106 ideas
| 15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
| Full Idea: The notion of a function evolved gradually from wanting to see what curves can be represented as trigonometric series. The study of arbitrary functions led Cantor to the ordinal numbers, which led to set theory. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I |
| 15946 | Cantor developed sets from a progression into infinity by addition, multiplication and exponentiation [Cantor, by Lavine] |
| Full Idea: Cantor's development of set theory began with his discovery of the progression 0, 1, ....∞, ∞+1, ∞+2, ..∞x2, ∞x3, ...∞^2, ..∞^3, ...∞^∞, ...∞^∞^∞..... | |
| From: report of George Cantor (Grundlagen (Foundations of Theory of Manifolds) [1883]) by Shaughan Lavine - Understanding the Infinite VIII.2 |
| 9616 | A set is a collection into a whole of distinct objects of our intuition or thought [Cantor] |
| Full Idea: A set is any collection into a whole M of definite, distinct objects m ... of our intuition or thought. | |
| From: George Cantor (The Theory of Transfinite Numbers [1897], p.85), quoted by James Robert Brown - Philosophy of Mathematics Ch.2 | |
| A reaction: This is the original conception of a set, which hit trouble with Russell's Paradox. Cantor's original definition immediately invites thoughts about the status of vague objects. |
| 18098 | Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock] |
| Full Idea: Cantor's diagonalisation argument generalises to show that any set has more subsets than it has members. | |
| From: report of George Cantor (works [1880]) by David Bostock - Philosophy of Mathematics 4.5 | |
| A reaction: Thus three members will generate seven subsets. This means that 'there is no end to the series of cardinal numbers' (Bostock p.106). |
| 13444 | Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD] |
| Full Idea: Cantor's Theorem says that for any set x, its power set P(x) has more members than x. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 |
| 15505 | If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis] |
| Full Idea: Cantor taught that a set is 'a many, which can be thought of as one'. ...After a time the unfortunate beginner student is told that some classes - the singletons - have only a single member. Here is a just cause for student protest, if ever there was one. | |
| From: report of George Cantor (works [1880]) by David Lewis - Parts of Classes 2.1 | |
| A reaction: There is a parallel question, almost lost in the mists of time, of whether 'one' is a number. 'Zero' is obviously dubious, but if numbers are for counting, that needs units, so the unit is the precondition of counting, not part of it. |
| 10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
| Full Idea: Cantor's theories exhibited the contradictions others had claimed to derive from the supposition of infinite sets as confusions resulting from the failure to mark the necessary distinctions with sufficient clarity. | |
| From: report of George Cantor (works [1880]) by Michael Potter - Set Theory and Its Philosophy Intro 1 |
| 10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
| Full Idea: Cantor discovered that the continuum is the powerset of the integers. While adding or multiplying infinities didn't move up a level of complexity, multiplying a number by itself an infinite number of times did. | |
| From: report of George Cantor (works [1880]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.14 |
| 17831 | Cantor gives informal versions of ZF axioms as ways of getting from one set to another [Cantor, by Lake] |
| Full Idea: Cantor gives informal versions of the axioms of ZF as ways of getting from one set to another. | |
| From: report of George Cantor (Later Letters to Dedekind [1899]) by John Lake - Approaches to Set Theory 1.6 | |
| A reaction: Lake suggests that it should therefore be called CZF. |
| 13016 | The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy] |
| Full Idea: Cantor first stated the Union Axiom in a letter to Dedekind in 1899. It is nearly too obvious to deserve comment from most commentators. Justifications usually rest on 'limitation of size' or on the 'iterative conception'. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Believing the Axioms I §1.3 | |
| A reaction: Surely someone can think of some way to challenge it! An opportunity to become notorious, and get invited to conferences. |
| 14199 | Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley] |
| Full Idea: Cantor's definition of a set was a collection of its members into a whole, but within a few years Dedekind had the idea of a set as a container, enclosing its members like a sack. | |
| From: report of George Cantor (works [1880]) by Oliver,A/Smiley,T - What are Sets and What are they For? Intro | |
| A reaction: As the article goes on to show, these two view don't seem significantly different until you start to ask about the status of the null set and of singletons. I intuitively vote for Dedekind. Set theory is the study of brackets. |
| 10082 | There are infinite sets that are not enumerable [Cantor, by Smith,P] |
| Full Idea: Cantor's Theorem (1874) says there are infinite sets that are not enumerable. This is proved by his 1891 'diagonal argument'. | |
| From: report of George Cantor (works [1880]) by Peter Smith - Intro to Gödel's Theorems 2.3 | |
| A reaction: [Smith summarises the diagonal argument] |
| 13483 | Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD] |
| Full Idea: The problem of Cantor's Paradox is that the power set of the universe has to be both bigger than the universe (by Cantor's theorem) and not bigger (since it is a subset of the universe). | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 3 | |
| A reaction: Russell eliminates the 'universe' in his theory of types. I don't see why you can't just say that the members of the set are hypothetical rather than real, and that hypothetically the universe might contain more things than it does. |
| 8710 | The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend] |
| Full Idea: Cantor's Paradox says that the powerset of a set has a cardinal number strictly greater than the original set, but that means that the powerset of the set of all the cardinal numbers is greater than itself. | |
| From: report of George Cantor (works [1880]) by Michčle Friend - Introducing the Philosophy of Mathematics | |
| A reaction: Friend cites this with the Burali-Forti paradox and the Russell paradox as the best examples of the problems of set theory in the early twentieth century. Did this mean that sets misdescribe reality, or that we had constructed them wrongly? |
