76 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 |
| 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). |
| 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 |
| 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. |
| 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 |
| 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 |
| 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 |
| 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. |
| 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 |
| 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. |
| 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. |
| 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. |
| 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? |
| 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'. |
| 7401 | Heat and colour don't exist, so cannot mislead about the external world [Galileo, by Tuck] |
| Full Idea: Galileo argued that there is no such thing as heat (and hence also as colour) in the external world, so there is no reason to conclude from colour-blindness that we cannot know the truth about the world. | |
| From: report of Galileo Galilei (Il Saggiatore ('The Assayer') [1623]) by Richard Tuck - Hobbes Ch.1 | |
| A reaction: This key idea, taken up by Gassendi, Descartes and Locke, seems to me to be one of the most important (and, in retrospect, rather obvious) facts ever worked out by the human mind. Why does anyone still doubt it? |
| 5454 | Tastes, odours and colours only reside in consciousness, and would disappear with creatures [Galileo] |
| Full Idea: I think tastes, odours, colours, and so on are mere names as far as the objects are concerned, and only reside in consciousness. Hence if the living creature were removed, all these qualities would be wiped away and annihilated. | |
| From: Galileo Galilei (Il Saggiatore ('The Assayer') [1623]), quoted by Brian Ellis - The Philosophy of Nature: new essentialism Ch.3 | |
| A reaction: A nice bold assertion of the primary/secondary distinction from the first great scientist. I agree, and to disagree (and hence side with Berkeley and Hume) is to head for metaphsical and epistemological confusion. |
| 16560 | Galileo introduced geometrico-mechanical explanation, based on Archimedes [Galileo, by Machamer/Darden/Craver] |
| Full Idea: The modern idea of explaining with mechanisms became current in the 17th century when Galileo articulated a geometrico-mechanical form of explanation based on Archimedes' simple machines. This became the 'mechanical philosophy'. | |
| From: report of Galileo Galilei (Il Saggiatore ('The Assayer') [1623]) by Machamer,P/Darden,L/Craver,C - Thinking About Mechanisms 5.2 | |
| A reaction: So is Archimedes the source? I would say that mechanical explanation is just commonsense, and is predominant in all human thinking, even in tiny infants. |
| 4363 | The word 'person' is useless in ethics, because what counts as a good or bad self-conscious being? [Hursthouse] |
| Full Idea: An excellent reason for keeping the word 'person' out of ethics is that it is usually so thinly defined that it cannot generate any sense of 'good person'. If a person is just a self-conscious being, what would count as a good or bad one? | |
| From: Rosalind Hursthouse (On Virtue Ethics [1999], Ch.9 n20) | |
| A reaction: A nice point. Locke's concept of a person (rational self-conscious being) lacks depth and individuality, and Hitler fulfils the criteria as well as any saint. But if Hitler wasn't a 'bad person', what was he bad at being? |
| 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. |
| 4355 | There may be inverse akrasia, where the agent's action is better than their judgement recommends [Hursthouse] |
| Full Idea: There seem to be cases of 'inverse akrasia', in which the course of action actually followed is superior to the course of action recommended by the agent's best judgement. | |
| From: Rosalind Hursthouse (On Virtue Ethics [1999], Ch.7) | |
| A reaction: This must occur, as when an assassin lets his victim off, and then regrets the deed. It strengthens the case against Socrates, and in favour of their being two parts of the soul which compete to motivate our actions. |
| 4325 | Must all actions be caused in part by a desire, or can a belief on its own be sufficient? [Hursthouse] |
| Full Idea: In contemporary philosophy of action, there is a fervid debate about whether any intentional action must be prompted in part by desire, or whether it is possible to be moved to action by a belief alone. | |
| From: Rosalind Hursthouse (On Virtue Ethics [1999], Intro) | |
| A reaction: I want a cool belief to be sufficient to produce an action, because it will permit at least a Kantian dimension to ethics, and make judgement central, and marginalise emotivism, which is the spawn of Satan. |
| 4351 | It is a fantasy that only through the study of philosophy can one become virtuous [Hursthouse] |
| Full Idea: It is a fantasy that only through the study of philosophy can one become virtuous. | |
