81 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 |
| 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 |
| 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. |
| 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 |
| 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. |
| 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. |
| 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? |
| 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 |
| 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? |
| 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'. |
| 12723 | The most primitive thing in substances is force, which leads to their actions and dispositions [Leibniz] |
| Full Idea: Since everything that one conceives in substances reduces to their actions and passions and to the dispositions that they have for this effect, I don't see how one can find there anything more primitive than the principle of all of this, which is force. | |
| From: Gottfried Leibniz (Letters to Jacques Lenfant [1693], 1693.11.25), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 4 | |
| A reaction: This is an attempt to connect Aristotelian essentialism with the notion of force in the new physics, and strikes me as an improvement on the original, and as good a basis for metaphysics as any I have heard of. |
| 23101 | Intuitions don't prove things; they just receptivity to interpretations [Kekes] |
| Full Idea: Appeal to intuitions cannot prove or disprove anything. They merely create receptivity to particular interpretations of particular cases. | |
| From: John Kekes (Against Liberalism [1997], 04.3) | |
| A reaction: A nice point, but more is needed. A gun to the head can create receptivity. What distinguishes good from bad intuitions? Why are intuitions different from mere whims or hopes? |
| 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. |
| 23086 | Liberals say we are only responsible for fully autonomous actions [Kekes] |
| Full Idea: The liberal view is that people can be held responsible only for actions that are in their control: actions that reflect the agents' unforced choices, evaluations, and understanding of their significance - that is, autonomous actions. | |
| From: John Kekes (Against Liberalism [1997], 01.5) | |
| A reaction: Kekes, who is a somewhat right wing anti-liberal, thinks people should be mainly held responsible for the consequences, unless they have a very good excuse. |
| 23100 | Collective responsibility conflicts with responsibility's requirement of authonomy [Kekes] |
| Full Idea: The ascription of collective responsibility is inconsistent with …the belief that people should be held responsible for only their own autonomous actions. | |
| From: John Kekes (Against Liberalism [1997], 04) | |
| A reaction: The autonomy would need to be displaced, from the decision to act to the decision of identify with the organisation. But if you invest in an evil group you are responsible for actions you never even knew occurred (never mind autonomy). |
| 23093 | Moral and causal responsibility are not clearly distinct [Kekes] |
| Full Idea: Moral and causal responsibility cannot be distinguished as clearly as the liberal strategy requires. | |
| From: John Kekes (Against Liberalism [1997], 03.2) | |
| A reaction: I take assessment to be a two-stage operation. It is usually easy to assign causal responsibility. Moral responsibiity is quite different. Our negligence can make us morally responsible for an event we didn’t cause. |
| 23096 | Morality should aim to prevent all evil actions, not just autonomous ones [Kekes] |
| Full Idea: If one main task of morality is to prevent evil, then morality must be concerned with all evil-producing actions, not just autonomous ones. | |
| From: John Kekes (Against Liberalism [1997], 03.3) | |
| A reaction: Hm. Is placing a railing next to a flight of steps a moral action? Possibly. |
| 23087 | Much human evil is not autonomous, so moral responsibility need not be autonomous [Kekes] |
| Full Idea: If much evil is due to nonautonomous actions, then liberals cannot be right in idenitfying the domain of moral responsibility with the domain of autonomy. | |
| From: John Kekes (Against Liberalism [1997], 02.1) | |
| A reaction: One might evade this anti-liberal thought by making responsibility directly proportional to degree of autonomy. Then the only counterexample would be genuine immorality that is entirely non-autonomous, but is there such a thing? |
| 23098 | Effects show the existence of moral responsibility, and mental states show the degree [Kekes] |
| Full Idea: Psychological states are relevant to the degree of an agent's moral responsibility, while the effects of their actions are relevant to whether the agents are liable to moral responsibility. | |
| From: John Kekes (Against Liberalism [1997], 03.5) | |
| A reaction: He has previously offered a problem case for this, where someone's social role makes them fully responsible whatever their mental state. I still think his distinction is helpful. 1) Whose fault is it, then 2) How far are they to blame? Normal practice. |
| 23089 | Evil people may not be autonomously aware, if they misjudge the situation [Kekes] |
| Full Idea: Agents who perform evil nonautonomously do not know what they are doing, because they have made a mistake in understanding or evaluating their own conduct. | |
| From: John Kekes (Against Liberalism [1997], 02.4) | |