| 15910 | Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine] |
| Full Idea: Cantor believed he had discovered that between the finite and the 'Absolute', which is 'incomprehensible to the human understanding', there is a third category, which he called 'the transfinite'. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.4 |
| 15905 | Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine] |
| Full Idea: In 1878 Cantor published the unexpected result that one can put the points on a plane, or indeed any n-dimensional space, into one-to-one correspondence with the points on a line. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.1 |
| 9983 | Cantor took the ordinal numbers to be primary [Cantor, by Tait] |
| Full Idea: Cantor took the ordinal numbers to be primary: in his generalization of the cardinals and ordinals into the transfinite, it is the ordinals that he calls 'numbers'. | |
| From: report of George Cantor (works [1880]) by William W. Tait - Frege versus Cantor and Dedekind VI | |
| A reaction: [Tait says Dedekind also favours the ordinals] It is unclear how the matter might be settled. Humans cannot give the cardinality of large groups without counting up through the ordinals. A cardinal gets its meaning from its place in the ordinals? |
| 17798 | Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry] |
| Full Idea: Cantor taught us to regard the totality of natural numbers, which was formerly thought to be infinite, as really finite after all. | |
| From: report of George Cantor (works [1880]) by John Mayberry - What Required for Foundation for Maths? p.414-2 | |
| A reaction: I presume this is because they are (by definition) countable. |
| 9971 | Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait] |
| Full Idea: Cantor introduced the distinction between cardinal and ordinal numbers. | |
| From: report of George Cantor (works [1880]) by William W. Tait - Frege versus Cantor and Dedekind Intro | |
| A reaction: This seems remarkably late for what looks like a very significant clarification. The two concepts coincide in finite cases, but come apart in infinite cases (Tait p.58). |
| 9892 | Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett] |
| Full Idea: Cantor's work revealed that the notion of an ordinal number is more fundamental than that of a cardinal number. | |
| From: report of George Cantor (works [1880]) by Michael Dummett - Frege philosophy of mathematics Ch.23 | |
| A reaction: Dummett makes it sound like a proof, which I find hard to believe. Is the notion that I have 'more' sheep than you logically prior to how many sheep we have? If I have one more, that implies the next number, whatever that number may be. Hm. |
| 15911 | Ordinals are generated by endless succession, followed by a limit ordinal [Cantor, by Lavine] |
| Full Idea: Ordinal numbers are generated by two principles: each ordinal has an immediate successor, and each unending sequence has an ordinal number as its limit (that is, an ordinal that is next after such a sequence). | |
| From: report of George Cantor (Grundlagen (Foundations of Theory of Manifolds) [1883]) by Shaughan Lavine - Understanding the Infinite III.4 |
| 14136 | A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor] |
| Full Idea: The cardinal number of M is the general idea which, by means of our active faculty of thought, is deduced from the collection M, by abstracting from the nature of its diverse elements and from the order in which they are given. | |
| From: George Cantor (works [1880]), quoted by Bertrand Russell - The Principles of Mathematics §284 | |
| A reaction: [Russell cites 'Math. Annalen, XLVI, §1'] See Fine 1998 on this. |
| 11015 | Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read] |
| Full Idea: Cantor's diagonal argument showed that all the infinite decimals between 0 and 1 cannot be written down even in a single never-ending list. | |
| From: report of George Cantor (works [1880]) by Stephen Read - Thinking About Logic Ch.6 |
| 15906 | Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine] |
| Full Idea: Cantor said he could show that every infinite set of points on the line could be placed into one-to-one correspondence with either the natural numbers or the real numbers - with no intermediate possibilies (the Continuum hypothesis). His proof failed. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.1 |
| 15903 | A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine] |
| Full Idea: Cantor's theory of Cauchy sequences defines a real number to be associated with an infinite set of infinite sequences of rational numbers. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II.6 | |
| A reaction: This sounds remarkably like the endless decimals we use when we try to write down an actual real number. |
| 18251 | Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine] |
| Full Idea: Cantor introduced irrationals to play the role of limits of Cauchy sequences of rational numbers. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite 4.2 |
| 15902 | Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine] |
| Full Idea: From the very nature of an irrational number, it seems necessary to understand the mathematical infinite thoroughly before an adequate theory of irrationals is possible. Infinite classes are obvious in the Dedekind Cut, but have logical difficulties | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II Intro | |
| A reaction: Almost the whole theory of analysis (calculus) rested on the irrationals, so a theory of the infinite was suddenly (in the 1870s) vital for mathematics. Cantor wasn't just being eccentric or mystical. |
| 15908 | It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine] |
| Full Idea: Cantor's 1891 diagonal argument revealed there are infinitely many infinite powers. Indeed, it showed more: it shows that given any set there is another of greater power. Hence there is an infinite power strictly greater than that of the set of the reals. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.2 |
| 13464 | Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD] |
| Full Idea: What we might call 'Cantor's Thesis' is that there won't be a potential infinity of any sort unless there is an actual infinity of some sort. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: This idea is nicely calculated to stop Aristotle in his tracks. |
| 10112 | The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman] |
