56 ideas
| 15945 | Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine] |
| Full Idea: Second-order set theory is just like first-order set-theory, except that we use the version of Replacement with a universal second-order quantifier over functions from set to sets. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VII.4) |
| 15914 | An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine] |
| Full Idea: A member m of M is an 'upper bound' of a subset N of M if m is not less than any member of N. A member m of M is a 'least upper bound' of N if m is an upper bound of N such that if l is any other upper bound of N, then m is less than l. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.4) | |
| A reaction: [if you don't follow that, you'll have to keep rereading it till you do] |
| 15921 | Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine] |
| Full Idea: Since combinatorial collections are enumerated, some multiplicities may be too large to be gathered into combinatorial collections. But the size of a multiplicity seems quite irrelevant to whether it forms a logical connection. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], IV.2) |
| 15937 | Those who reject infinite collections also want to reject the Axiom of Choice [Lavine] |
| Full Idea: Many of those who are skeptical about the existence of infinite combinatorial collections would want to doubt or deny the Axiom of Choice. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VI.2) |
| 15936 | The Power Set is just the collection of functions from one collection to another [Lavine] |
| Full Idea: The Power Set is just he codification of the fact that the collection of functions from a mathematical collection to a mathematical collection is itself a mathematical collection that can serve as a domain of mathematical study. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VI.1) |
| 15899 | Replacement was immediately accepted, despite having very few implications [Lavine] |
| Full Idea: The Axiom of Replacement (of Skolem and Fraenkel) was remarkable for its universal acceptance, though it seemed to have no consequences except for the properties of the higher reaches of the Cantorian infinite. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], I) |
| 15930 | Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine] |
| Full Idea: The Axiom of Foundation (Zermelo 1930) says 'Every (descending) chain in which each element is a member of the previous one is of finite length'. ..This forbids circles of membership, or ungrounded sets. ..The iterative conception gives this centre stage. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], V.4) |
| 15920 | Pure collections of things obey Choice, but collections defined by a rule may not [Lavine] |
| Full Idea: Combinatorial collections (defined just by the members) obviously obey the Axiom of Choice, while it is at best dubious whether logical connections (defined by a rule) do. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], IV.2) |
| 15898 | The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine] |
| Full Idea: The controversy was not about Choice per se, but about the correct notion of function - between advocates of taking mathematics to be about arbitrary functions and advocates of taking it to be about functions given by rules. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], I) |
| 15919 | The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine] |
| Full Idea: The Peano-Russell notion of class is the 'logical' notion, where each collection is associated with some kind of definition or rule that characterises the members of the collection. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], IV.1) |
| 15900 | The iterative conception of set wasn't suggested until 1947 [Lavine] |
| Full Idea: The iterative conception of set was not so much as suggested, let alone advocated by anyone, until 1947. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], I) |
| 15931 | The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine] |
| Full Idea: The iterative conception of sets does not tell us how far to iterate, and so we must start with an Axiom of Infinity. It also presupposes the notion of 'transfinite iteration'. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], V.5) |
| 15932 | The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine] |
| Full Idea: The iterative conception does not provide a conception that unifies the axioms of set theory, ...and it has had very little impact on what theorems can be proved. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], V.5) | |
| A reaction: He says he would like to reject the iterative conception, but it may turn out that Foundation enables new proofs in mathematics (though it hasn't so far). |
| 15933 | Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine] |
| Full Idea: Limitation of Size has it that if a collection is the same size as a set, then it is a set. The Axiom of Replacement is characteristic of limitation of size. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], V.5) |
| 15913 | A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine] |
