74 ideas
8013 | In the Reformation, morality became unconditional but irrational, individually autonomous, and secular [MacIntyre] |
8021 | The Levellers and the Diggers mark a turning point in the history of morality [MacIntyre] |
8060 | In the 17th-18th centuries morality offered a cure for egoism, through altruism [MacIntyre] |
8053 | Twentieth century social life is re-enacting eighteenth century philosophy [MacIntyre] |
8047 | Philosophy has been marginalised by its failure in the Enlightenment to replace religion [MacIntyre] |
8062 | Proof is a barren idea in philosophy, and the best philosophy never involves proof [MacIntyre] |
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] |
15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [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] |
8052 | To find empiricism and science in the same culture is surprising, as they are really incompatible [MacIntyre] |
10356 | Relativism can be seen as about the rationality of different cultural traditions [MacIntyre, by Kusch] |
8057 | Unpredictability doesn't entail inexplicability, and predictability doesn't entail explicability [MacIntyre] |
8054 | Social sciences discover no law-like generalisations, and tend to ignore counterexamples [MacIntyre] |
8006 | When Aristotle speaks of soul he means something like personality [MacIntyre] |
21050 | I can only make decisions if I see myself as part of a story [MacIntyre] |
8056 | AI can't predict innovation, or consequences, or external relations, or external events [MacIntyre] |
8059 | The good life for man is the life spent seeking the good life for man [MacIntyre] |
8034 | We still have the appearance and language of morality, but we no longer understand it [MacIntyre] |
8036 | Unlike expressions of personal preference, evaluative expressions do not depend on context [MacIntyre] |
8049 | Moral judgements now are anachronisms from a theistic age [MacIntyre] |
8045 | The failure of Enlightenment attempts to justify morality will explain our own culture [MacIntyre] |
8051 | Mention of 'intuition' in morality means something has gone wrong with the argument [MacIntyre] |
8048 | When 'man' is thought of individually, apart from all roles, it ceases to be a functional concept [MacIntyre] |
8035 | In trying to explain the type of approval involved, emotivists are either silent, or viciously circular [MacIntyre] |
8037 | The expression of feeling in a sentence is in its use, not in its meaning [MacIntyre] |
8040 | Emotivism cannot explain the logical terms in moral discourse ('therefore', 'if..then') [MacIntyre] |
8042 | Nowadays most people are emotivists, and it is embodied in our culture [MacIntyre] |
8002 | Sophists don't distinguish a person outside one social order from someone outside all order [MacIntyre] |
8012 | The value/fact logical gulf is misleading, because social facts involve values [MacIntyre] |
8005 | 'Happiness' is a bad translation of 'eudaimonia', which includes both behaving and faring well [MacIntyre] |
5954 | All inventions of the mind aim at pleasure, and those that don't are worthless [Metrodorus of Lamp., by Plutarch] |
8058 | Maybe we can only understand rules if we first understand the virtues [MacIntyre] |
7097 | Virtue is secondary to a role-figure, defined within a culture [MacIntyre, by Statman] |
8043 | Characters are the masks worn by moral philosophies [MacIntyre] |
8061 | If morality just is emotion, there are no external criteria for judging emotions [MacIntyre] |
8001 | 'Dikaiosune' is justice, but also fairness and personal integrity [MacIntyre] |
8023 | My duties depend on my identity, which depends on my social relations [MacIntyre] |
8038 | Since Moore thinks the right action produces the most good, he is a utilitarian [MacIntyre] |
8022 | I am naturally free if I am not tied to anyone by a contract [MacIntyre] |
8050 | There are no natural or human rights, and belief in them is nonsense [MacIntyre] |
23080 | Liberals debate how conservative or radical to be, but don't question their basics [MacIntyre] |
8031 | Fans of natural rights or laws can't agree on what the actual rights or laws are [MacIntyre] |
8055 | If God is omniscient, he confronts no as yet unmade decisions, so decisions are impossible [MacIntyre] |
8008 | The Bible is a story about God in which humans are incidental characters [MacIntyre] |