41 ideas
| 10859 | A set is 'well-ordered' if every subset has a first element [Clegg] |
| Full Idea: For a set to be 'well-ordered' it is required that every subset of the set has a first element. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10857 | Set theory made a closer study of infinity possible [Clegg] |
| Full Idea: Set theory made a closer study of infinity possible. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10864 | Any set can always generate a larger set - its powerset, of subsets [Clegg] |
| Full Idea: The idea of the 'power set' means that it is always possible to generate a bigger one using only the elements of that set, namely the set of all its subsets. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.14) |
| 10872 | Extensionality: Two sets are equal if and only if they have the same elements [Clegg] |
| Full Idea: Axiom of Extension: Two sets are equal if and only if they have the same elements. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10875 | Pairing: For any two sets there exists a set to which they both belong [Clegg] |
| Full Idea: Axiom of Pairing: For any two sets there exists a set to which they both belong. So you can make a set out of two other sets. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10876 | Unions: There is a set of all the elements which belong to at least one set in a collection [Clegg] |
| Full Idea: Axiom of Unions: For every collection of sets there exists a set that contains all the elements that belong to at least one of the sets in the collection. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10878 | Infinity: There exists a set of the empty set and the successor of each element [Clegg] |
| Full Idea: Axiom of Infinity: There exists a set containing the empty set and the successor of each of its elements. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |
| A reaction: This is rather different from the other axioms because it contains the notion of 'successor', though that can be generated by an ordering procedure. |
| 10877 | Powers: All the subsets of a given set form their own new powerset [Clegg] |
| Full Idea: Axiom of Powers: For each set there exists a collection of sets that contains amongst its elements all the subsets of the given set. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |
| A reaction: Obviously this must include the whole of the base set (i.e. not just 'proper' subsets), otherwise the new set would just be a duplicate of the base set. |
| 10879 | Choice: For every set a mechanism will choose one member of any non-empty subset [Clegg] |
| Full Idea: Axiom of Choice: For every set we can provide a mechanism for choosing one member of any non-empty subset of the set. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |
| A reaction: This axiom is unusual because it makes the bold claim that such a 'mechanism' can always be found. Cohen showed that this axiom is separate. The tricky bit is choosing from an infinite subset. |
| 10871 | Axiom of Existence: there exists at least one set [Clegg] |
| Full Idea: Axiom of Existence: there exists at least one set. This may be the empty set, but you need to start with something. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10874 | Specification: a condition applied to a set will always produce a new set [Clegg] |
| Full Idea: Axiom of Specification: For every set and every condition, there corresponds a set whose elements are exactly the same as those elements of the original set for which the condition is true. So the concept 'number is even' produces a set from the integers. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |
| A reaction: What if the condition won't apply to the set? 'Number is even' presumably won't produce a set if it is applied to a set of non-numbers. |
| 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. |
| 10880 | Mathematics can be 'pure' (unapplied), 'real' (physically grounded); or 'applied' (just applicable) [Clegg] |
| Full Idea: Three views of mathematics: 'pure' mathematics, where it doesn't matter if it could ever have any application; 'real' mathematics, where every concept must be physically grounded; and 'applied' mathematics, using the non-real if the results are real. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.17) | |
| A reaction: Very helpful. No one can deny the activities of 'pure' mathematics, but I think it is undeniable that the origins of the subject are 'real' (rather than platonic). We do economics by pretending there are concepts like the 'average family'. |
| 10861 | Beyond infinity cardinals and ordinals can come apart [Clegg] |
| Full Idea: With ordinary finite numbers ordinals and cardinals are in effect the same, but beyond infinity it is possible for two sets to have the same cardinality but different ordinals. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10860 | An ordinal number is defined by the set that comes before it [Clegg] |
