79 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] |
| 1502 | Parmenides was much more cautious about accepting ideas than his predecessors [Simplicius on Parmenides] |
| 17596 | Coherence problems have positive and negative restraints; solutions maximise constraint satisfaction [Thagard] |
| 17597 | Coherence is explanatory, deductive, conceptual, analogical, perceptual, and deliberative [Thagard] |
| 17598 | Explanatory coherence needs symmetry,explanation,analogy,data priority, contradiction,competition,acceptance [Thagard] |
| 17602 | Verisimilitude comes from including more phenomena, and revealing what underlies [Thagard] |
| 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] |
| 449 | Being is not divisible, since it is all alike [Parmenides] |
| 448 | No necessity could produce Being either later or earlier, so it must exist absolutely or not at all [Parmenides] |
| 447 | Being must be eternal and uncreated, and hence it is timeless [Parmenides] |
| 445 | The realm of necessary non-existence cannot be explored, because it is unknowable [Parmenides] |
| 1503 | There is no such thing as nothing [Parmenides] |
| 21820 | Parmenides at least saw Being as the same as Nous, and separate from the sensed realm [Parmenides, by Plotinus] |
| 452 | All our concepts of change and permanence are just names, not the truth [Parmenides] |
| 13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' [Hart,WD] |
| 1504 | Something must be unchanging to make recognition and knowledge possible [Aristotle on Parmenides] |
| 444 | The first way of enquiry involves necessary existence [Parmenides] |
| 450 | Necessity sets limits on being, in order to give it identity [Parmenides] |
| 451 | Thinking implies existence, because thinking depends on it [Parmenides] |
| 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] |
| 1506 | Parmenides treats perception and intellectual activity as the same [Theophrastus on Parmenides] |
| 3058 | Only reason can prove the truth of facts [Parmenides] |
| 17601 | Neither a priori rationalism nor sense data empiricism account for scientific knowledge [Thagard] |
| 17600 | Bayesian inference is forced to rely on approximations [Thagard] |
| 17599 | The best theory has the highest subjective (Bayesian) probability? [Thagard] |
| 13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
| 555 | People who say that the cosmos is one forget that they must explain movement [Aristotle on Parmenides] |
| 226 | The one is without any kind of motion [Parmenides] |
| 5081 | There could be movement within one thing, as there is within water [Aristotle on Parmenides] |
| 1509 | The one can't be divisible, because if it was it could be infinitely divided down to nothing [Parmenides, by Simplicius] |
| 20900 | Defenders of the One say motion needs the void - but that is not part of Being [Parmenides, by Aristotle] |
| 1505 | Reason sees reality as one, the senses see it as many [Aristotle on Parmenides] |
| 453 | Reality is symmetrical and balanced, like a sphere, with no reason to be greater one way rather than another [Parmenides] |
| 1792 | He taught that there are two elements, fire the maker, and earth the matter [Parmenides, by Diog. Laertius] |
| 5115 | It is feeble-minded to look for explanations of everything being at rest [Aristotle on Parmenides] |
| 13217 | The void can't exist, and without the void there can't be movement or separation [Parmenides, by Aristotle] |
| 22918 | What could have triggered the beginning [of time and being]? [Parmenides] |
| 1794 | He was the first to discover the identity of the Morning and Evening Stars [Parmenides, by Diog. Laertius] |
| 1791 | He was the first person to say the earth is spherical [Parmenides, by Diog. Laertius] |