73 ideas
| 8060 | In the 17th-18th centuries morality offered a cure for egoism, through altruism [MacIntyre] |
| Full Idea: It was in the seventeenth and eighteenth century that morality came generally to be understood as offering a solution to the problems posed by human egoism and that the content of morality came to be largely equated with altruism. | |
| From: Alasdair MacIntyre (After Virtue: a Study in Moral Theory [1981], Ch.16) | |
| A reaction: It was the elevation of altruism that caused Nietzsche's rebellion. The sixteenth century certainly looks striking cynical to modern eyes. The development was an attempt to secularise Jesus. Altruism has a paradox: it needs victims. |
| 8053 | Twentieth century social life is re-enacting eighteenth century philosophy [MacIntyre] |
| Full Idea: Twentieth century social life turns out in key part to be the concrete and dramatic re-enactment of eighteenth-century philosophy. | |
| From: Alasdair MacIntyre (After Virtue: a Study in Moral Theory [1981], Ch. 8) | |
| A reaction: This suggest a two hundred year lag between the philosophy and its impact on the culture. One might note the Victorian insistence on 'duty' (e.g. in George Eliot), alongside Mill's view that the Kantian account of it didn't work (Idea 3768). |
| 8047 | Philosophy has been marginalised by its failure in the Enlightenment to replace religion [MacIntyre] |
| Full Idea: The failure, in the Enlightenment, of philosophy to provide what religion could no longer furnish was an important cause of philosophy losing its central cultural role and becoming a marginal, narrowly academic subject. | |
| From: Alasdair MacIntyre (After Virtue: a Study in Moral Theory [1981], Ch. 4) | |
| A reaction: A strange way of presenting the situation. Philosophy has never aspired to furnish beliefs for the masses. Plato offered them myths. The refutation of religion was difficult and complex. There is no returning from there to a new folk simplicity. |
| 8062 | Proof is a barren idea in philosophy, and the best philosophy never involves proof [MacIntyre] |
| Full Idea: Arguments in philosophy rarely take the form of proofs; and the most successful arguments on topics central to philosophy never do. (The ideal of proof is a relatively barren one in philosophy). | |
| From: Alasdair MacIntyre (After Virtue: a Study in Moral Theory [1981], Ch.18) | |
| A reaction: He seems proud of this, but he must settle for something which is less than proof, which has to be vindicated to the mathematicians and scientists. I agree, though. Plato is the model, and the best philosophy builds a broad persuasive picture. |
| 15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
| Full Idea: The notion of a function evolved gradually from wanting to see what curves can be represented as trigonometric series. The study of arbitrary functions led Cantor to the ordinal numbers, which led to set theory. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I |
| 13444 | Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD] |
| Full Idea: Cantor's Theorem says that for any set x, its power set P(x) has more members than x. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 |
| 18098 | Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock] |
| Full Idea: Cantor's diagonalisation argument generalises to show that any set has more subsets than it has members. | |
| From: report of George Cantor (works [1880]) by David Bostock - Philosophy of Mathematics 4.5 | |
| A reaction: Thus three members will generate seven subsets. This means that 'there is no end to the series of cardinal numbers' (Bostock p.106). |
| 15505 | If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis] |
| Full Idea: Cantor taught that a set is 'a many, which can be thought of as one'. ...After a time the unfortunate beginner student is told that some classes - the singletons - have only a single member. Here is a just cause for student protest, if ever there was one. | |
| From: report of George Cantor (works [1880]) by David Lewis - Parts of Classes 2.1 | |
| A reaction: There is a parallel question, almost lost in the mists of time, of whether 'one' is a number. 'Zero' is obviously dubious, but if numbers are for counting, that needs units, so the unit is the precondition of counting, not part of it. |
| 10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
| Full Idea: Cantor's theories exhibited the contradictions others had claimed to derive from the supposition of infinite sets as confusions resulting from the failure to mark the necessary distinctions with sufficient clarity. | |
| From: report of George Cantor (works [1880]) by Michael Potter - Set Theory and Its Philosophy Intro 1 |