| Full Idea: Cantor showed that the complete totality of natural numbers cannot be mapped 1-1 onto the complete totality of the real numbers - so there are different sizes of infinity. | |
| From: report of George Cantor (works [1880]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.4 |
| 15896 | Cantor needed Power Set for the reals, but then couldn't count the new collections [Cantor, by Lavine] |
| Full Idea: Cantor grafted the Power Set axiom onto his theory when he needed it to incorporate the real numbers, ...but his theory was supposed to be theory of collections that can be counted, but he didn't know how to count the new collections. | |
| From: report of George Cantor (The Theory of Transfinite Numbers [1897]) by Shaughan Lavine - Understanding the Infinite I | |
| A reaction: I take this to refer to the countability of the sets, rather than the members of the sets. Lavine notes that counting was Cantor's key principle, but he now had to abandon it. Zermelo came to the rescue. |
| 17889 | CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner] |
| Full Idea: Cantor's Continuum Hypothesis (CH) says that for every infinite set X of reals there is either a one-to-one correspondence between X and the natural numbers, or between X and the real numbers. | |
| From: report of George Cantor (works [1880]) by Peter Koellner - On the Question of Absolute Undecidability 1.2 | |
| A reaction: Every single writer I read defines this differently, which drives me crazy, but is also helpfully illuminating. There is a moral there somewhere. |
| 10883 | Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten] |
| Full Idea: Cantor's Continuum Hypothesis states that there are no sets which are too large for there to be a one-to-one correspondence between the set and the natural numbers, but too small for there to exist a one-to-one correspondence with the real numbers. | |
| From: report of George Cantor (works [1880]) by Leon Horsten - Philosophy of Mathematics §5.1 |
| 8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
| Full Idea: Cantor's 'continuum hypothesis' is the assertion that there are no infinite cardinalities strictly between the size of the natural numbers and the size of the real numbers. | |
| From: report of George Cantor (works [1880]) by Stewart Shapiro - Thinking About Mathematics 2.4 | |
| A reaction: The tricky question is whether this hypothesis can be proved. |
| 13528 | Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS] |
| Full Idea: Cantor's conjecture (the Continuum Hypothesis) is that there are no sets between N and P(N). The 'generalized' version replaces N with an arbitrary infinite set. | |
| From: report of George Cantor (works [1880]) by Robert S. Wolf - A Tour through Mathematical Logic 2.2 | |
| A reaction: The initial impression is that there is a single gap in the numbers, like a hole in ozone layer, but the generalised version implies an infinity of gaps. How can there be gaps in the numbers? Weird. |
| 9555 | Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara] |
| Full Idea: Cantor's Continuum Hypothesis was that there is no cardinal number greater than aleph-null but less than the cardinality of the continuum. | |
| From: report of George Cantor (works [1880]) by Charles Chihara - A Structural Account of Mathematics 05.1 | |
| A reaction: I have no view on this (have you?), but the proposal that there are gaps in the number sequences has to excite all philosophers. |
| 13447 | Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD] |
| Full Idea: Cantor conjectured that there is no size between those of the naturals and the reals - called the 'continuum hypothesis'. The generalized version says that for no infinite set A is there a set larger than A but smaller than P(A). | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: Thus there are gaps between infinite numbers, and the power set is the next size up from any infinity. Much discussion as ensued about whether these two can be proved. |
| 15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
| Full Idea: Cantor's set theory was not of collections in some familiar sense, but of collections that can be counted using the indexes - the finite and transfinite ordinal numbers. ..He treated infinite collections as if they were finite. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I |
| 18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
| Full Idea: Cantor's second innovation was to extend the sequence of ordinal numbers into the transfinite, forming a handy scale for measuring infinite cardinalities. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: Struggling with this. The ordinals seem to locate the cardinals, but in what sense do they 'measure' them? |
| 18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy] |
| Full Idea: Cantor's first innovation was to treat cardinality as strictly a matter of one-to-one correspondence, so that the question of whether two infinite sets are or aren't of the same size suddenly makes sense. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: It makes sense, except that all sets which are infinite but countable can be put into one-to-one correspondence with one another. What's that all about, then? |
| 9992 | The 'extension of a concept' in general may be quantitatively completely indeterminate [Cantor] |
| Full Idea: The author entirely overlooks the fact that the 'extension of a concept' in general may be quantitatively completely indeterminate. Only in certain cases is the 'extension of a concept' quantitatively determinate. | |
| From: George Cantor (Review of Frege's 'Grundlagen' [1885], 1932:440), quoted by William W. Tait - Frege versus Cantor and Dedekind | |
| A reaction: Cantor presumably has in mind various infinite sets. Tait is drawing our attention to the fact that this objection long precedes Russell's paradox, which made the objection more formal (a language Frege could understand!). |
| 10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro] |
| Full Idea: Cantor's theorem entails that there are more property extensions than objects. So there are not enough objects in any domain to serve as extensions for that domain. So Frege's view that numbers are objects led to the Caesar problem. | |
| From: report of George Cantor (works [1880]) by Stewart Shapiro - Philosophy of Mathematics 4.6 | |
| A reaction: So the possibility that Caesar might have to be a number arises because otherwise we are threatening to run out of numbers? Is that really the problem? |
| 18176 | Pure mathematics is pure set theory [Cantor] |
| Full Idea: Pure mathematics ...according to my conception is nothing other than pure set theory. | |
