77 ideas
| 15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
| Full Idea: The notion of a function evolved gradually from wanting to see what curves can be represented as trigonometric series. The study of arbitrary functions led Cantor to the ordinal numbers, which led to set theory. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I |
| 15946 | Cantor developed sets from a progression into infinity by addition, multiplication and exponentiation [Cantor, by Lavine] |
| Full Idea: Cantor's development of set theory began with his discovery of the progression 0, 1, ....∞, ∞+1, ∞+2, ..∞x2, ∞x3, ...∞^2, ..∞^3, ...∞^∞, ...∞^∞^∞..... | |
| From: report of George Cantor (Grundlagen (Foundations of Theory of Manifolds) [1883]) by Shaughan Lavine - Understanding the Infinite VIII.2 |
| 9616 | A set is a collection into a whole of distinct objects of our intuition or thought [Cantor] |
| Full Idea: A set is any collection into a whole M of definite, distinct objects m ... of our intuition or thought. | |
| From: George Cantor (The Theory of Transfinite Numbers [1897], p.85), quoted by James Robert Brown - Philosophy of Mathematics Ch.2 | |
| A reaction: This is the original conception of a set, which hit trouble with Russell's Paradox. Cantor's original definition immediately invites thoughts about the status of vague objects. |
| 13444 | Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD] |
| Full Idea: Cantor's Theorem says that for any set x, its power set P(x) has more members than x. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 |
| 18098 | Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock] |
| Full Idea: Cantor's diagonalisation argument generalises to show that any set has more subsets than it has members. | |
| From: report of George Cantor (works [1880]) by David Bostock - Philosophy of Mathematics 4.5 | |
| A reaction: Thus three members will generate seven subsets. This means that 'there is no end to the series of cardinal numbers' (Bostock p.106). |
| 15505 | If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis] |
| Full Idea: Cantor taught that a set is 'a many, which can be thought of as one'. ...After a time the unfortunate beginner student is told that some classes - the singletons - have only a single member. Here is a just cause for student protest, if ever there was one. | |
| From: report of George Cantor (works [1880]) by David Lewis - Parts of Classes 2.1 | |
| A reaction: There is a parallel question, almost lost in the mists of time, of whether 'one' is a number. 'Zero' is obviously dubious, but if numbers are for counting, that needs units, so the unit is the precondition of counting, not part of it. |
| 10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
| Full Idea: Cantor's theories exhibited the contradictions others had claimed to derive from the supposition of infinite sets as confusions resulting from the failure to mark the necessary distinctions with sufficient clarity. | |
| From: report of George Cantor (works [1880]) by Michael Potter - Set Theory and Its Philosophy Intro 1 |
| 10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
| Full Idea: Cantor discovered that the continuum is the powerset of the integers. While adding or multiplying infinities didn't move up a level of complexity, multiplying a number by itself an infinite number of times did. | |
| From: report of George Cantor (works [1880]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.14 |
| 17831 | Cantor gives informal versions of ZF axioms as ways of getting from one set to another [Cantor, by Lake] |
| Full Idea: Cantor gives informal versions of the axioms of ZF as ways of getting from one set to another. | |
| From: report of George Cantor (Later Letters to Dedekind [1899]) by John Lake - Approaches to Set Theory 1.6 | |
| A reaction: Lake suggests that it should therefore be called CZF. |
| 13016 | The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy] |
| Full Idea: Cantor first stated the Union Axiom in a letter to Dedekind in 1899. It is nearly too obvious to deserve comment from most commentators. Justifications usually rest on 'limitation of size' or on the 'iterative conception'. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Believing the Axioms I §1.3 | |
| A reaction: Surely someone can think of some way to challenge it! An opportunity to become notorious, and get invited to conferences. |
| 14199 | Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley] |
| Full Idea: Cantor's definition of a set was a collection of its members into a whole, but within a few years Dedekind had the idea of a set as a container, enclosing its members like a sack. | |
| From: report of George Cantor (works [1880]) by Oliver,A/Smiley,T - What are Sets and What are they For? Intro | |
| A reaction: As the article goes on to show, these two view don't seem significantly different until you start to ask about the status of the null set and of singletons. I intuitively vote for Dedekind. Set theory is the study of brackets. |
