Combining Philosophers

Ideas for B Hale / C Wright, Baron,S/Miller,K and John Stuart Mill

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30 ideas

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 / 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 / g. Incompleteness of Arithmetic
10624 The incompletability of formal arithmetic reveals that logic also cannot be completely characterized [Hale/Wright]
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 / 5. Definitions of Number / d. Hume's Principle
8784 Neo-logicism founds arithmetic on Hume's Principle along with second-order logic [Hale/Wright]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
8787 The Julius Caesar problem asks for a criterion for the concept of a 'number' [Hale/Wright]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
10629 If structures are relative, this undermines truth-value and objectivity [Hale/Wright]
10628 The structural view of numbers doesn't fit their usage outside arithmetical contexts [Hale/Wright]
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]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
8788 Logicism is only noteworthy if logic has a privileged position in our ontology and epistemology [Hale/Wright]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
10622 The neo-Fregean is more optimistic than Frege about contextual definitions of numbers [Hale/Wright]
8783 Logicism might also be revived with a quantificational approach, or an abstraction-free approach [Hale/Wright]
12225 Neo-Fregeanism might be better with truth-makers, rather than quantifier commitment [Hale/Wright]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
12224 Are neo-Fregeans 'maximalists' - that everything which can exist does exist? [Hale/Wright]