115 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 Michčle 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'. |
| 22355 | In the realist view, the real external world explains how it (and perceptions of it) are possible [Williams,B] |
| Full Idea: The substance of the absolute conception [of external reality] lies in the idea that it could nonvacuously explain how it itself, and the various perspectival views of the world, are possible. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], p.139), quoted by Reiss,J/Spreger,J - Scientific Objectivity 2.1 | |
| A reaction: I like this. Explanation and understanding strike me as more important than justified truths, and I am struck by the complete inability of subjectivists, relativists and anti-realists to give any kinds of good explanation. |
| 23283 | Necessity implies possibility, but in experience it matters which comes first [Williams,B] |
| Full Idea: Any notion of necessity must carry with it a corresponding notion of impossibility, …but it can make a difference which one of them presents itself first and more naturally. | |
| From: Bernard Williams (Practical Necessity [1982], p.127) | |
| A reaction: I like this because it connects modality with experience, rather than with formal logic. It seems right that in life we immediately see either a necessity or an impossibility, and inferring the other case is an afterthought. |
| 4244 | It is very confused to deduce a nonrelativist morality of universal toleration from relativism [Williams,B] |
| Full Idea: Some people believe a properly relativist view requires you to be equally well disposed to everybody's ethical beliefs, but this is seriously confused, as relativism has led to a nonrelativist morality of universal toleration. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 9) | |
| A reaction: Good point. This need not stop a relativist from passionately defending tolerance - it is only that the lack of rational support for the passion must be recognised. |
| 4243 | Our ability to react to an alien culture shows that ethical thought extends beyond cultural boundaries [Williams,B] |
| Full Idea: The fact that people can and must react when confronted with another culture, and do so by applying existing notions, seems to show that ethical thought of a given culture can always stretch beyond its boundaries. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 9) | |
| A reaction: Hardly conclusive, but it does seem to show that there is an element of universalising in values, no matter how local you may consider them to be. |
| 3238 | 'Dead person' isn't a contradiction, so 'person' is somewhat vague [Williams,B] |
| Full Idea: If we say (in opposition to a physical view of identity) that when Jones dies 'Jones ceases to exist' but 'Jones' body does not cease to exist', this shouldn't be pressed too hard, because it would make 'dead person' a contradiction. | |
| From: Bernard Williams (Are Persons Bodies? [1970], p.74) | |
| A reaction: A good point, which nicely challenges the distinction between a 'human' and a 'person', but the problem case is much more the one where Jones gets advanced Alzheimer's, rather than dies. A dead body ceases as a mechanism, as well as as a personality. |
| 3239 | You can only really love a person as a token, not as a type [Williams,B] |
| Full Idea: If you love a person as a type instead of as a token (i.e. a "person", instead of a physical body) you might prefer a run-down copy of them to no person at all, but at this point our idea of loving a person begins to crack. | |
| From: Bernard Williams (Are Persons Bodies? [1970], p.81) | |
| A reaction: Very persuasive. If you love a person you can cope with them getting old. If you own an original watercolour, you can accept that it fades, but you would replace a reproduction of it if that faded. But what, then, is it that you love? |
| 7946 | The memory criterion has a problem when one thing branches into two things [Williams,B, by Macdonald,C] |
| Full Idea: The memory criterion for personal identity permits 'branching' (where two things can later meet the criteria of persistence of a single earlier thing), which presents it with serious problems. | |
| From: report of Bernard Williams (Personal Identity and Individuation [1956]) by Cynthia Macdonald - Varieties of Things Ch.4 | |
| A reaction: Of course, any notion of personal identity would have serious problem if people could branch into two, like fissioning amoeba. If that happened, we probably wouldn't have had a strong notion of personal identity in the first place. See Parfit. |
| 2181 | It is an absurd Kantian idea that at the limit rationality and freedom coincide [Williams,B] |
| Full Idea: There is a deluded Kantian idea that at the limit rationality and freedom will totally coincide. | |
| From: Bernard Williams (Shame and Necessity [1993], VI - p.158) |
| 2176 | There is only a problem of free will if you think the notion of 'voluntary' can be metaphysically deepened [Williams,B] |
| Full Idea: There is a problem of free will only for those who think that the notion of voluntary can be metaphysically deepened. | |