| From: Rosalind Hursthouse (On Virtue Ethics [1999], Ch.6) | |
| A reaction: I personally believe that philosophy is the best route yet devised to the achievement of virtue, but it is clearly not essential. All the philosophers I meet are remarkably virtuous, but that may be a chicken/egg thing. |
| 4329 | After a moral dilemma is resolved there is still a 'remainder', requiring (say) regret [Hursthouse] |
| Full Idea: When one moral requirement has overriden another in a dilemma, there is still a 'remainder', so that regret, or the recognition of some new requirement, are still appropriate. | |
| From: Rosalind Hursthouse (On Virtue Ethics [1999], Ch.2) | |
| A reaction: This is a powerful point on behalf of virtue ethics. There is a correct way to feel about the application of rules and calculations. Judges sleep well at night, but virtuous people may not. |
| 4330 | Deontologists resolve moral dilemmas by saying the rule conflict is merely apparent [Hursthouse] |
| Full Idea: With respect to resolvable dilemmas, the deontologist's strategy is to argue that the 'conflict' between the two rules which has generated the dilemma is merely apparent. | |
| From: Rosalind Hursthouse (On Virtue Ethics [1999], Ch.2) | |
| A reaction: This assumes that the rules can't conflict (because they come for God, or pure reason), but we might say that there are correct rules which do conflict. Morality isn't physics, or tennis. |
| 4341 | Involuntary actions performed in tragic dilemmas are bad because they mar a good life [Hursthouse] |
| Full Idea: The actions a virtuous agent is forced to in tragic dilemmas fail to be good actions because the doing of them, no matter how unwillingly or involuntarily, mars or ruins a good life. | |
| From: Rosalind Hursthouse (On Virtue Ethics [1999], Ch.3) | |
| A reaction: Of course, only virtuous people have their lives ruined by such things. For the cold or the wicked it is just water off a duck's back. |
| 4340 | You are not a dishonest person if a tragic dilemma forces you to do something dishonest [Hursthouse] |
| Full Idea: Doing what is, say, dishonest solely in the context of a tragic dilemma does not entail being dishonest, possessing that vice. | |
| From: Rosalind Hursthouse (On Virtue Ethics [1999], Ch.3 n8) | |
| A reaction: This seems right, although it mustn't be thought that the dishonesty is thereby excused. Virtuous people find being dishonest very painful. |
| 4358 | Virtue may be neither sufficient nor necessary for eudaimonia [Hursthouse] |
| Full Idea: Some critics say virtue is not necessary for eudaimonia (since the wicked sometimes flourish), and others say it is not sufficient (because virtuous behaviour sometimes ruins a life). | |
| From: Rosalind Hursthouse (On Virtue Ethics [1999], Ch.8) | |
| A reaction: Both criticisms seem wrong (the wicked don't 'flourish', and complete virtue never ruins lives, except in tragic dilemmas). But it is hard to prove them wrong. |
| 4337 | Teenagers are often quite wise about ideals, but rather stupid about consequences [Hursthouse] |
| Full Idea: Adolescents tend to be much more gormless about consequences than they are about ideals. | |
| From: Rosalind Hursthouse (On Virtue Ethics [1999], Ch.2 n12) | |
| A reaction: Very accurate, I'm afraid. But this cuts both ways. They seem to need education not in virtue, but simply in consequences. |
| 4324 | Animals and plants can 'flourish', but only rational beings can have eudaimonia [Hursthouse] |
| Full Idea: The trouble with 'flourishing' as a translation of 'eudaimonia' is that animals and even plants can flourish, but eudaimonia is possible only for rational beings. | |
| From: Rosalind Hursthouse (On Virtue Ethics [1999], Intro) | |
| A reaction: 'Flourishing' still seems better than 'happy', which is centrally used now to refer to a state of mind, not a situation. 'Well being' seems good, and plants are usually permitted that. |
| 4359 | When it comes to bringing up children, most of us think that the virtues are the best bet [Hursthouse] |
| Full Idea: If you think about bringing up children to prepare them for life, rather than converting the wicked or convincing the moral sceptic, isn't virtue the most reliable bet? | |
| From: Rosalind Hursthouse (On Virtue Ethics [1999], Ch.8) | |
| A reaction: A very convincing idea. They haven't the imagination to grasp consequences properly, or sufficient abstract thought to grasp principles, or the political cunning to negotiate contracts, but they can grasp ideals of what a good person is like. |
| 4336 | Any strict ranking of virtues or rules gets abandoned when faced with particular cases [Hursthouse] |
| Full Idea: Any codification ranking the virtues, like any codification ranking the rules, is bound to come up against cases where we will want to change the rankings. | |
| From: Rosalind Hursthouse (On Virtue Ethics [1999], Ch.2) | |
| A reaction: This seems right, and yet it feels like a slippery slope. Am I supposed to be virtuous and wise, but have no principles? Infinite flexibility can lead straight to wickedness. Even the wise need something to hang on to. |