| A reaction: So while liberals say that true evil must be autonomous, Kekes says it may result from factual or evaluative error, for which people are also responsible. |
| 23094 | Ought implies can means moral responsibility needs autonomy [Kekes] |
| Full Idea: Ought implies can translates into the claim that only autonomous agents are morally responsible. | |
| From: John Kekes (Against Liberalism [1997], 03.3) | |
| A reaction: Since Kekes identifies autonomy as the key to liberalism, he sees this also as a basic liberal claim (which he rejects). I ought to ring my mother, but my phone is broken (so I ought not to ring my mother?).. |
| 23095 | Why should moral responsibility depend on autonomy, rather than social role or experience? [Kekes] |
| Full Idea: Why should moral responsibility be made to depend on autonomy, rather than on intelligence, education, social role, experience, or whatever? | |
| From: John Kekes (Against Liberalism [1997], 03.3) | |
| A reaction: Social role seems a particularly good one to cite. 'I didn't really understand what I was doing.' 'But it's your job to understand!' |
| 23090 | Liberals assume people are naturally free, equal, rational, and morally good [Kekes] |
| Full Idea: The view of human nature at the core of the liberal faith is that human beings are by their nature free, equal, rational, and morally good. | |
| From: John Kekes (Against Liberalism [1997], 02.5) | |
| A reaction: These four claims are quite distinct, and should be evaluated separately. I think I'm something of a liberal, but I don't really accept any of them. Hm. I just want all people to have these attributes. |
| 23117 | Love should be partial, and discriminate in favour of its object [Kekes] |
| Full Idea: Love is personal and partial. It is not love if it does not discriminate in favor of its object. | |
| From: John Kekes (Against Liberalism [1997], 09.4) | |
| A reaction: I agree with that, mainly on the grounds that this is the natural form of human love. Generalised love of mankind seems like a distortion, even if it is well-meaning. |
| 23119 | Sentimental love distorts its object [Kekes] |
| Full Idea: Love is sentimental if it exaggerates the virtues and minimises the vices of its object. | |
| From: John Kekes (Against Liberalism [1997], 09.5) | |
| A reaction: Not sure about this. It implies that we should retain a streak of cold evaluative objectivity, even about the people we love most. There is difference between knowing a person's qualities, and the importance we attach to those qualities. Forgive vices! |
| 23088 | Evil is not deviation from the good, any more than good is a deviation from evil [Kekes] |
| Full Idea: There is no more reason to think of evil as deviation from the good than there is to think of the good as deviation from evil. | |
| From: John Kekes (Against Liberalism [1997], 02.2) | |
| A reaction: This is a political moderate right winger defending the concept of evil as a basic and inescapable component of existence, in contrast to liberals who tend to deny 'pure evil'. |
| 23097 | What matters for morality is the effects of action, not the psychological causes [Kekes] |
| Full Idea: What is crucial to morality are the good and evil effects of human actions, not their psychological causes. | |
| From: John Kekes (Against Liberalism [1997], 03.4) | |
| A reaction: The context is his attack on the liberal idea that morality only concerns the actions of autonomous agents. Kekes says he is not a full consequentialist. He just urges that consequences be given greater weight. Even Kant must care about that. |
| 23099 | It is said that if an agent is not autonomous then their evil actions don't reflect on their character [Kekes] |
| Full Idea: Liberals deny the reflexivity of evil, …to prevent the evil consequences of an agent's morally deplorable actions from redounding to their detriment. Evil actions are allowed to reflect on their agents only if the agents cause them autonomously. | |
| From: John Kekes (Against Liberalism [1997], 03.5) | |
| A reaction: A central question of morality is essentialising character. That is, when does an eater of carrots become a carrot-eater? When does a performer of wicked deeds become a wicked person? Never, say many liberals. Wrong, says Kekes. |
| 23118 | Awareness of others' suffering doesn't create an obligation to help [Kekes] |
| Full Idea: It is a mistaken assumption that knowledge of the sufferings of others creates an obligation to help them. | |
| From: John Kekes (Against Liberalism [1997], 09.4) | |
| A reaction: A nice question is when that knowledge does become an obligation. The obvious criteria are proximity to the suffering, and capacity to relieve it. But then a wealthy person couldn't walk down the street without such obigations. Hm. |
| 23109 | The veil of ignorance is only needed because people have bad motivations [Kekes] |
| Full Idea: If the darker aspects of human motivation did not exist, there would be no need for Rawls to place his people behind the veil of ignorance. | |
| From: John Kekes (Against Liberalism [1997], 07.2) | |
| A reaction: All the critics observe that Rawls's blind choosers are nothing like as simple as the mere specks of rationality he seems to imagine. The usual objection is that they are already liberals, but this objection says they are already benevolent. |
| 23114 | The chief function of the state is to arbitrate between contending visions of the good life [Kekes] |
