76 ideas
14179 | The finest branch of wisdom is justice and moderation in ordering states and families [Plato] |
13466 | We are all post-Kantians, because he set the current agenda for philosophy [Hart,WD] |
14255 | We understand things through their dependency relations [Fine,K] |
13477 | The problems are the monuments of philosophy [Hart,WD] |
14250 | Metaphysics deals with the existence of things and with the nature of things [Fine,K] |
13515 | To study abstract problems, some knowledge of set theory is essential [Hart,WD] |
14259 | Maybe two objects might require simultaneous real definitions, as with two simultaneous terms [Fine,K] |
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] |
13471 | Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD] |
14253 | An object's 'being' isn't existence; there's more to an object than existence, and its nature doesn't include existence [Fine,K] |
14251 | A natural modal account of dependence says x depends on y if y must exist when x does [Fine,K] |
14254 | Dependency is the real counterpart of one term defining another [Fine,K] |
14257 | An object depends on another if the second cannot be eliminated from the first's definition [Fine,K] |
14261 | There is 'weak' dependence in one definition, and 'strong' dependence in all the definitions [Fine,K] |
13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' [Hart,WD] |
1607 | Diotima said the Forms are the objects of desire in philosophical discourse [Plato, by Roochnik] |
14252 | We should understand identity in terms of the propositions it renders true [Fine,K] |
14256 | How do we distinguish basic from derived esssences? [Fine,K] |
14258 | Maybe some things have essential relationships as well as essential properties [Fine,K] |
14260 | An object only essentially has a property if that property follows from every definition of the object [Fine,K] |
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] |
174 | True opinion without reason is midway between wisdom and ignorance [Plato] |
181 | Only the gods stay unchanged; we replace our losses with similar acquisitions [Plato] |
180 | We call a person the same throughout life, but all their attributes change [Plato] |
13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
172 | Love of ugliness is impossible [Plato] |
4026 | Beauty is harmony with what is divine, and ugliness is lack of such harmony [Plato] |
173 | Beauty and goodness are the same [Plato] |
183 | Stage two is the realisation that beauty of soul is of more value than beauty of body [Plato] |
184 | Progress goes from physical beauty, to moral beauty, to the beauty of knowledge, and reaches absolute beauty [Plato] |
171 | Music is a knowledge of love in the realm of harmony and rhythm [Plato] |
14177 | Love assists men in achieving merit and happiness [Plato] |
179 | Love is desire for perpetual possession of the good [Plato] |
176 | Love follows beauty, wisdom is exceptionally beautiful, so love follows wisdom [Plato] |
177 | If a person is good they will automatically become happy [Plato] |
14178 | Happiness is secure enjoyment of what is good and beautiful [Plato] |
182 | The first step on the right path is the contemplation of physical beauty when young [Plato] |
170 | The only slavery which is not dishonourable is slavery to excellence [Plato] |
175 | Gods are not lovers of wisdom, because they are already wise [Plato] |