106 ideas
| 14857 | The highest wisdom has the guise of simplicity [Nietzsche] |
| 22087 | Philosophy fails to articulate the continual becoming of existence [Kierkegaard, by Carlisle] |
| 14854 | Deep thinkers know that they are always wrong [Nietzsche] |
| 14833 | Comedy is a transition from fear to exuberance [Nietzsche] |
| 14853 | Truth finds fewest champions not when it is dangerous, but when it is boring [Nietzsche] |
| 24075 | Convictions, more than lies, are the great enemy of truth [Nietzsche] |
| 5651 | Traditional views of truth are tautologies, and truth is empty without a subject [Kierkegaard, by Scruton] |
| 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] |
| 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] |
| 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] |
| 8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
| 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] |
| 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] |
| 18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
| 15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
| 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] |
| 14858 | Being certain presumes that there are absolute truths, and means of arriving at them [Nietzsche] |
| 14830 | Intuition only recognises what is possible, not what exists or is certain [Nietzsche] |
| 20378 | Just as skin hides the horrors of the body, vanity conceals the passions of the soul [Nietzsche] |
| 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] |
| 14820 | People always do what they think is right, according to the degree of their intellect [Nietzsche] |
| 14856 | Our judgment seems to cause our nature, but actually judgment arises from our nature [Nietzsche] |
| 14842 | Why are the strong tastes of other people so contagious? [Nietzsche] |
| 14835 | Artists are not especially passionate, but they pretend to be [Nietzsche] |
| 22473 | Nietzsche said the will doesn't exist, so it can't ground moral responsibility [Nietzsche, by Foot] |
| 14807 | The history of morality rests on an error called 'responsibility', which rests on an error called 'free will' [Nietzsche] |
| 14823 | Ceasing to believe in human responsibility is bitter, if you had based the nobility of humanity on it [Nietzsche] |
| 14824 | It is absurd to blame nature and necessity; we should no more praise actions than we praise plants or artworks [Nietzsche] |
| 14812 | Intellect is tied to morality, because it requires good memory and powerful imagination [Nietzsche] |
| 14810 | Originally it was the rulers who requited good for good and evil for evil who were called 'good' [Nietzsche] |
| 14831 | No one has ever done anything that was entirely for other people [Nietzsche] |
| 14855 | Simultaneous love and respect are impossible; love has no separation or rank, but respect admits power [Nietzsche] |
| 14815 | We get enormous pleasure from tales of noble actions [Nietzsche] |
| 14849 | We can only achieve happy moments, not happy eras [Nietzsche] |
| 14818 | First morality is force, then custom, then acceptance, then instinct, then a pleasure - and finally 'virtue' [Nietzsche] |
| 20103 | You are mastered by your own virtues, but you must master them, and turn them into tools [Nietzsche] |
| 14817 | The 'good' man does the moral thing as if by nature, easily and gladly, after a long inheritance [Nietzsche] |
| 14809 | All societies of good men give a priority to gratitude [Nietzsche] |
| 14816 | Justice (fairness) originates among roughly equal powers (as the Melian dialogues show) [Nietzsche] |
| 20112 | Pity consoles those who suffer, because they see that they still have the power to hurt [Nietzsche] |
| 14821 | Apart from philosophers, most people rightly have a low estimate of pity [Nietzsche] |
| 14841 | Many people are better at having good friends than being a good friend [Nietzsche] |
| 14843 | Women can be friends with men, but only some physical antipathy will maintain it [Nietzsche] |
| 14811 | In Homer it is the contemptible person, not the harmful person, who is bad [Nietzsche] |
| 20111 | We could live more naturally, relishing the spectacle, and not thinking we are special [Nietzsche] |
| 22090 | For me time stands still, and I with it [Kierkegaard, by Carlisle] |
| 14844 | People do not experience boredom if they have never learned to work properly [Nietzsche] |
| 9305 | The plebeians bore others; only the nobility bore themselves [Kierkegaard] |
| 5650 | Reason is just abstractions, so our essence needs a subjective 'leap of faith' [Kierkegaard, by Scruton] |
| 14808 | Over huge periods of time human character would change endlessly [Nietzsche] |
| 22095 | There are aesthetic, ethical and religious subjectivity [Kierkegaard, by Carlisle] |
| 20747 | What matters is not right choice, but energy, earnestness and pathos in the choosing [Kierkegaard] |
| 14822 | If self-defence is moral, then so are most expressions of 'immoral' egoism [Nietzsche] |
| 14838 | The state aims to protect individuals from one another [Nietzsche] |
| 14852 | Culture cannot do without passions and vices [Nietzsche] |
| 14846 | If we want the good life for the greatest number, we must let them decide on the good life [Nietzsche] |
| 22091 | Kierkegaard prioritises the inward individual, rather than community [Kierkegaard, by Carlisle] |
| 14819 | Slavery cannot be judged by our standards, because the sense of justice was then less developed [Nietzsche] |
| 14847 | Laws that are well thought out, or laws that are easy to understand? [Nietzsche] |
| 14814 | Execution is worse than murder, because we are using the victim, and really we are the guilty [Nietzsche] |
| 14836 | People will enthusiastically pursue an unwanted war, once sacrifices have been made [Nietzsche] |
| 14845 | Don't crush girls with dull Gymnasium education, the way we have crushed boys! [Nietzsche] |
| 14848 | Education in large states is mediocre, like cooking in large kitchens [Nietzsche] |
| 14839 | Interest in education gains strength when we lose interest in God [Nietzsche] |
| 14834 | Teachers only gather knowledge for their pupils, and can't be serious about themselves [Nietzsche] |
| 14825 | In religious thought nature is a complex of arbitrary acts by conscious beings [Nietzsche] |
| 14826 | Modern man wants laws of nature in order to submit to them [Nietzsche] |
| 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] |
| 14827 | The Greeks saw the gods not as their masters, but as idealised versions of themselves [Nietzsche] |
| 14813 | Science rejecting the teaching of Christianity in favour of Epicurus shows the superiority of the latter [Nietzsche] |
| 14832 | The Sermon on the Mount is vanity - praying to one part of oneself, and demonising the rest [Nietzsche] |
| 14850 | Christ was the noblest human being [Nietzsche] |
| 14837 | Christ seems warm hearted, and suppressed intellect in favour of the intellectually weak [Nietzsche] |
| 14828 | Religion is tempting if your life is boring, but you can't therefore impose it on the busy people [Nietzsche] |
| 22088 | Faith is like a dancer's leap, going up to God, but also back to earth [Kierkegaard, by Carlisle] |