94 ideas
| 15575 | Knowledge is not a static set of correct propositions, but a continuing search for better interpretations [Polt] |
| Full Idea: Thanks to Heidegger, hermeneutics has gained wider acceptance - that knowledge is not a static set of correct propositions, but a continuing search for better interpretations. | |
| From: Richard Polt (Heidegger: an introduction [1999], 3.§7) | |
| A reaction: I am not sure if I understand the notion of a search that has a refusal to actually find anything as one of its basic principles. |
| 15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
| Full Idea: The notion of a function evolved gradually from wanting to see what curves can be represented as trigonometric series. The study of arbitrary functions led Cantor to the ordinal numbers, which led to set theory. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I |
| 13444 | Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD] |
| Full Idea: Cantor's Theorem says that for any set x, its power set P(x) has more members than x. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 |
| 18098 | Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock] |
| Full Idea: Cantor's diagonalisation argument generalises to show that any set has more subsets than it has members. | |
| From: report of George Cantor (works [1880]) by David Bostock - Philosophy of Mathematics 4.5 | |
| A reaction: Thus three members will generate seven subsets. This means that 'there is no end to the series of cardinal numbers' (Bostock p.106). |
| 15505 | If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis] |
| Full Idea: Cantor taught that a set is 'a many, which can be thought of as one'. ...After a time the unfortunate beginner student is told that some classes - the singletons - have only a single member. Here is a just cause for student protest, if ever there was one. | |
| From: report of George Cantor (works [1880]) by David Lewis - Parts of Classes 2.1 | |
| A reaction: There is a parallel question, almost lost in the mists of time, of whether 'one' is a number. 'Zero' is obviously dubious, but if numbers are for counting, that needs units, so the unit is the precondition of counting, not part of it. |
| 10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
| Full Idea: Cantor's theories exhibited the contradictions others had claimed to derive from the supposition of infinite sets as confusions resulting from the failure to mark the necessary distinctions with sufficient clarity. | |
| From: report of George Cantor (works [1880]) by Michael Potter - Set Theory and Its Philosophy Intro 1 |
| 10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
| Full Idea: Cantor discovered that the continuum is the powerset of the integers. While adding or multiplying infinities didn't move up a level of complexity, multiplying a number by itself an infinite number of times did. | |
| From: report of George Cantor (works [1880]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.14 |
| 13016 | The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy] |
| Full Idea: Cantor first stated the Union Axiom in a letter to Dedekind in 1899. It is nearly too obvious to deserve comment from most commentators. Justifications usually rest on 'limitation of size' or on the 'iterative conception'. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Believing the Axioms I §1.3 | |
| A reaction: Surely someone can think of some way to challenge it! An opportunity to become notorious, and get invited to conferences. |
| 14199 | Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley] |
| Full Idea: Cantor's definition of a set was a collection of its members into a whole, but within a few years Dedekind had the idea of a set as a container, enclosing its members like a sack. | |
| From: report of George Cantor (works [1880]) by Oliver,A/Smiley,T - What are Sets and What are they For? Intro | |
| A reaction: As the article goes on to show, these two view don't seem significantly different until you start to ask about the status of the null set and of singletons. I intuitively vote for Dedekind. Set theory is the study of brackets. |
| 10082 | There are infinite sets that are not enumerable [Cantor, by Smith,P] |
| Full Idea: Cantor's Theorem (1874) says there are infinite sets that are not enumerable. This is proved by his 1891 'diagonal argument'. | |
| From: report of George Cantor (works [1880]) by Peter Smith - Intro to Gödel's Theorems 2.3 | |
| A reaction: [Smith summarises the diagonal argument] |
| 13483 | Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD] |
| Full Idea: The problem of Cantor's Paradox is that the power set of the universe has to be both bigger than the universe (by Cantor's theorem) and not bigger (since it is a subset of the universe). | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 3 | |
| A reaction: Russell eliminates the 'universe' in his theory of types. I don't see why you can't just say that the members of the set are hypothetical rather than real, and that hypothetically the universe might contain more things than it does. |
| 8710 | The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend] |
| Full Idea: Cantor's Paradox says that the powerset of a set has a cardinal number strictly greater than the original set, but that means that the powerset of the set of all the cardinal numbers is greater than itself. | |
| From: report of George Cantor (works [1880]) by Michèle Friend - Introducing the Philosophy of Mathematics | |
| A reaction: Friend cites this with the Burali-Forti paradox and the Russell paradox as the best examples of the problems of set theory in the early twentieth century. Did this mean that sets misdescribe reality, or that we had constructed them wrongly? |
| 15910 | Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine] |
| Full Idea: Cantor believed he had discovered that between the finite and the 'Absolute', which is 'incomprehensible to the human understanding', there is a third category, which he called 'the transfinite'. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.4 |
