39 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) |
| 16185 | Causality indicates which properties are real [Cartwright,N] |
| Full Idea: Causality is a clue to what properties are real. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], 9.3) | |
| A reaction: An interesting variant on the Shoemaker proposal that properties actually are causal. I'm not sure that there is anything more to causality that the expression in action of properties, which I take to be powers. Structures are not properties. |
| 16182 | Two main types of explanation are by causes, or by citing a theoretical framework [Cartwright,N] |
| Full Idea: In explaining a phenomenon one can cite the causes of that phenomenon; or one can set the phenomenon in a general theoretical framework. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], 4.1) | |
| A reaction: The thing is, you need to root an explanation in something taken as basic, and theoretical frameworks need further explanation, whereas causes seem to be basic. |
| 16184 | An explanation is a model that fits a theory and predicts the phenomenological laws [Cartwright,N] |
| Full Idea: To explain a phenomenon is to find a model that fits it into the basic framework of the theory and that thus allows us to derive analogues for the messy and complicated phenomenological laws that are true of it. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], 8.3) | |
| A reaction: This summarises the core of her view in this book. She is after models rather than laws, and the models are based on causes. |
| 16167 | Laws get the facts wrong, and explanation rests on improvements and qualifications of laws [Cartwright,N] |
| Full Idea: We explain by ceteris paribus laws, by composition of causes, and by approximations that improve on what the fundamental laws dictate. In all of these cases the fundamental laws patently do not get the facts right. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], Intro) | |
| A reaction: It is rather headline-grabbing to say in this case that laws do not get the facts right. If they were actually 'wrong' and 'lied', there wouldn't be much point in building explanations on them. |
| 16169 | Laws apply to separate domains, but real explanations apply to intersecting domains [Cartwright,N] |
| Full Idea: When different kinds of causes compose, we want to explain what happens in the intersection of different domains. But the laws we use are designed only to tell truly what happens in each domain separately. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], Intro) | |
| A reaction: Since presumably the laws are discovered through experiments which try to separate out a single domain, in those circumstances they actually are true, so they don't 'lie'. |
| 16176 | Covering-law explanation lets us explain storms by falling barometers [Cartwright,N] |
| Full Idea: Much criticism of the original covering-law model objects that it lets in too much. It seems we can explain Henry's failure to get pregnant by his taking birth control pills, and we can explain the storm by the falling barometer. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], 2.0) | |
| A reaction: I take these examples to show that true explanations must be largely causal in character. The physicality of causation is what matters, not 'laws'. I'd say the same of attempts to account for causation through counterfactuals. |
| 16177 | I disagree with the covering-law view that there is a law to cover every single case [Cartwright,N] |
| Full Idea: Covering-law theorists tend to think that nature is well-regulated; in the extreme, that there is a law to cover every case. I do not. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], 2.2) | |
| A reaction: The problem of coincidence is somewhere at the back of this thought. Innumerable events have their own explanations, but it is hard to explain their coincidence (see Aristotle's case of bumping into a friend in the market). |
| 16180 | You can't explain one quail's behaviour by just saying that all quails do it [Cartwright,N] |
| Full Idea: 'Why does that quail in the garden bob its head up and down in that funny way whenever it walks?' …'Because they all do'. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], 3.5) | |
| A reaction: She cites this as an old complaint against the covering-law model of explanation. It captures beautifully the basic error of the approach. We want to know 'why', rather than just have a description of the pattern. 'They all do' is useful information. |
| 16171 | The covering law view assumes that each phenomenon has a 'right' explanation [Cartwright,N] |
| Full Idea: The covering-law account supposes that there is, in principle, one 'right' explanation for each phenomenon. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], Intro) | |
| A reaction: Presumably the law is held to be 'right', but there must be a bit of flexibility in describing the initial conditions, and the explanandum itself. |
| 16183 | In science, best explanations have regularly turned out to be false [Cartwright,N] |
| Full Idea: There are a huge number of cases in the history of science where we now know our best explanations were false. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], 5.3) | |
| A reaction: [She cites Laudan 1981 for this] The Ptolemaic system and aether are the standard example cited for this. I believe strongly in the importance of best explanation. Only a fool would just accept the best explanation available. Coherence is needed. |
| 23110 | Human injustice is not a permanent feature of communities [Rawls] |
| Full Idea: Men's propensity to injustice is not a permanent aspect of community life. | |
| From: John Rawls (A Theory of Justice [1972], p.245), quoted by John Kekes - Against Liberalism | |