| From: George Cantor (works [1880], I.1), quoted by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: [an unpublished paper of 1884] So right at the beginning of set theory this claim was being made, before it was axiomatised, and so on. Zermelo endorsed the view, and it flourished unchallenged until Benacerraf (1965). |
| 8631 | Cantor says that maths originates only by abstraction from objects [Cantor, by Frege] |
| Full Idea: Cantor calls mathematics an empirical science in so far as it begins with consideration of things in the external world; on his view, number originates only by abstraction from objects. | |
| From: report of George Cantor (works [1880]) by Gottlob Frege - Grundlagen der Arithmetik (Foundations) §21 | |
| A reaction: Frege utterly opposed this view, and he seems to have won the day, but I am rather thrilled to find the great Cantor endorsing my own intuitions on the subject. The difficulty is to explain 'abstraction'. |
| 8715 | Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend] |
| Full Idea: Cantor (in his exploration of infinities) pushed the bounds of conceivability further than anyone before him. To discover what is conceivable, we have to enquire into the concept. | |
| From: report of George Cantor (works [1880]) by Michčle Friend - Introducing the Philosophy of Mathematics 6.5 | |
| A reaction: This remark comes during a discussion of Husserl's phenomenology. Intuitionists challenge Cantor's claim, and restrict what is conceivable to what is provable. Does possibility depend on conceivability? |
| 13454 | Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor] |
| Full Idea: Cantor thought that we abstract a number as something common to all and only those sets any one of which has as many members as any other. ...However one wants to see the logic of the inference. The irony is that set theory lays out this logic. | |
| From: comment on George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: The logic Hart has in mind is the notion of an equivalence relation between sets. This idea sums up the older and more modern concepts of abstraction, the first as psychological, the second as logical (or trying very hard to be!). Cf Idea 9145. |
| 9145 | We form the image of a cardinal number by a double abstraction, from the elements and from their order [Cantor] |
| Full Idea: We call 'cardinal number' the general concept which, by means of our active faculty of thought, arises when we make abstraction from an aggregate of its various elements, and of their order. From this double abstraction the number is an image in our mind. | |
| From: George Cantor (Beitrage [1915], §1), quoted by Kit Fine - Cantorian Abstraction: Recon. and Defence Intro | |
| A reaction: [compressed] This is the great Cantor, creator of set theory, endorsing the traditional abstractionism which Frege and his followers so despise. Fine offers a defence of it. The Frege view is platonist, because it refuses to connect numbers to the world. |
| 19956 | True goodness is political, and consists of love of and submission to the laws [Montesquieu] |
| Full Idea: The good man is he whose goodness is not Christian, but rather political, in the sense I have given. Such a man loves the laws of his land and is moved to act by them. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], Intro) | |
| A reaction: I take this to have a lot in common with Aristotle, whose simple slogan for virtue I take to as 'be a good citizen'. |
| 19962 | Men do not desire to subjugate one another; domination is a complex and advanced idea [Montesquieu] |
| Full Idea: It is unreasonable to impute to men, as Hobbes does, the desire to subjugate one another. The idea of sovereignty [l'empire] and domination is so complex and depends on so many other ideas, that it could not be the first to occur to men. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 01.02) |
| 19961 | Primitive people would be too vulnerable and timid to attack anyone, so peace would reign [Montesquieu] |
| Full Idea: A being concerned only with preservation would be very timid. In such a state every man would feel himself an inferior; he could scarcely imagine himself an equal. No one would seek to attack anyone else; peace would be the first law of nature. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 01.02) | |
| A reaction: Exactly the idea that Rousseau took up, and they both attack Hobbes for describing a more advanced stage of society, instead of focusing on the original state. A solitary individual would be crazy to launch attacks on other individuals. |
| 19963 | People are drawn into society by needs, shared fears, pleasure, and knowledge [Montesquieu] |
| Full Idea: To his sense of weakness, man would soon add his needs. Encouraged by indications that their fear was shared, men would soon come together. They would feel the pleasure (and sexual attraction) of their own species. Knowledge then draws them into society. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 01.02) | |
| A reaction: He doesn't make the point about 'knowledge' very clear. |
| 20008 | People are guided by a multitude of influences, from which the spirit of a nation emerges [Montesquieu] |
| Full Idea: Men are ruled by many causes: climate, religion, laws, maxims of government, examples drawn from the pasts, customs, manners. Out of them is formed the general spirit of a nation. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 19.04) | |
| A reaction: This is one step away from Rousseau's general will, which is an attempt to give precise expression to this 'spirit of a nation'. |
| 19993 | In small republics citizens identify with the public good, and abuses are fewer [Montesquieu] |
| Full Idea: In a small republic, the public good is more keenly felt, better known, closer to every citizen; abuses are spread less widely, and consequently, are less tolerated. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 08.16) | |
| A reaction: This idea of very small republics now seems outdated, but this idea still applies. Small states like the Baltic States (or Scotland?) have a better chance of the citizens identifying with the whole community. |
| 19992 | In a large republic there is too much wealth for individuals to manage it [Montesquieu] |
| Full Idea: In a large republic, there are large fortunes, and therefore but little moderation in the minds of men. its resources are too considerable to be entrusted to a citizen. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 08.16) |
