Combining Texts

All the ideas for 'fragments/reports', 'works' and 'The Theory of Transfinite Numbers'

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5 ideas

4. Formal Logic / F. Set Theory ST / 1. Set Theory

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.

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / f. Uncountable infinities

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.

10. Modality / A. Necessity / 8. Transcendental Necessity

490	Everything happens by reason and necessity [Leucippus]
	Full Idea: Nothing happens at random; everything happens out of reason and by necessity.
	From: Leucippus (fragments/reports [c.435 BCE], B002), quoted by (who?) - where?

14. Science / B. Scientific Theories / 1. Scientific Theory

22363	You have only begun to do real science when you can express it in numbers [Kelvin]
	Full Idea: When you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science, whatever the matter may be.
	From: Lord Kelvin (Wm Thomson) (works [1881]), quoted by Reiss,J/Spreger,J - Scientific Objectivity 4.1
	A reaction: [Popular Lectures 1 p.73] Clearly the writer is a physicist! Astronomers discover objects, geologists discover structures, biologists reveal mechanisms.

27. Natural Reality / A. Classical Physics / 2. Thermodynamics / a. Energy

20644	Energy has progressed from a mere formula, to a principle pervading all nature [Kelvin]
	Full Idea: The name 'energy', first used by Thomas Young, has come into use after it was raised from a mere formula of mathematical dynamics to become a principle pervading all nature, and guiding every field of science.
	From: Lord Kelvin (Wm Thomson) (works [1881]), quoted by Peter Watson - Convergence 01 'Principle'
	A reaction: [bit compressed] As far as I can see energy behaves exactly as if it were a substance, like water conserved in rainfalls, and yet it isn't a stuff, and seems to result from a process of abstraction. I take it to be one of the biggest mysteries in physics.