58 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] |
| 20391 | Feminists warn that ideologies use timeless objective definitions as a tool of repression [Davies,S] |
| 15945 | Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine] |
| 15914 | An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine] |
| 15921 | Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine] |
| 15937 | Those who reject infinite collections also want to reject the Axiom of Choice [Lavine] |
| 15936 | The Power Set is just the collection of functions from one collection to another [Lavine] |
| 15899 | Replacement was immediately accepted, despite having very few implications [Lavine] |
| 15930 | Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine] |
| 15920 | Pure collections of things obey Choice, but collections defined by a rule may not [Lavine] |
| 15898 | The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine] |
| 15919 | The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine] |
| 15900 | The iterative conception of set wasn't suggested until 1947 [Lavine] |
| 15931 | The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine] |
| 15932 | The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine] |
| 15933 | Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine] |
| 15913 | A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine] |
| 15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine] |
| 15934 | Mathematical proof by contradiction needs the law of excluded middle [Lavine] |
| 15907 | Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine] |
| 15942 | Every rational number, unlike every natural number, is divisible by some other number [Lavine] |
| 15922 | For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine] |
| 18250 | Cauchy gave a necessary condition for the convergence of a sequence [Lavine] |
| 15904 | The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine] |
| 15912 | Counting results in well-ordering, and well-ordering makes counting possible [Lavine] |
| 15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine] |
| 15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
| 15940 | The intuitionist endorses only the potential infinite [Lavine] |
| 15909 | 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine] |
| 15915 | Ordinals are basic to Cantor's transfinite, to count the sets [Lavine] |
| 15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine] |
| 15918 | Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine] |
| 15929 | Set theory will found all of mathematics - except for the notion of proof [Lavine] |
| 15935 | Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine] |
| 15928 | Intuitionism rejects set-theory to found mathematics [Lavine] |
| 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] |
| 1513 | The Egyptians were the first to say the soul is immortal and reincarnated [Herodotus] |