Combining Philosophers

Ideas for B Hale / C Wright, Anthony Quinton and David Bostock

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48 ideas

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]
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 / 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]
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 / 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 / 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]
8784 Neo-logicism founds arithmetic on Hume's Principle along with second-order logic [Hale/Wright]
18149 There are many criteria for the identity of numbers [Bostock]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
18143 Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set! [Bostock]
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
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]
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 / 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 / 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]
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 / 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]
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]
18147 Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number [Bostock]
10622 The neo-Fregean is more optimistic than Frege about contextual definitions of numbers [Hale/Wright]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
18146 If Hume's Principle is the whole story, that implies structuralism [Bostock]
12224 Are neo-Fregeans 'maximalists' - that everything which can exist does exist? [Hale/Wright]
18111 Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality [Bostock]
18129 Many crucial logicist definitions are in fact impredicative [Bostock]
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]