103 ideas
| 18730 | The history of philosophy only matters if the subject is a choice between rival theories [Wittgenstein] |
| 13466 | We are all post-Kantians, because he set the current agenda for philosophy [Hart,WD] |
| 18704 | Philosophy tries to be rid of certain intellectual puzzles, irrelevant to daily life [Wittgenstein] |
| 13477 | The problems are the monuments of philosophy [Hart,WD] |
| 18710 | Philosophers express puzzlement, but don't clearly state the puzzle [Wittgenstein] |
| 18732 | We don't need a theory of truth, because we use the word perfectly well [Wittgenstein] |
| 13515 | To study abstract problems, some knowledge of set theory is essential [Hart,WD] |
| 18714 | We already know what we want to know, and analysis gives us no new facts [Wittgenstein] |
| 17621 | What matters in mathematics is its objectivity, not the existence of the objects [Dummett] |
| 18706 | Words of the same kind can be substituted in a proposition without producing nonsense [Wittgenstein] |
| 18719 | Grammar says that saying 'sound is red' is not false, but nonsense [Wittgenstein] |
| 18735 | Talking nonsense is not following the rules [Wittgenstein] |
| 18731 | There is no theory of truth, because it isn't a concept [Wittgenstein] |
| 18707 | All thought has the logical form of reality [Wittgenstein] |
| 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] |
| 10537 | The ordered pairs <x,y> can be reduced to the class of sets of the form {{x},{x,y}} [Dummett] |
| 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] |
| 10542 | To associate a cardinal with each set, we need the Axiom of Choice to find a representative [Dummett] |
| 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] |
| 18724 | In logic nothing is hidden [Wittgenstein] |
| 18709 | Laws of logic are like laws of chess - if you change them, it's just a different game [Wittgenstein] |
| 18736 | Contradiction is between two rules, not between rule and reality [Wittgenstein] |
| 18723 | We may correctly use 'not' without making the rule explicit [Wittgenstein] |
| 18718 | Saying 'and' has meaning is just saying it works in a sentence [Wittgenstein] |
| 18727 | A person's name doesn't mean their body; bodies don't sit down, and their existence can be denied [Wittgenstein] |
| 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] |
| 18738 | We don't get 'nearer' to something by adding decimals to 1.1412... (root-2) [Wittgenstein] |
| 13446 | 19th century arithmetization of analysis isolated the real numbers from geometry [Hart,WD] |
| 18708 | Infinity is not a number, so doesn't say how many; it is the property of a law [Wittgenstein] |
| 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] |
| 10554 | Intuitionists find the Incompleteness Theorem unsurprising, since proof is intuitive, not formal [Dummett] |
| 13471 | Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD] |
| 10552 | Intuitionism says that totality of numbers is only potential, but is still determinate [Dummett] |
| 10540 | We can't say that light is concrete but radio waves abstract [Dummett] |
| 10515 | Ostension is possible for concreta; abstracta can only be referred to via other objects [Dummett, by Hale] |
| 10544 | The concrete/abstract distinction seems crude: in which category is the Mistral? [Dummett] |
| 10546 | We don't need a sharp concrete/abstract distinction [Dummett] |
| 13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' [Hart,WD] |
| 18737 | There are no positive or negative facts; these are just the forms of propositions [Wittgenstein] |
| 10548 | The context principle for names rules out a special philosophical sense for 'existence' [Dummett] |
| 10281 | The objects we recognise the world as containing depends on the structure of our language [Dummett] |
| 10532 | We can understand universals by studying predication [Dummett] |
| 18715 | Using 'green' is a commitment to future usage of 'green' [Wittgenstein] |
| 10534 | 'Nominalism' used to mean denial of universals, but now means denial of abstract objects [Dummett] |
| 10541 | Concrete objects such as sounds and smells may not be possible objects of ostension [Dummett] |
| 10545 | Abstract objects may not cause changes, but they can be the subject of change [Dummett] |
| 10555 | If we can intuitively apprehend abstract objects, this makes them observable and causally active [Dummett] |
| 10543 | Abstract objects must have names that fall within the range of some functional expression [Dummett] |
| 10320 | If a genuine singular term needs a criterion of identity, we must exclude abstract nouns [Dummett, by Hale] |
| 10547 | Abstract objects can never be confronted, and need verbal phrases for reference [Dummett] |
| 10531 | There is a modern philosophical notion of 'object', first introduced by Frege [Dummett] |
| 18726 | For each necessity in the world there is an arbitrary rule of language [Wittgenstein] |
| 18712 | Understanding is translation, into action or into other symbols [Wittgenstein] |
| 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] |
| 18280 | We live in sense-data, but talk about physical objects [Wittgenstein] |
| 18729 | Part of what we mean by stating the facts is the way we tend to experience them [Wittgenstein] |
| 18734 | If you remember wrongly, then there must be some other criterion than your remembering [Wittgenstein] |
| 18721 | Explanation and understanding are the same [Wittgenstein] |
| 18720 | Explanation gives understanding by revealing the full multiplicity of the thing [Wittgenstein] |
| 18716 | A machine strikes us as being a rule of movement [Wittgenstein] |
| 18713 | If an explanation is good, the symbol is used properly in the future [Wittgenstein] |
| 18717 | Thought is an activity which we perform by the expression of it [Wittgenstein] |
| 19168 | Concepts only have a 'functional character', because they map to truth values, not objects [Dummett, by Davidson] |
| 13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
| 10549 | Since abstract objects cannot be picked out, we must rely on identity statements [Dummett] |
| 18725 | A proposition draws a line around the facts which agree with it [Wittgenstein] |
| 18728 | The meaning of a proposition is the mode of its verification [Wittgenstein] |
| 18705 | Words function only in propositions, like levers in a machine [Wittgenstein] |
| 10516 | A realistic view of reference is possible for concrete objects, but not for abstract objects [Dummett, by Hale] |
| 18711 | A proposition is any expression which can be significantly negated [Wittgenstein] |
| 18733 | Laws of nature are an aspect of the phenomena, and are just our mode of description [Wittgenstein] |