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'The Limits of Abstraction', 'System of Logic' and 'Aquinas, Thomas'
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20 ideas
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units
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9801
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Numbers must be assumed to have identical units, as horses are equalised in 'horse-power' [Mill]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
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8742
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The only axioms needed are for equality, addition, and successive numbers [Mill, by Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic
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9800
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Arithmetic is based on definitions, and Sums of equals are equal, and Differences of equals are equal [Mill]
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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
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5201
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Mill says logic and maths is induction based on a very large number of instances [Mill, by Ayer]
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9360
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If two black and two white objects in practice produced five, what colour is the fifth one? [Lewis,CI on Mill]
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9888
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Mill mistakes particular applications as integral to arithmetic, instead of general patterns [Dummett on Mill]
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9794
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There are no such things as numbers in the abstract [Mill]
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9796
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Things possess the properties of numbers, as quantity, and as countable parts [Mill]
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9795
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Numbers have generalised application to entities (such as bodies or sounds) [Mill]
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9798
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Different parcels made from three pebbles produce different actual sensations [Mill]
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9797
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'2 pebbles and 1 pebble' and '3 pebbles' name the same aggregation, but different facts [Mill]
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9799
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3=2+1 presupposes collections of objects ('Threes'), which may be divided thus [Mill]
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9802
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Numbers denote physical properties of physical phenomena [Mill]
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9803
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We can't easily distinguish 102 horses from 103, but we could arrange them to make it obvious [Mill]
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9804
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Arithmetical results give a mode of formation of a given number [Mill]
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9805
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12 is the cube of 1728 means pebbles can be aggregated a certain way [Mill]
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8741
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Numbers must be of something; they don't exist as abstractions [Mill]
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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
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12411
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Mill is too imprecise, and is restricted to simple arithmetic [Kitcher on Mill]
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5656
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Empirical theories of arithmetic ignore zero, limit our maths, and need probability to get started [Frege on Mill]
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6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
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9624
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Numbers are a very general property of objects [Mill, by Brown,JR]
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