| A reaction: This attitude is dismissed by Kekes, with some justification, as naïve optimism. What could be Rawls's grounds for making such a claim? It couldn't be the facts of human history. |
| 15676 | Rawls defends the priority of right over good [Rawls, by Finlayson] |
| Full Idea: Rawls defends the thesis of the priority of the right over the good. | |
| From: report of John Rawls (A Theory of Justice [1972]) by James Gordon Finlayson - Habermas Ch.7:100 | |
| A reaction: It depends whether you are talking about actions, or about states of affairs. I don't see how any state of affairs can be preferred to the good one. It may be that the highest duty of action is to do what is right, rather than to achieve what is good. |
| 4123 | A fair arrangement is one that parties can agree to without knowing how it will benefit them personally [Rawls, by Williams,B] |
| Full Idea: Rawls's theory is an elaboration of a simple idea: a fair system of arrangements is one that the parties can agree to without knowing how it will benefit them personally. | |
| From: report of John Rawls (A Theory of Justice [1972]) by Bernard Williams - Ethics and the Limits of Philosophy Ch.5 | |
| A reaction: The essence of modern Kantian contractualism. It is an appealing principle for building a rational world, but I hear Nietzsche turning in his grave. |
| 3279 | Utilitarianism inappropriately scales up the individual willingness to make sacrifices [Rawls, by Nagel] |
| Full Idea: Rawls claims that utilitarianism applies to the problem of many interests a method appropriate for one individual. A single person may accept disadvantages in exchange for benefits, but in society other people get the benefits. | |
| From: report of John Rawls (A Theory of Justice [1972], p.74,104) by Thomas Nagel - Equality §7 |
| 3280 | Why does the rational agreement of the 'Original Position' in Rawls make it right? [Nagel on Rawls] |
| Full Idea: Why does what it is rational to agree to in Rawls' 'Original Position' determine what is right? | |
| From: comment on John Rawls (A Theory of Justice [1972]) by Thomas Nagel - Equality §7 |
| 20552 | The original position models the idea that citizens start as free and equal [Rawls, by Swift] |
| Full Idea: The original position is presented by Rawls as modelling the sense in which citizens are to be understood as free and equal. | |
| From: report of John Rawls (A Theory of Justice [1972]) by Adam Swift - Political Philosophy (3rd ed) 3 'Strikes' | |
| A reaction: In other words, Rawls's philosophy is not a demonstration of why we should be liberals, but a guidebook for how liberals should go about organising society. |
| 18636 | Choose justice principles in ignorance of your own social situation [Rawls] |
| Full Idea: The principles of justice are chosen behind a veil of ignorance. ...Since all are similarly situated and no one is able to design principles to favor his particular condition, the principles of justice are the rest of a fair agreement or bargain. | |
| From: John Rawls (A Theory of Justice [1972], §03) | |
| A reaction: A famous idea. It tries to impose a Kantian impartiality onto the assessment of political principles. It is a beautifully simple idea, and saying that such impartiality never occurs is no objection to it. Think of a planet far far away. |
| 18631 | All desirable social features should be equal, unless inequality favours the disadvantaged [Rawls] |
| Full Idea: All social primary goods - liberty and opportunity, income and wealth, and the bases of self-respect - are to be distributed equally unless an unequal distribution of any or all of these goods is to the advantage of the least favoured. | |
| From: John Rawls (A Theory of Justice [1972], §46) | |
| A reaction: In the wholehearted capitalism of the 21st century this sounds like cloud-cuckoo land. As an 'initial position' (just as in the 'Republic') the clean slate brings out some interesting principles. Actual politics takes vested interests as axiomatic. |
| 20538 | Utilitarians lump persons together; Rawls somewhat separates them; Nozick wholly separates them [Swift on Rawls] |
| Full Idea: Rawls objects to utilitarianism because it fails to take seriously the separateness of persons (because there is no overall person to enjoy the overall happiness). But Nozick thinks Rawls does not take the separateness of persons seriously enough. | |
| From: comment on John Rawls (A Theory of Justice [1972]) by Adam Swift - Political Philosophy (3rd ed) 1 'Nozick' | |
| A reaction: In this sense, Nozick seems to fit our picture of a liberal more closely than Rawls does. I think they both exaggerate the separateness of persons, based on a false concept of human nature. |
| 9277 | Rawls's account of justice relies on conventional fairness, avoiding all moral controversy [Gray on Rawls] |
| Full Idea: Rawls's account of justice works only with widely accepted intuitions of fairness and relies at no point on controversial positions in ethics. The fruit of this modesty is a pious commentary on conventional moral beliefs. | |
| From: comment on John Rawls (A Theory of Justice [1972]) by John Gray - Straw Dogs 3.6 | |
| A reaction: Presumably this is the thought which provoked Nozick to lob his grenade on the subject. It resembles the charges of Schopenhauer and Nietzsche against Kant, that he was just dressing up conventional morality. Are 'controversial' ethics good? |
| 20527 | Liberty Principle: everyone has an equal right to liberties, if compatible with others' liberties [Rawls] |
| Full Idea: First Principle [Liberty]: Each person is to have an equal right to the most extensive total system of equal basic liberties compatible with a similar system of liberty for all. | |
| From: John Rawls (A Theory of Justice [1972], 46) | |
