63 ideas
| 15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
| Full Idea: The notion of a function evolved gradually from wanting to see what curves can be represented as trigonometric series. The study of arbitrary functions led Cantor to the ordinal numbers, which led to set theory. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I |
| 13444 | Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD] |
| Full Idea: Cantor's Theorem says that for any set x, its power set P(x) has more members than x. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 |
| 18098 | Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock] |
| Full Idea: Cantor's diagonalisation argument generalises to show that any set has more subsets than it has members. | |
| From: report of George Cantor (works [1880]) by David Bostock - Philosophy of Mathematics 4.5 | |
| A reaction: Thus three members will generate seven subsets. This means that 'there is no end to the series of cardinal numbers' (Bostock p.106). |
| 15505 | If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis] |
| Full Idea: Cantor taught that a set is 'a many, which can be thought of as one'. ...After a time the unfortunate beginner student is told that some classes - the singletons - have only a single member. Here is a just cause for student protest, if ever there was one. | |
| From: report of George Cantor (works [1880]) by David Lewis - Parts of Classes 2.1 | |
| A reaction: There is a parallel question, almost lost in the mists of time, of whether 'one' is a number. 'Zero' is obviously dubious, but if numbers are for counting, that needs units, so the unit is the precondition of counting, not part of it. |
| 10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
| Full Idea: Cantor's theories exhibited the contradictions others had claimed to derive from the supposition of infinite sets as confusions resulting from the failure to mark the necessary distinctions with sufficient clarity. | |
| From: report of George Cantor (works [1880]) by Michael Potter - Set Theory and Its Philosophy Intro 1 |
| 10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
| Full Idea: Cantor discovered that the continuum is the powerset of the integers. While adding or multiplying infinities didn't move up a level of complexity, multiplying a number by itself an infinite number of times did. | |
| From: report of George Cantor (works [1880]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.14 |
| 13016 | The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy] |
| Full Idea: Cantor first stated the Union Axiom in a letter to Dedekind in 1899. It is nearly too obvious to deserve comment from most commentators. Justifications usually rest on 'limitation of size' or on the 'iterative conception'. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Believing the Axioms I §1.3 | |
| A reaction: Surely someone can think of some way to challenge it! An opportunity to become notorious, and get invited to conferences. |
| 14199 | Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley] |
| Full Idea: Cantor's definition of a set was a collection of its members into a whole, but within a few years Dedekind had the idea of a set as a container, enclosing its members like a sack. | |
| From: report of George Cantor (works [1880]) by Oliver,A/Smiley,T - What are Sets and What are they For? Intro | |
| A reaction: As the article goes on to show, these two view don't seem significantly different until you start to ask about the status of the null set and of singletons. I intuitively vote for Dedekind. Set theory is the study of brackets. |
| 10082 | There are infinite sets that are not enumerable [Cantor, by Smith,P] |
| Full Idea: Cantor's Theorem (1874) says there are infinite sets that are not enumerable. This is proved by his 1891 'diagonal argument'. | |
| From: report of George Cantor (works [1880]) by Peter Smith - Intro to Gödel's Theorems 2.3 | |
| A reaction: [Smith summarises the diagonal argument] |
| 13483 | Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD] |
| Full Idea: The problem of Cantor's Paradox is that the power set of the universe has to be both bigger than the universe (by Cantor's theorem) and not bigger (since it is a subset of the universe). | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 3 | |
| A reaction: Russell eliminates the 'universe' in his theory of types. I don't see why you can't just say that the members of the set are hypothetical rather than real, and that hypothetically the universe might contain more things than it does. |
| 8710 | The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend] |
| Full Idea: Cantor's Paradox says that the powerset of a set has a cardinal number strictly greater than the original set, but that means that the powerset of the set of all the cardinal numbers is greater than itself. | |
| From: report of George Cantor (works [1880]) by Michèle Friend - Introducing the Philosophy of Mathematics | |
| A reaction: Friend cites this with the Burali-Forti paradox and the Russell paradox as the best examples of the problems of set theory in the early twentieth century. Did this mean that sets misdescribe reality, or that we had constructed them wrongly? |
| 15910 | Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine] |
| Full Idea: Cantor believed he had discovered that between the finite and the 'Absolute', which is 'incomprehensible to the human understanding', there is a third category, which he called 'the transfinite'. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.4 |
| 15905 | Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine] |
