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Ideas for 'fragments/reports', 'System of Logic' and 'What Numbers Could Not Be'

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43 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
9912 There are no such things as numbers [Benacerraf]
9901 Numbers can't be sets if there is no agreement on which sets they are [Benacerraf]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
9151 Benacerraf says numbers are defined by their natural ordering [Benacerraf, by Fine,K]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
13891 To understand finite cardinals, it is necessary and sufficient to understand progressions [Benacerraf, by Wright,C]
17904 A set has k members if it one-one corresponds with the numbers less than or equal to k [Benacerraf]
17906 To explain numbers you must also explain cardinality, the counting of things [Benacerraf]
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 / c. Counting procedure
9898 We can count intransitively (reciting numbers) without understanding transitive counting of items [Benacerraf]
17903 Someone can recite numbers but not know how to count things; but not vice versa [Benacerraf]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
9897 The application of a system of numbers is counting and measurement [Benacerraf]
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]
9900 For Zermelo 3 belongs to 17, but for Von Neumann it does not [Benacerraf]
9899 The successor of x is either x and all its members, or just the unit set of x [Benacerraf]
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 / b. Mathematics is not set theory
8697 Disputes about mathematical objects seem irrelevant, and mathematicians cannot resolve them [Benacerraf, by Friend]
8304 No particular pair of sets can tell us what 'two' is, just by one-to-one correlation [Benacerraf, by Lowe]
9906 If ordinal numbers are 'reducible to' some set-theory, then which is which? [Benacerraf]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
9907 If any recursive sequence will explain ordinals, then it seems to be the structure which matters [Benacerraf]
9908 The job is done by the whole system of numbers, so numbers are not objects [Benacerraf]
9909 The number 3 defines the role of being third in a progression [Benacerraf]
9911 Number words no more have referents than do the parts of a ruler [Benacerraf]
8925 Mathematical objects only have properties relating them to other 'elements' of the same structure [Benacerraf]
9938 How can numbers be objects if order is their only property? [Benacerraf, by Putnam]
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
9910 Number-as-objects works wholesale, but fails utterly object by object [Benacerraf]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
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]
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]
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]
9903 Number words are not predicates, as they function very differently from adjectives [Benacerraf]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
9904 The set-theory paradoxes mean that 17 can't be the class of all classes with 17 members [Benacerraf]