| 10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
| Full Idea: Cantor discovered that the continuum is the powerset of the integers. While adding or multiplying infinities didn't move up a level of complexity, multiplying a number by itself an infinite number of times did. | |
| From: report of George Cantor (works [1880]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.14 |
| 13016 | The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy] |
| Full Idea: Cantor first stated the Union Axiom in a letter to Dedekind in 1899. It is nearly too obvious to deserve comment from most commentators. Justifications usually rest on 'limitation of size' or on the 'iterative conception'. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Believing the Axioms I §1.3 | |
| A reaction: Surely someone can think of some way to challenge it! An opportunity to become notorious, and get invited to conferences. |
| 14199 | Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley] |
| Full Idea: Cantor's definition of a set was a collection of its members into a whole, but within a few years Dedekind had the idea of a set as a container, enclosing its members like a sack. | |
| From: report of George Cantor (works [1880]) by Oliver,A/Smiley,T - What are Sets and What are they For? Intro | |
| A reaction: As the article goes on to show, these two view don't seem significantly different until you start to ask about the status of the null set and of singletons. I intuitively vote for Dedekind. Set theory is the study of brackets. |
| 18812 | Split out the logical vocabulary, make an assignment to the rest. It's logical if premises and conclusion match [Tarski, by Rumfitt] |
| Full Idea: Tarski made a division of logical and non-logical vocabulary. He then defined a model as a non-logical assignment satisfying the corresponding sentential function. Then a conclusion follows logically if every model of the premises models the conclusion. | |
| From: report of Alfred Tarski (The Concept of Logical Consequence [1936]) by Ian Rumfitt - The Boundary Stones of Thought 3.2 | |
| A reaction: [compressed] This is Tarski's account of logical consequence, which follows on from his account of truth. 'Logical validity' is then 'true in every model'. Rumfitt doubts whether Tarski has given the meaning of 'logical consequence'. |
| 13344 | X follows from sentences K iff every model of K also models X [Tarski] |
| Full Idea: The sentence X follows logically from the sentences of the class K if and only if every model of the class K is also a model of the sentence X. | |
| From: Alfred Tarski (The Concept of Logical Consequence [1936], p.417) | |
| A reaction: [see Idea 13343 for his account of a 'model'] He is offering to define logical consequence in general, but this definition fits what we now call 'semantic consequence', written |=. This it is standard practice to read |= as 'models'. |
| 13343 | A 'model' is a sequence of objects which satisfies a complete set of sentential functions [Tarski] |
| Full Idea: An arbitrary sequence of objects which satisfies every sentential function of the sentences L' will be called a 'model' or realization of the class L of sentences. There can also be a model of a single sentence is this way. | |
| From: Alfred Tarski (The Concept of Logical Consequence [1936], p.417) | |
| A reaction: [L' is L with the constants replaced by variables] Tarski is the originator of model theory, which is central to modern logic. The word 'realization' is a helpful indicator of what he has in mind. A model begins to look like a possible world. |
| 10082 | There are infinite sets that are not enumerable [Cantor, by Smith,P] |
| Full Idea: Cantor's Theorem (1874) says there are infinite sets that are not enumerable. This is proved by his 1891 'diagonal argument'. | |
| From: report of George Cantor (works [1880]) by Peter Smith - Intro to Gödel's Theorems 2.3 | |
| A reaction: [Smith summarises the diagonal argument] |
| 13483 | Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD] |
| Full Idea: The problem of Cantor's Paradox is that the power set of the universe has to be both bigger than the universe (by Cantor's theorem) and not bigger (since it is a subset of the universe). | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 3 | |
| A reaction: Russell eliminates the 'universe' in his theory of types. I don't see why you can't just say that the members of the set are hypothetical rather than real, and that hypothetically the universe might contain more things than it does. |
| 8710 | The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend] |