| 10082 | There are infinite sets that are not enumerable [Cantor, by Smith,P] |
| Full Idea: Cantor's Theorem (1874) says there are infinite sets that are not enumerable. This is proved by his 1891 'diagonal argument'. | |
| From: report of George Cantor (works [1880]) by Peter Smith - Intro to Gödel's Theorems 2.3 | |
| A reaction: [Smith summarises the diagonal argument] |
| 13483 | Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD] |
| Full Idea: The problem of Cantor's Paradox is that the power set of the universe has to be both bigger than the universe (by Cantor's theorem) and not bigger (since it is a subset of the universe). | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 3 | |
| A reaction: Russell eliminates the 'universe' in his theory of types. I don't see why you can't just say that the members of the set are hypothetical rather than real, and that hypothetically the universe might contain more things than it does. |
| 8710 | The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend] |
| Full Idea: Cantor's Paradox says that the powerset of a set has a cardinal number strictly greater than the original set, but that means that the powerset of the set of all the cardinal numbers is greater than itself. | |
| From: report of George Cantor (works [1880]) by Michèle Friend - Introducing the Philosophy of Mathematics | |
| A reaction: Friend cites this with the Burali-Forti paradox and the Russell paradox as the best examples of the problems of set theory in the early twentieth century. Did this mean that sets misdescribe reality, or that we had constructed them wrongly? |
| 15910 | Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine] |
| Full Idea: Cantor believed he had discovered that between the finite and the 'Absolute', which is 'incomprehensible to the human understanding', there is a third category, which he called 'the transfinite'. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.4 |
| 15905 | Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine] |
| Full Idea: In 1878 Cantor published the unexpected result that one can put the points on a plane, or indeed any n-dimensional space, into one-to-one correspondence with the points on a line. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.1 |
| 9983 | Cantor took the ordinal numbers to be primary [Cantor, by Tait] |
| Full Idea: Cantor took the ordinal numbers to be primary: in his generalization of the cardinals and ordinals into the transfinite, it is the ordinals that he calls 'numbers'. | |
| From: report of George Cantor (works [1880]) by William W. Tait - Frege versus Cantor and Dedekind VI | |
| A reaction: [Tait says Dedekind also favours the ordinals] It is unclear how the matter might be settled. Humans cannot give the cardinality of large groups without counting up through the ordinals. A cardinal gets its meaning from its place in the ordinals? |
| 17798 | Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry] |
| Full Idea: Cantor taught us to regard the totality of natural numbers, which was formerly thought to be infinite, as really finite after all. | |
| From: report of George Cantor (works [1880]) by John Mayberry - What Required for Foundation for Maths? p.414-2 | |
| A reaction: I presume this is because they are (by definition) countable. |
| 9971 | Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait] |
| Full Idea: Cantor introduced the distinction between cardinal and ordinal numbers. | |
| From: report of George Cantor (works [1880]) by William W. Tait - Frege versus Cantor and Dedekind Intro | |
| A reaction: This seems remarkably late for what looks like a very significant clarification. The two concepts coincide in finite cases, but come apart in infinite cases (Tait p.58). |
| 9892 | Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett] |
| Full Idea: Cantor's work revealed that the notion of an ordinal number is more fundamental than that of a cardinal number. | |
| From: report of George Cantor (works [1880]) by Michael Dummett - Frege philosophy of mathematics Ch.23 | |
| A reaction: Dummett makes it sound like a proof, which I find hard to believe. Is the notion that I have 'more' sheep than you logically prior to how many sheep we have? If I have one more, that implies the next number, whatever that number may be. Hm. |
| 15911 | Ordinals are generated by endless succession, followed by a limit ordinal [Cantor, by Lavine] |
| Full Idea: Ordinal numbers are generated by two principles: each ordinal has an immediate successor, and each unending sequence has an ordinal number as its limit (that is, an ordinal that is next after such a sequence). | |
| From: report of George Cantor (Grundlagen (Foundations of Theory of Manifolds) [1883]) by Shaughan Lavine - Understanding the Infinite III.4 |
| 14136 | A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor] |
| Full Idea: The cardinal number of M is the general idea which, by means of our active faculty of thought, is deduced from the collection M, by abstracting from the nature of its diverse elements and from the order in which they are given. | |
