14 ideas
| 12170 | Amusement rests on superiority, or relief, or incongruity [Scruton] |
| Full Idea: There are three common accounts of amusement: superiority theories (Hobbes's 'sudden glory'), 'relief from restraint' (Freud on jokes), and 'incongruity' theories (Schopenhauer). | |
| From: Roger Scruton (Laughter [1982], §5) | |
| A reaction: All three contain some truth. But one need not feel superior to laugh, and one may already be in a state of unrestraint. Schopenhauer seems closest to a good general account. |
| 12173 | The central object of amusement is the human [Scruton] |
| Full Idea: There are amusing buildings, but not amusing rocks and cliffs. If I were to propose a candidate for the formal object of amusement, then the human would be my choice, ...or at least emphasise its centrality. | |
| From: Roger Scruton (Laughter [1982], §9) | |
| A reaction: Sounds good. Animal behaviour only seems to amuse if it evokes something human. Plants would have to look a bit human to be funny. |
| 12169 | Since only men laugh, it seems to be an attribute of reason [Scruton] |
| Full Idea: Man is the only animal that laughs, so a starting point for all enquiries into laughter must be the hypothesis that it is an attribute of reason (though that gets us no further than our definition of reason). | |
| From: Roger Scruton (Laughter [1982], §1) | |
| A reaction: I would be inclined to say that both our capacity for reason and our capacity for laughter (and, indeed, our capacity for language) are a consequence of our evolved capacity for meta-thought. |
| 12172 | Objects of amusement do not have to be real [Scruton] |
| Full Idea: It is a matter of indifference whether the object of amusement be thought to be real. | |
| From: Roger Scruton (Laughter [1982], §7) | |
| A reaction: Sort of. If I say 'wouldn't it be funny if someone did x?', it is probably much less funny than if I say 'apparently he really did x'. The fantasy case has to be much funnier to evoke the laughter. |
| 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. |
| 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) |
| 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) |
| 12174 | Only rational beings are attentive without motive or concern [Scruton] |
| Full Idea: It is only rational beings who can be attentive without a motive; only rational beings who can be interested in that in which they have no interest. | |
| From: Roger Scruton (Laughter [1982], §12) | |
| A reaction: Rational beings make long term plans, so they cannot prejudge which things may turn out to be of interest to them. Scruton (a Kantian) makes it sound a little loftier than it actually is. |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice. | |
| From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where? |