Combining Philosophers

Ideas for Friedrich Schlegel, John Stuart Mill and Theophrastus

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20 ideas

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units
Numbers must be assumed to have identical units, as horses are equalised in 'horse-power' [Mill]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
The only axioms needed are for equality, addition, and successive numbers [Mill, by Shapiro]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic
Arithmetic is based on definitions, and Sums of equals are equal, and Differences of equals are equal [Mill]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
Mill says logic and maths is induction based on a very large number of instances [Mill, by Ayer]
If two black and two white objects in practice produced five, what colour is the fifth one? [Lewis,CI on Mill]
Mill mistakes particular applications as integral to arithmetic, instead of general patterns [Dummett on Mill]
Things possess the properties of numbers, as quantity, and as countable parts [Mill]
There are no such things as numbers in the abstract [Mill]
Numbers have generalised application to entities (such as bodies or sounds) [Mill]
'2 pebbles and 1 pebble' and '3 pebbles' name the same aggregation, but different facts [Mill]
Different parcels made from three pebbles produce different actual sensations [Mill]
3=2+1 presupposes collections of objects ('Threes'), which may be divided thus [Mill]
We can't easily distinguish 102 horses from 103, but we could arrange them to make it obvious [Mill]
Numbers denote physical properties of physical phenomena [Mill]
Arithmetical results give a mode of formation of a given number [Mill]
12 is the cube of 1728 means pebbles can be aggregated a certain way [Mill]
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
Mill is too imprecise, and is restricted to simple arithmetic [Kitcher on Mill]
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
Numbers are a very general property of objects [Mill, by Brown,JR]