87 ideas
14857 | The highest wisdom has the guise of simplicity [Nietzsche] |
14854 | Deep thinkers know that they are always wrong [Nietzsche] |
14833 | Comedy is a transition from fear to exuberance [Nietzsche] |
17774 | Definitions make our intuitions mathematically useful [Mayberry] |
17773 | Proof shows that it is true, but also why it must be true [Mayberry] |
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] |
17796 | There is a semi-categorical axiomatisation of set-theory [Mayberry] |
17795 | Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation [Mayberry] |
17800 | The misnamed Axiom of Infinity says the natural numbers are finite in size [Mayberry] |
17801 | The set hierarchy doesn't rely on the dubious notion of 'generating' them [Mayberry] |
17803 | Limitation of size is part of the very conception of a set [Mayberry] |
17786 | The mainstream of modern logic sees it as a branch of mathematics [Mayberry] |
17788 | First-order logic only has its main theorems because it is so weak [Mayberry] |
17791 | Only second-order logic can capture mathematical structure up to isomorphism [Mayberry] |
17787 | Big logic has one fixed domain, but standard logic has a domain for each interpretation [Mayberry] |
17790 | No Löwenheim-Skolem logic can axiomatise real analysis [Mayberry] |
17779 | 'Classificatory' axioms aim at revealing similarity in morphology of structures [Mayberry] |
17778 | Axiomatiation relies on isomorphic structures being essentially the same [Mayberry] |
17780 | 'Eliminatory' axioms get rid of traditional ideal and abstract objects [Mayberry] |
17789 | No logic which can axiomatise arithmetic can be compact or complete [Mayberry] |
17784 | Real numbers can be eliminated, by axiom systems for complete ordered fields [Mayberry] |
17782 | Greek quantities were concrete, and ratio and proportion were their science [Mayberry] |
17781 | Real numbers were invented, as objects, to simplify and generalise 'quantity' [Mayberry] |
17799 | Cantor's infinite is an absolute, of all the sets or all the ordinal numbers [Mayberry] |
17797 | Cantor extended the finite (rather than 'taming the infinite') [Mayberry] |
17775 | If proof and definition are central, then mathematics needs and possesses foundations [Mayberry] |
17776 | The ultimate principles and concepts of mathematics are presumed, or grasped directly [Mayberry] |
17777 | Foundations need concepts, definition rules, premises, and proof rules [Mayberry] |
17804 | Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms [Mayberry] |
17792 | 1st-order PA is only interesting because of results which use 2nd-order PA [Mayberry] |
17793 | It is only 2nd-order isomorphism which suggested first-order PA completeness [Mayberry] |
17794 | Set theory is not just first-order ZF, because that is inadequate for mathematics [Mayberry] |
17802 | We don't translate mathematics into set theory, because it comes embodied in that way [Mayberry] |
17805 | Set theory is not just another axiomatised part of mathematics [Mayberry] |
17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
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] |
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] |
14807 | The history of morality rests on an error called 'responsibility', which rests on an error called 'free will' [Nietzsche] |
14824 | It is absurd to blame nature and necessity; we should no more praise actions than we praise plants or artworks [Nietzsche] |
14823 | Ceasing to believe in human responsibility is bitter, if you had based the nobility of humanity on it [Nietzsche] |
22473 | Nietzsche said the will doesn't exist, so it can't ground moral responsibility [Nietzsche, by Foot] |
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] |
5271 | Prejudice apart, push-pin has equal value with music and poetry [Bentham] |
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] |
14844 | People do not experience boredom if they have never learned to work properly [Nietzsche] |
14808 | Over huge periods of time human character would change endlessly [Nietzsche] |
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] |
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] |
14839 | Interest in education gains strength when we lose interest in God [Nietzsche] |
14848 | Education in large states is mediocre, like cooking in large kitchens [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] |
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] |