| From: Bernard Williams (Shame and Necessity [1993], III - p.68) | |
| A reaction: Years later, I now see that his refers to a pet hate of mine in discussions of free will, which is the idea that a person can have something called 'ultimate' responsibility for an action (which is the 'deep' version of 'you did it'). |
| 24008 | Reference to a person's emotions is often essential to understanding their actions [Williams,B] |
| Full Idea: The reference to a man's emotions has a significance for our understanding of his moral sincerity, not as a substitute for or addition to how he acts, but as, on occasion, underlying our understanding of how he acts. | |
| From: Bernard Williams (Morality and the emotions [1965], p.223) | |
| A reaction: Williams aims to rescue emotion from the emotivists, and replace it at the centre of traditional modes of moral judgement. I suppose we could assess one rogue robot as behaving 'badly' in a community of robots. |
| 24009 | Moral education must involve learning about various types of feeling towards things [Williams,B] |
| Full Idea: If moral education does not revolve around what to fear, to be angry about, to despise, and where to draw the line between kindness and a stupid sentimentality - I do not know what it is. (Though there are principles, of truth-telling and justice). | |
| From: Bernard Williams (Morality and the emotions [1965], p.225) | |
| A reaction: He cites Aristotle as the obvious source of this correct idea. The examples of principle both require us to place a high value on truth and justice, and not just follow rules in the style of arithmetic. |
| 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 Michčle 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. |
| 9381 | If some inferences are needed to fix meaning, but we don't know which, they are all relevant [Fodor/Lepore, by Boghossian] |
| Full Idea: The Master Argument for linguistic holism is: Some of an expression's inferences are relevant to fixing its meaning; there is no way to distinguish the inferences that are constitutive (from Quine); so all inferences are relevant to fixing meaning. | |
| From: report of J Fodor / E Lepore (Holism: a Shopper's Guide [1993], §III) by Paul Boghossian - Analyticity Reconsidered | |
| A reaction: This would only be if you thought that the pattern of inferences is what fixes the meanings, but how can you derive inferences before you have meanings? The underlying language of thought generates the inferences? Meanings are involved! |
| 4317 | We judge weakness of will by an assessment after the event is concluded [Williams,B, by Cottingham] |
| Full Idea: Williams has shown that whether an action was weakness of will depends on an evaluation after the event, as in the question of whether Gauguin was right to abandon his family to pursue his art. | |
| From: report of Bernard Williams (Shame and Necessity [1993]) by John Cottingham - Reason, Emotions and Good Life p.1 | |
| A reaction: The 'Gauguin Problem' is that Gauguin's actions only become weakness of will if the pictures are no good, and he can't know that till he's painted them. Good point. |
| 9284 | Reasons are 'internal' if they give a person a motive to act, but 'external' otherwise [Williams,B] |
| Full Idea: Someone has 'internal reasons' to act when the person has some motive which will be served or furthered by the action; if this turns out not to be so, the reason is false. Reasons are 'external' when there is no such condition. | |
| From: Bernard Williams (Internal and External Reasons [1980], p.101) | |
| A reaction: [compressed] An external example given is a family tradition of joining the army, if the person doesn't want to. Williams says (p.111) external reason statements are actually false, and a misapplication of the concept of a 'reason to act'. See Idea 8815. |
| 2174 | Responsibility involves cause, intention, state of mind, and response after the event [Williams,B] |
| Full Idea: The four elements of any conception of responsibility are cause, intention, state of mind, and response after the event. | |
| From: Bernard Williams (Shame and Necessity [1993], III - p.53) |
| 22455 | Many ethical theories neglect the power of regretting the ought not acted upon [Williams,B] |
| Full Idea: It is a fundamental criticism of many ethical theories that their accounts of moral conflict and its resolution do not do justice to the facts of regret...: basically because they eliminate from the scene the ought that is not acted upon. | |
| From: Bernard Williams (Ethical consistency [1965], p.175), quoted by Philippa Foot - Moral Realism and Moral Dilemma p.39 | |
| A reaction: [p.175 in Problems of the Self] Williams seems to have initiated this idea. It doesn't matter much for Kantians and Utilitarians (any more than a wrong answer in maths), but it matters if character is the focus. The virtuous have regrets. |
| 4114 | Philosophers try to produce ethical theories because they falsely assume that ethics can be simple [Williams,B] |
| Full Idea: If there is a truth about the subject matter of ethics, why should it be simple? ..I shall argue that philosophy should not try to produce ethical theories. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 1) | |