| 20005 | The rich would never submit to a lottery deciding which part of their society should be slaves [Montesquieu] |
| Full Idea: I do not believe that anyone of [that small part of a nation that is rich and voluptuous] would submit to a lottery determining which part of the nation would be free, and which slave. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 15.09) | |
| A reaction: Wonderful! This is exactly Rawls's 'initial position' and 'veil of ignorance'. It is used here to deconstruct implausible arguments in favour of slavery. |
| 19995 | All states aim at preservation, and then have distinctive individual purposes [Montesquieu] |
| Full Idea: Although all states share the same general objective, to preserve themselves, each has its own particular purpose (such as aggrandisement, war, religion, commerce, tranquillity, navigation, liberty, pleasures of the ruler, glory, individual independence). | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 11.05) | |
| A reaction: [he gives examples for each of the list in brackets] I'm trying to think of the distinctive purpose of the UK, and can't get beyond sport, music gigs and comedy shows. |
| 19964 | The natural power of a father suggests rule by one person, but that authority can be spread [Montesquieu] |
| Full Idea: Some have thought that because nature has established the power of the parent, the most natural government is that of a single person. But the example of paternal power proves nothing. The inheritance by a father's brothers would support rule by the many. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 01.03) | |
| A reaction: [last bit compressed] Locke pointed out that the mother has similar entitlement, and he and Rousseau agree in rejecting this idea. |
| 19972 | The nobility are an indispensable part of a monarchy [Montesquieu] |
| Full Idea: In a sense, nobility is one part of the essence of monarchy, whose fundamenta maxim is: 'without a monarchy, no nobility; without a nobility, no monarchy'. There are, of course, despots, but they are something else. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 02.04) | |
| A reaction: Hence the worst vice associated with a monarchy is patronage, even when the monarch is weak and 'constitutional'. |
| 19986 | Monarchies can act more quickly, because one person is in charge [Montesquieu] |
| Full Idea: Monarchical government has the great advantage that, since public business is guided by a single person, the executive power can operate more speedily. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 05.10) | |
| A reaction: Liberal democracies are particularly hopeless at quick action, because so many views have to be heard. |
| 19974 | Monarchs must not just have links to the people; they need a body which maintains the laws [Montesquieu] |
| Full Idea: In a monarchy, it is not enough to have intermediary ranks; there must also be a body that is a depositary of laws. They must announce the laws when they are made, and recall them to the public's attention when they are forgotten. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 02.04) | |
| A reaction: This is the crucial difference between a monarch and a despot, because the monarch must be subservient to the law. |
| 19976 | Ambition is good in a monarchy, because the monarch can always restrain it [Montesquieu] |
| Full Idea: In a republic ambition is pernicious, but in a monarchy it has a good effect; it gives life to that type of government. Its advantage lies in that it is not dangerous, because a monarchy can continue to restrain it. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 03.07) | |
| A reaction: That is sometimes offered as a defence of the very weak British monarchy. |
| 19978 | In monarchies, men's actions are judged by their grand appearance, not their virtues [Montesquieu] |
| Full Idea: In monarchies, men's actions are judged, not by whether they are good, but whether they appear attractive [belles]; not by whether they are just, but whether they appear grand; not by whether they are reasonable, but by whether they appear extraordinary. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 04.02) | |
| A reaction: A person that comes to mind is the Duke of Buckingham under James I and Charles I. Or the Earl of Essex under Elizabeth I. |
| 19985 | In a monarchy, the nobility must be hereditary, to bind them together [Montesquieu] |
| Full Idea: In a monarchy, the laws must make the nobility hereditary, not to serve as the boundary between the power of the ruler and the weakness of the people, but as the tie that binds them together.... | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 05.09) | |
| A reaction: This seems rather disingenuous. If the nobility are bound together in some tight manner, this immediately serves as a sharp boundary between them and the rest of the people. Monarchs are bound to want the strict boundary. |
| 19989 | The will of a despot is an enigma, so magistrates can only follow their own will [Montesquieu] |
| Full Idea: Under despotism, the law is nothing more than the will of the ruler. Even if the despot were wise, how could a magistrate follow a will unknown to him? He has no choice but to follow his own. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 05.16) |
| 19988 | A despot's agents must be given power, so they inevitably become corrupt [Montesquieu] |
| Full Idea: A government cannot be unjust without putting some power in the hands of its agents; it is impossible that they not profit from their position. Embezzlement is, therefore, natural to such governments. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 05.15) |
| 19975 | Despots are always lazy and ignorant, so they always delegate their power to a vizier [Montesquieu] |
| Full Idea: Anyone whom his senses inform continually that he is everything, and others nothing, is naturally lazy, voluptuous, and ignorant. Hence the establishment of a vizier, with power the same as his own, is a law fundamental to a despotic state. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 02.05) |
| 19977 | Despotism and honour are incompatible, because honour scorns his power, and lives by rules [Montesquieu] |
| Full Idea: How could a despot permit honour? Honour depends upon scorning life; the despot has power only because he can deprive men of life. How could honour tolerate the despot? Honour has fixed rules, ...but the despot has no rule. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 03.08) | |
| A reaction: The old German aristocracy seem to have been utterly alienated and isolated by the Nazi regime. |