| Full Idea: A collection M is 'well-ordered' by a relation < if < linearly orders M with a least element, and every subset of M that has an upper bound not in it has an immediate successor. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.4) |
| 15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine] |
| Full Idea: The distinctive feature of second-order logic is that it presupposes that, given a domain, there is a fact of the matter about what the relations on it are, so that the range of the second-order quantifiers is fixed as soon as the domain is fixed. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], V.3) | |
| A reaction: This sounds like a rather large assumption, which is open to challenge. I am not sure whether it was the basis of Quine's challenge to second-order logic. He seems to have disliked its vagueness, because it didn't stick with 'objects'. |
| 15934 | Mathematical proof by contradiction needs the law of excluded middle [Lavine] |
| Full Idea: The Law of Excluded Middle is (part of) the foundation of the mathematical practice of employing proofs by contradiction. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VI.1) | |
| A reaction: This applies in a lot of logic, as well as in mathematics. Come to think of it, it applies in Sudoku. |
| 15907 | Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine] |
| Full Idea: Mathematics is today thought of as the study of abstract structure, not the study of quantity. That point of view arose directly out of the development of the set-theoretic notion of abstract structure. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.2) | |
| A reaction: It sounds as if Structuralism, which is a controversial view in philosophy, is a fait accompli among mathematicians. |
| 15942 | Every rational number, unlike every natural number, is divisible by some other number [Lavine] |
| Full Idea: One reason to introduce the rational numbers is that it simplifes the theory of division, since every rational number is divisible by every nonzero rational number, while the analogous statement is false for the natural numbers. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VI.3) | |
| A reaction: That is, with rations every division operation has an answer. |
| 15922 | For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine] |
| Full Idea: The chief importance of the Continuum Hypothesis for Cantor (I believe) was that it would show that the real numbers form a set, and hence that they were encompassed by his theory. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], IV.2) |
| 18250 | Cauchy gave a necessary condition for the convergence of a sequence [Lavine] |
| Full Idea: The Cauchy convergence criterion for a sequence: the sequence S0,S1,... has a limit if |S(n+r) - S(n)| is less than any given quantity for every value of r and sufficiently large values of n. He proved this necessary, but not sufficient. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], 2.5) |
| 15904 | The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine] |
| Full Idea: Roughly speaking, the upper and lower parts of the Dedekind cut correspond to the commensurable ratios greater than and less than a given incommensurable ratio. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], II.6) | |
| A reaction: Thus there is the problem of whether the contents of the gap are one unique thing, or many. |
| 15912 | Counting results in well-ordering, and well-ordering makes counting possible [Lavine] |
| Full Idea: Counting a set produces a well-ordering of it. Conversely, if one has a well-ordering of a set, one can count it by following the well-ordering. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.4) | |
| A reaction: Cantor didn't mean that you could literally count the set, only in principle. |
| 15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine] |
| Full Idea: The indiscernibility of indefinitely large sizes will be a critical part of the theory of indefinitely large sizes. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VIII.2) |
| 15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
| Full Idea: My proposal is that the concept of the infinite began with an extrapolation from the experience of indefinitely large size. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VIII.2) | |
| A reaction: I think it might be better to talk of an 'abstraction' than an 'extrapolition', since the latter is just more of the same, which doesn't get you to concept. Lavine spends 100 pages working out his proposal. |
| 15940 | The intuitionist endorses only the potential infinite [Lavine] |
| Full Idea: The intuitionist endorse the actual finite, but only the potential infinite. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VI.2) |
| 15909 | 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine] |
| Full Idea: The symbol 'aleph-nought' denotes the cardinal number of the set of natural numbers. The symbol 'aleph-one' denotes the next larger cardinal number. 'Aleph-omega' denotes the omega-th cardinal number. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.3) |
| 15915 | Ordinals are basic to Cantor's transfinite, to count the sets [Lavine] |
| Full Idea: The ordinals are basic because the transfinite sets are those that can be counted, or (equivalently for Cantor), those that can be numbered by an ordinal or are well-ordered. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.4) | |