| From: George Cantor (works [1880]), quoted by Bertrand Russell - The Principles of Mathematics §284 | |
| A reaction: [Russell cites 'Math. Annalen, XLVI, §1'] See Fine 1998 on this. |
| 15906 | Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine] |
| Full Idea: Cantor said he could show that every infinite set of points on the line could be placed into one-to-one correspondence with either the natural numbers or the real numbers - with no intermediate possibilies (the Continuum hypothesis). His proof failed. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.1 |
| 11015 | Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read] |
| Full Idea: Cantor's diagonal argument showed that all the infinite decimals between 0 and 1 cannot be written down even in a single never-ending list. | |
| From: report of George Cantor (works [1880]) by Stephen Read - Thinking About Logic Ch.6 |
| 15903 | A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine] |
| Full Idea: Cantor's theory of Cauchy sequences defines a real number to be associated with an infinite set of infinite sequences of rational numbers. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II.6 | |
| A reaction: This sounds remarkably like the endless decimals we use when we try to write down an actual real number. |
| 18251 | Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine] |
| Full Idea: Cantor introduced irrationals to play the role of limits of Cauchy sequences of rational numbers. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite 4.2 |
| 15902 | Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine] |
| Full Idea: From the very nature of an irrational number, it seems necessary to understand the mathematical infinite thoroughly before an adequate theory of irrationals is possible. Infinite classes are obvious in the Dedekind Cut, but have logical difficulties | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II Intro | |
| A reaction: Almost the whole theory of analysis (calculus) rested on the irrationals, so a theory of the infinite was suddenly (in the 1870s) vital for mathematics. Cantor wasn't just being eccentric or mystical. |
| 15908 | It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine] |
| Full Idea: Cantor's 1891 diagonal argument revealed there are infinitely many infinite powers. Indeed, it showed more: it shows that given any set there is another of greater power. Hence there is an infinite power strictly greater than that of the set of the reals. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.2 |
| 13464 | Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD] |
| Full Idea: What we might call 'Cantor's Thesis' is that there won't be a potential infinity of any sort unless there is an actual infinity of some sort. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: This idea is nicely calculated to stop Aristotle in his tracks. |
| 10112 | The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman] |
| Full Idea: Cantor showed that the complete totality of natural numbers cannot be mapped 1-1 onto the complete totality of the real numbers - so there are different sizes of infinity. | |
| From: report of George Cantor (works [1880]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.4 |
| 15896 | Cantor needed Power Set for the reals, but then couldn't count the new collections [Cantor, by Lavine] |
| Full Idea: Cantor grafted the Power Set axiom onto his theory when he needed it to incorporate the real numbers, ...but his theory was supposed to be theory of collections that can be counted, but he didn't know how to count the new collections. | |
| From: report of George Cantor (The Theory of Transfinite Numbers [1897]) by Shaughan Lavine - Understanding the Infinite I | |
| A reaction: I take this to refer to the countability of the sets, rather than the members of the sets. Lavine notes that counting was Cantor's key principle, but he now had to abandon it. Zermelo came to the rescue. |
| 17889 | CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner] |
| Full Idea: Cantor's Continuum Hypothesis (CH) says that for every infinite set X of reals there is either a one-to-one correspondence between X and the natural numbers, or between X and the real numbers. | |
| From: report of George Cantor (works [1880]) by Peter Koellner - On the Question of Absolute Undecidability 1.2 | |
| A reaction: Every single writer I read defines this differently, which drives me crazy, but is also helpfully illuminating. There is a moral there somewhere. |
| 13447 | Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD] |
| Full Idea: Cantor conjectured that there is no size between those of the naturals and the reals - called the 'continuum hypothesis'. The generalized version says that for no infinite set A is there a set larger than A but smaller than P(A). | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: Thus there are gaps between infinite numbers, and the power set is the next size up from any infinity. Much discussion as ensued about whether these two can be proved. |
