105 ideas
3722 | Metaphysics goes beyond the empirical, so doesn't need examples [Kant] |
3738 | The hallmark of rationality is setting itself an end [Kant] |
15945 | Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine] |
15914 | An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine] |
15921 | Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine] |
15937 | Those who reject infinite collections also want to reject the Axiom of Choice [Lavine] |
15936 | The Power Set is just the collection of functions from one collection to another [Lavine] |
15899 | Replacement was immediately accepted, despite having very few implications [Lavine] |
15930 | Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine] |
15920 | Pure collections of things obey Choice, but collections defined by a rule may not [Lavine] |
15898 | The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine] |
15919 | The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine] |
15900 | The iterative conception of set wasn't suggested until 1947 [Lavine] |
15931 | The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine] |
15932 | The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine] |
15933 | Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine] |
15913 | A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine] |
15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine] |
15934 | Mathematical proof by contradiction needs the law of excluded middle [Lavine] |
15907 | Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine] |
15942 | Every rational number, unlike every natural number, is divisible by some other number [Lavine] |
15922 | For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine] |
18250 | Cauchy gave a necessary condition for the convergence of a sequence [Lavine] |
15904 | The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine] |
15912 | Counting results in well-ordering, and well-ordering makes counting possible [Lavine] |
15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine] |
15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
15940 | The intuitionist endorses only the potential infinite [Lavine] |
15909 | 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine] |
15915 | Ordinals are basic to Cantor's transfinite, to count the sets [Lavine] |
15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine] |
15918 | Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine] |
15929 | Set theory will found all of mathematics - except for the notion of proof [Lavine] |
15935 | Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine] |
15928 | Intuitionism rejects set-theory to found mathematics [Lavine] |
3726 | The categorical imperative is a practical synthetic a priori proposition [Kant] |
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] |
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] |
7105 | If 'maxims' are deeper underlying intentions, Kant can be read as a virtue theorist [Kant, by Statman] |
7625 | We can ask how rational goodness is, but also why is rationality good [Putnam on Kant] |
3715 | Other causes can produce nice results, so morality must consist in the law, found only in rational beings [Kant] |
20715 | It is basic that moral actions must be done from duty [Kant] |
4024 | Kant follows Rousseau in defining freedom and morality in terms of each other [Taylor,C on Kant] |
3710 | The only purely good thing is a good will [Kant] |
3737 | The will is good if its universalised maxim is never in conflict with itself [Kant] |
3718 | Telling the truth from duty is quite different from doing so to avoid inconvenience [Kant] |
3723 | There are no imperatives for a holy will, as the will is in harmony with moral law [Kant] |
3735 | Men are subject to laws which are both self-made and universal [Kant] |
3724 | A categorical imperative sees an action as necessary purely for its own sake [Kant] |
3714 | Dutiful actions are judged not by purpose, but by the maxim followed [Kant] |
5295 | Kant was happy with 'good will', even if it had no result [Kant, by Marx/Engels] |
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] |
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] |
3719 | If lying were the universal law it would make promises impossible [Kant] |
3762 | Why couldn't all rational beings accept outrageously immoral rules of conduct? [Mill on Kant] |
4413 | The categorical imperative smells of cruelty [Nietzsche on Kant] |
3716 | Act according to a maxim you can will as a universal law [Kant] |
3727 | Act as if your maxim were to become a universal law of nature [Kant] |
22009 | Morality is the creation of the laws that enable a Kingdom of Ends [Kant] |
22050 | The maxim of an action is chosen, and not externally imposed [Kant, by Bowie] |
6694 | Always treat humanity as an end and never as a means only [Kant] |
3731 | Rational beings necessarily conceive their own existence as an end in itself [Kant] |
4345 | For Kant, even a person who lacks all sympathy for others still has a motive for benevolence [Kant, by Hursthouse] |
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] |
3711 | Only a good will makes us worthy of happiness [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] |
4344 | Actions where people spread happiness because they enjoy it have no genuine moral worth [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] |
13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
3730 | Non-rational beings only have a relative value, as means rather than as ends [Kant] |
20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
8046 | We can only know we should obey God if we already have moral standards for judging God [Kant, by MacIntyre] |
3721 | We judge God to be good by a priori standards of moral perfection [Kant] |
20714 | God is not proved by reason, but is a postulate of moral thinking [Kant, by Davies,B] |