| 20007 | Tyranny is either real violence, or the imposition of unpopular legislation [Montesquieu] |
| Full Idea: There are two sorts of tyranny: that which is real and consists of the violence of government; and the tyranny of opinion, when those who govern institute things contrary to a nation's mode of thought. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 19.03) | |
| A reaction: By this reckoning the abolition of the death penalty by the UK partliament was tyrannous, as it went against popular enthusiasm for it. Representative democracy is always in danger of drifting towards mild tyranny. |
| 19970 | If the nobility is numerous, the senate is the artistocracy, and the nobles are a democracy [Montesquieu] |
| Full Idea: When the nobility is numerous, there must be a senate to regulate those matters which the body of nobles is incapable of deciding. Thus aristocracy of a kind resides in the senate, democracy in the body of nobles, while the people is nothing. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 02.03) | |
| A reaction: This presumes that the body of nobles elects the senate. |
| 19971 | Aristocracy is democratic if they resemble the people, but not if they resemble the monarch [Montesquieu] |
| Full Idea: Aristocratic families ought to be, as much as possible, members of the people. The more an aristocracy resembles a democracy, the more perfect it is; the more it resembles a monarchy, the more imperfect. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 02.03) | |
| A reaction: Aristocrats far from the big cities seem remarkably like the rest of the people. As soon as they approach the monarch's court, they aspire to dignity and power, and begin to spurn the citizens. |
| 19984 | Great inequality between aristocrats and the rest is bad - and also among aristocrats themselves [Montesquieu] |
| Full Idea: In aristocratic state there are two main sources of disorders: excessive inequality between those who govern and those who are governed, and the same degree of inequality among the different members of the ruling group [corps]. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 05.08) | |
| A reaction: This sounds like a very historically accurate observation, since aristocrats are always at one another's throats. But maybe junior aristocrats just need to be kept more firmly in their place. |
| 19980 | If a government is to be preserved, it must first be loved [Montesquieu] |
| Full Idea: Government is like everything else in this world: if it is to be preserved, it must first be loved. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 04.05) | |
| A reaction: Nice one! Right now there seems to be a declining love for representative democracy, even though almost everyone endorses it. |
| 19996 | A government has a legislature, an international executive, and a domestic executive [Montesquieu] |
| Full Idea: In every government there are three sorts of powers: the legislative; the executivem in regard to those matters determined by the laws of nations; and the executive, in regard to those matters determined by civil law. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 11.06) |
| 19997 | The judiciary must be separate from the legislature, to avoid arbitrary power [Montesquieu] |
| Full Idea: Were the judicial power joined to the legislative, the life and liberty of the citizens would be subject to arbitrary power. For the judge would then be the legislator. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 11.06) | |
| A reaction: This is the key 'separation of powers', which seems to be a mantra for nearly all theories of the state. |
| 19965 | The fundamental laws of a democracy decide who can vote [Montesquieu] |
| Full Idea: The laws fundamental to a democracy are those that establish who is eligible to vote. (p.118 No less fundamental is the method of voting itself). | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 02.02) | |
| A reaction: Tricky groups now are teenagers, convicted criminals, people with damaged brains, and citizens who live abroad. Maybe people who evade paying tax should lose the right to vote. |
| 19968 | It is basic to a democracy that the people themselves must name their ministers [Montesquieu] |
| Full Idea: A democratic people may be said to have ministers only when these have been named by the people itself. Thus it is a maxim fundamental to this type of government that the people must name its ministers. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 02.02) | |
| A reaction: In the UK we don't do this. We elect local representatives, usually of a preferred party, and then they chose the ministers, and even the leader. The people who run our country are a long way from direct democracy. |
| 19969 | Voting should be public, so the lower classes can be influenced by the example of notable people [Montesquieu] |
| Full Idea: When the people votes it should do so in public. ...For it is necessary that the lower classes be enlightened by those of higher rank, that the precipitous qualities of the lower classes be held in check by the grave example of certain notables. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 02.02) | |
| A reaction: This sounds shocking to us, but the lower classes were largely illiterate. Nowadays we have television to tell us how the notables are going to vote, and the less notables seem to be increasingly unimpressed. |
| 19999 | All citizens (apart from the very humble poor) should choose their representatives [Montesquieu] |
| Full Idea: All citizens ought to have the right to choose their representatives by election. The only exception concerns those whose condition is so base that they are considered to have no will of their own. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 11.06) | |
| A reaction: This is an amazingly liberal view of the franchise for its time (though he may not be including women), but with a rather breathtaking coda! It may be hard for us now to grasp the very humble state of an illiterate peasant. |
| 19967 | In a democracy the people should manage themselves, and only delegate what they can't do [Montesquieu] |
| Full Idea: In a democracy, the people, which holds the sovereign power, ought itself to do everything it can do well; that which it cannot do well must be done by its ministers. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 02.02) | |
| A reaction: This is just what you see when a group of residents manages their own building. Citizens in representative democracies become utterly lazy about running their society, so that they won't even pick up litter, or report communal problems. |