| 15905 | Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine] |
| Full Idea: In 1878 Cantor published the unexpected result that one can put the points on a plane, or indeed any n-dimensional space, into one-to-one correspondence with the points on a line. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.1 |
| 9983 | Cantor took the ordinal numbers to be primary [Cantor, by Tait] |
| Full Idea: Cantor took the ordinal numbers to be primary: in his generalization of the cardinals and ordinals into the transfinite, it is the ordinals that he calls 'numbers'. | |
| From: report of George Cantor (works [1880]) by William W. Tait - Frege versus Cantor and Dedekind VI | |
| A reaction: [Tait says Dedekind also favours the ordinals] It is unclear how the matter might be settled. Humans cannot give the cardinality of large groups without counting up through the ordinals. A cardinal gets its meaning from its place in the ordinals? |
| 17798 | Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry] |
| Full Idea: Cantor taught us to regard the totality of natural numbers, which was formerly thought to be infinite, as really finite after all. | |
| From: report of George Cantor (works [1880]) by John Mayberry - What Required for Foundation for Maths? p.414-2 | |
| A reaction: I presume this is because they are (by definition) countable. |
| 9971 | Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait] |
| Full Idea: Cantor introduced the distinction between cardinal and ordinal numbers. | |
| From: report of George Cantor (works [1880]) by William W. Tait - Frege versus Cantor and Dedekind Intro | |
| A reaction: This seems remarkably late for what looks like a very significant clarification. The two concepts coincide in finite cases, but come apart in infinite cases (Tait p.58). |
| 9892 | Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett] |
| Full Idea: Cantor's work revealed that the notion of an ordinal number is more fundamental than that of a cardinal number. | |
| From: report of George Cantor (works [1880]) by Michael Dummett - Frege philosophy of mathematics Ch.23 | |
| A reaction: Dummett makes it sound like a proof, which I find hard to believe. Is the notion that I have 'more' sheep than you logically prior to how many sheep we have? If I have one more, that implies the next number, whatever that number may be. Hm. |
| 14136 | A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor] |
| Full Idea: The cardinal number of M is the general idea which, by means of our active faculty of thought, is deduced from the collection M, by abstracting from the nature of its diverse elements and from the order in which they are given. | |
| From: George Cantor (works [1880]), quoted by Bertrand Russell - The Principles of Mathematics §284 | |
| A reaction: [Russell cites 'Math. Annalen, XLVI, §1'] See Fine 1998 on this. |
| 15906 | Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine] |
| Full Idea: Cantor said he could show that every infinite set of points on the line could be placed into one-to-one correspondence with either the natural numbers or the real numbers - with no intermediate possibilies (the Continuum hypothesis). His proof failed. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.1 |
| 11015 | Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read] |
| Full Idea: Cantor's diagonal argument showed that all the infinite decimals between 0 and 1 cannot be written down even in a single never-ending list. | |
| From: report of George Cantor (works [1880]) by Stephen Read - Thinking About Logic Ch.6 |
| 15903 | A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine] |
| Full Idea: Cantor's theory of Cauchy sequences defines a real number to be associated with an infinite set of infinite sequences of rational numbers. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II.6 | |
| A reaction: This sounds remarkably like the endless decimals we use when we try to write down an actual real number. |
| 18251 | Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine] |
| Full Idea: Cantor introduced irrationals to play the role of limits of Cauchy sequences of rational numbers. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite 4.2 |
| 15902 | Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine] |
| Full Idea: From the very nature of an irrational number, it seems necessary to understand the mathematical infinite thoroughly before an adequate theory of irrationals is possible. Infinite classes are obvious in the Dedekind Cut, but have logical difficulties | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II Intro | |
| A reaction: Almost the whole theory of analysis (calculus) rested on the irrationals, so a theory of the infinite was suddenly (in the 1870s) vital for mathematics. Cantor wasn't just being eccentric or mystical. |
| 15908 | It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine] |
| Full Idea: Cantor's 1891 diagonal argument revealed there are infinitely many infinite powers. Indeed, it showed more: it shows that given any set there is another of greater power. Hence there is an infinite power strictly greater than that of the set of the reals. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.2 |
| 13464 | Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD] |
| Full Idea: What we might call 'Cantor's Thesis' is that there won't be a potential infinity of any sort unless there is an actual infinity of some sort. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: This idea is nicely calculated to stop Aristotle in his tracks. |
| 10112 | The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman] |
| Full Idea: Cantor showed that the complete totality of natural numbers cannot be mapped 1-1 onto the complete totality of the real numbers - so there are different sizes of infinity. | |