| Full Idea: In 1878 Cantor published the unexpected result that one can put the points on a plane, or indeed any n-dimensional space, into one-to-one correspondence with the points on a line. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.1 |
| 9983 | Cantor took the ordinal numbers to be primary [Cantor, by Tait] |
| Full Idea: Cantor took the ordinal numbers to be primary: in his generalization of the cardinals and ordinals into the transfinite, it is the ordinals that he calls 'numbers'. | |
| From: report of George Cantor (works [1880]) by William W. Tait - Frege versus Cantor and Dedekind VI | |
| A reaction: [Tait says Dedekind also favours the ordinals] It is unclear how the matter might be settled. Humans cannot give the cardinality of large groups without counting up through the ordinals. A cardinal gets its meaning from its place in the ordinals? |
| 17798 | Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry] |
| Full Idea: Cantor taught us to regard the totality of natural numbers, which was formerly thought to be infinite, as really finite after all. | |
| From: report of George Cantor (works [1880]) by John Mayberry - What Required for Foundation for Maths? p.414-2 | |
| A reaction: I presume this is because they are (by definition) countable. |
| 9971 | Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait] |
| Full Idea: Cantor introduced the distinction between cardinal and ordinal numbers. | |
| From: report of George Cantor (works [1880]) by William W. Tait - Frege versus Cantor and Dedekind Intro | |
| A reaction: This seems remarkably late for what looks like a very significant clarification. The two concepts coincide in finite cases, but come apart in infinite cases (Tait p.58). |
| 9892 | Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett] |
| Full Idea: Cantor's work revealed that the notion of an ordinal number is more fundamental than that of a cardinal number. | |
| From: report of George Cantor (works [1880]) by Michael Dummett - Frege philosophy of mathematics Ch.23 | |
| A reaction: Dummett makes it sound like a proof, which I find hard to believe. Is the notion that I have 'more' sheep than you logically prior to how many sheep we have? If I have one more, that implies the next number, whatever that number may be. Hm. |
| 14136 | A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor] |
| Full Idea: The cardinal number of M is the general idea which, by means of our active faculty of thought, is deduced from the collection M, by abstracting from the nature of its diverse elements and from the order in which they are given. | |
| From: George Cantor (works [1880]), quoted by Bertrand Russell - The Principles of Mathematics §284 | |
| A reaction: [Russell cites 'Math. Annalen, XLVI, §1'] See Fine 1998 on this. |
| 15906 | Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine] |
| Full Idea: Cantor said he could show that every infinite set of points on the line could be placed into one-to-one correspondence with either the natural numbers or the real numbers - with no intermediate possibilies (the Continuum hypothesis). His proof failed. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.1 |
| 11015 | Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read] |
| Full Idea: Cantor's diagonal argument showed that all the infinite decimals between 0 and 1 cannot be written down even in a single never-ending list. | |
| From: report of George Cantor (works [1880]) by Stephen Read - Thinking About Logic Ch.6 |
| 15903 | A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine] |
| Full Idea: Cantor's theory of Cauchy sequences defines a real number to be associated with an infinite set of infinite sequences of rational numbers. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II.6 | |
| A reaction: This sounds remarkably like the endless decimals we use when we try to write down an actual real number. |
| 18251 | Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine] |
| Full Idea: Cantor introduced irrationals to play the role of limits of Cauchy sequences of rational numbers. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite 4.2 |
| 15902 | Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine] |
| Full Idea: From the very nature of an irrational number, it seems necessary to understand the mathematical infinite thoroughly before an adequate theory of irrationals is possible. Infinite classes are obvious in the Dedekind Cut, but have logical difficulties | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II Intro | |
| A reaction: Almost the whole theory of analysis (calculus) rested on the irrationals, so a theory of the infinite was suddenly (in the 1870s) vital for mathematics. Cantor wasn't just being eccentric or mystical. |
| 15908 | It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine] |
| Full Idea: Cantor's 1891 diagonal argument revealed there are infinitely many infinite powers. Indeed, it showed more: it shows that given any set there is another of greater power. Hence there is an infinite power strictly greater than that of the set of the reals. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.2 |
| 13464 | Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD] |
| Full Idea: What we might call 'Cantor's Thesis' is that there won't be a potential infinity of any sort unless there is an actual infinity of some sort. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: This idea is nicely calculated to stop Aristotle in his tracks. |
| 10112 | The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman] |
| Full Idea: Cantor showed that the complete totality of natural numbers cannot be mapped 1-1 onto the complete totality of the real numbers - so there are different sizes of infinity. | |
| From: report of George Cantor (works [1880]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.4 |
