Combining Philosophers

Ideas for Anaxarchus, John Mayberry and J.L. Austin

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5 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers

17784	Real numbers can be eliminated, by axiom systems for complete ordered fields [Mayberry]
	Full Idea: We eliminate the real numbers by giving an axiomatic definition of the species of complete ordered fields. These axioms are categorical (mutually isomorphic), and thus are mathematically indistinguishable.
	From: John Mayberry (What Required for Foundation for Maths? [1994], p.408-2)
	A reaction: Hence my clever mathematical friend says that it is a terrible misunderstanding to think that mathematics is about numbers. Mayberry says the reals are one ordered field, but mathematics now studies all ordered fields together.

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / b. Quantity

17782	Greek quantities were concrete, and ratio and proportion were their science [Mayberry]
	Full Idea: Quantities for Greeks were concrete things - lines, surfaces, solids, times, weights. At the centre of their science of quantity was the beautiful theory of ratio and proportion (...in which the notion of number does not appear!).
	From: John Mayberry (What Required for Foundation for Maths? [1994], p.407-2)
	A reaction: [He credits Eudoxus, and cites Book V of Euclid]

17781	Real numbers were invented, as objects, to simplify and generalise 'quantity' [Mayberry]
	Full Idea: The abstract objects of modern mathematics, the real numbers, were invented by the mathematicians of the seventeenth century in order to simplify and to generalize the Greek science of quantity.
	From: John Mayberry (What Required for Foundation for Maths? [1994], p.407-2)

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite

17799	Cantor's infinite is an absolute, of all the sets or all the ordinal numbers [Mayberry]
	Full Idea: In Cantor's new vision, the infinite, the genuine infinite, does not disappear, but presents itself in the guise of the absolute, as manifested in the species of all sets or the species of all ordinal numbers.
	From: John Mayberry (What Required for Foundation for Maths? [1994], p.414-2)

17797	Cantor extended the finite (rather than 'taming the infinite') [Mayberry]
	Full Idea: We may describe Cantor's achievement by saying, not that he tamed the infinite, but that he extended the finite.
	From: John Mayberry (What Required for Foundation for Maths? [1994], p.414-2)