| From: report of George Cantor (works [1880]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.4 |
| 8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
| Full Idea: Cantor's 'continuum hypothesis' is the assertion that there are no infinite cardinalities strictly between the size of the natural numbers and the size of the real numbers. | |
| From: report of George Cantor (works [1880]) by Stewart Shapiro - Thinking About Mathematics 2.4 | |
| A reaction: The tricky question is whether this hypothesis can be proved. |
| 17889 | CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner] |
| Full Idea: Cantor's Continuum Hypothesis (CH) says that for every infinite set X of reals there is either a one-to-one correspondence between X and the natural numbers, or between X and the real numbers. | |
| From: report of George Cantor (works [1880]) by Peter Koellner - On the Question of Absolute Undecidability 1.2 | |
| A reaction: Every single writer I read defines this differently, which drives me crazy, but is also helpfully illuminating. There is a moral there somewhere. |
| 13447 | Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD] |
| Full Idea: Cantor conjectured that there is no size between those of the naturals and the reals - called the 'continuum hypothesis'. The generalized version says that for no infinite set A is there a set larger than A but smaller than P(A). | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: Thus there are gaps between infinite numbers, and the power set is the next size up from any infinity. Much discussion as ensued about whether these two can be proved. |
| 10883 | Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten] |
| Full Idea: Cantor's Continuum Hypothesis states that there are no sets which are too large for there to be a one-to-one correspondence between the set and the natural numbers, but too small for there to exist a one-to-one correspondence with the real numbers. | |
| From: report of George Cantor (works [1880]) by Leon Horsten - Philosophy of Mathematics §5.1 |
| 13528 | Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS] |
| Full Idea: Cantor's conjecture (the Continuum Hypothesis) is that there are no sets between N and P(N). The 'generalized' version replaces N with an arbitrary infinite set. | |
| From: report of George Cantor (works [1880]) by Robert S. Wolf - A Tour through Mathematical Logic 2.2 | |
| A reaction: The initial impression is that there is a single gap in the numbers, like a hole in ozone layer, but the generalised version implies an infinity of gaps. How can there be gaps in the numbers? Weird. |
| 9555 | Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara] |
| Full Idea: Cantor's Continuum Hypothesis was that there is no cardinal number greater than aleph-null but less than the cardinality of the continuum. | |
| From: report of George Cantor (works [1880]) by Charles Chihara - A Structural Account of Mathematics 05.1 | |
| A reaction: I have no view on this (have you?), but the proposal that there are gaps in the number sequences has to excite all philosophers. |
| 18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
| Full Idea: Cantor's second innovation was to extend the sequence of ordinal numbers into the transfinite, forming a handy scale for measuring infinite cardinalities. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: Struggling with this. The ordinals seem to locate the cardinals, but in what sense do they 'measure' them? |
| 15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
| Full Idea: Cantor's set theory was not of collections in some familiar sense, but of collections that can be counted using the indexes - the finite and transfinite ordinal numbers. ..He treated infinite collections as if they were finite. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I |
| 18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy] |
| Full Idea: Cantor's first innovation was to treat cardinality as strictly a matter of one-to-one correspondence, so that the question of whether two infinite sets are or aren't of the same size suddenly makes sense. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: It makes sense, except that all sets which are infinite but countable can be put into one-to-one correspondence with one another. What's that all about, then? |
| 10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro] |
| Full Idea: Cantor's theorem entails that there are more property extensions than objects. So there are not enough objects in any domain to serve as extensions for that domain. So Frege's view that numbers are objects led to the Caesar problem. | |
| From: report of George Cantor (works [1880]) by Stewart Shapiro - Philosophy of Mathematics 4.6 | |
| A reaction: So the possibility that Caesar might have to be a number arises because otherwise we are threatening to run out of numbers? Is that really the problem? |
| 18176 | Pure mathematics is pure set theory [Cantor] |
| Full Idea: Pure mathematics ...according to my conception is nothing other than pure set theory. | |
| From: George Cantor (works [1880], I.1), quoted by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: [an unpublished paper of 1884] So right at the beginning of set theory this claim was being made, before it was axiomatised, and so on. Zermelo endorsed the view, and it flourished unchallenged until Benacerraf (1965). |
| 8631 | Cantor says that maths originates only by abstraction from objects [Cantor, by Frege] |
| Full Idea: Cantor calls mathematics an empirical science in so far as it begins with consideration of things in the external world; on his view, number originates only by abstraction from objects. | |
| From: report of George Cantor (works [1880]) by Gottlob Frege - Grundlagen der Arithmetik (Foundations) §21 | |
| A reaction: Frege utterly opposed this view, and he seems to have won the day, but I am rather thrilled to find the great Cantor endorsing my own intuitions on the subject. The difficulty is to explain 'abstraction'. |