| 8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
| Full Idea: Cantor's 'continuum hypothesis' is the assertion that there are no infinite cardinalities strictly between the size of the natural numbers and the size of the real numbers. | |
| From: report of George Cantor (works [1880]) by Stewart Shapiro - Thinking About Mathematics 2.4 | |
| A reaction: The tricky question is whether this hypothesis can be proved. |
| 17889 | CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner] |
| Full Idea: Cantor's Continuum Hypothesis (CH) says that for every infinite set X of reals there is either a one-to-one correspondence between X and the natural numbers, or between X and the real numbers. | |
| From: report of George Cantor (works [1880]) by Peter Koellner - On the Question of Absolute Undecidability 1.2 | |
| A reaction: Every single writer I read defines this differently, which drives me crazy, but is also helpfully illuminating. There is a moral there somewhere. |
| 13447 | Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD] |
| Full Idea: Cantor conjectured that there is no size between those of the naturals and the reals - called the 'continuum hypothesis'. The generalized version says that for no infinite set A is there a set larger than A but smaller than P(A). | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: Thus there are gaps between infinite numbers, and the power set is the next size up from any infinity. Much discussion as ensued about whether these two can be proved. |
| 10883 | Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten] |
| Full Idea: Cantor's Continuum Hypothesis states that there are no sets which are too large for there to be a one-to-one correspondence between the set and the natural numbers, but too small for there to exist a one-to-one correspondence with the real numbers. | |
| From: report of George Cantor (works [1880]) by Leon Horsten - Philosophy of Mathematics §5.1 |
| 13528 | Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS] |
| Full Idea: Cantor's conjecture (the Continuum Hypothesis) is that there are no sets between N and P(N). The 'generalized' version replaces N with an arbitrary infinite set. | |
| From: report of George Cantor (works [1880]) by Robert S. Wolf - A Tour through Mathematical Logic 2.2 | |
| A reaction: The initial impression is that there is a single gap in the numbers, like a hole in ozone layer, but the generalised version implies an infinity of gaps. How can there be gaps in the numbers? Weird. |
| 9555 | Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara] |
| Full Idea: Cantor's Continuum Hypothesis was that there is no cardinal number greater than aleph-null but less than the cardinality of the continuum. | |
| From: report of George Cantor (works [1880]) by Charles Chihara - A Structural Account of Mathematics 05.1 | |
| A reaction: I have no view on this (have you?), but the proposal that there are gaps in the number sequences has to excite all philosophers. |
| 18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
| Full Idea: Cantor's second innovation was to extend the sequence of ordinal numbers into the transfinite, forming a handy scale for measuring infinite cardinalities. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: Struggling with this. The ordinals seem to locate the cardinals, but in what sense do they 'measure' them? |
| 15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
| Full Idea: Cantor's set theory was not of collections in some familiar sense, but of collections that can be counted using the indexes - the finite and transfinite ordinal numbers. ..He treated infinite collections as if they were finite. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I |
| 18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy] |
| Full Idea: Cantor's first innovation was to treat cardinality as strictly a matter of one-to-one correspondence, so that the question of whether two infinite sets are or aren't of the same size suddenly makes sense. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: It makes sense, except that all sets which are infinite but countable can be put into one-to-one correspondence with one another. What's that all about, then? |
| 10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro] |
| Full Idea: Cantor's theorem entails that there are more property extensions than objects. So there are not enough objects in any domain to serve as extensions for that domain. So Frege's view that numbers are objects led to the Caesar problem. | |
| From: report of George Cantor (works [1880]) by Stewart Shapiro - Philosophy of Mathematics 4.6 | |
| A reaction: So the possibility that Caesar might have to be a number arises because otherwise we are threatening to run out of numbers? Is that really the problem? |
| 18176 | Pure mathematics is pure set theory [Cantor] |
| Full Idea: Pure mathematics ...according to my conception is nothing other than pure set theory. | |
| From: George Cantor (works [1880], I.1), quoted by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: [an unpublished paper of 1884] So right at the beginning of set theory this claim was being made, before it was axiomatised, and so on. Zermelo endorsed the view, and it flourished unchallenged until Benacerraf (1965). |
| 8631 | Cantor says that maths originates only by abstraction from objects [Cantor, by Frege] |
| Full Idea: Cantor calls mathematics an empirical science in so far as it begins with consideration of things in the external world; on his view, number originates only by abstraction from objects. | |
| From: report of George Cantor (works [1880]) by Gottlob Frege - Grundlagen der Arithmetik (Foundations) §21 | |