| 10883 | Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten] |
| Full Idea: Cantor's Continuum Hypothesis states that there are no sets which are too large for there to be a one-to-one correspondence between the set and the natural numbers, but too small for there to exist a one-to-one correspondence with the real numbers. | |
| From: report of George Cantor (works [1880]) by Leon Horsten - Philosophy of Mathematics §5.1 |
| 13528 | Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS] |
| Full Idea: Cantor's conjecture (the Continuum Hypothesis) is that there are no sets between N and P(N). The 'generalized' version replaces N with an arbitrary infinite set. | |
| From: report of George Cantor (works [1880]) by Robert S. Wolf - A Tour through Mathematical Logic 2.2 | |
| A reaction: The initial impression is that there is a single gap in the numbers, like a hole in ozone layer, but the generalised version implies an infinity of gaps. How can there be gaps in the numbers? Weird. |
| 9555 | Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara] |
| Full Idea: Cantor's Continuum Hypothesis was that there is no cardinal number greater than aleph-null but less than the cardinality of the continuum. | |
| From: report of George Cantor (works [1880]) by Charles Chihara - A Structural Account of Mathematics 05.1 | |
| A reaction: I have no view on this (have you?), but the proposal that there are gaps in the number sequences has to excite all philosophers. |
| 8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
| Full Idea: Cantor's 'continuum hypothesis' is the assertion that there are no infinite cardinalities strictly between the size of the natural numbers and the size of the real numbers. | |
| From: report of George Cantor (works [1880]) by Stewart Shapiro - Thinking About Mathematics 2.4 | |
| A reaction: The tricky question is whether this hypothesis can be proved. |
| 15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
| Full Idea: Cantor's set theory was not of collections in some familiar sense, but of collections that can be counted using the indexes - the finite and transfinite ordinal numbers. ..He treated infinite collections as if they were finite. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I |
| 18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
| Full Idea: Cantor's second innovation was to extend the sequence of ordinal numbers into the transfinite, forming a handy scale for measuring infinite cardinalities. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: Struggling with this. The ordinals seem to locate the cardinals, but in what sense do they 'measure' them? |
| 18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy] |
| Full Idea: Cantor's first innovation was to treat cardinality as strictly a matter of one-to-one correspondence, so that the question of whether two infinite sets are or aren't of the same size suddenly makes sense. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: It makes sense, except that all sets which are infinite but countable can be put into one-to-one correspondence with one another. What's that all about, then? |
| 9992 | The 'extension of a concept' in general may be quantitatively completely indeterminate [Cantor] |
| Full Idea: The author entirely overlooks the fact that the 'extension of a concept' in general may be quantitatively completely indeterminate. Only in certain cases is the 'extension of a concept' quantitatively determinate. | |
| From: George Cantor (Review of Frege's 'Grundlagen' [1885], 1932:440), quoted by William W. Tait - Frege versus Cantor and Dedekind | |
| A reaction: Cantor presumably has in mind various infinite sets. Tait is drawing our attention to the fact that this objection long precedes Russell's paradox, which made the objection more formal (a language Frege could understand!). |
| 10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro] |
| Full Idea: Cantor's theorem entails that there are more property extensions than objects. So there are not enough objects in any domain to serve as extensions for that domain. So Frege's view that numbers are objects led to the Caesar problem. | |
| From: report of George Cantor (works [1880]) by Stewart Shapiro - Philosophy of Mathematics 4.6 | |
| A reaction: So the possibility that Caesar might have to be a number arises because otherwise we are threatening to run out of numbers? Is that really the problem? |
| 18176 | Pure mathematics is pure set theory [Cantor] |
| Full Idea: Pure mathematics ...according to my conception is nothing other than pure set theory. | |
| From: George Cantor (works [1880], I.1), quoted by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: [an unpublished paper of 1884] So right at the beginning of set theory this claim was being made, before it was axiomatised, and so on. Zermelo endorsed the view, and it flourished unchallenged until Benacerraf (1965). |
| 8631 | Cantor says that maths originates only by abstraction from objects [Cantor, by Frege] |
