69 ideas
20388 | 'Necessary' conditions are requirements, and 'sufficient' conditions are guarantees [Davies,S] |
20389 | A definition of a thing gives all the requirements which add up to a guarantee of it [Davies,S] |
8721 | An 'impredicative' definition seems circular, because it uses the term being defined [Friend] |
8680 | Classical definitions attempt to refer, but intuitionist/constructivist definitions actually create objects [Friend] |
20391 | Feminists warn that ideologies use timeless objective definitions as a tool of repression [Davies,S] |
3678 | Reductio ad absurdum proves an idea by showing that its denial produces contradiction [Friend] |
8705 | Anti-realists see truth as our servant, and epistemically contrained [Friend] |
8713 | In classical/realist logic the connectives are defined by truth-tables [Friend] |
8708 | Double negation elimination is not valid in intuitionist logic [Friend] |
8694 | Free logic was developed for fictional or non-existent objects [Friend] |
8665 | A 'proper subset' of A contains only members of A, but not all of them [Friend] |
8672 | A 'powerset' is all the subsets of a set [Friend] |
8677 | Set theory makes a minimum ontological claim, that the empty set exists [Friend] |
8666 | Infinite sets correspond one-to-one with a subset [Friend] |
8682 | Major set theories differ in their axioms, and also over the additional axioms of choice and infinity [Friend] |
8709 | The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false [Friend] |
8711 | Intuitionists read the universal quantifier as "we have a procedure for checking every..." [Friend] |
8675 | Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets' [Friend] |
8674 | The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal [Friend] |
8667 | The 'integers' are the positive and negative natural numbers, plus zero [Friend] |
8668 | The 'rational' numbers are those representable as fractions [Friend] |
8670 | A number is 'irrational' if it cannot be represented as a fraction [Friend] |
8661 | The natural numbers are primitive, and the ordinals are up one level of abstraction [Friend] |
8664 | Cardinal numbers answer 'how many?', with the order being irrelevant [Friend] |
8671 | The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps [Friend] |
8663 | Raising omega to successive powers of omega reveal an infinity of infinities [Friend] |
8662 | The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend] |
8669 | Between any two rational numbers there is an infinite number of rational numbers [Friend] |
8676 | Is mathematics based on sets, types, categories, models or topology? [Friend] |
8678 | Most mathematical theories can be translated into the language of set theory [Friend] |
8701 | The number 8 in isolation from the other numbers is of no interest [Friend] |
8702 | In structuralism the number 8 is not quite the same in different structures, only equivalent [Friend] |
8699 | Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? [Friend] |
8696 | Structuralist says maths concerns concepts about base objects, not base objects themselves [Friend] |
8695 | Structuralism focuses on relations, predicates and functions, with objects being inessential [Friend] |
8700 | 'In re' structuralism says that the process of abstraction is pattern-spotting [Friend] |
8681 | The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? [Friend] |
8712 | Mathematics should be treated as true whenever it is indispensable to our best physical theory [Friend] |
8716 | Formalism is unconstrained, so cannot indicate importance, or directions for research [Friend] |
8706 | Constructivism rejects too much mathematics [Friend] |
8707 | Intuitionists typically retain bivalence but reject the law of excluded middle [Friend] |
16062 | A necessary relation between fact-levels seems to be a further irreducible fact [Lynch/Glasgow] |
16061 | If some facts 'logically supervene' on some others, they just redescribe them, adding nothing [Lynch/Glasgow] |
16060 | Nonreductive materialism says upper 'levels' depend on lower, but don't 'reduce' [Lynch/Glasgow] |
16064 | The hallmark of physicalism is that each causal power has a base causal power under it [Lynch/Glasgow] |
8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
8685 | Studying biology presumes the laws of chemistry, and it could never contradict them [Friend] |
8688 | Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend] |
20387 | Aesthetic experience involves perception, but also imagination and understanding [Davies,S] |
20385 | The faculty of 'taste' was posited to explain why only some people had aesthetic appreciation [Davies,S] |
20386 | The sublime is negative in awareness of insignificance, and positive in showing understanding [Davies,S] |
20384 | The idea that art forms are linked into a single concept began in the 1740s [Davies,S] |
20390 | Defining art as representation or expression or form were all undermined by the avant-garde [Davies,S] |
20392 | 'Aesthetic functionalism' says art is what is intended to create aesthetic experiences [Davies,S] |
20405 | Music may be expressive by being 'associated' with other emotional words or events [Davies,S] |
20403 | It seems unlikely that sad music expresses a composer's sadness; it takes ages to write [Davies,S] |
20393 | The 'institutional' theory says art is just something appropriately placed in the 'artworld' [Davies,S] |
20402 | Music is too definite to be put into words (not too indefinite!) [Davies,S] |
20395 | The title of a painting can be vital, and the artist decrees who the portrait represents [Davies,S] |
20396 | We must know what the work is meant to be, to evaluate the artist's achievement [Davies,S] |
20399 | Intentionalism says either meaning just is intention, or ('moderate') meaning is successful intention [Davies,S] |
20401 | The meaning is given by the audience's best guess at the author's intentions [Davies,S] |
20397 | If we could perfectly clone the Mona Lisa, the original would still be special [Davies,S] |
20398 | Art that is multiply instanced may require at least one instance [Davies,S] |
20404 | Music isn't just sad because it makes the listener feel sad [Davies,S] |
22705 | If the depiction of evil is glorified, that is an artistic flaw [Davies,S] |
22707 | It is an artistic defect if excessive moral outrage distorts the story, and narrows our sympathies [Davies,S] |
22706 | A work which seeks approval for immorality, but alienates the audience, is a failure [Davies,S] |
22704 | Immorality may or may not be an artistic defect [Davies,S] |