78 ideas
13466 | We are all post-Kantians, because he set the current agenda for philosophy [Hart,WD] |
5300 | Philosophers have interpreted the world, but the point is to change it [Marx] |
13477 | The problems are the monuments of philosophy [Hart,WD] |
13515 | To study abstract problems, some knowledge of set theory is essential [Hart,WD] |
5297 | Whether human thinking can be 'true' must be decided in practice, not theory [Marx] |
13469 | Tarski showed how we could have a correspondence theory of truth, without using 'facts' [Hart,WD] |
13504 | Truth for sentences is satisfaction of formulae; for sentences, either all sequences satisfy it (true) or none do [Hart,WD] |
13503 | A first-order language has an infinity of T-sentences, which cannot add up to a definition of truth [Hart,WD] |
13500 | Conditional Proof: infer a conditional, if the consequent can be deduced from the antecedent [Hart,WD] |
13502 | ∃y... is read as 'There exists an individual, call it y, such that...', and not 'There exists a y such that...' [Hart,WD] |
13456 | Set theory articulates the concept of order (through relations) [Hart,WD] |
13497 | Nowadays ZFC and NBG are the set theories; types are dead, and NF is only useful for the whole universe [Hart,WD] |
13443 | ∈ relates across layers, while ⊆ relates within layers [Hart,WD] |
13442 | Without the empty set we could not form a∩b without checking that a and b meet [Hart,WD] |
13493 | In the modern view, foundation is the heart of the way to do set theory [Hart,WD] |
13495 | Foundation Axiom: an nonempty set has a member disjoint from it [Hart,WD] |
13461 | We can choose from finite and evident sets, but not from infinite opaque ones [Hart,WD] |
13462 | With the Axiom of Choice every set can be well-ordered [Hart,WD] |
13516 | If we accept that V=L, it seems to settle all the open questions of set theory [Hart,WD] |
13441 | Naïve set theory has trouble with comprehension, the claim that every predicate has an extension [Hart,WD] |
13494 | The iterative conception may not be necessary, and may have fixed points or infinitely descending chains [Hart,WD] |
13457 | A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets [Hart,WD] |
13458 | A partial ordering becomes 'total' if any two members of its field are comparable [Hart,WD] |
13460 | 'Well-ordering' must have a least member, so it does the natural numbers but not the integers [Hart,WD] |
13490 | Von Neumann defines α<β as α∈β [Hart,WD] |
13481 | Maybe sets should be rethought in terms of the even more basic categories [Hart,WD] |
13506 | The universal quantifier can't really mean 'all', because there is no universal set [Hart,WD] |
13512 | Modern model theory begins with the proof of Los's Conjecture in 1962 [Hart,WD] |
13505 | Model theory studies how set theory can model sets of sentences [Hart,WD] |
13511 | Model theory is mostly confined to first-order theories [Hart,WD] |
13513 | Models are ways the world might be from a first-order point of view [Hart,WD] |
13496 | First-order logic is 'compact': consequences of a set are consequences of a finite subset [Hart,WD] |
13484 | Berry's Paradox: we succeed in referring to a number, with a term which says we can't do that [Hart,WD] |
13482 | The Burali-Forti paradox is a crisis for Cantor's ordinals [Hart,WD] |
13507 | The machinery used to solve the Liar can be rejigged to produce a new Liar [Hart,WD] |
9117 | The smallest heap has four objects: three on the bottom, one on the top [Hart,WD, by Sorensen] |
13459 | The less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers [Hart,WD] |
13491 | The axiom of infinity with separation gives a least limit ordinal ω [Hart,WD] |
13463 | There are at least as many infinite cardinals as transfinite ordinals (because they will map) [Hart,WD] |
13492 | Von Neumann's ordinals generalise into the transfinite better, because Zermelo's ω is a singleton [Hart,WD] |
13446 | 19th century arithmetization of analysis isolated the real numbers from geometry [Hart,WD] |
13509 | We can establish truths about infinite numbers by means of induction [Hart,WD] |
13474 | Euclid has a unique parallel, spherical geometry has none, and saddle geometry has several [Hart,WD] |
13471 | Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD] |
13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' [Hart,WD] |
13480 | Fregean self-evidence is an intrinsic property of basic truths, rules and definitions [Hart,WD] |
13476 | The failure of key assumptions in geometry, mereology and set theory throw doubt on the a priori [Hart,WD] |
20653 | Six reduction levels: groups, lives, cells, molecules, atoms, particles [Putnam/Oppenheim, by Watson] |
22598 | The authentic self exists at the level of class, rather than the individual [Marx, by Dunt] |
13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
5298 | The human essence is not found in individuals but in social relations [Marx] |
23874 | Armies and businesses create moralities in which their activity can do no wrong [Marx, by Weil] |
22001 | The real will of the cooperative will replace the 'will of the people' [Marx] |
21991 | The middle class gain freedom through property, but workers can only free all of humanity [Marx, by Singer] |
21990 | Theory is as much a part of a revolution as material force is [Marx] |
22971 | In moving from capitalism to communism a revolutionary dictatorship of the proletariat is needed [Marx] |
18662 | Liberal freedom is the right to be separate, and ignores the union of man with man [Marx] |
23372 | Liberals want the right to be separate, rather than for people to be united [Marx] |
20576 | Early Marx anticipates communitarian objections to liberalism [Marx, by Oksala] |
21989 | Man is dominated by money, which is the essence of his alienation [Marx] |
22000 | From each according to his ability, to each according to his need [Marx] |
23862 | By saying the material dialectic of history aspires to the best, Marx agreed with capitalism [Weil on Marx] |
21999 | False consciousness results from concealment by the superstructure [Marx, by Singer] |
23875 | Marx says force is everything, and that the weak will become strong, while remaining the weak [Weil on Marx] |
21995 | Must production determine superstructure, or could it be the other way round? [Singer on Marx] |
18653 | Marx rejected equal rights because they never actually treat people as equals [Marx, by Kymlicka] |
20577 | Even decently paid workers still have their produce bought with money stolen from them [Marx] |
22969 | People who only have their labour power are the slaves of those permitting them to work [Marx] |
21996 | Freedom only comes when labour is no longer necessary [Marx] |
22970 | Freedom is making the state subordinate to its society [Marx] |
21994 | The handmill gives feudalism, the steam mill capitalism [Marx] |
20958 | Capitalism changes the world, by socialising the idea of a commodity [Marx, by Bowie] |
23876 | The essence of capitalism is the subordination of people to things [Marx, by Weil] |
20960 | Marx thought capitalism was partly liberating, and could make labour and ownership more humane [Marx, by Bowie] |
22972 | Bourgeois 'freedom of conscience' just tolerates all sorts of religious intolerance [Marx] |
20519 | Marxists say liberal rights are confrontational, and liberal equality is a sham [Marx, by Wolff,J] |
7128 | Religion is the opium of the people, and real happiness requires its abolition [Marx] |
5299 | Religious feeling is social in origin [Marx] |