| Full Idea: Cantor calls mathematics an empirical science in so far as it begins with consideration of things in the external world; on his view, number originates only by abstraction from objects. | |
| From: report of George Cantor (works [1880]) by Gottlob Frege - Grundlagen der Arithmetik (Foundations) §21 | |
| A reaction: Frege utterly opposed this view, and he seems to have won the day, but I am rather thrilled to find the great Cantor endorsing my own intuitions on the subject. The difficulty is to explain 'abstraction'. |
| 14280 | The probability of two events is the first probability times the second probability assuming the first [Bayes] |
| Full Idea: The probability that two events will both happen is the probability of the first [multiplied by] the probability of the second on the supposition that the first happens. | |
| From: Thomas Bayes (Essay on a Problem in the Doctrine of Chances [1763]), quoted by Dorothy Edgington - Conditionals (Stanf) 3.1 |
| 8715 | Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend] |
| Full Idea: Cantor (in his exploration of infinities) pushed the bounds of conceivability further than anyone before him. To discover what is conceivable, we have to enquire into the concept. | |
| From: report of George Cantor (works [1880]) by Michèle Friend - Introducing the Philosophy of Mathematics 6.5 | |
| A reaction: This remark comes during a discussion of Husserl's phenomenology. Intuitionists challenge Cantor's claim, and restrict what is conceivable to what is provable. Does possibility depend on conceivability? |
| 13454 | Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor] |
| Full Idea: Cantor thought that we abstract a number as something common to all and only those sets any one of which has as many members as any other. ...However one wants to see the logic of the inference. The irony is that set theory lays out this logic. | |
| From: comment on George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: The logic Hart has in mind is the notion of an equivalence relation between sets. This idea sums up the older and more modern concepts of abstraction, the first as psychological, the second as logical (or trying very hard to be!). Cf Idea 9145. |
| 9145 | We form the image of a cardinal number by a double abstraction, from the elements and from their order [Cantor] |
| Full Idea: We call 'cardinal number' the general concept which, by means of our active faculty of thought, arises when we make abstraction from an aggregate of its various elements, and of their order. From this double abstraction the number is an image in our mind. | |
| From: George Cantor (Beitrage [1915], §1), quoted by Kit Fine - Cantorian Abstraction: Recon. and Defence Intro | |
| A reaction: [compressed] This is the great Cantor, creator of set theory, endorsing the traditional abstractionism which Frege and his followers so despise. Fine offers a defence of it. The Frege view is platonist, because it refuses to connect numbers to the world. |
| 22849 | Rawls's theory cannot justify liberalism, since it presupposes free and equal participants [Charvet] |
| Full Idea: Rawls's theory presupposes that the contractors are conceived, and conceive themselves, to be free and equal persons. Consequently, the theory cannot be presented as a justificatory theory of liberalism. | |
| From: John Charvet (Liberalism: the basics [2019], 14) | |
| A reaction: Nice. If you imagine diverse groups with many strong beliefs coming together to form a society, Rawls is asking them all to become liberals before they all decide how to live together. |
| 22848 | People with strong prior beliefs would have nothing to do with a veil of ignorance [Charvet] |
| Full Idea: Why would a group of people with strong beliefs (e.g. religious beliefs) agree to debate the problem of what norms should govern their association from behind a veil of ignorance? …They would not accept the veil of ignorance as fair. | |
| From: John Charvet (Liberalism: the basics [2019], 14) | |
| A reaction: Nice. Rawls's experiment assumes liberal people with very few beliefs. No racial supremacist is going to enter a society in which they may be of a different race. Charvet says the entrants would all need to be pluralists about the good. |
| 22838 | Societies need shared values, so conservatism is right if rational discussion of values is impossible [Charvet] |
| Full Idea: Were it true that rational discussion of values is impossible, then a conservative attitude would seem to be the only viable position. Some set of common values is necessary to maintain the unity of a political society. | |
| From: John Charvet (Liberalism: the basics [2019], 07) | |
| A reaction: Better to say that the less values can be both discussed and changed the stronger is the case for a degree of conservatism. Conservatives tend to favour values asserted by authority, rather than by popular (undiscussed) consensus. |
| 22846 | The universalism of utilitarianism implies a world state [Charvet] |
