93 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] |
| 9161 | Maybe reasonableness requires circular justifications - that is one coherentist view [Field,H] |
| 10825 | The notion of truth is to help us make use of the utterances of others [Field,H] |
| 10820 | In the early 1930s many philosophers thought truth was not scientific [Field,H] |
| 13469 | Tarski showed how we could have a correspondence theory of truth, without using 'facts' [Hart,WD] |
| 13499 | Tarski reduced truth to reference or denotation [Field,H, by Hart,WD] |
| 10818 | Tarski really explained truth in terms of denoting, predicating and satisfied functions [Field,H] |
| 10817 | Tarski just reduced truth to some other undefined semantic notions [Field,H] |
| 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] |
| 9570 | In Field's Platonist view, set theory is false because it asserts existence for non-existent things [Field,H, by Chihara] |
| 13481 | Maybe sets should be rethought in terms of the even more basic categories [Hart,WD] |
| 10260 | Logical consequence is defined by the impossibility of P and ¬q [Field,H, by Shapiro] |
| 13506 | The universal quantifier can't really mean 'all', because there is no universal set [Hart,WD] |
| 10819 | Tarski gives us the account of truth needed to build a group of true sentences in a model [Field,H] |
| 13512 | Modern model theory begins with the proof of Los's Conjecture in 1962 [Hart,WD] |
| 10827 | Model theory is unusual in restricting the range of the quantifiers [Field,H] |
| 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] |
| 6007 | If you know your father, but don't recognise your father veiled, you know and don't know the same person [Eubulides, by Dancy,R] |
| 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] |
| 6006 | If you say truly that you are lying, you are lying [Eubulides, by Dancy,R] |
| 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] |
| 6008 | Removing one grain doesn't destroy a heap, so a heap can't be destroyed [Eubulides, by Dancy,R] |
| 9226 | If mathematical theories conflict, it may just be that they have different subject matter [Field,H] |
| 8958 | In Field's version of science, space-time points replace real numbers [Field,H, by Szabó] |
| 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] |
| 18221 | 'Metric' axioms uses functions, points and numbers; 'synthetic' axioms give facts about space [Field,H] |
| 13474 | Euclid has a unique parallel, spherical geometry has none, and saddle geometry has several [Hart,WD] |
| 8757 | The Indispensability Argument is the only serious ground for the existence of mathematical entities [Field,H] |
| 18212 | Nominalists try to only refer to physical objects, or language, or mental constructions [Field,H] |
| 10261 | The application of mathematics only needs its possibility, not its truth [Field,H, by Shapiro] |
| 18218 | Hilbert explains geometry, by non-numerical facts about space [Field,H] |
| 9623 | Field needs a semantical notion of second-order consequence, and that needs sets [Brown,JR on Field,H] |
| 18215 | It seems impossible to explain the idea that the conclusion is contained in the premises [Field,H] |
| 13471 | Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD] |
| 18216 | Abstractions can form useful counterparts to concrete statements [Field,H] |
| 18214 | Mathematics is only empirical as regards which theory is useful [Field,H] |
| 8714 | Fictionalists say 2+2=4 is true in the way that 'Oliver Twist lived in London' is true [Field,H] |
| 18210 | Why regard standard mathematics as truths, rather than as interesting fictions? [Field,H] |
| 13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' [Hart,WD] |
| 18211 | You can reduce ontological commitment by expanding the logic [Field,H] |
| 8959 | Field presumes properties can be eliminated from science [Field,H, by Szabó] |
| 18213 | Abstract objects are only applicable to the world if they are impure, and connect to the physical [Field,H] |
| 13480 | Fregean self-evidence is an intrinsic property of basic truths, rules and definitions [Hart,WD] |
| 9160 | Lots of propositions are default reasonable, but the a priori ones are empirically indefeasible [Field,H] |
| 9164 | We treat basic rules as if they were indefeasible and a priori, with no interest in counter-evidence [Field,H] |
| 13476 | The failure of key assumptions in geometry, mereology and set theory throw doubt on the a priori [Hart,WD] |
| 9165 | Reliability only makes a rule reasonable if we place a value on the truth produced by reliable processes [Field,H] |
| 9162 | Believing nothing, or only logical truths, is very reliable, but we want a lot more than that [Field,H] |
| 9166 | People vary in their epistemological standards, and none of them is 'correct' [Field,H] |
| 9163 | If we only use induction to assess induction, it is empirically indefeasible, and hence a priori [Field,H] |
| 18222 | Beneath every extrinsic explanation there is an intrinsic explanation [Field,H] |
| 10826 | 'Valence' and 'gene' had to be reduced to show their compatibility with physicalism [Field,H] |
| 13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
| 9917 | 'Abstract' is unclear, but numbers, functions and sets are clearly abstract [Field,H] |
| 22244 | 'Partial reference' is when the subject thinks two objects are one object [Field,H, by Recanati] |
| 7615 | Field says reference is a causal physical relation between mental states and objects [Field,H, by Putnam] |
| 8404 | Explain single events by general rules, or vice versa, or probability explains both, or they are unconnected [Field,H] |
| 8401 | Physical laws are largely time-symmetric, so they make a poor basis for directional causation [Field,H] |
| 8400 | Identifying cause and effect is not just conventional; we explain later events by earlier ones [Field,H] |
| 8402 | The only reason for adding the notion of 'cause' to fundamental physics is directionality [Field,H] |
| 18223 | In theories of fields, space-time points or regions are causal agents [Field,H] |
| 18220 | Both philosophy and physics now make substantivalism more attractive [Field,H] |
| 18219 | Relational space is problematic if you take the idea of a field seriously [Field,H] |