Combining Texts

Ideas for 'The Evolution of Modern Metaphysics', 'The Theory of Transfinite Numbers' and 'Grundgesetze der Arithmetik 2 (Basic Laws)'

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3 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
9886 Cardinals say how many, and reals give measurements compared to a unit quantity [Frege]
     Full Idea: The cardinals and the reals are completely disjoint domains. The cardinal numbers answer the question 'How many objects of a given kind are there?', but the real numbers are for measurement, saying how large a quantity is compared to a unit quantity.
     From: Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §157), quoted by Michael Dummett - Frege philosophy of mathematics Ch.19
     A reaction: We might say that cardinals are digital and reals are analogue. Frege is unusual in totally separating them. They map onto one another, after all. Cardinals look like special cases of reals. Reals are dreams about the gaps between cardinals.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
9889 Real numbers are ratios of quantities [Frege, by Dummett]
     Full Idea: Frege fixed on construing real numbers as ratios of quantities (in agreement with Newton).
     From: report of Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903]) by Michael Dummett - Frege philosophy of mathematics Ch.20
     A reaction: If 3/4 is the same real number as 6/8, which is the correct ratio? Why doesn't the square root of 9/16 also express it? Why should irrationals be so utterly different from rationals? In what sense are they both 'numbers'?
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.