| 15568 | When we consider possibilities, there must be something we are considering [Polt] |
| Full Idea: We would hardly want to say that a possibility is nothing, since surely we are considering something when we consider possibilities. | |
| From: Richard Polt (Heidegger: an introduction [1999], 1) | |
| A reaction: A nice contribution to the issue of whether modality is a feature of actuality. I would prefer to say that we can self-evidently utter truths and falsehoods about what is or is not possible, in nature, in logic, and maybe in metaphysics. |
| 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. |
| 19906 | All countries are in a mutual state of nature [Locke] |
| Full Idea: All commonwealths are in a state of Nature one with another. | |
| From: John Locke (Second Treatise of Government [1690], 153) | |
| A reaction: A striking remark. It is easy to think that the state of nature no longer exists. International law attempts to rectify this, but diplomacy is much more like negotiations in nature than it is like obedience to laws. |
| 19882 | We are not created for solitude, but are driven into society by our needs [Locke] |
| Full Idea: God, having made man such a creature that, in His own judgement, it was not good for him to be alone, put him under strong obligations of necessity, convenience, and inclination, to drive him into society. | |
| From: John Locke (Second Treatise of Government [1690], 077) | |
| A reaction: This is almost Aristotelian, apart from the individualistic assumption that we are 'driven' into society. The only time I see other people looking generally happy is when they are sitting around at leisure and talking to other people. |
| 19864 | In nature men can dispose of possessions and their persons in any way that is possible [Locke] |
| Full Idea: The estate all men are naturally in is perfect freedom to order their actions, and dispose of their possessions and persons as they think fit, within the bounds of the laws of nature. | |
| From: John Locke (Second Treatise of Government [1690], 004) | |
| A reaction: Note that they have possessions, so property is not an invention of society, but something which society should protect. Presumably Locke thinks they could sell themselves into slavery, which Rousseau rejects. |
| 19865 | There is no subjection in nature, and all creatures of the same species are equal [Locke] |
| Full Idea: Creatures of the same species and rank, promiscuously born to all the same advantages of Nature, are also equal one among another, without subordination or subjection. | |
| From: John Locke (Second Treatise of Government [1690], 004) | |
| A reaction: The birds in my garden don't behave as if that were true. Physical strength is surely a natural inequality. |
| 19866 | The rational law of nature says we are all equal and independent, and should show mutual respect [Locke] |
| Full Idea: The state of Nature has a law of Nature to govern it, which obliges everyone, and reason, which is that law, teaches mankind that all being equal and independent, no one ought to harm another in his life, health, liberty or possessions. | |
| From: John Locke (Second Treatise of Government [1690], 006) | |
| A reaction: He adds that this is because we are all the property of God. Locke is more optimistic than Hobbes or Rousseau about this, since he thinks we have a natural obligation to be nice. |
| 19872 | The animals and fruits of the earth belong to mankind [Locke] |
| Full Idea: All the fruits the earth naturally produces, and beasts it feeds, belong to mankind in common, as they are produced by the spontaneous hand of Nature. | |
| From: John Locke (Second Treatise of Government [1690], 026) | |
| A reaction: Not a popular view among 21st century ecologists, I guess, but this remains the implicit belief of anyone who goes hunting in the woods, and our enclosed gardens seem to endorse the idea. |
| 19907 | There is a natural right to inheritance within a family [Locke] |
| Full Idea: Every man is born with a right before any other man, to inherit, with his brethren, his father's goods. | |
| From: John Locke (Second Treatise of Government [1690], 190) | |
| A reaction: If a child is fully grown, they may well have drifted into a state of partial ownership of the goods of the parent, of which it would be hard then to deprive them. It is hard to see this as a natural right of tiny orphaned infants. |
| 19863 | Politics is the right to make enforceable laws to protect property and the state, for the common good [Locke] |
| Full Idea: Political power is the right of making laws, with penalties up to death, for the preserving of property, employing the force of community in the execution of such laws, in defence of the commonwealth, and only for the common good. | |
| From: John Locke (Second Treatise of Government [1690], 003) | |
| A reaction: Since political power can be used for selfish corruption and genocide, this isn't very accurate, so I take it this is how power ought to be exercised! Notice that defence gets equal billing with his famous defence of property. |
| 5654 | The Second Treatise explores the consequences of the contractual view of the state [Locke, by Scruton] |
| Full Idea: In his second Treatise, Locke gave us perhaps the first extended account of the true logical consequences of Hobbes's contractual view of the state. | |
| From: report of John Locke (Second Treatise of Government [1690]) by Roger Scruton - Short History of Modern Philosophy Ch.14 | |
| A reaction: The issue seems to boil down to an opposition between the Cartesian and the Aristotelian view of the individual, with Locke following Descartes. The alternative, endorsed by Hegel, which I prefer, is that the state is part of human nature. |
| 19888 | A society only begins if there is consent of all the individuals to join it [Locke] |
| Full Idea: The beginning of politic society depends upon the consent of the individuals to join into and make one society. | |