| Full Idea: The chief function of the state is seen to be to maintain what is referred to as the dialogue or conversation among the contending visions of how life should be lived. | |
| From: John Kekes (Against Liberalism [1997], 08.4) | |
| A reaction: This is Kekes's defence of 'pluralism'. It is not liberal, because liberal freedom, autonomy and equality is only one of the competing visions of the good life. Almost every state suppresses some such visions. |
| 23116 | Citizenship is easier than parenthood [Kekes] |
| Full Idea: It is much easier to be a good citizen than it is to be a good parent. | |
| From: John Kekes (Against Liberalism [1997], 09.4) | |
| A reaction: A nice observation. It is shocking how many people are bad citizens, given the limited demands. I think philosophers have some responsibility for beliefs and values which people bring to their citizenship. Parents need communal support. |
| 23103 | Power is meant to be confined to representatives, and subsequent delegation [Kekes] |
| Full Idea: Universal adult suffrage and representative government are intended to give everyone equal initial political power, and assure that delegation is the only legitimate means to acquiring greater power. | |
| From: John Kekes (Against Liberalism [1997], 05.1) | |
| A reaction: The delegation bit is where it all goes wrong. Once you've packed your representative off to the capital, you lose nearly all control over what sort of delegation happens next. It is hard to trust representatives voters have barely met. |
| 23107 | Prosperity is a higher social virtue than justice [Kekes] |
| Full Idea: If social institutions were to have a first virtue, …prosperity would be a much stronger candidate that justice. | |
| From: John Kekes (Against Liberalism [1997], 06.3) | |
| A reaction: Kekes occasionally pays lip service to ecological issues, but this shows he is not serious. Endless economic growth will kill our planet, so it should never be our prime virtue. Also the impplication that you can't be too prosperous is plainly false. |
| 23081 | Liberal basics are pluralism, freedom, rights, equality, and distributive justice - for autonomy [Kekes] |
| Full Idea: The basic liberal values are pluralism, freedom, rights, equality, and distributive justice. What makes them basically valuable is that they enable individuals to live autonomously. | |
| From: John Kekes (Against Liberalism [1997], 01.2) | |
| A reaction: Helpful. Kekes identifies respect for autonomy as the single value which unites all liberal doctrines (and he traces it back to Kant). |
| 23085 | The key liberal values are explained by the one core value, which is autonomy [Kekes] |
| Full Idea: Liberals regard pluralism, freedom, rights, equality and distributive justice as basic …but this particular group of values is explained by the true core of liberalism, the inner citadel for whose protection all the liberal battles are waged: autonomy. | |
| From: John Kekes (Against Liberalism [1997], 01.5) | |
| A reaction: Given that children, soldiers, monks and nuns, and people in old folks homes have very limited autonomy, it is reasonable to query whether it really is so important. I like autonomy if I have external power over my life; not so good when in hospital. |
| 23092 | Agents have little control over the capacities needed for liberal autonomy [Kekes] |
| Full Idea: It is important [for liberals] to realise that agents have no control over their possession of the capacities and opportunities on which their autonomy depends. | |
| From: John Kekes (Against Liberalism [1997], 03.2) | |
| A reaction: It can be replied to Kekes that they also have little control over the capacities upon which his prized 'desert' depends. It may be an axiom of all modern political thought that people have less control than we imagine. |
| 23102 | Liberals are egalitarians, but in varying degrees [Kekes] |
| Full Idea: All liberals are egalitarians, though they may be more or less so. | |
| From: John Kekes (Against Liberalism [1997], 05.1) | |
| A reaction: In the broadest view, this may be the one thing which distinguishes generalised liberals from the rest. To reject it needs a basis for the rejection, and every basis for its flat rejection is anathema to liberals. |
| 23084 | Are egalitarians too coercive, or not egalitarian enough, or lax over morality? [Kekes] |
| Full Idea: Egalitarian liberalism is criticised by classical [freedom] liberals for its coercive redistribution, by socialist liberals for not being egalitarian enough, and by conservative liberals for abandoning moral standards in the guise of neutrality. | |
| From: John Kekes (Against Liberalism [1997], 01.4) | |
| A reaction: Income tax is 'coercive' distribution, but it is done with general consent in most liberal democracies. An interesting line between the needs of the state and the needs of its most needy citizens. |
| 23079 | Liberal justice ignores desert, which is the essence of justice [Kekes] |
| Full Idea: The liberal conception of justice …excludes the essence of justice: desert. | |
| From: John Kekes (Against Liberalism [1997], Pref) | |
| A reaction: Certainly our normal concept of justice includes such thoughts as 'serves him right'. The trouble with the Kekes view is his society is continually morally judging people, and most people's grounds for that are fairly irrational. It's why we have courts. |
| 23091 | Why do liberals not see a much wider range of values as basic? [Kekes] |