| A reaction: This is the result of consensus after the initial ignorant position of assessment. It is characteristic of liberalism. I'm struggling to think of a disagreement. |
| 21018 | The social contract has problems with future generations, national boundaries, disabilities and animals [Rawls, by Nussbaum] |
| Full Idea: Rawls saw four difficulties for justice in the social contract approach: future generations; justice across national boundaries; fair treatment of people with disabilities; and moral issues involving non-human animals. | |
| From: report of John Rawls (A Theory of Justice [1972]) by Martha Nussbaum - Creating Capabilities 4 | |
| A reaction: These are all classic examples of groups who do not have sufficient power to negotiate contracts. |
| 21041 | Justice concerns not natural distributions, or our born location, but what we do about them [Rawls] |
| Full Idea: The natural distribution is neither just nor unjust; nor is it unjust that persons are born into society at some particular position. These are simply natural facts. What is just and unjust is the way that institutions deal with these facts. | |
| From: John Rawls (A Theory of Justice [1972], 17) | |
| A reaction: Lovely quotation. There is no point in railing against the given, and that includes what is given by history, as well as what is given by nature. It comes down to intervening, in history and in nature. How much intervention will individuals tolerate? |
| 23583 | If an aggression is unjust, the constraints on how it is fought are much stricter [Rawls] |
| Full Idea: When a country's right to war is questionable and uncertain, the constraints on the means it can use are all the more severe. | |
| From: John Rawls (A Theory of Justice [1972], p.379), quoted by Michael Walzer - Just and Unjust Wars 14 | |
| A reaction: This is Rawls opposing the idea that combatants are moral equals. The restraints are, of course, moral. In practice aggressors are usually the worst behaved. |
| 16175 | A cause won't increase the effect frequency if other causes keep interfering [Cartwright,N] |
| Full Idea: A cause ought to increase the frequency of the effect, but this fact may not show up in the probabilities if other causes are at work. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], 1.1) | |
| A reaction: [She cites Patrick Suppes for this one] Presumably in experimental situations you can weed out the interference, but that threatens to eliminate mere 'probability' entirely. |
| 6781 | There are fundamental explanatory laws (false!), and phenomenological laws (regularities) [Cartwright,N, by Bird] |
| Full Idea: Nancy Cartwright distinguishes between 'fundamental explanatory laws', which we should not believe, and 'phenomenological laws', which are regularities established on the basis of observation. | |
| From: report of Nancy Cartwright (How the Laws of Physics Lie [1983]) by Alexander Bird - Philosophy of Science Ch.4 | |
| A reaction: The distinction is helpful, so that we can be clearer about what everyone is claiming. We can probably all agree on the phenomenological laws, which are epistemological. Personally I claim truth for the best fundamental explanatory laws. |
| 16166 | Laws of appearances are 'phenomenological'; laws of reality are 'theoretical' [Cartwright,N] |
| Full Idea: Philosophers distinguish phenomenological from theoretical laws. Phenomenological laws are about appearances; theoretical ones are about the reality behind the appearances. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], Intro) | |
| A reaction: I'm suspecting that Humeans only really believe in the phenomenological kind. I'm only interested in the theoretical kind, and I take inference to the best explanation to be the bridge between the two. Cartwright rejects the theoretical laws. |
| 16179 | Good organisation may not be true, and the truth may not organise very much [Cartwright,N] |
| Full Idea: There is no reason to think that the principles that best organise will be true, nor that the principles that are true will organise much. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], 2.5) | |
| A reaction: This is aimed at the Mill-Ramsey-Lewis account of laws, as axiomatisations of the observed patterns in nature. |
| 16170 | To get from facts to equations, we need a prepared descriptions suited to mathematics [Cartwright,N] |
| Full Idea: To get from a detailed factual knowledge of a situation to an equation, we must prepare the description of the situation to meet the mathematical needs of the theory. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], Intro) | |
| A reaction: She is clearly on to something here, as Galileo is blatantly wrong in his claim that the book of nature is written in mathematics. Mathematics is the best we can manage in getting a grip on the chaos. |
| 16181 | Simple laws have quite different outcomes when they act in combinations [Cartwright,N] |
| Full Idea: For explanation simple laws must have the same form when they act together as when they act singly. ..But then what the law states cannot literally be true, for the consequences that occur if it acts alone are not what occurs when they act in combination. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], 3.6) | |
| A reaction: This is Cartwright's basic thesis. Her point is that the laws 'lie', because they claim to predict a particular outcome which never ever actually occurs. She says we could know all the laws, and still not be able to explain anything. |
| 16178 | There are few laws for when one theory meets another [Cartwright,N] |
| Full Idea: Where theories intersect, laws are usually hard to come by. | |
| From: Nancy Cartwright (How the Laws of Physics Lie [1983], 2.3) | |
| A reaction: There are attempts at so-called 'bridge laws', to get from complex theories to simple ones, but her point is well made about theories on the same 'level'. |