| From: John Locke (Second Treatise of Government [1690], 106) | |
| A reaction: This is the dramatic new political idea (originating with Hobbes), that all of the members must (at some point) consent to the state. In practice we are all born into a state, so it is not clear what this means in real life. |
| 6702 | If anyone enjoys the benefits of government (even using a road) they give tacit assent to its laws [Locke] |
| Full Idea: Every man, that hath an possession, or enjoyment, of any part of the dominions of any government, doth thereby give his tacit consent, and is obliged to obedience to the laws, ..whether it be barely travelling on the highway. | |
| From: John Locke (Second Treatise of Government [1690], 119), quoted by Gordon Graham - Eight Theories of Ethics Ch.8 | |
| A reaction: Locke's famous assertion of an unspoken and inescapable contract, to which we are all subject. Hume gave an effective reply (Idea 6703). Locke has a point though. The more you accept, the more obliged you are. I accept the law more as I get older. |
| 19909 | A politic society is created from a state of nature by a unanimous agreement [Locke] |
| Full Idea: That which makes the community, and brings men out of the loose state of Nature into one politic society, is the agreement that everyone has with the rest to incorporate and act as one body. | |
| From: John Locke (Second Treatise of Government [1690], 211) | |
| A reaction: Geography usually keeps commonwealths in place once they have been established, but some of them become disfunctional hell holes because they are trapped in perpetual disagreement. |
| 19910 | A single will creates the legislature, which is duty-bound to preserve that will [Locke] |
| Full Idea: The essence and union of the society consisting in having one will; the legislative, when once established by the majority, has the declaring and, as it were, keeping of that will. | |
| From: John Locke (Second Treatise of Government [1690], 212) | |
| A reaction: Not far from Rousseau's big idea, apart from the emphasis on the 'majority'. Rousseau reduced the role of the general will to preliminaries and basics, but wanted close to unanimity, so that everyone accepts being a subject, to government and law. |
| 19893 | Anyone who enjoys the benefits of a state has given tacit consent to be part of it [Locke] |
| Full Idea: Every man that hath any possession or enjoyment of any part of the dominions of any government doth thereby give his tacit consent, and is as far forth obliged to obedience to the laws of that government, during such enjoyment. | |
| From: John Locke (Second Treatise of Government [1690], 119) | |
| A reaction: I wondered at the age of about 18 whether I had given tacit consent to be a British citizen. Locke says you only have to travel freely down the highways to give consent! We are all free, of course, to apply for citizenship elsewhere. But Idea 19894. |
| 19894 | You can only become an actual member of a commonwealth by an express promise [Locke] |
| Full Idea: Nothing can make any man a subject or member of a commonwealth but his actually entering into it by positive engagement, and express promise and compact. | |
| From: John Locke (Second Treatise of Government [1690], 122) | |
| A reaction: In practice the indigenous population never do this. But it a clear distinction for foreign residents in any country. States cannot induct resident foreigners into their army, or allow them to vote. |
| 19892 | Children are not born into citizenship of a state [Locke] |
| Full Idea: It is plain, by the practices of governments themselves, as well as by the laws of right reason, that a child is born a subject of no country nor government. | |
| From: John Locke (Second Treatise of Government [1690], 118) | |
| A reaction: At what age do they become citizens, given that there is no induction ceremony? If a small British child were attacked overseas, we would expect the British government to defend its rights. |
| 19885 | Absolute monarchy is inconsistent with civil society [Locke] |
| Full Idea: Absolute monarchy, which by some men is counted for the only government in the world, is inconsistent with civil society, and so can be no form of civil government at all. | |
| From: John Locke (Second Treatise of Government [1690], 090) | |
| A reaction: This is because citizens do not have a 'decisive' power to appeal for redress of injuries. Rousseau thought that there could be an absolute monarchy, as long as the general will agreed it, and its term of office could be brought to an end by the assembly. |
| 19886 | The idea that absolute power improves mankind is confuted by history [Locke] |
| Full Idea: He that thinks absolute power purifies men's blood, and corrects the baseness of human nature, need but read the history of this, or any other age, to be convinced to the contrary. | |
| From: John Locke (Second Treatise of Government [1690], 092) | |
| A reaction: I can't imagine who proposed the view that Locke is attacking, but it will have been some real 17th century thinker. Attitudes to monarchy changed drastically in England, but Louis XIV was still ruling in France. |
| 19903 | Despotism is arbitrary power to kill, based neither on natural equality, nor any social contract [Locke] |
| Full Idea: Despotical power is an absolute, arbitrary power one man over another, to take away his life whenever he pleases; and this is a power which neither Nature gives, for it has made no such distinction between one man and another, nor compact can convey. | |
| From: John Locke (Second Treatise of Government [1690], 172) | |
| A reaction: Colonies of seals, walruses and apes seem to display despotism, based on physical strength, though that is largely to do with mating. There could be such a compact, but Locke would regard it as invalid. |
| 19905 | People stripped of their property are legitimately subject to despotism [Locke] |