| 4334 | Virtue ethics is open to the objection that it fails to show priority among the virtues [Hursthouse] |
| Full Idea: One criticism of virtue ethics is that it lamentably fails to come up with a priority ranking of the virtues. | |
| From: Rosalind Hursthouse (On Virtue Ethics [1999], Ch.2) | |
| A reaction: However, one might refer to man's essential function, or characteristic function, and one might derive the virtues of a good citizen from the nature of a good society. |
| 4361 | Good animals can survive, breed, feel characteristic pleasure and pain, and contribute to the group [Hursthouse] |
| Full Idea: A good social animal is well fitted for 1) individual survival, 2) continuance of its species, 3) characteristic freedom from pain and enjoyment, and 4) good characteristic functioning of its social group. | |
| From: Rosalind Hursthouse (On Virtue Ethics [1999], Ch.9) | |
| A reaction: This feels right, but brings out the characteristic conservativism of virtue theory. A squirrel which can recite Shakespeare turns out to be immoral. |
| 4349 | Virtuous people may not be fully clear about their reasons for action [Hursthouse] |
| Full Idea: Virtue must surely be compatible with a fair amount of inarticulacy about one's reasons for action. | |
| From: Rosalind Hursthouse (On Virtue Ethics [1999], Ch.6) | |
| A reaction: Virtuous people may be unclear, but we are entitled to hope for clarification from moral philosophers. The least we can hope for is some distinction between virtue and vice. |
| 4352 | Performing an act simply because it is virtuous is sufficient to be 'morally motivated' or 'dutiful' [Hursthouse] |
| Full Idea: Acting virtuously, in the way the virtuous agent acts, namely from virtue, is sufficient for being 'morally motivated' or acting 'from a sense of duty'. | |
| From: Rosalind Hursthouse (On Virtue Ethics [1999], Ch.7) | |
| A reaction: Fine, but it invites the question of WHY virtue is motivating, just as one can ask this of maximum happiness, or duty, or even satisfaction of selfish desires. |
| 4353 | If moral motivation is an all-or-nothing sense of duty, how can children act morally? [Hursthouse] |
| Full Idea: If you are inclined to think that 'moral motivation', acting because you think it is right, must be an all-or-nothing matter, its presence determined by the agent's mind at the moment of acting, do, please, remember children. | |
| From: Rosalind Hursthouse (On Virtue Ethics [1999], Ch.7) | |
| A reaction: I agree about the vital importance of remembering children when discussing morality. However, Kantians might legitimately claim that when a child is simply trained to behave well, it has not yet reached the age of true morality. |
| 4346 | The emotions of sympathy, compassion and love are no guarantee of right action or acting well [Hursthouse] |
| Full Idea: The emotions of sympathy, compassion and love are no guarantee of right action or acting well. | |
| From: Rosalind Hursthouse (On Virtue Ethics [1999], Ch.4) | |
| A reaction: This is a critique of Hume, and of utlitarianism. It pushes us either to the concept of duty, or the concept of virtue (independent of right feeling). |
| 4339 | According to virtue ethics, two agents may respond differently, and yet both be right [Hursthouse] |
| Full Idea: According to virtue ethics, in a given situation two different agents may do what is right, what gets a tick of approval, despite the fact that each fails to do what the other did. | |
| From: Rosalind Hursthouse (On Virtue Ethics [1999], Ch.3) | |
| A reaction: You could certainly have great respect for two entirely different decisions about a medical dilemma, if they both showed integrity and good will, even if one had worse consequences than the other. |
| 4354 | Maybe in a deeply poisoned character none of their milder character traits could ever be a virtue [Hursthouse] |
| Full Idea: I am prepare to stick my neck out and say that extreme Nazis or racists (say) have poisoned characters to such an extent that none of their character traits could ever count as a virtue. | |
| From: Rosalind Hursthouse (On Virtue Ethics [1999], Ch.7) | |
| A reaction: Hard to justify, but it is hard to respect a mass murderer because they seem to love their dog or the beauty of music or flowers. They can't possibly appreciate the Platonic Form of love or beauty? |
| 4356 | We are puzzled by a person who can show an exceptional virtue and also behave very badly [Hursthouse] |
| Full Idea: That we have some intuitive belief in the unity of the virtues is shown by our reaction to stories of a person who has shown an exceptional virtue, but also done something morally repellent. | |
| From: Rosalind Hursthouse (On Virtue Ethics [1999], Ch.7) | |
| A reaction: A nice observation, but not enough to establish the unity of virtue. People tend to love all virtue, but it is not obviously impossible to love selected virtues and despise others (e.g. love courage, and despise charity). |
| 4364 | Being unusually virtuous in some areas may entail being less virtuous in others [Hursthouse] |