| A reaction: Lavine observes (p.55) that for Cantor 'countable' meant 'countable by God'! |
| 15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine] |
| Full Idea: The paradox of the largest ordinal (the 'Burali-Forti') is that the class of all ordinal numbers is apparently well-ordered, and so it has an ordinal number as order type, which must be the largest ordinal - but all ordinals can be increased by one. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.5) |
| 15918 | Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine] |
| Full Idea: The paradox of the largest cardinal ('Cantor's Paradox') says the diagonal argument shows there is no largest cardinal, but the class of all individuals (including the classes) must be the largest cardinal number. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.5) |
| 15929 | Set theory will found all of mathematics - except for the notion of proof [Lavine] |
| Full Idea: Every theorem of mathematics has a counterpart with set theory - ...but that theory cannot serve as a basis for the notion of proof. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], V.3) |
| 15935 | Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine] |
| Full Idea: In modern mathematics virtually all work is only up to isomorphism and no one cares what the numbers or points and lines 'really are'. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VI.1) | |
| A reaction: At least that leaves the field open for philosophers, because we do care what things really are. So should everybody else, but there is no persuading some people. |
| 15928 | Intuitionism rejects set-theory to found mathematics [Lavine] |
| Full Idea: Intuitionism in philosophy of mathematics rejects set-theoretic foundations. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], V.3 n33) |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
| Full Idea: The six perfections are of giving, morality, patience, vigour, meditation, and wisdom. | |
| From: Nagarjuna (Mahaprajnaparamitashastra [c.120], 88) | |
| A reaction: What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate'). |
| 23369 | Some liberals thinks checks and balances are enough, without virtuous citizens [Kymlicka] |
| Full Idea: Many classical liberals believed that a liberal democracy could function effectively even in the absence of an especially virtuous citizenry, by creating checks and balances. …One set of private interests would check another set of private interests. | |
| From: Will Kymlicka (Contemporary Political Philosophy (2nd edn) [2002], 7) | |
| A reaction: This seems to be the view of those who think a completely free market will evolve into a flourishing and just society. There is a basic debate about the importance of the character of the citizens in any polity. Marxists say they are entangled. |
| 23370 | Good citizens need civic virtues of loyalty, independence, diligence, respect, etc. [Kymlicka] |
| Full Idea: Galston says responsible citizenship requires four types of civic virtue: general (law-abiding, loyal), social (independent, open-minded), economic (diligent, restrained, adaptable), and political (respect, sensible, judgement, engagement). | |
| From: Will Kymlicka (Contemporary Political Philosophy (2nd edn) [2002], 7) | |
| A reaction: [Galston's 'Liberal Purposes' 1991] (compressed) This immediately seems to be asking too much, especially for those who know little, or are short of money. |
| 23373 | Liberals accept that people need society, but Aristotelians must show that they need political activity [Kymlicka] |
| Full Idea: To defend Aristotelian republicanism it is not enough to show that individual require society - liberals do not deny this. They must also show that individuals need to be politically active. | |
| From: Will Kymlicka (Contemporary Political Philosophy (2nd edn) [2002], 7) | |
| A reaction: Interesting. People are not just inactive because they have been rendered powerless. In any group of people there are some who are keen to have a voice, or lead, and others who are largely happy to follow. |
| 23375 | Minimal liberal citizenship needs common civility, as well as mere non-interference [Kymlicka] |
| Full Idea: Minimal citizenship is often seen as simply requiring non-interference with others, but that ignores a basic requirement of liberal citizenship, which is the social virtue of 'civility' or 'decency'. | |
| From: Will Kymlicka (Contemporary Political Philosophy (2nd edn) [2002], 7) | |
| A reaction: He makes the point that the minimal requirement has to be given up when there is a crisis, which needs much more involvement. This largely describes modern Britain, prior to the Brexit rift. |
| 23376 | Modern non-discrimination obliges modern citizens to treat each other as equals [Kymlicka] |
| Full Idea: The extension of non-discrimination from government to civil society …involves a radical extension of the obligations of liberal citizenship. The obligation to treat people as equal citizens now applies to everyday decisions. | |
| From: Will Kymlicka (Contemporary Political Philosophy (2nd edn) [2002], 7) | |