| Full Idea: Forfeiture gives despotical power to lords for their own benefit over those who are stripped of all property. ...Despotical power is over such as have no property at all. | |
| From: John Locke (Second Treatise of Government [1690], 173) | |
| A reaction: Nasty! Shylock is stripped of his property by Venice, so these things happened. This is taking the significance of property a long way beyond its role at the beginning of Locke's book. Property is the start of society, but then becomes your passport. |
| 19904 | Legitimate prisoners of war are subject to despotism, because that continues the state of war [Locke] |
| Full Idea: Captives, taken in a just and lawful war, and such only, are subject to a despotical power, which, as it arises not from compact, so neither is it capable of any, but is the state of war continued. | |
| From: John Locke (Second Treatise of Government [1690], 205) | |
| A reaction: How long after a war finishes is such despotism legitimate? What happened to the German prisoners in Russia in 1945? Locke defined despotism as the right to kill, but that is expressly contrary to the rules of war, look you. |
| 19895 | Even the legislature must be preceded by a law which gives it power to make laws [Locke] |
| Full Idea: The first and fundamental positive law of all commonwealths is the establishing of the legislative power, as the first and fundamental natural law which is to govern even the legislative. | |
| From: John Locke (Second Treatise of Government [1690], 134) | |
| A reaction: I think Rousseau says that there cannot be a law which enables the general will to set up legislative powers. It just seems to be something which happens. Locke is threatened with an infinite regress. What legitimises the enabling law? |
| 19900 | The executive must not be the legislature, or they may exempt themselves from laws [Locke] |
| Full Idea: It may be too great temptation to human frailty, apt to grasp at power, for the same persons to have the power of making laws to also have in their hands the power to execute them, whereby they may exempt themselves. | |
| From: John Locke (Second Treatise of Government [1690], 143) | |
| A reaction: The main principles of modern constitutions are devised to avoid corruption. If people were incorruptible (yeah, right) the world would presumably be run very differently, and rather more efficiently, like a good family. |
| 19902 | Any obstruction to the operation of the legislature can be removed forcibly by the people [Locke] |
| Full Idea: Having erect a legislative with the power of making laws, when they are hindered by any force from what is so necessary to society, and wherein the safety and preservation of the people consists, the people have a right to remove it by force. | |
| From: John Locke (Second Treatise of Government [1690], 155) | |
| A reaction: I doubt if he was thinking of the French Revolution, but this will clearly have application to the English events of 1642. The Speaker of the Commons was held down in his chair in the 1620s, so that some legislation could be enacted. |
| 19908 | Rebelling against an illegitimate power is no sin [Locke] |
| Full Idea: It is plain that shaking off a power which force, and not right, hath set over any one, though it have the name of rebellion, yet it is no offence against God. | |
| From: John Locke (Second Treatise of Government [1690], 196) | |
| A reaction: [He cites Hezekiah at 2 Kings 18.7] At this time the English Civil War was referred to as the 'Great Rebellion' (so this is an interesting and brave remark of Locke's), though few people would think that Charles I had illegitimate power. |
| 19911 | If legislators confiscate property, or enslave people, they are no longer owed obedience [Locke] |
| Full Idea: Whenever the legislators endeavour to take away and destroy the property of the people, or reduce them to slavery under arbitrary power, they put themselves into a state of war with the people, who are thereupon absolved from any further obedience. | |
| From: John Locke (Second Treatise of Government [1690], 222) | |
| A reaction: This might fit Louis XVI in 1788. Locke was certainly not averse to consideration the situations in which revolution might be justified. He was trying to be even-handed about 1642. Locke seems to think that without property you ARE a slave. |
| 19901 | The people have supreme power, to depose a legislature which has breached their trust [Locke] |
| Full Idea: There remains still in the people a supreme power to remove or alter the legislative, when they find the legislative act contrary to the trust reposed in them. | |
| From: John Locke (Second Treatise of Government [1690], 149) | |
| A reaction: This seems to be the most important aspect of representative democracy. It is not the power of people to make decisions, but the power to get rid of bad rulers. |
| 19887 | Unanimous consent makes a united community, which is then ruled by the majority [Locke] |
| Full Idea: When any number of men have, by the consent of every individual, made a community, they have thereby made that community into one body, with a power to act as one body, which is only by the will and determination of the majority. | |
| From: John Locke (Second Treatise of Government [1690], 096) | |
| A reaction: This seems to be presume democracy without discussion, although the formation of the community is by universal consent, which is the 'general will'. Rousseau has the constitution also made almost unanimously, not by a majority. |
| 19913 | A master forfeits ownership of slaves he abandons [Locke] |
| Full Idea: A master forfeits the dominion over his slaves whom he hath abandoned. | |
| From: John Locke (Second Treatise of Government [1690], 237) | |
| A reaction: How often did slave owners take a day off, I wonder? Presumably slaves will take back their freedom, even if the masters haven't 'forfeited' their ownership, so Locke's point is fairly academic. |