| Full Idea: Utilitarianism is a universalist ethic, so the political realisation of this ethic would seem to be a world state seeking to maximise happiness for the world's population. | |
| From: John Charvet (Liberalism: the basics [2019], 12) | |
| A reaction: It certainly doesn't seem to favour the citizens of the state where it is implemented, since miserable people just across the border would have priority, and all miserable migrants must be welcomed. There is no loyalty to citizens. |
| 22835 | Liberals value freedom and equality, but the society itself must decide on its values [Charvet] |
| Full Idea: While freedom and equality are liberal values …they are fundamental regulative ideas of an independent society that is self-regulating …and decides what its own social and political arrangements should be. | |
| From: John Charvet (Liberalism: the basics [2019], 06) | |
| A reaction: So the central political activity is persuasion, not enforcement. Illiberal societies all contain liberal individuals. |
| 22831 | Modern libertarian societies still provide education and some housing [Charvet] |
| Full Idea: No society today is libertarian in the extreme sense. Even the freest economically, such as Singapore have their governments provide education services and public housing. | |
| From: John Charvet (Liberalism: the basics [2019], 05) | |
| A reaction: There is a good argument that many other services should be provided by a libertarian state, on the grounds that it is more efficient, and the services must otherwise paid for by much higher salaries. |
| 22839 | Liberalism needs people to either have equal autonomy, or everyone to have enough autonomy [Charvet] |
| Full Idea: To get a liberal society one would have to claim that either everyone possesses autonomy to an equal degree or that everyone possesses a threshold level of the capacity that entitles them to enjoy the full liberal rights. | |
| From: John Charvet (Liberalism: the basics [2019], 07) | |
| A reaction: This leaves out the more right-wing attitude that people can increase their capacity for autonomy if they are forced to stand on their own feet. A liberal society must decide how to treat persons incapable of proper autonomy. |
| 22847 | Kant places a higher value on the universal rational will than on the people asserting it [Charvet] |
| Full Idea: For Kant what is of absolute worth is the universal rational will which become an individual's actual will. Insofar as the individual fails to will the universal, they have no absolute worth, so whether or not they exist is unimportant. | |
| From: John Charvet (Liberalism: the basics [2019], 14) | |
| A reaction: A lovely demolition of the claims of Kant to be the patriarch of liberalism! Liberalism must place supreme value on each individual, not on some abstracted realm of pure reason and moral good. Liberals are motivated by love, not reason. |
| 22821 | Liberalism asserts maximum freedom, but that must be equal for all participants [Charvet] |
| Full Idea: Liberalism attaches fundamental value to leaving individuals as free as possible … - but there is another fundamental value implicit in this idea - the equal status of the participants in the practice. By this I mean that they all have the same rights. | |
| From: John Charvet (Liberalism: the basics [2019], Intro) | |
| A reaction: Libertarian liberalism (e.g. Nozick) only asserts the fundament principle of freedom, but such a society swiftly deprives most of its members of those very freedoms. Egalitarian Liberalism should be our default political ideology. |
| 22834 | Egalitarian liberals prefer equality (either of input or outcome) to liberty [Charvet] |
| Full Idea: Rather than libertarianism, egalitarian liberals promote equality, either of outcomes (of happiness or of well-being), or of inputs (such as opportunities, capacities or resources), which they favour ahead of freedom. | |
| From: John Charvet (Liberalism: the basics [2019], 06) | |
| A reaction: This is my team, I think. I think I'm a liberal who thinks liberty is a bit overrated. Equal outcome according to capacity (promoted by Nussbaum) seems attractive. |
| 22822 | Liberals promote community and well-being - because all good societies need them [Charvet] |
| Full Idea: Community and well-being are not specifically liberal values. They are values any independent political society must pursue whether it is a liberal society or not. | |
| From: John Charvet (Liberalism: the basics [2019], Intro) | |
| A reaction: This seems, at a stroke, to undermine the familiar debate between liberals and communitarians. I've switched to the former from the latter, because communitarians is potentially too paternalistic and conservative. Persuade individuals to be communal! |
| 22841 | Identity multiculturalism emerges from communitarianism, preferring community to humanity [Charvet] |
| Full Idea: Identity-based multiculturalism developed from communitarianism. …People come to consciousness of themselves as members of some community before they identify themselves as members of the human race. | |