| A reaction: Bizarrely defeatist - in parallel with Mysterians about the mind like McGinn. Is there any point in thinking at all? I suggest the aim of life as the best starting point. |
| 22453 | Moral conflicts have a different feeling and structure from belief conflicts [Williams,B, by Foot] |
| Full Idea: Williams insisted that the feelings we have in situations of moral conflict show that the 'structure' of moral judgements is unlike that of assertions expressing beliefs. | |
| From: report of Bernard Williams (Ethical consistency [1965]) by Philippa Foot - Moral Realism and Moral Dilemma p.36 | |
| A reaction: Foot presents this as a key reason for the non-cognitivist view of ethics, and her paper attacks it. I don't usually react to moral disagreement with the same vigour I have when I think a belief is untrue. It may just be uncertainty, though. |
| 22454 | We tolerate inconsistency in ethics but not in other beliefs (which reflect an independent order) [Williams,B, by Foot] |
| Full Idea: Williams argued that we can tolerate inconsistency in moral principles though not in assertions, and that this is explained by the fact that it is the concern of the latter but not of the former to reflect an 'independent order of things'. | |
| From: report of Bernard Williams (Consistency and realism (with 1972 note) [1966]) by Philippa Foot - Moral Realism and Moral Dilemma p.37 | |
| A reaction: Put like this, Williams seems to beg the question, which is whether there is an independent moral order to things. There seems to be an easy answer, which is that we are only intolerant of inconsistency when we are confident about it. |
| 22450 | If moral systems can't judge other moral systems, then moral relativism is true [Williams,B, by Foot] |
| Full Idea: If some societies with divergent moral systems merely confront each other, having no use for the assertion that their own systems are true and the others false except to mark the system to which they adhere, then relativism is a true theory of morality. | |
| From: report of Bernard Williams (The Truth in Relativism [1974]) by Philippa Foot - Moral Relativism p.3 | |
| A reaction: 'Having no use for' an assertion is not the same as the assertion being impossible. Some liberal cultures refuse to criticise others because their highest value is tolerance, even when the target culture wholly contradicts the critics' other values. |
| 2178 | In bad actions, guilt points towards victims, and shame to the agent [Williams,B] |
| Full Idea: In what I have done, the guilt points in one direction towards what has happened to others, and the shame in another direction to what I am. | |
| From: Bernard Williams (Shame and Necessity [1993], IV - p.92) | |
| A reaction: Not convinced. I think shame has the fear of being observed as an inescapable component. Even when alone shame involves imagining what others might think. |
| 20168 | Blame usually has no effect if the recipient thinks it unjustified [Williams,B] |
| Full Idea: One of the most obvious facts about blame is that in many cases it is effective only if the recipient thinks that it is justified. | |
| From: Bernard Williams (How free does the will need to be? [1985], 5) | |
| A reaction: The point of the blame might not be reform of the agent, but a public justification for punishment as deterrence, in which case who cares what the agent thinks? Is blame attribution of causes, or reasons to punish? |
| 20167 | Blame partly rests on the fiction that blamed agents always know their obligations [Williams,B] |
| Full Idea: Blame rests, in part, on a fiction; the idea that ethical reasons, in particular the special kind of ethical reasons that are obligations, must, really, be available to the blamed agent. | |
| From: Bernard Williams (How free does the will need to be? [1985], 5) | |
| A reaction: In blaming someone, you may be telling them that they should know their obligations, rather than assuming that they do know them. How else can we give children a moral education? |
| 4128 | Intuitionism has been demolished by critics, and no longer looks interesting [Williams,B] |
| Full Idea: Intuitionism in ethics has been demolished by a succession of critics, and the ruins of it that remain above ground are not impressive enough to invite much history of what happened to it. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 6) | |
| A reaction: Why does intuitionism have such appeal to beginners in moral philosophy? There is a truth buried in it somewhere. See 'Sources of the Self' by Charles Taylor. |
| 4366 | We can't accept Aristotle's naturalism about persons, because it is normative and unscientific [Williams,B, by Hursthouse] |
| Full Idea: Williams has expressed pessimism about the project of Aristotelian naturalism on the grounds that his conception of nature, and thereby of human nature, was normative, and that, in a scientific age, this is not a conception that we can take on board. | |
| From: report of Bernard Williams (works [1971]) by Rosalind Hursthouse - On Virtue Ethics Ch.11 | |