| Full Idea: You can think of an ordinal number as being defined by the set that comes before it, so, in the non-negative integers, ordinal 5 is defined as {0, 1, 2, 3, 4}. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10854 | Transcendental numbers can't be fitted to finite equations [Clegg] |
| Full Idea: The 'transcendental numbers' are those irrationals that can't be fitted to a suitable finite equation, of which π is far and away the best known. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch. 6) |
| 10858 | By adding an axis of imaginary numbers, we get the useful 'number plane' instead of number line [Clegg] |
| Full Idea: The realisation that brought 'i' into the toolkit of physicists and engineers was that you could extend the 'number line' into a new dimension, with an imaginary number axis at right angles to it. ...We now have a 'number plane'. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.12) |
| 10853 | Either lack of zero made early mathematics geometrical, or the geometrical approach made zero meaningless [Clegg] |
| Full Idea: It is a chicken-and-egg problem, whether the lack of zero forced forced classical mathematicians to rely mostly on a geometric approach to mathematics, or the geometric approach made 0 a meaningless concept, but the two remain strongly tied together. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch. 6) |
| 10866 | Cantor's account of infinities has the shaky foundation of irrational numbers [Clegg] |
| Full Idea: As far as Kronecker was concerned, Cantor had built a whole structure on the irrational numbers, and so that structure had no foundation at all. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10869 | The Continuum Hypothesis is independent of the axioms of set theory [Clegg] |
| Full Idea: Paul Cohen showed that the Continuum Hypothesis is independent of the axioms of set theory. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10862 | The 'continuum hypothesis' says aleph-one is the cardinality of the reals [Clegg] |
| Full Idea: The 'continuum hypothesis' says that aleph-one is the cardinality of the rational and irrational numbers. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.14) |
| 16062 | A necessary relation between fact-levels seems to be a further irreducible fact [Lynch/Glasgow] |
| Full Idea: It seems unavoidable that the facts about logically necessary relations between levels of facts are themselves logically distinct further facts, irreducible to the microphysical facts. | |
| From: Lynch,MP/Glasgow,JM (The Impossibility of Superdupervenience [2003], C) | |
| A reaction: I'm beginning to think that rejecting every theory of reality that is proposed by carefully exposing some infinite regress hidden in it is a rather lazy way to do philosophy. Almost as bad as rejecting anything if it can't be defined. |
| 16061 | If some facts 'logically supervene' on some others, they just redescribe them, adding nothing [Lynch/Glasgow] |
| Full Idea: Logical supervenience, restricted to individuals, seems to imply strong reduction. It is said that where the B-facts logically supervene on the A-facts, the B-facts simply re-describe what the A-facts describe, and the B-facts come along 'for free'. | |
| From: Lynch,MP/Glasgow,JM (The Impossibility of Superdupervenience [2003], C) | |
| A reaction: This seems to be taking 'logically' to mean 'analytically'. Presumably an entailment is logically supervenient on its premisses, and may therefore be very revealing, even if some people think such things are analytic. |
| 16060 | Nonreductive materialism says upper 'levels' depend on lower, but don't 'reduce' [Lynch/Glasgow] |
| Full Idea: The root intuition behind nonreductive materialism is that reality is composed of ontologically distinct layers or levels. …The upper levels depend on the physical without reducing to it. | |
| From: Lynch,MP/Glasgow,JM (The Impossibility of Superdupervenience [2003], B) | |
| A reaction: A nice clear statement of a view which I take to be false. This relationship is the sort of thing that drives people fishing for an account of it to use the word 'supervenience', which just says two things seem to hang out together. Fluffy materialism. |
| 16064 | The hallmark of physicalism is that each causal power has a base causal power under it [Lynch/Glasgow] |
| Full Idea: Jessica Wilson (1999) says what makes physicalist accounts different from emergentism etc. is that each individual causal power associated with a supervenient property is numerically identical with a causal power associated with its base property. | |
| From: Lynch,MP/Glasgow,JM (The Impossibility of Superdupervenience [2003], n 11) | |
| A reaction: Hence the key thought in so-called (serious, rather than self-evident) 'emergentism' is so-called 'downward causation', which I take to be an idle daydream. |
| 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? |
| 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. |