| From: John Charvet (Liberalism: the basics [2019], 08) | |
| A reaction: This is 'identity politics', which Carvet sees as a problem from liberalism. Is it more important to be a woman or a Muslim or a Scot than to be a human being? It seems to create institutional antagonisms. |
| 22842 | For communitarians it seems that you must accept the culture you are born into [Charvet] |
| Full Idea: Communitarians have difficulty avoiding the relativist trap. It seems they must claim that if one is born into a liberal society one cannot but be a liberal, and if one is born into a communist society one cannot but be a communist. | |
| From: John Charvet (Liberalism: the basics [2019], 08) | |
| A reaction: Anyone who accepts the Hegelian view of history and culture seems doomed to such relativism, and Hegel is a communitarian precursor. This is a good reason for me to reject communitarianism, after a long flirtation. We can criticise our own culture. |
| 22830 | Give by ability and receive by need, rather than a free labour market [Charvet] |
| Full Idea: Only the most extreme collective socialism denies the freedom to sell one's labour power and buy that of others, under the communist slogan 'from each according to his ability, and to each according to his needs'. | |
| From: John Charvet (Liberalism: the basics [2019], 05) | |
| A reaction: [He cites Marx 'Critique of the Gotha Programme'] I would guess that this practice is not abnormal in old traditional villages, though a community would be tempted to reward highly a very successful member. |
| 22829 | Allowing defamatory speech is against society's interests, by blurring which people are trustworthy [Charvet] |
| Full Idea: The argument for restricting defamatory speech is that unrestricted speech makes it impossible, or too difficult, to distinguish between those who deserve a trustworthy reputation and those who don't - a distinction in society's best interests. | |
| From: John Charvet (Liberalism: the basics [2019], 03) | |
| A reaction: A nice example of appeal to the common good, in opposition to the normal freedoms of liberalism. An example of the Prisoner's Dilemma. Should assertion of the common good of a group be a prime value of liberalism? |
| 22836 | 'Freedom from' is an empty idea, if the freedom is not from impediments to my desires [Charvet] |
| Full Idea: Berlin's distinction of 'freedom from' and 'freedom to' is worthless …because to say that I want to be free from something for absolutely no reason makes no sense. Unfreedom is being blocked from what I want to do, which ceases if I no longer want it. | |
| From: John Charvet (Liberalism: the basics [2019], 07) | |
| A reaction: [compressed] The government could guarantee us against attacks by albatrosses, but we would hardly have a national holiday to celebrate the freedom. Still, there is freedom from incoming troubles, and freedom to output things. |
| 22837 | Positive freedom can lead to coercion, if you are forced to do what you chose to do [Charvet] |
| Full Idea: Berlin saw positive freedom as a justification for illiberal coercion. If I am positively free only in doing X, then if I am forced to do X, I will still be free. | |
| From: John Charvet (Liberalism: the basics [2019], 07) | |
| A reaction: I suppose Berlin is thinking of Russian farmers, who wanted to farm, but then found they were forced to do what they were going to do anyway. It's better than being forced to do what you didn't want to do. Forcing clearly isn't freedom. |
| 22844 | First level autonomy is application of personal values; second level is criticising them [Charvet] |
| Full Idea: First level autonomy is being able to apply one's scheme of values to one's actions and life; second level autonomy is being able to subject those values to critical evaluation. | |
| From: John Charvet (Liberalism: the basics [2019], 10) | |
| A reaction: Charvet sees this as a key issue for liberalism. How do you treat citizens who cannot advance beyond the first level? He mentions the elitism of Plato's Republic that results. |
| 22840 | Mere equality, as in two trees being the same height, has no value at all [Charvet] |
| Full Idea: That the relation of equality might be considered a value in itself is an absurdity. Would the equality of blinding the only sighted person in a blind society be good? Is it inherently good that two trees are the same height? This is nonsense. | |
| From: John Charvet (Liberalism: the basics [2019], 08) | |
| A reaction: He cites Temkin 1993 as defending the blinding example! Obviously equality is only possible in certain respects (though electrons might be equal in all respects). So the point is to identify the important respects. The rest is rhetoric. |
| 22843 | Inequalities are worse if they seem to be your fault, rather than social facts [Charvet] |
| Full Idea: Inequality is worse in a meritocracy than in a stratified society, because everyone enjoys a formal equality of status and your position in the social order is due to your merit or lack of merit, so you have only yourself to blame for being at the bottom. | |
