Combining Philosophers

Ideas for Penelope Maddy, John Stuart Mill and Hippias

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51 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 / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
18190 Completed infinities resulted from giving foundations to calculus [Maddy]
18171 Cantor and Dedekind brought completed infinities into mathematics [Maddy]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
17615 Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
18196 An 'inaccessible' cardinal cannot be reached by union sets or power sets [Maddy]
18172 Infinity has degrees, and large cardinals are the heart of set theory [Maddy]
18175 For any cardinal there is always a larger one (so there is no set of all sets) [Maddy]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / l. Limits
18187 Theorems about limits could only be proved once the real numbers were understood [Maddy]
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 / 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 / c. Fregean numbers
18182 The extension of concepts is not important to me [Maddy]
18177 In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets [Maddy]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
18164 Frege solves the Caesar problem by explicitly defining each number [Maddy]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
17825 Set theory (unlike the Peano postulates) can explain why multiplication is commutative [Maddy]
17826 Standardly, numbers are said to be sets, which is neat ontology and epistemology [Maddy]
17828 Numbers are properties of sets, just as lengths are properties of physical objects [Maddy]
10718 A natural number is a property of sets [Maddy, by Oliver]
18184 Making set theory foundational to mathematics leads to very fruitful axioms [Maddy]
18185 Unified set theory gives a final court of appeal for mathematics [Maddy]
18183 Set theory brings mathematics into one arena, where interrelations become clearer [Maddy]
18186 Identifying geometric points with real numbers revealed the power of set theory [Maddy]
18188 The line of rationals has gaps, but set theory provided an ordered continuum [Maddy]
17618 Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy]
18163 Mathematics rests on the logic of proofs, and on the set theoretic axioms [Maddy]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
17830 Number theory doesn't 'reduce' to set theory, because sets have number properties [Maddy]
17827 Sets exist where their elements are, but numbers are more like universals [Maddy]
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
17823 If mathematical objects exist, how can we know them, and which objects are they? [Maddy]
6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics
8756 Intuition doesn't support much mathematics, and we should question its reliability [Maddy, by Shapiro]
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]
17733 We know mind-independent mathematical truths through sets, which rest on experience [Maddy, by Jenkins]
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 / b. Indispensability of mathematics
18204 Scientists posit as few entities as possible, but set theorist posit as many as possible [Maddy]
18207 Maybe applications of continuum mathematics are all idealisations [Maddy]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
17614 The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy]
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
17829 Number words are unusual as adjectives; we don't say 'is five', and numbers always come first [Maddy]
9624 Numbers are a very general property of objects [Mill, by Brown,JR]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
18167 We can get arithmetic directly from HP; Law V was used to get HP from the definition of number [Maddy]