113 ideas
| 13466 | We are all post-Kantians, because he set the current agenda for philosophy [Hart,WD] |
| 13477 | The problems are the monuments of philosophy [Hart,WD] |
| 13515 | To study abstract problems, some knowledge of set theory is essential [Hart,WD] |
| 3807 | Reason is and ought to be the slave of the passions [Hume] |
| 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] |
| 13462 | With the Axiom of Choice every set can be well-ordered [Hart,WD] |
| 13461 | We can choose from finite and evident sets, but not from infinite opaque ones [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] |
| 13460 | 'Well-ordering' must have a least member, so it does the natural numbers but not the integers [Hart,WD] |
| 13458 | A partial ordering becomes 'total' if any two members of its field are comparable [Hart,WD] |
| 13457 | A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets [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] |
| 13513 | Models are ways the world might be from a first-order point of view [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] |
| 13512 | Modern model theory begins with the proof of Los's Conjecture in 1962 [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] |
| 13459 | The less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers [Hart,WD] |
| 13463 | There are at least as many infinite cardinals as transfinite ordinals (because they will map) [Hart,WD] |
| 13491 | The axiom of infinity with separation gives a least limit ordinal ω [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] |
| 8649 | Two numbers are equal if all of their units correspond to one another [Hume] |
| 13471 | Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD] |
| 21291 | There is no medium state between existence and non-existence [Hume] |
| 13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' [Hart,WD] |
| 11942 | Power is the possibility of action, as discovered by experience [Hume] |
| 11949 | There may well be powers in things, with which we are quite unacquainted [Hume] |
| 11950 | We have no idea of powers, because we have no impressions of them [Hume] |
| 11941 | The distinction between a power and its exercise is entirely frivolous [Hume] |
| 11098 | Momentary impressions are wrongly identified with one another on the basis of resemblance [Hume, by Quine] |
| 7954 | If we see a resemblance among objects, we apply the same name to them, despite their differences [Hume] |
| 21293 | Individuation is only seeing that a thing is stable and continuous over time [Hume] |
| 12048 | The only meaning we have for substance is a collection of qualities [Hume] |
| 13424 | Aristotelians propose accidents supported by substance, but they don't understand either of them [Hume] |
| 21299 | Changing a part can change the whole, not absolutely, but by its proportion of the whole [Hume] |
| 21300 | A change more obviously destroys an identity if it is quick and observed [Hume] |
| 1321 | If identity survives change or interruption, then resemblance, contiguity or causation must unite the parts of it [Hume] |
| 1330 | If a republic can retain identity through many changes, so can an individual [Hume] |
| 21302 | If a ruined church is rebuilt, its relation to its parish makes it the same church [Hume] |
| 21303 | We accept the identity of a river through change, because it is the river's nature [Hume] |
| 21301 | The purpose of the ship makes it the same one through all variations [Hume] |
| 1207 | Both number and unity are incompatible with the relation of identity [Hume] |
| 21290 | Multiple objects cannot convey identity, because we see them as different [Hume] |
| 21289 | 'An object is the same with itself' is meaningless; it expresses unity, not identity [Hume] |
| 21292 | Saying an object is the same with itself is only meaningful over a period of time [Hume] |
| 9428 | Nothing we clearly imagine is absolutely impossible [Hume] |
| 4766 | Necessity only exists in the mind, and not in objects [Hume] |
| 6526 | Hume says objects are not a construction, but an imaginative leap [Hume, by Robinson,H] |
| 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] |
| 6489 | Associationism results from having to explain intentionality just with sense-data [Robinson,H on Hume] |
| 6182 | Even Hume didn't include mathematics in his empiricism [Hume, by Kant] |
| 12417 | Mathematicians only accept their own proofs when everyone confims them [Hume] |
| 5548 | Hume became a total sceptic, because he believed that reason was a deception [Hume, by Kant] |
| 7446 | The idea of inductive evidence, around 1660, made Hume's problem possible [Hume, by Hacking] |
| 21806 | Memory, senses and understanding are all founded on the imagination [Hume] |
| 3819 | Hume's 'bundle' won't distinguish one mind with ten experiences from ten minds [Searle on Hume] |
| 1317 | A person is just a fast-moving bundle of perceptions [Hume] |
| 1331 | The parts of a person are always linked together by causation [Hume] |
| 1388 | Hume gives us an interesting sketchy causal theory of personal identity [Perry on Hume] |
| 21297 | A person is simply a bundle of continually fluctuating perceptions [Hume] |
| 1316 | Introspection always discovers perceptions, and never a Self without perceptions [Hume] |
| 1333 | Memory only reveals personal identity, by showing cause and effect [Hume] |
| 1332 | We use memory to infer personal actions we have since forgotten [Hume] |
| 21305 | Memory not only reveals identity, but creates it, by producing resemblances [Hume] |
| 21307 | Who thinks that because you have forgotten an incident you are no longer that person? [Hume] |
| 21306 | Causation unites our perceptions, by producing, destroying and modifying each other [Hume] |
| 21294 | A continuous lifelong self must be justified by a single sustained impression, which we don't have [Hume] |
| 21295 | When I introspect I can only observe my perceptions, and never a self which has them [Hume] |
| 21298 | We pretend our perceptions are continuous, and imagine a self to fill the gaps [Hume] |
| 21304 | Identity in the mind is a fiction, like that fiction that plants and animals stay the same [Hume] |
| 13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
| 20030 | If one event causes another, the two events must be wholly distinct [Hume, by Wilson/Schpall] |
| 6692 | For Hume, practical reason has little force, because we can always modify our desires [Hume, by Graham] |
| 3546 | 'Phronesis' should translate as 'practical intelligence', not as prudence [Annas] |
| 8257 | Reason alone can never be a motive to any action of the will [Hume] |
| 22374 | You can only hold people responsible for actions which arise out of their character [Hume] |
| 22382 | We cannot discover vice by studying a wilful murder; that only arises from our own feelings [Hume] |
| 4008 | Modern science has destroyed the Platonic synthesis of scientific explanation and morality [Hume, by Taylor,C] |
| 8067 | The problem of getting to 'ought' from 'is' would also apply in getting to 'owes' or 'needs' [Anscombe on Hume] |
| 4578 | You can't move from 'is' to 'ought' without giving some explanation or reason for the deduction [Hume] |
| 3650 | Total selfishness is not irrational [Hume] |
| 3547 | Epicureans achieve pleasure through character development [Annas] |
| 3543 | Cyrenaics pursue pleasure, but don't equate it with happiness [Annas] |
| 3541 | Ancient ethics uses attractive notions, not imperatives [Annas] |
| 3550 | Principles cover life as a whole, where rules just cover actions [Annas] |
| 3551 | Virtue theory tries to explain our duties in terms of our character [Annas] |
| 3552 | If excessively good actions are admirable but not required, then duty isn't basic [Annas] |
| 3542 | We should do good when necessary, not maximise it [Annas] |
| 14301 | We have no good concept of solidity or matter, because accounts of them are all circular [Hume] |
| 8382 | For Hume a constant conjunction is both necessary and sufficient for causation [Hume, by Crane] |
| 19274 | Hume seems to presuppose necessary connections between mental events [Kripke on Hume] |
| 21296 | If all of my perceptions were removed by death, nothing more is needed for total annihilation [Hume] |