| From: John Charvet (Liberalism: the basics [2019], 10) | |
| A reaction: This is the simple point that it is worse to lack some good if you might have possessed it, rather than it being entirely out of reach. It also makes the false assumption that people are largely responsible for their merit or lack of it (ignoring luck). |
| 22845 | Money allows unlimited inequalities, and we obviously all agree to money [Charvet] |
| Full Idea: The introduction of money allows people to accumulate wealth without limit. Since money only works through everyone's agreement …everyone can be taken to have agreed to the consequences of money in the unequal distribution of wealth. | |
| From: John Charvet (Liberalism: the basics [2019], 11) | |
| A reaction: [Locke] Presumably large inequalities of possessions and territory were possible before money, but there was at least an upper limit. The current owner of Amazon may end up with more wealth than the whole of the rest of humanity combined. |
| 22823 | The rule of law is mainly to restrict governments [Charvet] |
| Full Idea: The rule of law is directed at the restriction of the power of governments as much, if not more, then the power of private individuals. | |
| From: John Charvet (Liberalism: the basics [2019], 02) | |
| A reaction: The more powerful you are the more restricting is the rule of law. Every government is tempted to change the law to expand its powers. The UK government has just legislated to restrict public demonstrations. Law is the people's weapon against autocrats. |
| 22825 | The 1689 Bill of Rights denied the monarch new courts, or the right to sit as judge [Charvet] |
| Full Idea: The 1689 Bill of Rights said the monarch could not create new courts of law, or act as a judge at law. | |
| From: John Charvet (Liberalism: the basics [2019], 02) | |
| A reaction: The background was the abolition of the court of Star Chamber in 1641, which had been secret, severe, and controlled by the monarch. Is it possible to create a new type of court, or are we stuck with the current ones? |
| 22826 | From 1701 only parliament could remove judges, whose decisions could not be discussed [Charvet] |
| Full Idea: In 1701 UK judges were given secure tenure, being removable only by parliament which at the same time undertook to follow a convention not to discuss particular judicial decisions. | |
| From: John Charvet (Liberalism: the basics [2019], 02) | |
| A reaction: In recent years the UK Daily Mail published the pictures of three judges, and labelled them 'traitors' because of their verdict about leaving the European Union. |
| 22827 | Justice superior to the rule of law is claimed on behalf of the workers, or the will of the nation [Charvet] |
| Full Idea: Communist leaders justify themselves as the embodiment of the people's will as workers, and fascist leaders as expressing the will of the nation. Both believe their policies contain a superior justice on this basis. | |
| From: John Charvet (Liberalism: the basics [2019], 02) | |
| A reaction: [compressed] A neat summary of why the rule of law might be rejected (other than by simple tyrrany justified only by force). In modern democracies recent right-wing governments have pushed back the law and attacked justice on this basis. |
| 22828 | The rule of law mainly benefits those with property and liberties [Charvet] |
| Full Idea: A rule of law regime will primarily benefit those possessing property and liberty rights. | |
| From: John Charvet (Liberalism: the basics [2019], 02) | |
| A reaction: Important. It's no good fighting for the law if the law doesn't protect what you have got, or if you have got nothing to protect. Important steps must precede assertion of the rule of law. |
| 22832 | Welfare is needed if citizens are to accept the obligations of a liberal state [Charvet] |
| Full Idea: The welfare state provides the background conditions under which it is reasonable to expect one's fellow citizens to commit to liberal principles of interaction, even if those conditions can only be achieved through a degree of compulsion. | |
| From: John Charvet (Liberalism: the basics [2019], 05) | |
| A reaction: You cannot expect people to accept the role of 'free' citizen if that is likely to result in swift misery. A liberal state will only command loyalty if it has a safety net. Fully committed liberalism implies modest socialism. |
| 10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
| Full Idea: Cantor proved that one-dimensional space has exactly the same number of points as does two dimensions, or our familiar three-dimensional space. | |
| From: report of George Cantor (works [1880]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.14 |
| 13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |
| Full Idea: Cantor said that only God is absolutely infinite. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: We are used to the austere 'God of the philosophers', but this gives us an even more austere 'God of the mathematicians'. |