| A reaction: This is very difficult for an older generation who felt their 'entitlement' as leading citizens, or who routinely favoured their local traditional community. But they just have to 'get over it'! |
| 23377 | The right wing sees citizenship in terms of responsibility to earn a living, rather than rights [Kymlicka] |
| Full Idea: According to the New Right, to promote active citizenship-for-all or entitlements, we must focus instead on people's responsibility to earn a living. | |
| From: Will Kymlicka (Contemporary Political Philosophy (2nd edn) [2002], 7) | |
| A reaction: Every creature has to earn a living, but one method is to successfully sponge off others. A cushy job is a sort of sponging. An excessively well paid job is a sort of sponging. Citizenship must involve responsibilities of some sort. |
| 23371 | Modern democratic theory focuses on talk, not votes, because we need consensus or compromise [Kymlicka] |
| Full Idea: Modern discussion has shifted from 'vote-centric' (or 'aggregative') to 'talk-centric' democracy. The vote-centric model has no mechanism for developing a consensus, or shaping public opinion, or even formulating an honourable compromise. | |
| From: Will Kymlicka (Contemporary Political Philosophy (2nd edn) [2002], 7) | |
| A reaction: I'm struck by the fact that a person's preferences betweent these two is a reflection of character, or basic attitudes to morality. Some people think democratically about their relationships, and others very obviously don't. |
| 23390 | In a liberal democracy all subjects of authority have a right to determine the authority [Kymlicka] |
| Full Idea: A liberal-democratic system is one in which those people who are subject to political authority have a right to participate in determining that authority. | |
| From: Will Kymlicka (Contemporary Political Philosophy (2nd edn) [2002], 8.4) | |
| A reaction: This applies to immigrants. The most anti-democratic move in recent democracies is the strategy of trying to make it more difficult to vote, perhaps by demanding identification documents, or creating huge queues. |
| 23374 | We have become attached to private life because that has become greatly enriched [Kymlicka] |
| Full Idea: Our attachment to private life, I believe, is the result not (or not only) of the impoverishment of public life, but the enrichment of private life. | |
| From: Will Kymlicka (Contemporary Political Philosophy (2nd edn) [2002], 7) | |
| A reaction: Interesting. Perhaps a sentiment expected more from a university lecturer than from a poorly-paid labourer. Does he mean watching innumerable TV shows instead of having sing-songs in the local pub? Increased leisure is indisputable. |
| 23387 | Liberals must avoid an official culture, as well as an official religion [Kymlicka] |
| Full Idea: Just as liberalism precludes the establishment of an official religion, so too there cannot be official cultures that have preferred status. | |
| From: Will Kymlicka (Contemporary Political Philosophy (2nd edn) [2002], 8.3) | |
| A reaction: This becomes tricky in schools, where the old way of teaching national literature and particular types of music has been eroded in modern times. But once wide diversity is allowed there is no single story which can be taught. |
| 23388 | Liberals need more than freedom; they must build a nation, through a language and institutions [Kymlicka] |
| Full Idea: Liberals need to replace the idea of 'benign neglect', and recognise the central role of nation-building in a democracy. …This means promoting a common language, and equal access to institutions operating in that language. | |
| From: Will Kymlicka (Contemporary Political Philosophy (2nd edn) [2002], 8.3) | |
| A reaction: 'Benign neglect' is non-interference with citizens' lives. Obviously the institutions include education, but is a state health service implied? Can equal access by guaranteed to private institutions? |
| 23380 | Some individuals can gain citizenship as part of a group, rather than as mere individuals [Kymlicka] |
| Full Idea: On the view of 'differentiated citizenship', members of certain groups would be incorporated into the community, not only as individuals, but also through the group, and their rights would depend in part on their group membership. | |
| From: Will Kymlicka (Contemporary Political Philosophy (2nd edn) [2002], 8) | |
| A reaction: This is obviously a strategy to enable marginalised individuals to be fully included in society. The downside is that individuals gain their social identity through a label, rather than through themselves, which pure liberals dislike. 'Identity politics'. |
| 23381 | The status hierarchy is independent of the economic hierarchy [Kymlicka] |
| Full Idea: The evidence suggests that (contrary to the Marxist view) the status hierarchy is not reducible to the economic hierarchy. | |
| From: Will Kymlicka (Contemporary Political Philosophy (2nd edn) [2002], 8) | |
| A reaction: Kymlicka is particularly thinking of racism, which lowers the status of certain groups, even if they are economically successful. I console myself for my modest economic status by getting lots of education. |
| 23383 | Some multiculturalists defended the rights of cohesive minorities against liberal individualism [Kymlicka] |
