79 ideas
| 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] |
| 16185 | Causality indicates which properties are real [Cartwright,N] |
| 16182 | Two main types of explanation are by causes, or by citing a theoretical framework [Cartwright,N] |
| 16184 | An explanation is a model that fits a theory and predicts the phenomenological laws [Cartwright,N] |
| 16167 | Laws get the facts wrong, and explanation rests on improvements and qualifications of laws [Cartwright,N] |
| 16169 | Laws apply to separate domains, but real explanations apply to intersecting domains [Cartwright,N] |
| 16176 | Covering-law explanation lets us explain storms by falling barometers [Cartwright,N] |
| 16177 | I disagree with the covering-law view that there is a law to cover every single case [Cartwright,N] |
| 16180 | You can't explain one quail's behaviour by just saying that all quails do it [Cartwright,N] |
| 16171 | The covering law view assumes that each phenomenon has a 'right' explanation [Cartwright,N] |
| 16183 | In science, best explanations have regularly turned out to be false [Cartwright,N] |
| 18658 | The 'Kantian' self steps back from commitment to its social situation [Kymlicka] |
| 18650 | Teleological theories give the good priority over concern for people [Kymlicka] |
| 18664 | Maybe the particularist moral thought of women is better than the impartial public thinking of men [Kymlicka] |
| 18624 | Utilitarianism is not a decision-procedure; choice of the best procedure is an open question [Kymlicka] |
| 18626 | One view says start with equality, and infer equal weight to interests, and hence maximum utility [Kymlicka] |
| 18627 | A second view says start with maximising the good, implying aggregation, and hence equality [Kymlicka] |
| 18625 | To maximise utility should we double the population, even if life somewhat deteriorates? [Kymlicka] |
| 18638 | The difference principles says we must subsidise the costs of other people's choices [Kymlicka] |
| 18635 | Social contract theories are usually rejected because there never was such a contract [Kymlicka] |
| 18630 | Utilitarianism is no longer a distinctive political position [Kymlicka] |
| 18623 | The quest of the general good is partly undermined by people's past entitlements [Kymlicka] |
| 18628 | We shouldn't endorse preferences which reject equality, and show prejudice and selfishness [Kymlicka] |
| 18629 | Using utilitarian principles to make decisions encourages cold detachment from people [Kymlicka] |
| 18637 | Utilitarianism is irrational if it tells you to trade in your rights and resources just for benefits [Kymlicka] |
| 18663 | Modern liberalism has added personal privacy to our personal social lives [Kymlicka] |
| 18632 | Liberalism tends to give priority to basic liberties [Kymlicka] |
| 18656 | Marxists say liberalism is unjust, because it allows exploitation in the sale of labour [Kymlicka] |
| 18659 | The 'Kantian' view of the self misses the way it is embedded or situated in society [Kymlicka] |
| 18660 | Communitarians say we should pay more attention to our history [Kymlicka] |
| 18657 | Communitarian states only encourage fairly orthodox ideas of the good life [Kymlicka] |
| 18649 | If everyone owned himself, that would prevent slavery [Kymlicka] |
| 18640 | Libertarians like the free market, but they also think that the free market is just [Kymlicka] |
| 18651 | The most valuable liberties to us need not be the ones with the most freedom [Kymlicka] |
| 18661 | Ancient freedom was free participation in politics, not private independence of life [Kymlicka] |
| 18633 | Equal opportunities seems fair, because your fate is from your choices, not your circumstances [Kymlicka] |
| 18634 | Equal opportunity arbitrarily worries about social circumstances, but ignores talents [Kymlicka] |
| 18654 | Marxists say justice is unneeded in the truly good community [Kymlicka] |
| 18652 | The Lockean view of freedom depends on whether you had a right to what is restricted [Kymlicka] |
| 18655 | Justice corrects social faults, but also expresses respect to individuals as ends [Kymlicka] |
| 16175 | A cause won't increase the effect frequency if other causes keep interfering [Cartwright,N] |
| 6781 | There are fundamental explanatory laws (false!), and phenomenological laws (regularities) [Cartwright,N, by Bird] |
| 16166 | Laws of appearances are 'phenomenological'; laws of reality are 'theoretical' [Cartwright,N] |
| 16179 | Good organisation may not be true, and the truth may not organise very much [Cartwright,N] |
| 16170 | To get from facts to equations, we need a prepared descriptions suited to mathematics [Cartwright,N] |
| 16181 | Simple laws have quite different outcomes when they act in combinations [Cartwright,N] |
| 16178 | There are few laws for when one theory meets another [Cartwright,N] |