51 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) |
| 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) |
| 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) |
| 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) |
| 21122 | Liberal Nationalism says welfare states and democracy needed a shared sense of nationality [Shorten] |
| Full Idea: The Liberal Nationalist argument is that if we want to have welfare states or vibrant democracies, then we will need the kind of solidarity that shared nationality fosters. …Unwelcome democratic decisions are more acceptable when made by co-nationals. | |
| From: Andrew Shorten (Contemporary Political Theory [2016], 02) | |
| A reaction: We've just experienced this with Brexit (2016), where perfectly sensible decisions were being made in Brussels, but the popular press whipped up hostility because the British had a restricted role in the decisions. Prefer our idiots to their sages. |
| 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. |
| 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. |
| 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) |
| 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) |
| 21136 | Utilitarians conflate acts and omissions; causing to drown and failing to save are the same [Shorten] |
| Full Idea: Most uitlitarians do not distinguish between acts and omissions, and see no morally relevant difference between walking past a drowning child and pushing a child into a pond. | |
| From: Andrew Shorten (Contemporary Political Theory [2016], 09) | |
| A reaction: He cites Peter Singer as an instance. The notorious Trolley Problem focuses on such issues. Michael Sandel in 'Justice' is good on that. If motive and intention matter, the two cases could be very different. Too timid to push, but also too timid to help? |
| 5954 | All inventions of the mind aim at pleasure, and those that don't are worthless [Metrodorus of Lamp., by Plutarch] |
| Full Idea: Metrodorus says that all the wonderful, ingenious and brilliant inventions of the mind have been contrived for the sake of pleasure of the flesh or for the sake of looking forward to it, and any accomplishment not leading to this end is worthless. | |
| From: report of Metrodorus (Lamp) (fragments/reports [c.291 BCE], Fr 6) by Plutarch - 74: Reply to Colotes §1125 | |
| A reaction: It is very hard to think of counterexamples! Would anyone bother to work out the theorems of number theory if they didn't enjoy doing it? Would any sensible person make great sacrifices if they didn't think that increased happiness would result? |
| 21135 | There are eight different ways in which groups of people can be oppressed [Shorten, by PG] |
| Full Idea: Groups can be oppressed in seven different ways: by violence, marginalisation, powerlessness, cultural domination, exploitation, stigmatisation, neglect of interests, and lack of egalitarian ethos. | |
| From: report of Andrew Shorten (Contemporary Political Theory [2016], 08) by PG - Db (ideas) | |
| A reaction: [my summary of Shorten's summary] These headings seem to overlap somewhat. It strengthens my growing view that if one builds a political philosophy around the supreme virtue of respect, then all of these modes of oppression are undermined. |
| 21118 | Constitutional Patriotism unites around political values (rather than national identity) [Shorten] |
| Full Idea: 'Constitutional patriots' favour a 'post-national' form of political identity in which members share common political values, but not necessarily a common national identity. | |
| From: Andrew Shorten (Contemporary Political Theory [2016], 02) | |
| A reaction: Interesting. Not sure if you can keep political values distinct from community values. In theory it is an approach designed for cultural pluralism. But if the political values are liberal that implies cultural freedoms for (e.g.) women. |
| 21129 | Democracy is a method of selection, or it involves participation, or it concerns public discussion [Shorten] |
| Full Idea: Competitive democrats believe that democracy is simply a method for selecting political leaders …Participatory democrats associate the democratic ideal with living in a participatory society …Deliberative democrats identify public reasoning as key. | |
| From: Andrew Shorten (Contemporary Political Theory [2016], 05) | |
| A reaction: Personally I would favour public discussion, but that is the last thing leaders want, especially if they are not very knowledgeable or clever. |
| 21130 | Some say democracy is intrinsically valuable, others that it delivers good outcomes [Shorten] |
| Full Idea: Some theorist think that democracy is intrinsically valuable, but others believe that it is valuable because it delivers good outcomes. | |
| From: Andrew Shorten (Contemporary Political Theory [2016], 05) | |
| A reaction: It is hard to see how the majority having a dictatorship over the minority could be an intrinsic good. If we start with respect as the supreme social virtue, then participation and public discussion might be intrinsic goods. |
| 21126 | Representative should be either obedient, or sensible, or typical [Shorten] |
| Full Idea: Mandate Representation says they are delegates who should not deviate from instructions; Trustee says they use their discretion and judgement; Descriptive says they share group characteristics. | |
| From: Andrew Shorten (Contemporary Political Theory [2016], 04) | |
| A reaction: [compressed] There is also being a representative because you have an audience (such as celebrity campains). The second type was famously defended by Edmund Burke. The third implies being the same colour, or gender, or religion. |
| 21128 | There is 'mirror representation' when the institution statistically reflects the population [Shorten] |
| Full Idea: The general theory of 'mirror representation' says that a representative body or institution should be a statistically accurate sample of the wider society it represents. | |