| A reaction: I think there is a compromise here. The existentialist denial of intrinsic human nature seems daft, but Aristotelians must grasp the enormous flexibility that is possible to human behaviour because of the open nature of rationality. |
| 4132 | The category of person is a weak basis for ethics, because it is not fixed but comes in degrees [Williams,B] |
| Full Idea: The category of person is a poor foundation for ethical thought, because it looks like a sortal or classificatory notion while in fact it signals characteristics that almost all come in degrees (responsibility, self-reflection etc). | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 6) | |
| A reaction: On the contrary, it must be the basis of moral theory, and its shifting character is strong support for Aristotle's approach to moral growth and responsibility. |
| 24007 | Emotivism saw morality as expressing emotions, and influencing others' emotions [Williams,B] |
| Full Idea: Emotivism held that there were two purposes of moral judgements: to express the emotions of the speaker, and to influence the emotions of his hearers. | |
| From: Bernard Williams (Morality and the emotions [1965], p.209) | |
| A reaction: I take Ayer to be typical of the first project, and Hare of the second. The theory is much more plausible when the second aim is added. Would we ever utter a moral opinion if we didn't hope to influence someone? |
| 4134 | The weakness of prescriptivism is shown by "I simply don't like staying at good hotels" [Williams,B] |
| Full Idea: That "I simply don't like staying at good hotels" is intelligible brings out the basic weakness of prescriptive accounts of the evaluative. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 7) | |
| A reaction: This might be an elision of two different prescriptions, mine and most people's. In what sense do I think the hotel good, as opposed to other people? |
| 4135 | Some ethical ideas, such as 'treachery' and 'promise', seem to express a union of facts and values [Williams,B] |
| Full Idea: Some 'thicker' ethical notions, such as 'treachery', 'promise', 'brutality' and 'courage', seem to express a union of facts and values. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 8) | |
| A reaction: The onus does seem to be on the followers of Hume to disentangle what the rest of us have united. They may, of course, manage it. |
| 22410 | Maybe the unthinkable is a moral category, and considering some options is dishonourable or absurd [Williams,B] |
| Full Idea: One might have the idea that the unthinkable was itself a moral category. ...Regarding certain things even as alternatives is itself something to be regarded as dishonourable or morally absurd. | |
| From: Bernard Williams (A Critique of Utilitarianism [1973], 2) | |
| A reaction: He's very tentative about this, but I think it is a powerful moral idea. See Kekes. He is particularly aiming at utilitarians, who happily assess vile possibilities. |
| 22408 | Consequentialism assumes that situations can be compared [Williams,B] |
| Full Idea: The emphasis on the necessary comparability of situations is a peculiar feature of consequentialism in general. | |
| From: Bernard Williams (A Critique of Utilitarianism [1973], 2) | |
| A reaction: A nice point. Utilitarians might achieve comparison by totting up the happiness in each situation, but once you include the consequences of the consequences the problems are obvious. Was 1789 a good thing? Too early to say. |
| 22411 | For a consequentialist massacring 7 million must be better than massacring 7 million and one [Williams,B] |
| Full Idea: Making the best of a bad job is a consequentialist maxim, and it will have something to say even pn the difference between massacring seven million and massacring seven million and one. | |
| From: Bernard Williams (A Critique of Utilitarianism [1973], 2) | |
| A reaction: If every life counts, the consequentialists have got something right here. Not caring exactly how many were massacred is a sort of callousness (even when the number can't be established). |
| 4120 | It is an error of consequentialism to think we just aim at certain states of affairs; we also want to act [Williams,B] |
| Full Idea: We do not merely want the world to contain certain states of affairs (it is a deep error of consequentialism to believe that this is all we want). Among the things we basically want is to act in certain ways. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 4) | |
| A reaction: A key objection. Does it matter whether Hiroshima is destroyed by earthquake or bombing? |
| 23282 | If all that matters in morality is motive and intention, that makes moral luck irrelevant [Williams,B] |
| Full Idea: The idea that one's whole life can be immune to luck has not prevailed (e.g. in Christianity), …but its place has been taken by the idea that moral value can be immune, …if it is motive that counts, and in actions it is not worldly changes but intention. | |
| From: Bernard Williams (Moral Luck [1976], p.20) | |
| A reaction: [compressed] That is, that Kant offers a way to make luck irrelevant to morality. Williams disagrees, but says at least Kant offers 'solace to a sense of the world's unfairness'. |