| 19966 | A democratic assembly must have a fixed number, to see whether everyone has spoken [Montesquieu] |
| Full Idea: It is essential to fix the number of citizens who can participate in assemblies. Otherwise it would be uncertain whether all the people had spoken, or only a part of it. At Sparta the number was fixed at ten thousand. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 02.02) | |
| A reaction: This looks like an immediate injustice to the citizen who came 10,001 in the rankings. 10,000 is just a smallish football crowd, so we could manage it today. We could pick the 10,000 by sortition (by lot). Most people are fairly sensible! |
| 19998 | If deputies represent people, they are accountable, but less so if they represent places [Montesquieu] |
| Full Idea: When the deputies represent the body or estate of the people, as in Holland, they ought to be accountable to their constituents. When the deputies represent boroughs, as in England, the situation is not the same. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 11.06) | |
| A reaction: Not sure how this works. Modern UK MPs are accountable to the residents of their borough. Did the Dutch actually name the citizens that a deputy represented? |
| 20000 | Slavery is entirely bad; the master abandons the virtues, and they are pointless in the slave [Montesquieu] |
| Full Idea: There is nothing good about the nature of slavery. The slave can achieve nothing by being virtuous. The master acquires all sorts of bad habits, and is accustomed to behaving with a total lack of moral virtues. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 15.01) | |
| A reaction: Most slavery of that time took place in colonies, far remote from the moral judgments of the mother country. The temptations of such power over others are far too great for most masters to live virtuously. |
| 20003 | Slaves are not members of the society, so no law can forbid them to run away [Montesquieu] |
| Full Idea: What civil law could prevent a slave from running away? Since he is not a member of society, why should the laws of society concern him? | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 15.02) | |
| A reaction: Hm. Does this apply to children, who can't vote or stand for office? |
| 20006 | The demand for slavery is just the masters' demand for luxury [Montesquieu] |
| Full Idea: The demand for slavery is the demand for luxury and voluptuousness; it has nothing to do with concern for public felicity. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 15.09) | |
| A reaction: True monarchists and aristocratic elitists presumably think that a society should have one part which lives in great luxury. Where else are the fine arts and wonderful buildings going to come from? |
| 20009 | Freedom of speech and writing, within the law, is essential to preserve liberty [Montesquieu] |
| Full Idea: If a state is to enjoy and preserve liberty, everyone must be able to say what he thinks. In a free state, therefore, a citizen may speak and write anything not expressly forbidden by the laws. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 19.27) | |
| A reaction: A commonplace now, but fairly bold then. I blame Freeborn John Lilburne for wild ideas like these. |
| 19994 | Freedom in society is ability to do what is right, and not having to do what is wrong [Montesquieu] |
| Full Idea: In a society where laws exist, liberty can consist only in being able to do what one ought to will, and in not being contrained to do what one ought not to will. ...If a citizen could do what the law prohibits, all others would have the same power. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 11.03) | |
| A reaction: This sounds pretty quaint in 2017, but I love it. |
| 19981 | No one even thinks of equality in monarchies and despotism; they all want superiority [Montesquieu] |
| Full Idea: In monarchies and despotic states, no one aspires to equality. Not even the idea occurs; everyone aspires to superiority. People of the very lowest rank only wish to rise in order to become masters of others. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 05.04) |
| 19991 | Equality is not command by everyone or no one, but command and obedience among equals [Montesquieu] |
| Full Idea: The spirit of true equality consists, not in creating a situation in which everyone commands, or in which no one is commanded, but rather in obeying or commanding only our equals. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 08.03) | |
| A reaction: I love this idea, but it is so easy to feel superior when you command, or to feel inferior when you are commanded. I take the solution to be the appointment of everyone in authority by those they will command (but fat chance of that). |
| 19990 | Democracy is corrupted by lack of equality, or by extreme equality (between rulers and ruled) [Montesquieu] |
| Full Idea: Democracy is corrupted in two ways: when it loses the spirit of equality, and when the spirit of equality becomes extreme, that is, when everyone wishes to be the equal of those he has chosen to command him. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 08.02) | |
| A reaction: The latter seems to be what happens when a referendum is called (as in Brexit 2016). The winners come to despise the elected representatives, if the latter disagree with the outcome. |
| 19982 | Some equality can be achieved by social categories, combined with taxes and poor relief [Montesquieu] |
| Full Idea: Equality is so difficult that exactitude is not possible. It is enough to place citizens by a census within categories that reduce or fix differences. Then laws compensate for inequalities by taxes on the rich and relief given to the poor. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 05.05) | |
| A reaction: [compressed] Placing citizens within categories (e.g. 'nobility') has long gone out of fashion. He doesn't say whether you tax the capital or the income of the rich. |
| 19983 | Democracies may sometimes need to restrict equality [Montesquieu] |
| Full Idea: In some cases, equality among citizens may be denied by democracy for the utility of democracy. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 05.05) | |
| A reaction: He cites people who make sacrifices for the public, and lower orders who are getting above themselves! The desire for equality quickly comes into conflict with other values. |
| 19959 | Prior to positive laws there is natural equity, of obedience, gratitude, dependence and merit [Montesquieu] |