| From: Andrew Shorten (Contemporary Political Theory [2016], 04) | |
| A reaction: How fine-grained should this be in accuracy. Should every small minority have at least one rep? Can't reps be trusted to speak for people a bit different from themselves? Maybe not! He quotes Mirabeau in support of this idea. |
| 21127 | In a changed situation a Mandated Representative can't keep promises and fight for constituents [Shorten] |
| Full Idea: An important tension in Mandate Representation seemingly requires politicians to both uphold their electoral promises and promote the interests of their constituents. These can conflict, with changed circumstances or information. | |
| From: Andrew Shorten (Contemporary Political Theory [2016], 04 Box 4.1) | |
| A reaction: So be careful what you promise, and don't take on a party loyalty that conflicts with your constituents' interests. Easy. |
| 21117 | Liberal citizens have a moral requirement to respect freedom and equality [Shorten] |
| Full Idea: The liberal theory of political community contains a moral thesis which says that members should share a moral concern for one another as free and equal citizens. …Citizens are not required to have much else in common with one another. | |
| From: Andrew Shorten (Contemporary Political Theory [2016], 02) | |
| A reaction: A key thought. Liberal hearts swell with pride at the first half, but anti-liberals are interested in the second bit. If my neighbour lives in miserable poverty, should I only ask whether they are 'equal and free'? Respect everything! |
| 21134 | Maybe the rational autonomous liberal individual is merely the result of domination [Shorten] |
| Full Idea: On a radical reading of Foucault, the very ideal of a rational, autonomous moral agent that lies at the heart of liberal governmentality is nothing more than the effect of a particular form of domination. | |
| From: Andrew Shorten (Contemporary Political Theory [2016], 06) | |
| A reaction: [Apologies for the word 'governmentality'; I'm just the messenger] Presumably Foucault's philosophy is also the result of domination, so it is hard to know where to start. The status of rationality is the central issue. |
| 21113 | Liberal equality concerns rights, and liberal freedom concerns choice of ends [Shorten] |
| Full Idea: A liberal society treats people as equals by equipping them with the same set of rights, and it respects their freedom by allowing them to choose their own freely chosen ends. | |
| From: Andrew Shorten (Contemporary Political Theory [2016], 01) | |
| A reaction: Equality of rights is fairly standard in any modern society (at least in principle). Freedom of ends is trickier. You can dismiss someone sleeping in the gutter as living a life that resulted from their choices. How many people have clear goals in life? |
| 21121 | Liberal Nationalism encourages the promotion of nationalistic values [Shorten] |
| Full Idea: 'Liberal nationalists' say liberalism is compatible with promoting nationality, by teaching national history and literature and supporting its language. Compatriot priority adds that the needs of compatriots can override those of foreigners. | |
| From: Andrew Shorten (Contemporary Political Theory [2016], 02) | |
| A reaction: [compressed] As a teacher of literature I always preferred to teach the literature of my own country, but without considering the reasons for it. But it was a combination of pride in my people's achievements, and a desire to strengthen social bonds. |
| 21115 | Liberalism should not make assumptions such as the value of choosing your own life plan [Shorten] |
| Full Idea: Communitarians say that liberalism could only justified by appealing to controversial assumptions that are not universally shared, such as the significance of choosing one's own plan of life. | |
| From: Andrew Shorten (Contemporary Political Theory [2016], 01) | |
| A reaction: In the past, at least, huge numbers of people have been perfectly happy living a life designed for them by their parents. It is not much consolation for a disastrous life that at least you planned it yourself. Liberal values are not self-evident. |
| 21114 | Liberals treat individuals as mutual strangers, rather than as social beings [Shorten] |
| Full Idea: Communitarians say that liberalism treats individuals as strangers to one another, and underestimates the extent to which individuals are 'constituted' by their societies and social memberships. | |
| From: Andrew Shorten (Contemporary Political Theory [2016], 01) | |
| A reaction: On the other hand you can have 'too much community'. Surely the test for any political system is the quality of lives led by individual citizens? There can never be a wonderful community full of miserable citizens. |
| 21123 | Liberal Nationalism is more communitarian, and Constitutional Patriotism more cosmopolitan [Shorten] |
| Full Idea: While Liberal Nationalists push liberalism in a particularist and communitarian direction, Constitutional Patriots emphasise its universalistic and cosmopolitan aspects. | |
| From: Andrew Shorten (Contemporary Political Theory [2016], 02) | |
| A reaction: So many attractive qualities to choose from! A tolerant community ought to be cosmopolitan. Being universalistic should not entail a neglect of the particular. Etc. |
| 21124 | Religious toleration has been institutionalised by the separation of church and state [Shorten] |
| Full Idea: One historically influential solution to the discord unleashed by the fact of religious diversity was to institutionalise the principle of toleration by separating church and state. | |
| From: Andrew Shorten (Contemporary Political Theory [2016], 03) | |
| A reaction: In 2018 Britain we still have an established religion (Anglicanism - Episcopalianism in the US), but toleration has arrived with the decline of religious belief. It must still be tough for Muslims, Jews etc to see a different religion as the official one. |