| Full Idea: It may well be that being particularly well endowed with respect to some virtues inevitably involves being not very well endowed in others. | |
| From: Rosalind Hursthouse (On Virtue Ethics [1999], Ch.9) | |
| A reaction: Maybe, but this sound a bit like an excuse. Newton wasn't very nice, but Einstein was. I can't believe in a finite reservoir of virtue. |
| 4327 | Deontologists do consider consequences, because they reveal when a rule might apply [Hursthouse] |
| Full Idea: Though it is sometimes said that deontologists 'take no account of consequences', this is manifestly false, for many actions we deliberate about only fall under rules or principles when we bring in their predicted consequences. | |
| From: Rosalind Hursthouse (On Virtue Ethics [1999], Ch.1) | |
| A reaction: An important defence of deontology, which otherwise is vulnerable to the 'well-meaning fool' problem. It is no good having a good will, but refusing to think about consequences. |
| 4335 | 'Codifiable' morality give rules for decisions which don't require wisdom [Hursthouse] |
| Full Idea: If morality is strongly 'codifiable', it should consist of rules which provide a decision procedure, and it should be equally applicable by the virtuous and the non-virtuous, without recourse to wisdom. | |
| From: Rosalind Hursthouse (On Virtue Ethics [1999], Ch.2) | |
| A reaction: A key idea. Religions want obedience, and Kant wants morality to be impersonal, and most people want morality which simple uneducated people can follow. And yet how can wisdom ever be irrelevant? |
| 4328 | Preference utilitarianism aims to be completely value-free, or empirical [Hursthouse] |
| Full Idea: There are some forms of utilitarianism which aim to be entirely 'value-free' or empirical, such as those which define happiness in terms of the satisfaction of actual desires or preferences, regardless of their content. | |
| From: Rosalind Hursthouse (On Virtue Ethics [1999], Ch.1) | |
| A reaction: This point makes it clear that preference utilitarianism is a doomed enterprise. For a start I can prefer not to be a utilitarian. You can only maximise something if you value if. Are preferences valuable? |
| 4338 | Deontologists usually accuse utilitarians of oversimplifying hard cases [Hursthouse] |
| Full Idea: Deontologists characteristically maintain that utilitarians have made out a particular hard case to be too simple. | |
| From: Rosalind Hursthouse (On Virtue Ethics [1999], Ch.3) | |
| A reaction: Utilitarianism certainly seems to ignore the anguish of hard dilemmas, but that is supposed to be its appeal. If you think for too long, every dilemma begins to seem hopeless. |
| 4343 | We are torn between utilitarian and deontological views of lying, depending on the examples [Hursthouse] |
| Full Idea: Utilitarianism says there is nothing intrinsically wrong with lying, but examples of bare-faced lying to increase happiness drive us to deontology; but then examples where telling the truth has appalling consequences drive us back to utilitarianism again. | |
| From: Rosalind Hursthouse (On Virtue Ethics [1999], Ch.3) | |
| A reaction: A nice illustration of why virtue theory suddenly seemed appealing. Deontology can cope, though, by seeing other duties when the consequences are dreadful. |
| 4365 | We are distinct from other animals in behaving rationally - pursuing something as good, for reasons [Hursthouse] |
| Full Idea: Our characteristic way of going on, which distinguishes us from all the other species of animals, is a rational way, which is any way we can rightly see as good, as something we have reason to do. | |
| From: Rosalind Hursthouse (On Virtue Ethics [1999], Ch10) | |
| A reaction: Some people more than others, and none of us all the time. Romantics see rationality as a restraint on the authentic emotional and animal life. 'Be a good animal'. However, I agree. |
| 3645 | To understand the universe mathematics is essential [Galileo] |
| Full Idea: The great book of the universe cannot be understood unless one can understand the language in which it is written - the language of mathematics. | |
| From: Galileo Galilei (Il Saggiatore ('The Assayer') [1623], VI.232) | |
| A reaction: Nice, though one might say that humans created the language of maths to help them discuss the patterns they perceived in nature. Maybe what is special is order, and all order can be described mathematically. |
| 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'. |
| 4350 | If people are virtuous in obedience to God, would they become wicked if they lost their faith? [Hursthouse] |
| Full Idea: If people perform virtuous actions simply because they are commanded by God, would they cease to perform such actions if they lost their faith in God? | |
| From: Rosalind Hursthouse (On Virtue Ethics [1999], Ch.6) | |
| A reaction: To be consistent, the answer might be 'yes', but that invites the response that only intrinsically evil people need to be Christians. The rest of us can be good without it. |