| 4252 | Promise keeping increases reliability, by making deliberation focus on something which would be overlooked [Williams,B] |
| Full Idea: The institution of promise keeping operates to provide portable reliability, by offering a formula that will confer high deliberative priority on what might not otherwise receive it. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch.10) | |
| A reaction: This is a bit pessimistic. We do not perceive promise keeping as a mere suggestion that we should bear something in mind when making a decision. 'May I rot in hell if I fail you'. |
| 4116 | A weakness of contractual theories is the position of a person of superior ability and power [Williams,B] |
| Full Idea: A particular weakness of the contractual theory is that it is unstable with respect to a superior agent, one more intelligent and resourceful and persuasive than the rest. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 3) | |
| A reaction: The very weak are equally a problem. Democratic societies produce fewer inequalities. Hierarchical societies are miserable (I expect..). |
| 2169 | Greek moral progress came when 'virtue' was freed from social status [Williams,B] |
| Full Idea: There was moral progress in the ancient Greek world, notably to the extent that the idea of areté, human excellence, was freed to some extent from determination by social position. | |
| From: Bernard Williams (Shame and Necessity [1993], I - p.7) |
| 4112 | A crucial feature of moral thought is second-order desire - the desire to have certain desires [Williams,B] |
| Full Idea: Recently there has been much emphasis on the importance of our capacity to have second-order desires - the desire to have certain desires - and its significance for ethical reflection and the practical consciousness. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 1) | |
| A reaction: This is a crucial point if we are to defend a reasonably rational view of morality against (say) emotivism. I agree that it is crucial to morality. |
| 24010 | An admirable human being should have certain kinds of emotional responses [Williams,B] |
| Full Idea: One's conception of an admirable human being implies that he should be disposed to certain kinds of emotional response, and not to others. | |
| From: Bernard Williams (Morality and the emotions [1965], p.225) | |
| A reaction: So are the good emotions an indicator of being a good person, or is that what their goodness consists of? The goodness must be cashed out in actions, and presumably good emotions both promise good actions, and motivate them. |
| 23279 | It is important that a person can change their character, and not just be successive 'selves' [Williams,B] |
| Full Idea: I want to emphasise the basic importance of the ordinary idea of a self or person which undergoes changes of character, as opposed to dissolving a changing person into a series of 'selves'. | |
| From: Bernard Williams (Persons, Character and Morality [1976], II) | |
| A reaction: [compressed] He mentions Derek Parfit for the rival view. Williams has the Aristotelian view, that a person has an essential nature, which endures through change, and explains that change. But that needs some non-essential character traits. |
| 23280 | Kantians have an poor account of individuals, and insist on impartiality, because they ignore character [Williams,B] |
| Full Idea: The Kantians' omission of character is a condition of their ultimate insistence on the demands of impartial morality, just as it is a reason to find inadequate their account of the individual. | |
| From: Bernard Williams (Persons, Character and Morality [1976], II) | |
| A reaction: This is also why the Kantian account of virtue is inadequate, in comparison with the Aristotelian view. |
| 3236 | Equality of opportunity without equality of respect would create a very inhuman society [Williams,B] |
| Full Idea: A highly rational, efficient and unmitigated application of the idea of equality of opportunity, while abandoning the idea of equality of respect as vague and nostalgic, would lead to a quite inhuman society. | |
| From: Bernard Williams (The Idea of Equality [1962], §3) |
| 4113 | 'Deon' in Greek means what one must do; there was no word meaning 'duty' [Williams,B] |
| Full Idea: There is no ancient Greek word for duty; the word 'deon' (the basis of 'deontology') means what one must do. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 1) | |
| A reaction: Presumably it covered compulsions which were not duties, such as the need to eat or drink. Greeks thought morally, but lacked a good moral vocabulary? |
| 4250 | The concept of a 'duty to myself' is fraudulent [Williams,B] |
| Full Idea: The concept of a 'duty to myself' is fraudulent. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch.10) | |
| A reaction: The only person who can offer a rebuttal of this is Aristotle. With the magnet of the Platonic Form of the Good, I can perceive the natural excellences of which I am capable, and feel a duty to pursue them. |