| A reaction: Frege utterly opposed this view, and he seems to have won the day, but I am rather thrilled to find the great Cantor endorsing my own intuitions on the subject. The difficulty is to explain 'abstraction'. |
| 15148 | Powers give explanations, without being necessary for some class membership [Chakravartty] |
| Full Idea: Powers explain behaviours regardless of whether they are necessary for membership in a particular class of things. | |
| From: Anjan Chakravarrty (Inessential Aristotle: Powers without Essences [2012], 3) | |
| A reaction: This seems right, and is important for driving a wedge between powers and essences. If there are essences, they are not simply some bunch of powers. |
| 15145 | A kind essence is the necessary and sufficient properties for membership of a class [Chakravartty] |
| Full Idea: The modern concept of a kind essence is a set of intrinsic properties that are individually necessary and jointly sufficient for the membership of something in a class of things, or 'kind'. | |
| From: Anjan Chakravarrty (Inessential Aristotle: Powers without Essences [2012], 2) | |
| A reaction: I am always struck by the problem that the kind itself is constructed from the individuals, so circularity always seems to loom. |
| 15147 | Cluster kinds are explained simply by sharing some properties, not by an 'essence' [Chakravartty] |
| Full Idea: The fact that members of some cluster kinds are subjects of causal generalizations reflects the degree to which they share causally efficacious properties, not the fact that they may be composed of essence kinds per se. | |
| From: Anjan Chakravarrty (Inessential Aristotle: Powers without Essences [2012], 2) | |
| A reaction: I think this is right. I am a fan of individual essences, but not of kind essences. I take kinds, and kind explanations, to be straightforward inductive generalisations from individuals. Extreme stabilities give the illusion of a kind essence. |
| 15144 | Explanation of causal phenomena concerns essential kinds - but also lack of them [Chakravartty] |
| Full Idea: Scientific practices such as prediction and explanation regarding causal phenomena are concerned not merely with kinds having essences, but also with kinds lacking them. | |
| From: Anjan Chakravarrty (Inessential Aristotle: Powers without Essences [2012], 1) | |
| A reaction: Not quite clear what he has in mind, but explanation should certainly involve a coherent picture, and not just the citation of some underlying causal mechanism. |
| 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. |
| 22673 | Wherever there is a small community, the association of the people is natural [Tocqueville] |
| Full Idea: The village or township is the only association which is so perfectly natural that, wherever a number of men are collected, it seems to constitute itself. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.04) | |
| A reaction: Seems like a chicken and egg issue. I would have thought that association precedes the development of a village. |
| 22676 | The people are just individuals, and only present themselves as united to foreigners [Tocqueville] |
| Full Idea: The people in themselves are only individuals; and the special reason why they need to be united under one government is that they may appear to advantage before foreigners. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.07) | |
| A reaction: I take this to be an observation on 1830s America, rather than a universal truth. It fits modern western societies rather well though. |
| 22679 | Vast empires are bad for well-being and freedom, though they may promote glory [Tocqueville] |
| Full Idea: Nothing is more opposed to the well-being and the freedom of men than vast empires. …But there is a love of glory in those who regard the applause of a great people as a worthy reward. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.07) | |
| A reaction: Presumably the main the problem is the central dominance over distant colonies. There may also be some freedom in being distant from the centres, especially in 1830. The Wild West. |
| 22680 | People would be much happier and freer in small nations [Tocqueville] |
| Full Idea: If none but small nations existed, I do not doubt that mankind would be more happy and more free. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.07) | |
| A reaction: In modern times many small states have appeared in Europe (in the Balkans and on the Baltic), and it looks to me a good thing. The prospect of Scottish independence may currently be looming, and De Tocqueville would approve. |
| 22675 | In American judges rule according to the Constitution, not the law [Tocqueville] |
| Full Idea: The Americans have acknowledged the right of judges to found their decisions on the Constitution, rather than on the laws. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.05) | |
| A reaction: Obviously the Constitution is one short document, so the details must be enshrined in the laws (which presumably defer to the Constitution). |
| 22677 | A monarchical family is always deeply concerned with the interests of the state [Tocqueville] |
| Full Idea: The advantages of a monarchy are that the private interests of a family are connected with the interests of the state, …and at least there is always someone available to conduct the affairs of a monarchy. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.07) | |
| A reaction: The second one is not much of a reason! The same defence can be given for the dominance of the Mafia. His defences are deliberately feeble, I suspect. England had plenty of monarchs who showed limited interest. |