| Full Idea: Cantor's Paradox says that the powerset of a set has a cardinal number strictly greater than the original set, but that means that the powerset of the set of all the cardinal numbers is greater than itself. | |
| From: report of George Cantor (works [1880]) by Michèle Friend - Introducing the Philosophy of Mathematics | |
| A reaction: Friend cites this with the Burali-Forti paradox and the Russell paradox as the best examples of the problems of set theory in the early twentieth century. Did this mean that sets misdescribe reality, or that we had constructed them wrongly? |
| 15910 | Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine] |
| Full Idea: Cantor believed he had discovered that between the finite and the 'Absolute', which is 'incomprehensible to the human understanding', there is a third category, which he called 'the transfinite'. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.4 |
| 15905 | Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine] |
| Full Idea: In 1878 Cantor published the unexpected result that one can put the points on a plane, or indeed any n-dimensional space, into one-to-one correspondence with the points on a line. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.1 |
| 9983 | Cantor took the ordinal numbers to be primary [Cantor, by Tait] |
| Full Idea: Cantor took the ordinal numbers to be primary: in his generalization of the cardinals and ordinals into the transfinite, it is the ordinals that he calls 'numbers'. | |
| From: report of George Cantor (works [1880]) by William W. Tait - Frege versus Cantor and Dedekind VI | |
| A reaction: [Tait says Dedekind also favours the ordinals] It is unclear how the matter might be settled. Humans cannot give the cardinality of large groups without counting up through the ordinals. A cardinal gets its meaning from its place in the ordinals? |
| 17798 | Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry] |
| Full Idea: Cantor taught us to regard the totality of natural numbers, which was formerly thought to be infinite, as really finite after all. | |
| From: report of George Cantor (works [1880]) by John Mayberry - What Required for Foundation for Maths? p.414-2 | |
| A reaction: I presume this is because they are (by definition) countable. |
| 9971 | Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait] |
| Full Idea: Cantor introduced the distinction between cardinal and ordinal numbers. | |
| From: report of George Cantor (works [1880]) by William W. Tait - Frege versus Cantor and Dedekind Intro | |
| A reaction: This seems remarkably late for what looks like a very significant clarification. The two concepts coincide in finite cases, but come apart in infinite cases (Tait p.58). |
| 9892 | Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett] |
| Full Idea: Cantor's work revealed that the notion of an ordinal number is more fundamental than that of a cardinal number. | |
| From: report of George Cantor (works [1880]) by Michael Dummett - Frege philosophy of mathematics Ch.23 | |
| A reaction: Dummett makes it sound like a proof, which I find hard to believe. Is the notion that I have 'more' sheep than you logically prior to how many sheep we have? If I have one more, that implies the next number, whatever that number may be. Hm. |
| 14136 | A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor] |
| Full Idea: The cardinal number of M is the general idea which, by means of our active faculty of thought, is deduced from the collection M, by abstracting from the nature of its diverse elements and from the order in which they are given. | |
| From: George Cantor (works [1880]), quoted by Bertrand Russell - The Principles of Mathematics §284 | |
| A reaction: [Russell cites 'Math. Annalen, XLVI, §1'] See Fine 1998 on this. |
| 15906 | Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine] |
| Full Idea: Cantor said he could show that every infinite set of points on the line could be placed into one-to-one correspondence with either the natural numbers or the real numbers - with no intermediate possibilies (the Continuum hypothesis). His proof failed. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.1 |
| 11015 | Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read] |
| Full Idea: Cantor's diagonal argument showed that all the infinite decimals between 0 and 1 cannot be written down even in a single never-ending list. | |
| From: report of George Cantor (works [1880]) by Stephen Read - Thinking About Logic Ch.6 |
| 15903 | A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine] |
| Full Idea: Cantor's theory of Cauchy sequences defines a real number to be associated with an infinite set of infinite sequences of rational numbers. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II.6 | |