| 4110 | Obligation and duty look backwards (because of a promise or job), although the acts are in the future [Williams,B] |
| Full Idea: Obligation and duty look backwards; the acts they require lie in the future, but the reasons for those acts lie in the fact that I have already promised, the job I have undertaken, the position I am already in. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 1) | |
| A reaction: Maybe the central issue in morality is forwards versus backwards. It reflects two types of human temperament. Tomorrow is another day. Spilt milk. |
| 4248 | Not all moral deliberations lead to obligations; some merely reveal what 'may' be done [Williams,B] |
| Full Idea: Not every conclusion of moral deliberation expresses an obligation; for example, some moral conclusions merely announce that you 'may' do something. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch.10) | |
| A reaction: An important point for any deontological ethics. It may be possible to translate what 'may' be done into some form of duty, but it will probably involve contortions. |
| 4249 | "Ought implies can" is a famous formula in connection with moral obligation [Williams,B] |
| Full Idea: "Ought implies can" is a famous formula in connection with moral obligation. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch.10) | |
| A reaction: Williams says it is true in particular instances, but is not generally true of 'ought'. Maybe you 'ought' before you know whether you 'can'. |
| 2172 | The modern idea of duty is unknown in archaic Greece [Williams,B] |
| Full Idea: Duty in some abstract modern sense is largely unknown to the Greeks, in particular to archaic Greeks. | |
| From: Bernard Williams (Shame and Necessity [1993], II - p.41) |
| 4121 | Why should I think of myself as both the legislator and the citizen who follows the laws? [Williams,B] |
| Full Idea: Why should I think of myself as a legislator and at the same time a citizen of a republic governed by some notional laws? | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 4) | |
| A reaction: Kant's answer is supposed to be 'because you are rational, and hence must want consistency'. If we were all rational, Kant would be right. |
| 22409 | We don't have a duty to ensure that others do their duty [Williams,B] |
| Full Idea: If the goodness of the world were to consist in people's fulfilling their obligations, it would by no means follow that one of my obligations was to bring it about that other people kept their obligations | |
| From: Bernard Williams (A Critique of Utilitarianism [1973], 2) | |
| A reaction: If the maxim of my action is 'ensure that everyone does their duty', presumably that can be universalised. Nelson thought so. It just sounds like a hideous world of self-righteous interference. |
| 4122 | If the self becomes completely impartial, it no longer has enough identity to worry about its interests [Williams,B] |
| Full Idea: How can an 'I' that has taken on the perspective of impartiality be left with enough identity to live a life that respects its own interests? | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 4) | |
| A reaction: Not a big problem. Thought constantly flips between objective and subjective, as Nagel has shown us. Compare Nagel in Idea 6446. |
| 2180 | If reason cannot lead people to good, we must hope they have an internal voice [Williams,B] |
| Full Idea: If we think the power of reason is not enough by itself to distinguish good and bad, then we would hope that people have limited autonomy, that there is an internalised other in them that carries some social weight. | |
| From: Bernard Williams (Shame and Necessity [1993], IV - p.100) |
| 24012 | Kant's love of consistency is too rigid, and it even overrides normal fairness [Williams,B] |
| Full Idea: There is a certain moral woodenness or even insolence in Kant's blank regard for consistency. It smacks of Keynes's Principle of Unfairness - that if you can't do a good turn to everybody, you shouldn't do it to anybody. | |
| From: Bernard Williams (Morality and the emotions [1965], p.226) | |
| A reaction: He says it also turns each of us into a Supreme Legislator, which deifies man. It is clearly not the case that morality consists entirely of rules and principles, but Williams recognises their role, in truth-telling for example. |
| 2179 | If the moral self is seen as characterless, then other people have a very limited role in our moral lives [Williams,B] |
| Full Idea: The conception of the moral self as characterless leaves only a limited positive role to other people in one's moral life. | |
| From: Bernard Williams (Shame and Necessity [1993], IV - p.95) |
| 22407 | Utilitarianism cannot make any serious sense of integrity [Williams,B] |
| Full Idea: Utilitarianism cannot hope to make sense, at any serious level, of integrity. | |
| From: Bernard Williams (A Critique of Utilitarianism [1973], 1) | |
| A reaction: There will be obvious problems with this. 'My whole platoon got killed, but looking on the bright side, I preserved my integrity'. Once a theory commits entirely to one value, it then has no way to make sense of rival values. |