| 22683 | Despots like to see their own regulations ignored, by themselves and their agents [Tocqueville] |
| Full Idea: In despotic states the sovereign is so much attached to his power that he dislikes the constraints even of his own regulations, and likes to see his agents acting irregularly. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.11) | |
| A reaction: A nice observation. What would Machiavelli say? At least the citizens can see where the real power resides. |
| 22669 | Aristocracy is constituted by inherited landed property [Tocqueville] |
| Full Idea: Land is the basis of an aristocracy; …it is by landed property handed down from generation to generation that an aristocracy is constituted. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.01) | |
| A reaction: Presumably there can be aristocrats by mere royal patronage, who have perhaps gambled away their land. They need protection by the other aristocrats. |
| 22674 | In Europe it is thought that local government is best handled centrally [Tocqueville] |
| Full Idea: The partisans of centralisation in Europe are wont to maintain that the government can administer the affairs of each locality better than the citizens can do it for themselves. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.04) | |
| A reaction: In the modern UK we have lots of local government, which is thoroughly starved of funds by the central government. He is contrasting it with the strong local system in the U.S. |
| 22678 | An election, and its lead up time, are always a national crisis [Tocqueville] |
| Full Idea: The period which immediately precedes an election, and that during which the election is taking place, must always be considered as a national crisis. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.07) | |
| A reaction: Rousseau said something similar. Election day in modern Britain is very peaceful and civilised, but it used to be chaotic. The weeks preceding it are invariably nasty. |
| 22682 | Universal suffrage is no guarantee of wise choices [Tocqueville] |
| Full Idea: Universal suffrage is by no means a guarantee of the wisdom of the popular choice. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.11) | |
| A reaction: This was precisely Plato's fear about democracy. There seems no way at all of preventing the people from electing representatives on superficial grounds of personality. |
| 22670 | Slavery undermines the morals and energy of a society [Tocqueville] |
| Full Idea: Slavery dishonours labour; it introduces idleness into society, and with idleness, ignorance and pride, luxury and distress. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.01) | |
| A reaction: A pretty feeble reason (in the 1830s) for disliking slavery. He seems only concerned with the adverse effects on the slave-owning society, and shows no interest in the slaves themselves. |
| 22681 | The liberty of the press is more valuable for what it prevents than what it promotes [Tocqueville] |
| Full Idea: I approve of the liberty of the press from a consideration more of the evils it prevents than of the advantages it ensures. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.10) | |
| A reaction: He accepts the freedom of the press as inevitable in a democracy, but he found U.S. newspapers to be nearly as bad then as they are now. |
| 22672 | It is admirable to elevate the humble to the level of the great, but the opposite is depraved [Tocqueville] |
| Full Idea: One manly and lawful passion for equality elevates the humble to the rank of the great. But there exists also a depraved taste for equality, which impels the weak to attempt to lower the powerful to their own level. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.02) | |
| A reaction: There is a distinction in modern political rhetoric between 'levelling down' and 'levelling up'. Since levelling down is just destructive, and levelling up is unaffordable, it seems obvious that true equality needs to be a compromise. |
| 22671 | Equality can only be established by equal rights for all (or no rights for anyone) [Tocqueville] |
| Full Idea: I know of only two methods of establishing equality in the political world; rights must be given to every citizen, or none at all to anyone. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.02) | |
| A reaction: We may have a vague concept of 'natural' rights, but primarily they are a tool of social engineering. You could grant equal rights on inheritance, for example, which turn out in practice to hugely favour the rich. |
| 15146 | Some kinds, such as electrons, have essences, but 'cluster kinds' do not [Chakravartty] |
| Full Idea: Many of the kinds we theorize about and experiment on today simply do not have essences. We can distinguish 'essence kinds', such as electrons, and 'cluster kinds', such as biological species. | |
| From: Anjan Chakravarrty (Inessential Aristotle: Powers without Essences [2012], 2) | |
| A reaction: This is an important point for essentialists. He offers a strict criterion, in Idea 15145, for mind membership, but we might allow species to have essences by just relaxing the criteria a bit, and acknowledging some vagueness, especially over time. |
| 15151 | Many causal laws do not refer to kinds, but only to properties [Chakravartty] |
| Full Idea: Causal laws often do not make reference to kinds of objects at all, but rather summarize relations between quantitative, causally efficacious properties of objects. | |
| From: Anjan Chakravarrty (Inessential Aristotle: Powers without Essences [2012], 3) | |
| A reaction: This would only be a serious challenge if it was not possible to translate talk of properties into talk of kinds, and vice versa. |
| 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'. |