| Full Idea: The relations of equity precede the positive laws that establish them. It is right to conform to laws in a society; intelligent beings should be grateful for benefits; we remain dependent on those who create us; an injury merits the same in return. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 01.01) | |
| A reaction: [the examples are compressed] A nice statement of the idea of natural law. It doesn't follow that because an injury merits retaliation, that it should be implemented (just that no one can complain if it happens). |
| 19960 | Sensation gives animals natural laws, but knowledge can make them break them [Montesquieu] |
| Full Idea: Animals have natural laws because they are united by sensation, ...but they do not invariably follow thieir natural laws; these are better observed by vegetables, which have neither knowledge nor sensation. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 01.02) | |
| A reaction: With the example of vegetables the concept of natural law is drifting into the laws of nature, and evidently Montesquie makes no sharp distinction here. |
| 20002 | The death penalty is permissible, because its victims enjoyed the protection of that law [Montesquieu] |
| Full Idea: It is permissible to put a criminal to death because the law that punishes him was made to protect him. For example, a murderer has enjoyed the benefits of the law by which he is condemned. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 15.02) | |
| A reaction: Dubious! We could add torture, and life imprisonment for parking offences, if this argument is sufficient justification. |
| 20010 | If religion teaches determinism, penalties must be severe; if free will, then that is different [Montesquieu] |
| Full Idea: When religion teaches that human actions are predetermined, penalties imposed by law ought to be more severe, for without these measures men would behave with complete abandon. If the dogma of religion is free will, the situation is altogether different. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 24.14) | |
| A reaction: Presumably persuasion and influence come into the free will picture. Calvinist Geneva was determinist, and Catholic France for free will. |
| 20001 | The only right victors have over captives is the protection of the former [Montesquieu] |
| Full Idea: War can confer only one right over captives, and that is to ensure that they no longer harm victors. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 15.02) | |
| A reaction: He is arguing against both the killing of captives, and their enslavement. |
| 19973 | The clergy are essential to a monarchy, but dangerous in a republic [Montesquieu] |
| Full Idea: The power of the clergy is as dangerous in a republic, as it is appropriate to a monarchy. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 02.04) | |
| A reaction: This makes me look at the UK in a new light, with the clergy hovering around when the monarch is crowned, and the bishops sitting by right in the House of Lords. |
| 20011 | Religion can support the state when the law fails to do so [Montesquieu] |
| Full Idea: Religion can support the state when the laws themselves lack the power to do so. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 24.16) | |
| A reaction: A thought which didn't occur to Spinoza, but then the thought merely confirms that religion offers a rival to the rule of law. |
| 19987 | Religion has the most influence in despotic states, and reinforces veneration for the ruler [Montesquieu] |
| Full Idea: In these [despotic] states, religion has more influence than anywhere else; it is fear added to fear. The peoples of the Mohammedan empires in part derive from their religion their extraordinary veneration for their rulers. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 05.14) | |
| A reaction: I suppose religions have submission to authority built into them. |
| 20004 | French slavery was accepted because it was the best method of religious conversion [Montesquieu] |
| Full Idea: Louis XIII was made extremely uneasy by the law that enslaved all the negroes in his colonies. But when told that this was the most efficacious way of converting them, he gave his consent. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 15.04) | |
| A reaction: That is a spectaculary bad advert for giving an established religion a leading role in society. It is relevant to the upbringing of children, as well as to slaves. |
| 19979 | In monarchies education ennobles people, and in despotisms it debases them [Montesquieu] |
| Full Idea: Just as the purpose of education in monarchies is to ennoble men's hearts, so its purpose in despotic states is to debase them. In despotic states education must be servile. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 04.03) | |
| A reaction: This is an early insight into the way that all social institutions, such as education, are largely pawns of a larger political system. |
| 19957 | Teaching is the best practice of the general virtue that leads us to love everyone [Montesquieu] |
| Full Idea: It is when we instruct others that we can best practice that general virtue which teaches us to love everyone. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], Preface) | |
| A reaction: A very nice thought. One tricky issue is that some people dislike, and even resent, being taught. If we all just adored both teaching and learning, we would be in a sort of paradise, but it doesn't seem to happen. |
| 6005 | Animals are dangerous and nourishing, and can't form contracts of justice [Hermarchus, by Sedley] |
| Full Idea: Hermarchus said that animal killing is justified by considerations of human safety and nourishment and by animals' inability to form contractual relations of justice with us. | |
| From: report of Hermarchus (fragments/reports [c.270 BCE]) by David A. Sedley - Hermarchus | |
| A reaction: Could the last argument be used to justify torturing animals? Or could we eat a human who was too brain-damaged to form contracts? |
| 19958 | Laws are the necessary relations that derive from the nature of things [Montesquieu] |
| Full Idea: Laws, in the broadest meaning of the term, are the necessary relations that derive from the nature of things. | |
| From: Baron de Montesquieu (The Spirit of the Laws (rev. 1757) [1748], 01.01) | |
| A reaction: Montesquieu is about to discuss social laws, but this is the clearest statement I have ever met of the essentialist view of the laws of nature. |
| 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'. |