| A reaction: This sounds remarkably like the endless decimals we use when we try to write down an actual real number. |
| 18251 | Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine] |
| Full Idea: Cantor introduced irrationals to play the role of limits of Cauchy sequences of rational numbers. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite 4.2 |
| 15902 | Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine] |
| Full Idea: From the very nature of an irrational number, it seems necessary to understand the mathematical infinite thoroughly before an adequate theory of irrationals is possible. Infinite classes are obvious in the Dedekind Cut, but have logical difficulties | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II Intro | |
| A reaction: Almost the whole theory of analysis (calculus) rested on the irrationals, so a theory of the infinite was suddenly (in the 1870s) vital for mathematics. Cantor wasn't just being eccentric or mystical. |
| 15908 | It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine] |
| Full Idea: Cantor's 1891 diagonal argument revealed there are infinitely many infinite powers. Indeed, it showed more: it shows that given any set there is another of greater power. Hence there is an infinite power strictly greater than that of the set of the reals. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.2 |
| 13464 | Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD] |
| Full Idea: What we might call 'Cantor's Thesis' is that there won't be a potential infinity of any sort unless there is an actual infinity of some sort. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: This idea is nicely calculated to stop Aristotle in his tracks. |
| 10112 | The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman] |
| Full Idea: Cantor showed that the complete totality of natural numbers cannot be mapped 1-1 onto the complete totality of the real numbers - so there are different sizes of infinity. | |
| From: report of George Cantor (works [1880]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.4 |
| 8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
| Full Idea: Cantor's 'continuum hypothesis' is the assertion that there are no infinite cardinalities strictly between the size of the natural numbers and the size of the real numbers. | |
| From: report of George Cantor (works [1880]) by Stewart Shapiro - Thinking About Mathematics 2.4 | |
| A reaction: The tricky question is whether this hypothesis can be proved. |
| 17889 | CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner] |
| Full Idea: Cantor's Continuum Hypothesis (CH) says that for every infinite set X of reals there is either a one-to-one correspondence between X and the natural numbers, or between X and the real numbers. | |
| From: report of George Cantor (works [1880]) by Peter Koellner - On the Question of Absolute Undecidability 1.2 | |
| A reaction: Every single writer I read defines this differently, which drives me crazy, but is also helpfully illuminating. There is a moral there somewhere. |
| 13447 | Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD] |
| Full Idea: Cantor conjectured that there is no size between those of the naturals and the reals - called the 'continuum hypothesis'. The generalized version says that for no infinite set A is there a set larger than A but smaller than P(A). | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: Thus there are gaps between infinite numbers, and the power set is the next size up from any infinity. Much discussion as ensued about whether these two can be proved. |
| 10883 | Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten] |
| Full Idea: Cantor's Continuum Hypothesis states that there are no sets which are too large for there to be a one-to-one correspondence between the set and the natural numbers, but too small for there to exist a one-to-one correspondence with the real numbers. | |
| From: report of George Cantor (works [1880]) by Leon Horsten - Philosophy of Mathematics §5.1 |
| 13528 | Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS] |
| Full Idea: Cantor's conjecture (the Continuum Hypothesis) is that there are no sets between N and P(N). The 'generalized' version replaces N with an arbitrary infinite set. | |
| From: report of George Cantor (works [1880]) by Robert S. Wolf - A Tour through Mathematical Logic 2.2 | |
| A reaction: The initial impression is that there is a single gap in the numbers, like a hole in ozone layer, but the generalised version implies an infinity of gaps. How can there be gaps in the numbers? Weird. |
| 9555 | Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara] |