| 19883 | Slaves captured in a just war have no right to property, so are not part of civil society [Locke] |
| Full Idea: Slave are captives taken in a just war, and by right of Nature subjected to the absolute dominion and arbitrary power of their masters. ...Being not capable of any property, they cannot in that state be considered any part of civil society. | |
| From: John Locke (Second Treatise of Government [1690], 085) | |
| A reaction: If the test of citizenship is being capable of holding property, presumably children and mentally damaged people (including the very old) will also fail to qualify. I see no principled reason why slaves should not be allowed to vote. Note 'just' war. |
| 19870 | If you try to enslave me, you have declared war on me [Locke] |
| Full Idea: He who makes an attempt to enslave me thereby puts himself into a state of war with me. | |
| From: John Locke (Second Treatise of Government [1690], 017) | |
| A reaction: So presumably actual slaves are in a state of permanent war with their owners. What of a woman who is enslaved by her husband? |
| 19871 | Freedom is not absence of laws, but living under laws arrived at by consent [Locke] |
| Full Idea: Liberty of man in society is to be under no other legislative power but that established by consent in the commonwealth. Freedom is not (as Filmer suggests) doing what you please while not tied by any laws. | |
| From: John Locke (Second Treatise of Government [1690], 022) | |
| A reaction: That sounds plausible if the consent is unanimous, but a minority is not free if the laws made by a large majority are a sort of persecution. |
| 19880 | All value depends on the labour involved [Locke] |
| Full Idea: It is labour that puts the difference of value on everything. ...Whatever bread is worth more than acorns, wine than water, that is wholly owing to labour and industry. | |
| From: John Locke (Second Treatise of Government [1690], 040) | |
| A reaction: In capitalism this is nonsense. Supply and demand fix all the values. Locke has slid from labour bestowing ownership to labour bestowing value. No one gets paid on the basis of how hard they work, except on piece rates. |
| 19884 | There is only a civil society if the members give up all of their natural executive rights [Locke] |
| Full Idea: Wherever any number of men so unite into one society as to quite every one his executive power of the law of Nature, and to resign it to the public, there and there only is a civil society. | |
| From: John Locke (Second Treatise of Government [1690], 089) | |
| A reaction: This seems to mean that you must give up your active ('executive') natural rights, but not your passive ones (which are inviolable). |
| 19873 | We all own our bodies, and the work we do is our own [Locke] |
| Full Idea: Every man has a 'property' in his own 'person'. This nobody has any right to it but himself. The 'labour' of his body and the 'work' of his hands, we may say, are properly his. | |
| From: John Locke (Second Treatise of Government [1690], 027) | |
| A reaction: He doesn't have any grounds for this claim. Why doesn't a cow own its body? He slides from my ownership of my laborious efforts to my ownership of what I have been working on. I can't acquire your car by servicing it. |
| 6580 | Locke (and Marx) held that ownership of objects is a natural relation, based on the labour put into it [Locke, by Fogelin] |
| Full Idea: Locke thought that property ownership reflected a natural relationship; for him the primordial notion of the ownership of an object is a function of the labour that one puts into it; Marx held a similar view. | |
| From: report of John Locke (Second Treatise of Government [1690]) by Robert Fogelin - Walking the Tightrope of Reason Ch.3 | |
| A reaction: Marx would have to think that, in order to believe that capitalist ownership of the means of production used by the workers was a fundamental injustice. A deeper Marxism might see the whole idea of 'ownership' as a capitalist (or feudal) conspiracy. |
| 20520 | Locke says 'mixing of labour' entitles you to land, as well as nuts and berries [Wolff,J on Locke] |
| Full Idea: The great advantage of Locke's 'labour-mixing' argument is that it seems it can justify the appropriation of land, as well as nuts and berries. | |
| From: comment on John Locke (Second Treatise of Government [1690]) by Jonathan Wolff - An Introduction to Political Philosophy (Rev) 5 'Locke' | |
| A reaction: The argument is dubious at best, and plausibly downright wicked. How much labour achieves ownership? What of previous people who worked the land but never thought to claim 'ownership'? Suppose I do more labour than you on 'your' land? |
| 19875 | A man's labour gives ownership rights - as long as there are fair shares for all [Locke] |
| Full Idea: The 'labour' being the unquestionable property of the labourer, no man but he can have a right to what that is once joined to, at least where there is enough, and as good left in common for others. | |
| From: John Locke (Second Treatise of Government [1690], 027) | |
| A reaction: The qualification at the end is a crucial (and problematic) addition to his theory. What is the situation when an area of wilderness is 98% owned? What of the single source of water? Who gets the best parts? Getting there first seems crucial. |
| 19874 | If a man mixes his labour with something in Nature, he thereby comes to own it [Locke] |
| Full Idea: Whatever a man removes out of the state that Nature hath provided and left it in, he hath mixed his labour with it, and joined something to it that is his own, and thereby makes it his property. ...This excludes the common right of other men. | |
| From: John Locke (Second Treatise of Government [1690], 027) | |
| A reaction: This is Locke's famous Labour Theory of Value. Does picking it up count as labour? Putting a fence round it? Paying someone else to do the labour? Do bees own their honey? Settlers in the wilderness own nothing on day one? |
