19 ideas
| 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) |
| 21515 | Incoherence may be more important for enquiry than coherence [Olsson] |
| Full Idea: While coherence may lack the positive role many have assigned to it, ...incoherence plays an important negative role in our enquiries. | |
| From: Erik J. Olsson (Against Coherence [2005], 10.1) | |
| A reaction: [He cites Peirce as the main source for this idea] We can hardly by deeply impressed by incoherence if we have no sense of coherence. Incoherence is just one of many markers for theory failure. Missing the target, bad concepts... |
| 21514 | Coherence is the capacity to answer objections [Olsson] |
| Full Idea: According to Lehrer, coherence should be understood in terms of the capacity to answer objections. | |
| From: Erik J. Olsson (Against Coherence [2005], 9) | |
| A reaction: [Keith Lehrer 1990] We can connect this with the Greek requirement of being able to give an account [logos], which is the hallmark of understanding. I take coherence to be the best method of achieving understanding. Any understanding meets Lehrer's test. |
| 21496 | Mere agreement of testimonies is not enough to make truth very likely [Olsson] |
| Full Idea: Far from guaranteeing a high likelihood of truth by itself, testimonial agreement can apparently do so only if the circumstances are favourable as regards independence, prior probability, and individual credibility. | |
| From: Erik J. Olsson (Against Coherence [2005], 1) | |
| A reaction: This is Olson's main thesis. His targets are C.I.Lewis and Bonjour, who hoped that a mere consensus of evidence would increase verisimilitude. I don't see a problem for coherence in general, since his favourable circumstances are part of it. |
| 21499 | Coherence is only needed if the information sources are not fully reliable [Olsson] |
| Full Idea: An enquirer who is fortunate enough to have at his or her disposal fully reliable information sources has no use for coherence, the need for which arises only in the context of less than fully reliable informations sources. | |
| From: Erik J. Olsson (Against Coherence [2005], 2.6.2) | |
| A reaction: I take this to be entirely false. How do you assess reliability? 'I've seen it with my own eyes'. Why trust your eyes? In what visibility conditions do you begin to doubt your eyes? Why do rational people mistrust their intuitions? |
| 21502 | A purely coherent theory cannot be true of the world without some contact with the world [Olsson] |
| Full Idea: The Input Objection says a pure coherence theory would seem to allow that a system of beliefs be justified in spite of being utterly out of contact with the world it purports to describe, so long as it is, to a sufficient extent, coherent. | |
| From: Erik J. Olsson (Against Coherence [2005], 4.1) | |
| A reaction: Olson seems impressed by this objection, but I don't see how a system could be coherently about the world if it had no known contact with the world. Olson seems to ignore meta-coherence, which evaluates the status of the system being studied. |
| 21512 | Extending a system makes it less probable, so extending coherence can't make it more probable [Olsson] |
| Full Idea: Any non-trivial extension of a belief system is less probable than the original system, but there are extensions that are more coherent than the original system. Hence more coherence does not imply a higher probability. | |
| From: Erik J. Olsson (Against Coherence [2005], 6.4) | |
| A reaction: [Olson cites Klein and Warfield 1994; compressed] The example rightly says the extension could have high internal coherence, but not whether the extension is coherent with the system being extended. |
| 22098 | Socrates neglects the gap between knowing what is good and doing good [Kierkegaard, by Carlisle] |
| Full Idea: There is a fundamental weakness in Socrates, that he does not take into account the gap between knowing what is good and actually putting this into action. | |
| From: report of Søren Kierkegaard (The Concept of Dread (/Anxiety) [1844]) by Clare Carlisle - Kierkegaard: a guide for the perplexed 5 | |
| A reaction: This rejects Socrates's intellectualism about weakness of will. It is perhaps a better criticism that Aristotle's view that desires sometimes overcome the will. It is also the problem of motivation in Kantian deontology. Or utilitarianism. |
| 22096 | Anxiety is not a passing mood, but a response to human freedom [Kierkegaard, by Carlisle] |
| Full Idea: For Kierkegaard anxiety is not simply a mood or an emotion that certain people experience at certain times, but a basic response to freedom that is part of the human condition. | |
| From: report of Søren Kierkegaard (The Concept of Dread (/Anxiety) [1844]) by Clare Carlisle - Kierkegaard: a guide for the perplexed 5 | |
| A reaction: Outside of Christianity, this may be Kierkegaard's most influential idea - since existential individualism is floating around in the romantic movement. But the Byronic hero experiences a sort of anxiety. If you can't face anxiety, become a monk or nun. |
| 22097 | The ultimate in life is learning to be anxious in the right way [Kierkegaard] |
| Full Idea: Every human being must learn to be anxious in order that he might not perish either by never having been in anxiety or by succumbing in anxiety. Whoever has learned to be anxious in the right way has learnt the ultimate. | |
| From: Søren Kierkegaard (The Concept of Dread (/Anxiety) [1844], p.154), quoted by Clare Carlisle - Kierkegaard: a guide for the perplexed 5 | |
| A reaction: I think this is the most existentialist quotation I have found in Kierkegaard. It sounds circular. You must be in anxiety because otherwise you won't be able to cope with anxiety? I suppose anxiety is facing up to his concept of truth. |
| 21909 | Ultimate knowledge is being anxious in the right way [Kierkegaard] |
| Full Idea: Whoever learns to be anxious in the right way has learned the ultimate. | |
| From: Søren Kierkegaard (The Concept of Dread (/Anxiety) [1844], p.187), quoted by Alastair Hannay - Kierkegaard 06 | |
| A reaction: This shows us that Kierkegaard had a rather bizarre mental life which the rest of us have little chance of penetrating. I'll have a go at cataloguing my types of anxiety, but I'm not hopeful. |
| 20758 | Anxiety is staring into the yawning abyss of freedom [Kierkegaard] |
| Full Idea: One may liken anxiety to dizziness. He whose eyes chance to look down into a yawning abyss becomes dizzy. Anxiety is the dizziness of freedom which is when freedom gazes down into its own possibility, grasping at finiteness to sustain itself. | |
| From: Søren Kierkegaard (The Concept of Dread (/Anxiety) [1844], p.55), quoted by Kevin Aho - Existentialism: an introduction 6 'Moods' | |
| A reaction: Most of us rapidly retreat from the thought of the infinity of things we might choose. Choosing bizarrely merely to assert one's freedom is simple stupidity. |