| Full Idea: Cantor's Continuum Hypothesis was that there is no cardinal number greater than aleph-null but less than the cardinality of the continuum. | |
| From: report of George Cantor (works [1880]) by Charles Chihara - A Structural Account of Mathematics 05.1 | |
| A reaction: I have no view on this (have you?), but the proposal that there are gaps in the number sequences has to excite all philosophers. |
| 18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
| Full Idea: Cantor's second innovation was to extend the sequence of ordinal numbers into the transfinite, forming a handy scale for measuring infinite cardinalities. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: Struggling with this. The ordinals seem to locate the cardinals, but in what sense do they 'measure' them? |
| 15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
| Full Idea: Cantor's set theory was not of collections in some familiar sense, but of collections that can be counted using the indexes - the finite and transfinite ordinal numbers. ..He treated infinite collections as if they were finite. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I |
| 18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy] |
| Full Idea: Cantor's first innovation was to treat cardinality as strictly a matter of one-to-one correspondence, so that the question of whether two infinite sets are or aren't of the same size suddenly makes sense. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: It makes sense, except that all sets which are infinite but countable can be put into one-to-one correspondence with one another. What's that all about, then? |
| 10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro] |
| Full Idea: Cantor's theorem entails that there are more property extensions than objects. So there are not enough objects in any domain to serve as extensions for that domain. So Frege's view that numbers are objects led to the Caesar problem. | |
| From: report of George Cantor (works [1880]) by Stewart Shapiro - Philosophy of Mathematics 4.6 | |
| A reaction: So the possibility that Caesar might have to be a number arises because otherwise we are threatening to run out of numbers? Is that really the problem? |
| 18176 | Pure mathematics is pure set theory [Cantor] |
| Full Idea: Pure mathematics ...according to my conception is nothing other than pure set theory. | |
| From: George Cantor (works [1880], I.1), quoted by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: [an unpublished paper of 1884] So right at the beginning of set theory this claim was being made, before it was axiomatised, and so on. Zermelo endorsed the view, and it flourished unchallenged until Benacerraf (1965). |
| 8631 | Cantor says that maths originates only by abstraction from objects [Cantor, by Frege] |
| Full Idea: Cantor calls mathematics an empirical science in so far as it begins with consideration of things in the external world; on his view, number originates only by abstraction from objects. | |
| From: report of George Cantor (works [1880]) by Gottlob Frege - Grundlagen der Arithmetik (Foundations) §21 | |
| A reaction: Frege utterly opposed this view, and he seems to have won the day, but I am rather thrilled to find the great Cantor endorsing my own intuitions on the subject. The difficulty is to explain 'abstraction'. |
| 8052 | To find empiricism and science in the same culture is surprising, as they are really incompatible [MacIntyre] |
| Full Idea: There is something extraordinary in the coexistence of empiricism and natural science in the same culture, for they represent radically different and incompatible ways of approaching the world. | |
| From: Alasdair MacIntyre (After Virtue: a Study in Moral Theory [1981], Ch. 7) | |
| A reaction: I would say that science is commitment to an ontology, and empiricism is a commitment to epistemology. It is a very nice point, given the usual assumption that science is an empirical activity. See Idea 7621. Strict empiricism distorts science. |
| 8057 | Unpredictability doesn't entail inexplicability, and predictability doesn't entail explicability [MacIntyre] |
| Full Idea: Just as unpredictability does not entail inexplicability, so predictability does not entail explicability. | |
| From: Alasdair MacIntyre (After Virtue: a Study in Moral Theory [1981], Ch. 8) | |
| A reaction: The second half is not quite as obvious as the first. The location of lightning strikes is an example of the first. He gives examples of the second, but they all seem to be very complex cases which might be explained, if only we knew enough. |
| 8054 | Social sciences discover no law-like generalisations, and tend to ignore counterexamples [MacIntyre] |
