14 ideas
| 13206 | A 'linear or total ordering' must be transitive and satisfy trichotomy [Enderton] |
| Full Idea: A 'linear ordering' (or 'total ordering') on A is a binary relation R meeting two conditions: R is transitive (of xRy and yRz, the xRz), and R satisfies trichotomy (either xRy or x=y or yRx). | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 3:62) |
| 13201 | ∈ says the whole set is in the other; ⊆ says the members of the subset are in the other [Enderton] |
| Full Idea: To know if A ∈ B, we look at the set A as a single object, and check if it is among B's members. But if we want to know whether A ⊆ B then we must open up set A and check whether its various members are among the members of B. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 1:04) | |
| A reaction: This idea is one of the key ideas to grasp if you are going to get the hang of set theory. John ∈ USA ∈ UN, but John is not a member of the UN, because he isn't a country. See Idea 12337 for a special case. |
| 13204 | The 'ordered pair' <x,y> is defined to be {{x}, {x,y}} [Enderton] |
| Full Idea: The 'ordered pair' <x,y> is defined to be {{x}, {x,y}}; hence it can be proved that <u,v> = <x,y> iff u = x and v = y (given by Kuratowski in 1921). ...The definition is somewhat arbitrary, and others could be used. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 3:36) | |
| A reaction: This looks to me like one of those regular cases where the formal definitions capture all the logical behaviour of the concept that are required for inference, while failing to fully capture the concept for ordinary conversation. |
| 13200 | Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ [Enderton] |
| Full Idea: Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ. A man with an empty container is better off than a man with nothing. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 1.03) |
| 13199 | The empty set may look pointless, but many sets can be constructed from it [Enderton] |
| Full Idea: It might be thought at first that the empty set would be a rather useless or even frivolous set to mention, but from the empty set by various set-theoretic operations a surprising array of sets will be constructed. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 1:02) | |
| A reaction: This nicely sums up the ontological commitments of mathematics - that we will accept absolutely anything, as long as we can have some fun with it. Sets are an abstraction from reality, and the empty set is the very idea of that abstraction. |
| 13203 | The singleton is defined using the pairing axiom (as {x,x}) [Enderton] |
| Full Idea: Given any x we have the singleton {x}, which is defined by the pairing axiom to be {x,x}. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 2:19) | |
| A reaction: An interesting contrivance which is obviously aimed at keeping the axioms to a minimum. If you can do it intuitively with a new axiom, or unintuitively with an existing axiom - prefer the latter! |
| 13202 | Fraenkel added Replacement, to give a theory of ordinal numbers [Enderton] |
| Full Idea: It was observed by several people that for a satisfactory theory of ordinal numbers, Zermelo's axioms required strengthening. The Axiom of Replacement was proposed by Fraenkel and others, giving rise to the Zermelo-Fraenkel (ZF) axioms. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 1:15) |
| 13205 | We can only define functions if Choice tells us which items are involved [Enderton] |
| Full Idea: For functions, we know that for any y there exists an appropriate x, but we can't yet form a function H, as we have no way of defining one particular choice of x. Hence we need the axiom of choice. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 3:48) |
| 20416 | By 1790 aestheticians were mainly trying to explain individual artistic genius [Kemp] |
| Full Idea: By 1790 the idea that a central task for the aesthetician was to explain or at least adequately to describe the phenomenon of the individual artistic genius had definitely taken hold. | |
| From: Gary Kemp (Croce and Collingwood [2012], Intro) | |
| A reaction: Hence when Kant and Hegel write about art, though are only really thinking of the greatest art (which might be in touch with the sublime or Spirit etc.). Nowadays I think we expect accounts of art to cover modest amateur efforts as well. |
| 20417 | Expression can be either necessary for art, or sufficient for art (or even both) [Kemp] |
| Full Idea: Seeing art as expression has two components: 1) if something is a work of art, then it is expressive, 2) if something is expressive, then it is a work of art. So expression can be necessary or sufficient for art. (or both, for Croce and Collingwood). | |
| From: Gary Kemp (Croce and Collingwood [2012], 1) | |
| A reaction: I take the idea that art 'expresses' the feelings of an artist to be false. Artists are more like actors. Nearly all art has some emotional impact, which is of major importance, but I don't think 'expression' is a very good word for that. |
| 20418 | The horror expressed in some works of art could equallly be expressed by other means [Kemp] |
| Full Idea: The horror or terror of Edvard Much's 'The Scream' could in principle be expressed by different paintings, or even by works of music. | |
| From: Gary Kemp (Croce and Collingwood [2012], 1) | |
| A reaction: A very good simple point against the idea that the point of art is expression. It leaves out the very specific nature of each work of art! |
| 20419 | We don't already know what to express, and then seek means of expressing it [Kemp] |
| Full Idea: One cannot really know, or be conscious of, what it is that one is going to express, and then set about expressing it; indeed if one is genuinely conscious of it then one has already expressed it. | |
| From: Gary Kemp (Croce and Collingwood [2012], 1) | |
| A reaction: That pretty conclusively demolishes the idea that art is expression. I picture Schubert composing at the piano: he doesn't feel an emotion, and then hunt for its expression on the keyboard; he seeks out expressive phrases by playing. |
| 23812 | Force is what turns man into a thing, and ultimately into a corpse [Weil] |
| Full Idea: To define 'force' - it is that x that turns anybody who is subjected to it into a thing. Exercised to the limit, it turns man into a thing in the most literal sense: it makes a corpse out of him. | |
| From: Simone Weil (The Iliad or the Poem of Force [1940], p.183) | |
| A reaction: She celebrates The Iliad as the great examination of force in human affairs. I have felt that sense of reduction to a thing whenever anyone above me in the hierarchy has arbitrarily exerted their power over me. |
| 23813 | Only people who understand force, and don't respect it, are capable of justice [Weil] |
| Full Idea: Only he who has measured the dominion of force, and knows how not to respect it, is capable of love and justice. | |
| From: Simone Weil (The Iliad or the Poem of Force [1940], p.212) | |
| A reaction: There are, of course, occasions when we are grateful to people who exercise appropriate force on our behalf. I think she was concerned with what is inappropriate. |