| Full Idea: Why are prosperity, order, civility, peace, a healthy environment, security, happiness, and law-abidingness not as important as those thought of by liberals as basic? | |
| From: John Kekes (Against Liberalism [1997], 02.5) | |
| A reaction: This presumes that liberals only see a narrow core of values as basic to the structure of the society. Presumably every society should be well disposed towards the nice features listed here. Would their absence wreck the society? |
| 23112 | Liberals ignore contingency, and think people are good and equal, and institutions cause evil [Kekes] |
| Full Idea: Liberals comfortably believe that autonomy minimises contingency, that humans are disposed to the good, that wickedness is due to remediable institutions, and that humans are morally equal because of their autonomy. | |
| From: John Kekes (Against Liberalism [1997], 07.4) | |
| A reaction: In a nutshell, Kekes thinks liberals are naïve. That institutions cause evil sounds more Marxist than liberal. When individuals become evil, it is reasonable for us to think that this need not have been the case. |
| 23082 | Liberal distribution cares more about recipients than donors [Kekes] |
| Full Idea: Liberal distribution cares more about the rights of the recipients than the rights of the donors. | |
| From: John Kekes (Against Liberalism [1997], 01.2) | |
| A reaction: Even if you are very left wing indeed, this is an important point. A society dominated by a powerful Robin Hood (steal from the rich, for the poor) is quite likely to end in civil war. But should society allow huge individual wealths to accumulate? |
| 23106 | To rectify the undeserved equality, we should give men longer and women shorter lives [Kekes] |
| Full Idea: Redistribution ought to aim to equalise the life expectancy of men and women, by making men have longer and women shorter lives. | |
| From: John Kekes (Against Liberalism [1997], 05.4) | |
| A reaction: This is a nice satirical counterexample to the Rawlsian claim that 'undeserved inequalities should somehow be compensated for' [Rawls 1971: 100]. See also Kurt Vonnegut's story 'Harrison Bergeron'. |
| 23121 | It is just a fact that some people are morally better than others [Kekes] |
| Full Idea: It is an obviolus fact that some people are morally better than others and that some are morally worse. | |
| From: John Kekes (Against Liberalism [1997], 10.4) | |
| A reaction: This could be conceded, without then asserting that the moral ones are superior, or more deserving. That is a social strategy, rather than a fact. We can challenge the criteria for 'morally better', but we can't deny a rankng once it is agreed. |
| 23105 | It is not deplorable that billionaires have more than millionaires [Kekes] |
| Full Idea: It is certainly not intuitively deplorable that billionaires have more money than millionaires. | |
| From: John Kekes (Against Liberalism [1997], 05.3) | |
| A reaction: Nice point. His claim is that sufficiency is the important feature, and equality is largely irrelevant. The reality, though, is that the billionaires, unlike the millionaires, could solve the insufficiency problem. |
| 23120 | The problem is basic insufficiency of resources, not their inequality [Kekes] |
| Full Idea: If everyone has sufficient resources, it is not objectionable that some have more than others. What is objectionable is that some do not have enough. | |
| From: John Kekes (Against Liberalism [1997], 10.3) | |
| A reaction: Reasonable, but there seems to be sharp disagreement between the haves and the have-nots over what counts as 'enough'. In an affluent country, does enough include a car, restaurant dining, and foreign holidays? Or just food and shelter? |
| 23108 | Justice combines consistency and desert; treat likes alike, judging likeness by desert [Kekes] |
| Full Idea: Justice is a combination of consistency and desert. Like cases should be treated alike, and likenesses should be evaluated according to desert. | |
| From: John Kekes (Against Liberalism [1997], 06.3) | |
| A reaction: [compressed] He needs to add that at least the desert should be relevant to the events being assessed. Should people not get a fair trial if they are branded as generally 'undeserving'? Hence the case must be judged before the desert is identified. |
| 23083 | Liberal welfare focuses on need rather than desert [Kekes] |
| Full Idea: In welfare legislation, liberals concentrate on what people need rather than on what they deserve. | |
| From: John Kekes (Against Liberalism [1997], 01,2) | |
| A reaction: He makes assessing what people 'deserve' sound easy. Do drowning people deserve to be rescued? Do billionaires deserve their wealth (which is not the same as 'did they acquire it legally')? What do rude people deserve? |
| 23113 | Sexual morality doesn't require monogamy, but it needs a group of sensible regulations [Kekes] |
| Full Idea: A moral tradition need not be committed to monogamy, but it must regulate sexual conduct to prevent inbreeding, protect the sexually immature, prohibit some forms of coercion, and assign responsibility for raising children. | |
| From: John Kekes (Against Liberalism [1997], 08.1) | |
| A reaction: Wise words, I would say. The sexual liberation which arose with the contraceptive pill rather swamped thoughts of this type. These are just sensible responses to the facts of life. |
| 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'. |