| Full Idea: Social sciences have discovered no law-like generalisations whatsoever, ...and for the most part they adopt a very tolerant attitude to counter-examples. | |
| From: Alasdair MacIntyre (After Virtue: a Study in Moral Theory [1981], Ch. 8) | |
| A reaction: I suspect that this is as much to do with a narrow and rigid view of what 'science' is supposed to be, as a failure of the social sciences. Have such sciences explained anything? I suspect that they have explained a lot, often after the facts. |
| 21050 | I can only make decisions if I see myself as part of a story [MacIntyre] |
| Full Idea: I can only answer the question 'What am I to do?' if I can answer the prior question 'Of what story or stories do I find myself a part?'. | |
| From: Alasdair MacIntyre (After Virtue: a Study in Moral Theory [1981], p.201), quoted by Michael J. Sandel - Justice: What's the right thing to do? 09 | |
| A reaction: MacIntyre is a great champion of the narrative view of the Self. Does this mean that if you had total amnesia, but retained other faculties, you could make no decisions? Can you start a new story whenever you like? |
| 8056 | AI can't predict innovation, or consequences, or external relations, or external events [MacIntyre] |
| Full Idea: AI machines have four types of unpredictability: they can't predict radical innovation or future maths proofs; they couldn't predict the outcome of their own decisions; their relations with other computers would be a game-theory tangle; and power failure. | |
| From: Alasdair MacIntyre (After Virtue: a Study in Moral Theory [1981], Ch. 8) | |
| A reaction: This isn't an assertion that they lack 'free will', just a very accurate observation of how the super new machines would face exactly the same problems that we ourselves face. |
| 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. |
| 13345 | Sentences are 'analytical' if every sequence of objects models them [Tarski] |
| Full Idea: A class of sentences can be called 'analytical' if every sequence of objects is a model of it. | |
| From: Alfred Tarski (The Concept of Logical Consequence [1936], p.418) | |
| A reaction: See Idea 13344 and Idea 13343 for the context of this assertion. |
| 8059 | The good life for man is the life spent seeking the good life for man [MacIntyre] |
| Full Idea: The good life for man is the life spent in seeking for the good life for man. | |
| From: Alasdair MacIntyre (After Virtue: a Study in Moral Theory [1981], Ch.15) | |
| A reaction: This contains a self-evident paradox - that success would be failure. The proposal suits philosophers more than it would suit the folk. Less seeking and more getting on with it seems good, if the activity is a 'flourishing' one. |
| 8034 | We still have the appearance and language of morality, but we no longer understand it [MacIntyre] |
| Full Idea: We possess simulacra of morality, we continue to use many of the key expressions. But we have - very largely, if not entirely - lost our comprehension, both theoretical and practical, of morality. | |
| From: Alasdair MacIntyre (After Virtue: a Study in Moral Theory [1981], Ch. 1) | |
| A reaction: MacIntyre's famous (or notorious) assault on modern ethics. We obviously can't prove him wrong by spouting moral talk. Are we actually more wicked than our ancestors? There is, I think, a relativism problem in the 20th centurty, but that is different. |
| 8036 | Unlike expressions of personal preference, evaluative expressions do not depend on context [MacIntyre] |
| Full Idea: There are good reasons for distinguishing between expressions of personal preference and evaluative expressions, as the first depend on who utters them to whom, while the second are not dependent for reason-giving force on the context of utterance. | |
| From: Alasdair MacIntyre (After Virtue: a Study in Moral Theory [1981], Ch. 2) | |
| A reaction: The sceptics will simply say that in the second type of expression the speaker tries to adopt a tone of impersonal authority, but it is merely an unjustified attempt to elevate personal preferences. "Blue just IS the best colour". |
| 8049 | Moral judgements now are anachronisms from a theistic age [MacIntyre] |
| Full Idea: Moral judgements are linguistic survivals from the practices of classical theism which have lost the context provided by these practices. | |
| From: Alasdair MacIntyre (After Virtue: a Study in Moral Theory [1981], Ch. 5) | |
| A reaction: He is sort of right. Richard Taylor is less dramatic and more plausible on this (Ideas 5065, 5066, 5077). Big claims about 'duty' have become rather hollow, but the rights and wrongs of (e.g.) mistreating children don't seem to need theism. |
| 8045 | The failure of Enlightenment attempts to justify morality will explain our own culture [MacIntyre] |
