Combining Texts

All the ideas for 'works', 'Grundlagen (Foundations of Theory of Manifolds)' and 'The Tragic Sense of Life'

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4 ideas

4. Formal Logic / F. Set Theory ST / 1. Set Theory
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
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
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
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
13007 Archimedes defined a straight line as the shortest distance between two points [Archimedes, by Leibniz]
     Full Idea: Archimedes gave a sort of definition of 'straight line' when he said it is the shortest line between two points.
     From: report of Archimedes (fragments/reports [c.240 BCE]) by Gottfried Leibniz - New Essays on Human Understanding 4.13
     A reaction: Commentators observe that this reduces the purity of the original Euclidean axioms, because it involves distance and measurement, which are absent from the purest geometry.
24. Political Theory / A. Basis of a State / 1. A People / a. Human distinctiveness
20740 Maybe humans are distinguished from other animals by feelings, rather than reason [Unamuno]
     Full Idea: Man is said to be a reasoning animal. I do not know why he has not been defined as an affective or feeling animal. Perhaps that which differentiates him from other animals is feeling rather than reason.
     From: Miguel de Unamuno (The Tragic Sense of Life [1912], p.3), quoted by Kevin Aho - Existentialism: an introduction 2 'Problem'
     A reaction: Perfectly plausible, given that we presume that our feelings are startlingly different from other animals - even if we feel far more community with other mammals than we did in Unamuno's day.