| 19877 | Fountain water is everyone's, but a drawn pitcher of water has an owner [Locke] |
| Full Idea: Though the water running in the fountain be every one's, yet who can doubt but that in the pitcher is his only who drew it out? | |
| From: John Locke (Second Treatise of Government [1690], 029) | |
| A reaction: This would certainly be the normal consensus of a community, as long as there is plenty of water. The strong and fit gatherers get all the best firewood, so I suppose that is just tough on the others. |
| 19876 | Gathering natural fruits gives ownership; the consent of other people is irrelevant [Locke] |
| Full Idea: If the first gathering of acorns and apples made them not a man's, nothing else could. ...Will anyone say he had no right to them because he had not the consent of all mankind to make them his? | |
| From: John Locke (Second Treatise of Government [1690], 028) | |
| A reaction: The ideas of Nozick are all in this sentence. Does this idea justify the enclosure of common land? The first member of the community who thought of Locke's labour theory had a huge head's start. Liberal individualism rampant. |
| 19878 | Mixing labour with a thing bestows ownership - as long as the thing is not wasted [Locke] |
| Full Idea: How far has God given us all things 'to enjoy'? As much as any one can make use of to any advantage of his life before it spoils, so much he may by his labour fix a property in. | |
| From: John Locke (Second Treatise of Government [1690], 031) | |
| A reaction: This adds a very different value to Locke's theory, because the person seems to be answerable to fellow citizens if they harvest important resources and then waste them. Where do luxuries fit in? |
| 19898 | Soldiers can be commanded to die, but not to hand over their money [Locke] |
| Full Idea: The sergeant that can command a soldier to march up to the mouth of a cannon ...cannot command that soldier to give him one penny of his money. | |
| From: John Locke (Second Treatise of Government [1690], 139) | |
| A reaction: A very nice and accurate illustration of a principle which runs so deep that it does indeed look like a basis of society. |
| 19879 | A man owns land if he cultivates it, to the limits of what he needs [Locke] |
| Full Idea: As much land as a man tills, plants, improves, cultivates, and can use the product of, so much is his property. | |
| From: John Locke (Second Treatise of Government [1690], 032) | |
| A reaction: Industrial farming rather changes this picture. Does the man himself decide how much he can use the product of, or do the neighbours tell him where his boundaries must be? 'Reason not the need', as King Lear said. What if he stops cultivating it? |
| 19881 | The aim of law is not restraint, but to make freedom possible [Locke] |
| Full Idea: The end of law is not to abolish or restrain, but to preserve and enlarge freedom, for where there is no law there is no freedom. | |
| From: John Locke (Second Treatise of Government [1690], 057) | |
| A reaction: This fits both the liberal and the communitarian view of the matter. Talk of 'freedom' is commonplace in England by this date, where it is hardly mention 60 years earler. John Lilburne almost single-handedly brought this about. |
| 19868 | It is only by a law of Nature that we can justify punishing foreigners [Locke] |
| Full Idea: If by the law of Nature every man hath not a power to punish offences against [the state], as he soberly judges the case to require, I see not how the magistrates of any community can punish an alien of another country. | |
| From: John Locke (Second Treatise of Government [1690], 009) | |
| A reaction: This is a nice point. You can't expect to be above the law in a foreign country, but you have entered into no social contract, unless visiting a place is a sort of contract. Intrusions into air space are often accidental visits. |
| 19867 | Reparation and restraint are the only justifications for punishment [Locke] |
| Full Idea: Reparation and restraint are the only two reasons why one man may lawfully do harm to another, which is that we call punishment. | |
| From: John Locke (Second Treatise of Government [1690], 008) | |
| A reaction: But by 'reparation' does be mean retribution, or compensation? He doesn't rule out capital punishment, but that may qualify as maximum restraint. |
| 19912 | Self-defence is natural, but not the punishment of superiors by inferiors [Locke] |
| Full Idea: It is natural for us to defend life and limb, but that an inferior should punish a superior is against nature. | |
| From: John Locke (Second Treatise of Government [1690], 236) | |
| A reaction: He is obliquely referring to the execution of Charles I, even though he may have been legitimately overthrown. I wonder what exactly he means by 'superior' and 'inferior'. An idea from another age! |
| 19869 | Punishment should make crime a bad bargain, leading to repentance and deterrence [Locke] |
| Full Idea: Each transgression may be punished to that degree, and with so much severity, as will suffice to make it an ill bargain to the offender, give him cause to repent, and terrify others from doing the like. | |
| From: John Locke (Second Treatise of Government [1690], 012) | |
| A reaction: I gather that the consensus among experts is that the biggest deterrence is a high likelihood of being caught, rather than the severity of the punishment. |
| 19899 | The consent of the people is essential for any tax [Locke] |
| Full Idea: The legislative power must not raise taxes on the property of the people without the consent of the people given by themselves or their deputies. | |
| From: John Locke (Second Treatise of Government [1690], 142) | |
| A reaction: He will be thinking of the resistance to Ship Money in the 1630s, which was a step towards civil war. The people of Boston, Ma, may have read this sentence 80 years later! |
| 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'. |