Combining Texts

All the ideas for 'Cours d'Analyse', 'The Foundations of Mathematics (2nd ed)' and 'Of the Laws of Ecclesiastical Polity'

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7 ideas

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / k. Infinitesimals
18085 Values that approach zero, becoming less than any quantity, are 'infinitesimals' [Cauchy]
     Full Idea: When the successive absolute values of a variable decrease indefinitely in such a way as to become less than any given quantity, that variable becomes what is called an 'infinitesimal'. Such a variable has zero as its limit.
     From: Augustin-Louis Cauchy (Cours d'Analyse [1821], p.19), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.4
     A reaction: The creator of the important idea of the limit still talked in terms of infinitesimals. In the next generation the limit took over completely.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / l. Limits
18084 When successive variable values approach a fixed value, that is its 'limit' [Cauchy]
     Full Idea: When the values successively attributed to the same variable approach indefinitely a fixed value, eventually differing from it by as little as one could wish, that fixed value is called the 'limit' of all the others.
     From: Augustin-Louis Cauchy (Cours d'Analyse [1821], p.19), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.4
     A reaction: This seems to be a highly significan proposal, because you can now treat that limit as a number, and adds things to it. It opens the door to Cantor's infinities. Is the 'limit' just a fiction?
8. Modes of Existence / A. Relations / 4. Formal Relations / b. Equivalence relation
18465 An 'equivalence' relation is one which is reflexive, symmetric and transitive [Kunen]
     Full Idea: R is an equivalence relation on A iff R is reflexive, symmetric and transitive on A.
     From: Kenneth Kunen (The Foundations of Mathematics (2nd ed) [2012], I.7.1)
25. Social Practice / C. Rights / 1. Basis of Rights
19896 It is not a law if not endorsed by the public [Hooker,R]
     Full Idea: Laws they are not which public approbation hath not made so.
     From: Richard Hooker (Of the Laws of Ecclesiastical Polity [1593], I s.10), quoted by John Locke - Second Treatise of Government 134 n1
     A reaction: Margaret Thatcher's Poll Tax, rejected by public rebellion, illustrates the point.
25. Social Practice / D. Justice / 2. The Law / b. Rule of law
19891 Rule of law is superior to autonomy, because citizens can see what is expected [Hooker,R]
     Full Idea: Men saw that to live by one man's will became the cause of all men's misery. This contrained them to come unto laws wherein all men might see their duty beforehand, and know the penalties of transgressing them.
     From: Richard Hooker (Of the Laws of Ecclesiastical Polity [1593], I s.10), quoted by John Locke - Second Treatise of Government 111 n1
     A reaction: One British school has a single rule, that pupils 'shall always treat other people with respect'. Presumably the rulers, as well as the pupils, must decide when this is transgressed. The rule of law may be preferable.
25. Social Practice / D. Justice / 2. The Law / c. Natural law
19897 Human laws must accord with the general laws of Nature [Hooker,R]
     Full Idea: Laws human must be made according to the general laws of Nature.
     From: Richard Hooker (Of the Laws of Ecclesiastical Polity [1593], III s.9), quoted by John Locke - Second Treatise of Government
     A reaction: The point simply seems to be that they won't get assent from the public if they are not in accord with natural justice. Positivists say you can make any damned law you like.
26. Natural Theory / D. Laws of Nature / 8. Scientific Essentialism / c. Essence and laws
17005 Natural things observe certain laws, and things cannot do otherwise if they retain their forms [Hooker,R]
     Full Idea: Things natural …do so necessarily observe their certain laws, that as long as they keep those forms which give them their being they cannot possibly be apt or inclinable to do otherwise than they do.
     From: Richard Hooker (Of the Laws of Ecclesiastical Polity [1593], 1.3.4), quoted by Marc Lange - Laws and Lawmakers 1.2
     A reaction: Cited by some as the beginnings of the idea of 'laws of nature', but it is striking that Hooker says the laws are controlled by 'forms' (which are Aristotelian essences). This is an essentialist view of laws, not a regularity or divine power one.