| Full Idea: A central thesis of this book is that the breakdown of the project (of 1630 to 1850) of an independent rational justification of morality provided the historical background against which the predicaments of our own culture can become intelligible. | |
| From: Alasdair MacIntyre (After Virtue: a Study in Moral Theory [1981], Ch. 4) | |
| A reaction: Possibly the most important question of our times is whether the Enlightenment failed. MacIntyre's claim is followed by an appeal for a return to Aristotelian/Thomist virtues. Continentals seem to have responded by sliding into relativism. |
| 8051 | Mention of 'intuition' in morality means something has gone wrong with the argument [MacIntyre] |
| Full Idea: The introduction of the word 'intuition' by a moral philosopher is always a signal that something has gone badly wrong with an argument. | |
| From: Alasdair MacIntyre (After Virtue: a Study in Moral Theory [1981], Ch. 6) | |
| A reaction: For the alternative view, see Kripke (Idea 4948). If Kripke is right about logic, I don't see why the same view should have some force in morality. At the bottom of all morality is an intuition that life is worth the struggle. How do you prove that? |
| 8048 | When 'man' is thought of individually, apart from all roles, it ceases to be a functional concept [MacIntyre] |
| Full Idea: It is only when man is thought of as an individual prior to and apart from all roles that 'man' ceases to be a functional concept. | |
| From: Alasdair MacIntyre (After Virtue: a Study in Moral Theory [1981], Ch. 5) | |
| A reaction: This is the one key idea at the heart of the revival of virtue ethics in modern times. It pinpoints what may be the single biggest disaster in intellectual history - the isolation of the individual. Yet it led to freedom, rights, and lots of good things. |
| 8035 | In trying to explain the type of approval involved, emotivists are either silent, or viciously circular [MacIntyre] |
| Full Idea: In reply to the question of what kinds of approval are expressed by the feelings or attitudes of moral judgments, every version of emotivism either remains silent, or becomes viciously circular by identifying it as moral approval. | |
| From: Alasdair MacIntyre (After Virtue: a Study in Moral Theory [1981], Ch. 2) | |
| A reaction: There seems to be an underlying assumption that moral judgements are sharply separated from other judgements, of which I am not convinced. I approve of creating a beautiful mural for an old folks home free of charge, but it must be beautiful. |
| 8037 | The expression of feeling in a sentence is in its use, not in its meaning [MacIntyre] |
| Full Idea: Expression of feeling is not a function of the meaning of sentences, but of their use, as when a teacher shouts at a pupil "7 x 7 = 49!", where the expression of feeling or attitude has nothing whatsoever to do with its meaning. | |
| From: Alasdair MacIntyre (After Virtue: a Study in Moral Theory [1981], Ch. 2) | |
| A reaction: This point is what underlies the Frege-Geach problem for emotivism, and is a very telling point. Apart from in metaethics, no one has ever put forward a theory of meaning that says it is just emotion. ...Unless it concerns speakers' intentions? |
| 8040 | Emotivism cannot explain the logical terms in moral discourse ('therefore', 'if..then') [MacIntyre] |
| Full Idea: Analytical moral philosophers resist emotivism because moral reasoning does occur, but there can be logical linkages between various moral judgements of a kind that emotivism could not allow for ('therefore' and 'if...then' express no moral feelings). | |
| From: Alasdair MacIntyre (After Virtue: a Study in Moral Theory [1981], Ch. 2) | |
| A reaction: This is the 'Frege-Geach Problem', nicely expressed, and is the key reason why emotivism seems unacceptable - it is a theory about language, but it just doesn't explain moral discourse sufficiently. |
| 8042 | Nowadays most people are emotivists, and it is embodied in our culture [MacIntyre] |
| Full Idea: To a large degree people now think, talk and act as if emotivism was true, no matter what their avowed theoretical standpoint may be. Emotivism has become embodied in our culture. | |
| From: Alasdair MacIntyre (After Virtue: a Study in Moral Theory [1981], Ch. 2) | |
| A reaction: I suspect that it is moderately educated people who have swallowed emotivism, in the same way that they have swallowed relativism; it provides an excuse for neglectly the pursuit of beauty, goodness and truth, in favour of pleasure. |
| 8058 | Maybe we can only understand rules if we first understand the virtues [MacIntyre] |
