Combining Philosophers

Ideas for John Mayberry, John Stuart Mill and Stephen Davies

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34 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]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units
9801 Numbers must be assumed to have identical units, as horses are equalised in 'horse-power' [Mill]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / b. Quantity
17782 Greek quantities were concrete, and ratio and proportion were their science [Mayberry]
17781 Real numbers were invented, as objects, to simplify and generalise 'quantity' [Mayberry]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
17797 Cantor extended the finite (rather than 'taming the infinite') [Mayberry]
17799 Cantor's infinite is an absolute, of all the sets or all the ordinal numbers [Mayberry]
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
17775 If proof and definition are central, then mathematics needs and possesses foundations [Mayberry]
17776 The ultimate principles and concepts of mathematics are presumed, or grasped directly [Mayberry]
17777 Foundations need concepts, definition rules, premises, and proof rules [Mayberry]
17804 Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms [Mayberry]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
8742 The only axioms needed are for equality, addition, and successive numbers [Mill, by Shapiro]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
17792 1st-order PA is only interesting because of results which use 2nd-order PA [Mayberry]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
17793 It is only 2nd-order isomorphism which suggested first-order PA completeness [Mayberry]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic
9800 Arithmetic is based on definitions, and Sums of equals are equal, and Differences of equals are equal [Mill]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
17794 Set theory is not just first-order ZF, because that is inadequate for mathematics [Mayberry]
17802 We don't translate mathematics into set theory, because it comes embodied in that way [Mayberry]
17805 Set theory is not just another axiomatised part of mathematics [Mayberry]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
5201 Mill says logic and maths is induction based on a very large number of instances [Mill, by Ayer]
9360 If two black and two white objects in practice produced five, what colour is the fifth one? [Lewis,CI on Mill]
9888 Mill mistakes particular applications as integral to arithmetic, instead of general patterns [Dummett on Mill]
9794 There are no such things as numbers in the abstract [Mill]
9796 Things possess the properties of numbers, as quantity, and as countable parts [Mill]
9795 Numbers have generalised application to entities (such as bodies or sounds) [Mill]
9798 Different parcels made from three pebbles produce different actual sensations [Mill]
9797 '2 pebbles and 1 pebble' and '3 pebbles' name the same aggregation, but different facts [Mill]
9799 3=2+1 presupposes collections of objects ('Threes'), which may be divided thus [Mill]
9802 Numbers denote physical properties of physical phenomena [Mill]
9803 We can't easily distinguish 102 horses from 103, but we could arrange them to make it obvious [Mill]
9804 Arithmetical results give a mode of formation of a given number [Mill]
9805 12 is the cube of 1728 means pebbles can be aggregated a certain way [Mill]
8741 Numbers must be of something; they don't exist as abstractions [Mill]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
12411 Mill is too imprecise, and is restricted to simple arithmetic [Kitcher on Mill]
5656 Empirical theories of arithmetic ignore zero, limit our maths, and need probability to get started [Frege on Mill]
6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
9624 Numbers are a very general property of objects [Mill, by Brown,JR]