a priori proposition [Kant] |
22668 | Apprehension is a complex intellect grasping the essence of a complex object [Holt,L] |
3739 | Free will is a kind of causality which works independently of other causes [Kant] |
3741 | We shall never be able to comprehend how freedom is possible [Kant] |
3740 | We cannot conceive of reason as being externally controlled [Kant] |
5296 | Kant made the political will into a pure self-determined "free" will [Kant, by Marx/Engels] |
24011 | Kant thought emotions are too random and passive to be part of morality [Kant, by Williams,B] |
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] |
5074 | Kant united religion and philosophy, by basing obedience to law on reason instead of faith [Taylor,R on Kant] |
8024 | The categorical imperative says nothing about what our activities and ends should be [MacIntyre on Kant] |
22390 | Kant thought human nature was pure hedonism, so virtue is only possible via the categorical imperative [Foot on Kant] |
9750 | We must only value what others find acceptable [Kant, by Korsgaard] |
20160 | Kant focuses exclusively on human values, and neglects cultural and personal values [Kekes on Kant] |
9749 | Our rational choices confer value, arising from the sense that we ourselves are important [Kant, by Korsgaard] |
7671 | Values are created by human choices, and are not some intrinsic quality, out there [Kant, by Berlin] |
3720 | We may claim noble motives, but we cannot penetrate our secret impulses [Kant] |
3717 | Reverence is awareness of a value which demolishes my self-love [Kant] |
3712 | A good will is not good because of what it achieves [Kant] |
3725 | The good of an action is in the mind of the doer, not the consequences [Kant] |
3733 | The 'golden rule' cannot be a universal law as it implies no duties [Kant] |
3736 | Virtue lets a rational being make universal law, and share in the kingdom of ends [Kant] |
3544 | Kant thinks virtue becomes passive, and hence morally unaccountable [Kant, by Annas] |
7674 | Generosity and pity are vices, because they falsely imply one person's superiority to another [Kant, by Berlin] |
21029 | Kantian respect is for humanity and reason (not from love or sympathy or solidarity) [Kant, by Sandel] |
3715 | Other causes can produce nice results, so morality must consist in the law, found only in rational beings [Kant] |
4024 | Kant follows Rousseau in defining freedom and morality in terms of each other [Taylor,C on Kant] |
7105 | If 'maxims' are deeper underlying intentions, Kant can be read as a virtue theorist [Kant, by Statman] |
3710 | The only purely good thing is a good will [Kant] |
7625 | We can ask how rational goodness is, but also why is rationality good [Putnam on Kant] |
20715 | It is basic that moral actions must be done from duty [Kant] |
3737 | The will is good if its universalised maxim is never in conflict with itself [Kant] |
3724 | A categorical imperative sees an action as necessary purely for its own sake [Kant] |
3735 | Men are subject to laws which are both self-made and universal [Kant] |
3723 | There are no imperatives for a holy will, as the will is in harmony with moral law [Kant] |
5295 | Kant was happy with 'good will', even if it had no result [Kant, by Marx/Engels] |
3714 | Dutiful actions are judged not by purpose, but by the maxim followed [Kant] |
3718 | Telling the truth from duty is quite different from doing so to avoid inconvenience [Kant] |
6695 | Kant has to attribute high moral worth to some deeply unattractive human lives [Kant, by Graham] |
8028 | Kantian duty seems to imply conformism with authority [MacIntyre on Kant] |
8026 | Almost any precept can be consistently universalized [MacIntyre on Kant] |
15673 | The intuition behind the categorical imperative is that one ought not to make an exception of oneself [Kant, by Finlayson] |
8068 | Universalising a maxim needs to first stipulate the right description for the action [Anscombe on Kant] |
22009 | Morality is the creation of the laws that enable a Kingdom of Ends [Kant] |
4413 | The categorical imperative smells of cruelty [Nietzsche on Kant] |
8025 | The categorical imperative will not suggest maxims suitable for testing [MacIntyre on Kant] |
8027 | I can universalize a selfish maxim, if it is expressed in a way that only applies to me [MacIntyre on Kant] |
3728 | Suicide, false promises, neglected talent, and lack of charity all involve contradictions of principle [Kant, by PG] |
22008 | Always treat yourself and others as an end, and never simply as a means [Kant] |
3762 | Why couldn't all rational beings accept outrageously immoral rules of conduct? [Mill on Kant] |
3719 | If lying were the universal law it would make promises impossible [Kant] |
3727 | Act as if your maxim were to become a universal law of nature [Kant] |
3716 | Act according to a maxim you can will as a universal law [Kant] |
22050 | The maxim of an action is chosen, and not externally imposed [Kant, by Bowie] |
3731 | Rational beings necessarily conceive their own existence as an end in itself [Kant] |
6694 | Always treat humanity as an end and never as a means only [Kant] |
4345 | For Kant, even a person who lacks all sympathy for others still has a motive for benevolence [Kant, by Hursthouse] |
3711 | Only a good will makes us worthy of happiness [Kant] |
4251 | If we are required to give moral thought the highest priority, this gives morality no content [Williams,B on Kant] |
16004 | If Kant lives by self-administered laws, this is as feeble as self-administered punishments [Kierkegaard on Kant] |
4344 | Actions where people spread happiness because they enjoy it have no genuine moral worth [Kant] |
3713 | The function of reason is to produce a good will [Kant] |
3729 | Our inclinations are not innately desirable; in fact most rational beings would like to be rid of them [Kant] |
3732 | Rational beings have a right to share in the end of an action, not just be part of the means [Kant] |
7670 | Kant is the father of the notion of exploitation as an evil [Kant, by Berlin] |
7591 | Kant completed Grotius's project of a non-religious basis for natural law [Scruton on Kant] |
7673 | Retributive punishment is better than being sent to hospital for your crimes [Kant, by Berlin] |
3730 | Non-rational beings only have a relative value, as means rather than as ends [Kant] |
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] |
3721 | We judge God to be good by a priori standards of moral perfection [Kant] |
8046 | We can only know we should obey God if we already have moral standards for judging God [Kant, by MacIntyre] |
20714 | God is not proved by reason, but is a postulate of moral thinking [Kant, by Davies,B] |