| Full Idea: Maybe we need to attend to the virtues first in the first place in order to understand the function and authority of rules. | |
| From: Alasdair MacIntyre (After Virtue: a Study in Moral Theory [1981], Ch. 9) | |
| A reaction: I think MacIntyre's project is exactly right. Morality is about how humans should live their lives. A bunch of robots could implement a set of moral rules, or make contracts, or maximise one another's benefits. The idea of a human community comes first. |
| 7097 | Virtue is secondary to a role-figure, defined within a culture [MacIntyre, by Statman] |
| Full Idea: MacIntyre argues that the concept of virtue is secondary to that of a role-figure, where the latter is always defined by some particular tradition and culture. | |
| From: report of Alasdair MacIntyre (After Virtue: a Study in Moral Theory [1981]) by Daniel Statman - Introduction to Virtue Ethics §3 | |
| A reaction: MacIntyre is much more of a relativist than Aristotle. There must be some attempt to deal with the problem of a rotten culture which throws up a corrupt role-model. We need a concept of a good culture and of individual flourishing. |
| 8043 | Characters are the masks worn by moral philosophies [MacIntyre] |
| Full Idea: Characters are the masks worn by moral philosophies. | |
| From: Alasdair MacIntyre (After Virtue: a Study in Moral Theory [1981], Ch. 3) | |
| A reaction: This may be presenting character in an excessively moral way. Being lively, for example, is a very distinctive trait of character, but hardly moral. This tells us why philosophers are interested in character, but not why other people are. |
| 8061 | If morality just is emotion, there are no external criteria for judging emotions [MacIntyre] |
| Full Idea: If there is nothing to judgements of virtue and vice except the expression of feelings of approval and disapproval, there can be no criteria external to those feelings by appeal to which we may pass judgement upon them. | |
| From: Alasdair MacIntyre (After Virtue: a Study in Moral Theory [1981], Ch.16) | |
| A reaction: The idea that there can be right and wrong feelings may be the key idea in virtue theory. See Idea 5217. A good person would be ashamed to have a bad feeling. Some emotional responses are intrinsically wicked, apart from actions. |
| 8038 | Since Moore thinks the right action produces the most good, he is a utilitarian [MacIntyre] |
| Full Idea: Moore takes it that to call an action right is simply to say that of the available alternative actions it is the one which does or did as a matter of fact produce the most good. Moore is thus a utilitarian. | |
| From: Alasdair MacIntyre (After Virtue: a Study in Moral Theory [1981], Ch. 2) | |
| A reaction: Far be it from me to disagree with MacIntyre on this, but I would have thought that this made him a consequentialist, rather than a utilitarian. Moore doesn't remotely think that pure pleasure or happiness is the good. He's closer to Rashdall (Idea 6673). |
| 8050 | There are no natural or human rights, and belief in them is nonsense [MacIntyre] |
| Full Idea: There are no natural or human rights, and belief in them is one with belief in witches and in unicorns. | |
| From: Alasdair MacIntyre (After Virtue: a Study in Moral Theory [1981], Ch. 6) | |
| A reaction: His point is that the notion of 'rights' only arises out of a community. However, while you might criticise an individual for absurdly asserting all sorts of dubious rights, no one could criticise them if they asserted the right to defend their own life. |
| 10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
| Full Idea: Cantor proved that one-dimensional space has exactly the same number of points as does two dimensions, or our familiar three-dimensional space. | |
| From: report of George Cantor (works [1880]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.14 |
| 13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |
| Full Idea: Cantor said that only God is absolutely infinite. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: We are used to the austere 'God of the philosophers', but this gives us an even more austere 'God of the mathematicians'. |
| 8055 | If God is omniscient, he confronts no as yet unmade decisions, so decisions are impossible [MacIntyre] |
| Full Idea: Omniscience excludes the making of decisions. If God knows everything that will occur, he confronts no as yet unmade decisions. | |
| From: Alasdair MacIntyre (After Virtue: a Study in Moral Theory [1981], Ch. 8) | |
| A reaction: [He cites Aquinas on this] I find it very difficult to see how anyone could read the Bible (see Idea 8008) while keeping this point continually in mind, without seeing the whole book as a piece of blatant anthropomorphism. |