| 23278 | For utilitarians states of affairs are what have value, not matter who produced them [Williams,B] |
| Full Idea: The basic bearer of value for Utilitarianism is the state of affairs, and hence, when the relevant causal differences have been allowed for, it cannot make any further difference who produces a given state of affairs. | |
| From: Bernard Williams (Persons, Character and Morality [1976], I) | |
| A reaction: Which is morally better, that I water your bed of flowers, or that it rains? Which is morally better, that I water them from love, or because you threaten me with a whip? |
| 4124 | Utilitarian benevolence involves no particular attachments, and is immune to the inverse square law [Williams,B] |
| Full Idea: Utilitarian benevolence involves no particular attachments, and it is immune to the inverse square law. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 5) | |
| A reaction: Nicely put. The point is that the theory is inhuman, but Mill says it tells us what we should do, not what we actually tend to do. |
| 4245 | Ethical conviction must be to some extent passive, and can't just depend on the will and decisions [Williams,B] |
| Full Idea: The view that the only alternative to the intellect is the will, and the source of ethical conviction is decisions about principles and ways of life, cannot be right; ethical conviction, like any conviction, must to some extent come to you passively. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 9) | |
| A reaction: Seems right. We cannot choose our factual beliefs (look at the sun and believe it is cloudy!). Could I 'decide' that it was right to betray my family just for fun? |
| 4246 | Taking responsibility won't cure ethical uncertainty by; we are uncertain what to decide [Williams,B] |
| Full Idea: If ethics is a matter of decision, and we must face the responsibility and burden of those decisions, this ignores the obvious point that if we are uncertain, then we are uncertain what to decide. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 9) | |
| A reaction: Good point. The defence would be that the decision itself contains the seeds of certainty. Do something rather than nothing, and the sense of it will emerge. Modify as you go along. |
| 3233 | Equality implies that people are alike in potential as well as in needs [Williams,B] |
| Full Idea: Supporters of equality have asserted that people are alike in certain things they could do or achieve, as well as in the things that they need and could suffer. | |
| From: Bernard Williams (The Idea of Equality [1962], §2) |
| 3234 | Equality seems to require that each person be acknowledged as having a significant point of view [Williams,B] |
| Full Idea: Equality seems to require that each person is owed an effort at identification; they should not be seen as a surface to which a label can be applied, but one should try to see the world (including the label) from their point of view. | |
| From: Bernard Williams (The Idea of Equality [1962], §2) |
| 3235 | It is a mark of extreme exploitation that the sufferers do not realise their plight [Williams,B] |
| Full Idea: It is a mark of extreme exploitation or degradation that those who suffer it do NOT see themselves differently from the way they are seen by the exploiters. | |
| From: Bernard Williams (The Idea of Equality [1962], §2) |
| 4247 | It is a mark of our having ethical values that we aim to reproduce them in our children [Williams,B] |
| Full Idea: It is a mark of our having ethical values that we aim to reproduce them in our children. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 9) | |
| A reaction: Maybe beliefs imply education. A commitment to truth is an aspiration that others will agree, especially those over whom we have the greatest influence. |
| 4131 | Most women see an early miscarriage and a late stillbirth as being very different in character [Williams,B] |
| Full Idea: Few women see a spontaneous abortion or early miscarriage as the same thing as having a child who is stillborn or who dies very soon after birth. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 6) | |
| A reaction: This implies a theory about the nature of what is lost. Everyone sees the difference between potential and actual. |
| 4133 | Speciesism isn't like racism, because the former implies a viewpoint which belongs to no one [Williams,B] |
| Full Idea: Speciesism is falsely modelled on racism and sexism, which really are prejudices; ..our arguments have to be founded on the human point of view; they cannot be derived from a point of view that is no one's point of view at all. | |
| From: Bernard Williams (Ethics and the Limits of Philosophy [1985], Ch. 6) | |
| A reaction: This must be wrong. How else are we going to judge cruelty to animals as wrong? The 'point of view of the Universe' (Sidgwick) is not an empty concept. |
| 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'. |
| 2175 | There is a problem of evil only if you expect the world to be good [Williams,B] |
| Full Idea: There is a "problem of evil" only for those who expect the world to be good. | |
| From: Bernard Williams (Shame and Necessity [1993], III - p.68) |