Combining Philosophers

Ideas for David Bostock, Crispin Wright and Moses Maimonides

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59 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
13861 Number theory aims at the essence of natural numbers, giving their nature, and the epistemology [Wright,C]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
18101 Each addition changes the ordinality but not the cardinality, prior to aleph-1 [Bostock]
18100 ω + 1 is a new ordinal, but its cardinality is unchanged [Bostock]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
18102 A cardinal is the earliest ordinal that has that number of predecessors [Bostock]
13892 One could grasp numbers, and name sizes with them, without grasping ordering [Wright,C]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
18106 Aleph-1 is the first ordinal that exceeds aleph-0 [Bostock]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
18095 Instead of by cuts or series convergence, real numbers could be defined by axioms [Bostock]
18099 The number of reals is the number of subsets of the natural numbers [Bostock]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
18093 For Eudoxus cuts in rationals are unique, but not every cut makes a real number [Bostock]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / d. Counting via concepts
13867 Instances of a non-sortal concept can only be counted relative to a sortal concept [Wright,C]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / k. Infinitesimals
18110 Infinitesimals are not actually contradictory, because they can be non-standard real numbers [Bostock]
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
18156 Modern axioms of geometry do not need the real numbers [Bostock]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
18097 The Peano Axioms describe a unique structure [Bostock]
17441 Wright thinks Hume's Principle is more fundamental to cardinals than the Peano Axioms are [Wright,C, by Heck]
13862 There are five Peano axioms, which can be expressed informally [Wright,C]
17853 Number truths are said to be the consequence of PA - but it needs semantic consequence [Wright,C]
17854 What facts underpin the truths of the Peano axioms? [Wright,C]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
13358 Ordinary or mathematical induction assumes for the first, then always for the next, and hence for all [Bostock]
13359 Complete induction assumes for all numbers less than n, then also for n, and hence for all numbers [Bostock]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
13894 Sameness of number is fundamental, not counting, despite children learning that first [Wright,C]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
18148 Hume's Principle is a definition with existential claims, and won't explain numbers [Bostock]
18145 Many things will satisfy Hume's Principle, so there are many interpretations of it [Bostock]
10140 We derive Hume's Law from Law V, then discard the latter in deriving arithmetic [Wright,C, by Fine,K]
8692 Frege has a good system if his 'number principle' replaces his basic law V [Wright,C, by Friend]
17440 Wright says Hume's Principle is analytic of cardinal numbers, like a definition [Wright,C, by Heck]
13893 It is 1-1 correlation of concepts, and not progression, which distinguishes natural number [Wright,C]
18149 There are many criteria for the identity of numbers [Bostock]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
13888 If numbers are extensions, Frege must first solve the Caesar problem for extensions [Wright,C]
18143 Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set! [Bostock]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
18116 Numbers can't be positions, if nothing decides what position a given number has [Bostock]
18117 Structuralism falsely assumes relations to other numbers are numbers' only properties [Bostock]
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
13869 Number platonism says that natural number is a sortal concept [Wright,C]
6. Mathematics / C. Sources of Mathematics / 3. Mathematical Nominalism
18141 Nominalism about mathematics is either reductionist, or fictionalist [Bostock]
18157 Nominalism as based on application of numbers is no good, because there are too many applications [Bostock]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
13870 We can't use empiricism to dismiss numbers, if numbers are our main evidence against empiricism [Wright,C]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
18150 Actual measurement could never require the precision of the real numbers [Bostock]
6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
18158 Ordinals are mainly used adjectively, as in 'the first', 'the second'... [Bostock]
13873 Treating numbers adjectivally is treating them as quantifiers [Wright,C]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
18127 Simple type theory has 'levels', but ramified type theory has 'orders' [Bostock]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
18144 Neo-logicists agree that HP introduces number, but also claim that it suffices for the job [Bostock]
13899 The Peano Axioms, and infinity of cardinal numbers, are logical consequences of how we explain cardinals [Wright,C]
13896 The aim is to follow Frege's strategy to derive the Peano Axioms, but without invoking classes [Wright,C]
18147 Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number [Bostock]
7804 Wright has revived Frege's discredited logicism [Wright,C, by Benardete,JA]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
18146 If Hume's Principle is the whole story, that implies structuralism [Bostock]
18129 Many crucial logicist definitions are in fact impredicative [Bostock]
18111 Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality [Bostock]
13863 Logicism seemed to fail by Russell's paradox, Gödel's theorems, and non-logical axioms [Wright,C]
13895 The standard objections are Russell's Paradox, non-logical axioms, and Gödel's theorems [Wright,C]
6. Mathematics / C. Sources of Mathematics / 9. Fictional Mathematics
18159 Higher cardinalities in sets are just fairy stories [Bostock]
18155 A fairy tale may give predictions, but only a true theory can give explanations [Bostock]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
18140 The best version of conceptualism is predicativism [Bostock]
18138 Conceptualism fails to grasp mathematical properties, infinity, and objective truth values [Bostock]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
18131 If abstracta only exist if they are expressible, there can only be denumerably many of them [Bostock]
18134 Predicativism makes theories of huge cardinals impossible [Bostock]
18135 If mathematics rests on science, predicativism may be the best approach [Bostock]
18136 If we can only think of what we can describe, predicativism may be implied [Bostock]
18133 The usual definitions of identity and of natural numbers are impredicative [Bostock]
18132 The predicativity restriction makes a difference with the real numbers [Bostock]