| Full Idea: Defending multiculturalism initially involved endorsing the communitarian critique of liberalism, and viewed minority rights as defending cohesive minority groups against the encroachment of liberal individualism. | |
| From: Will Kymlicka (Contemporary Political Philosophy (2nd edn) [2002], 8.1) | |
| A reaction: Liberal individualists have to accept these criticisms from Marxists, communitarians and multiculturalists. The lone individual has no group that guarantees support, and individuals (especially the young) can easily sink. |
| 23384 | 'Culturalist' liberals say that even liberal individuals may need minority rights [Kymlicka] |
| Full Idea: The 'liberal culturalist' position is that minorities which share basic liberal principles nonetheless need minority rights. | |
| From: Will Kymlicka (Contemporary Political Philosophy (2nd edn) [2002], 8.2) | |
| A reaction: Disabled liberals are an obvious example. This strikes me as a promising version of liberalism, which accepts the criticisms of extreme individualism. |
| 23385 | Multiculturalism may entail men dominating women in minority groups [Kymlicka] |
| Full Idea: Many feminists express concern that multiculturalism in practice typically means giving male members of the group the power to control the women in the group. | |
| From: Will Kymlicka (Contemporary Political Philosophy (2nd edn) [2002], 8.2) | |
| A reaction: The way the young are treated might also be a problem. The underlying question is whether the minority group is more or less civilised than the central state. Liberalism always fights for the rights of the least powerful. |
| 23386 | Liberals must prefer minority right which are freedoms, not restrictions [Kymlicka] |
| Full Idea: Liberal defenders of multiculturalism must distinguish 'bad' minority rights which are restrictions from 'good' minority rights which supplement individual rights. | |
| From: Will Kymlicka (Contemporary Political Philosophy (2nd edn) [2002], 8.2) | |
| A reaction: Presumably no sensible liberal wants to remove all restrictions, so deeper values must be invoked to justify the mode of approved minority rights. A list of human goods seems needed. |
| 23389 | Why shouldn't national minorities have their own right to nation-build? [Kymlicka] |
| Full Idea: Why should national minorities not have the same powers of nation-building as the majority? | |
| From: Will Kymlicka (Contemporary Political Philosophy (2nd edn) [2002], 8.4) | |
| A reaction: A 'national minority' is marked by a different language, or a different religion, or both. No one doubts the majority's right to nation-build. Some further principle would be needed to deny that right to a minority. Maybe the minority was there first? |
| 23391 | Multiculturalism is liberal if it challenges inequality, conservative if it emphasises common good [Kymlicka] |
| Full Idea: Liberal multiculturalism challenges status inequalities while preserving individual freedom. …Conservative multiculturalism replaces liberal principles with a communitarian politics of the common good, at least at the local or group level. | |
| From: Will Kymlicka (Contemporary Political Philosophy (2nd edn) [2002], 8.6) | |
| A reaction: [compressed] This sounds a bit simplistic. Recent emphasis on 'the common good', in the face of white supremacists etc., seems admirable, but surely challenging inequalities promotes the common good? Minority cultures are often conservative. |
| 23379 | Rights are a part of nation-building, to build a common national identity and culture [Kymlicka] |
| Full Idea: Extending citizenship to include common social rights was a tool of nation-building, intended in part to construct and consolidate a sense of common national identity and culture. | |
| From: Will Kymlicka (Contemporary Political Philosophy (2nd edn) [2002], 8) | |
| A reaction: Kymlicka explains a lot of politics and society in terms of the desire of governments to 'build' their nation. You have to make people who are essentially powerless feel that they are at least in some way involved, and benefiting. |
| 23382 | Rights derived from group membership are opposed to the idea of state citizenship [Kymlicka] |
| Full Idea: The organisation of society on the basis of rights or claims that derive from group membership is sharply opposed to the concept of society based on citizenship. | |
| From: Will Kymlicka (Contemporary Political Philosophy (2nd edn) [2002], 8) | |
| A reaction: [from John Porter 1987] Does this imply that you might have rights as part of a group which you don't have as a state citizen? Positive discrimination, for example. Differential rights sounds like potential trouble. |
| 23378 | The welfare state helps to integrate the working classes into a national culture [Kymlicka] |
| Full Idea: The development of the welfare state has been quite successful in integrating the working classes into national cultures throughout the Western democracies. | |
| From: Will Kymlicka (Contemporary Political Philosophy (2nd edn) [2002], 8) | |
| A reaction: Hard-line capitalists tend to hate the welfare state, as unfair to high earners, but it not only makes workers feel involved, but also provides a healthier, happier, more knowledgeable work force for employers. |