Ideas from 'Philosophy of Mathematics' by David Bostock , by Theme Structure

[found in 'Philosophy of Mathematics: An Introduction' by Bostock,David [Wiley-Blackwell 2009,978-1-4051-8991-0]].

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2. Reason / D. Definition / 8. Impredicative Definition
 18137 Impredicative definitions are wrong, because they change the set that is being defined?
4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
 18122 Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism
4. Formal Logic / F. Set Theory ST / 1. Set Theory
 18114 There is no single agreed structure for set theory
4. Formal Logic / F. Set Theory ST / 3. Types of Set / a. Types of set
 18107 A 'proper class' cannot be a member of anything
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
 18115 We could add axioms to make sets either as small or as large as possible
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
 18139 The Axiom of Choice relies on reference to sets that we are unable to describe
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
 18105 Replacement enforces a 'limitation of size' test for the existence of sets
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
 18109 The completeness of first-order logic implies its compactness
 18108 First-order logic is not decidable: there is no test of whether any formula is valid
5. Theory of Logic / G. Quantification / 4. Substitutional Quantification
 18123 Substitutional quantification is just standard if all objects in the domain have a name
5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
 18120 The Deduction Theorem is what licenses a system of natural deduction
 18125 Berry's Paradox considers the meaning of 'The least number not named by this name'
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
 18100 ω + 1 is a new ordinal, but its cardinality is unchanged
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
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
 18106 Aleph-1 is the first ordinal that exceeds aleph-0
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
 18099 The number of reals is the number of subsets of the natural numbers
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
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
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
 18156 Modern axioms of geometry do not need the real numbers
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
 18097 The Peano Axioms describe a unique structure
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
 18145 Many things will satisfy Hume's Principle, so there are many interpretations of it
 18149 There are many criteria for the identity of numbers
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!
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
 18117 Structuralism falsely assumes relations to other numbers are numbers' only properties
6. Mathematics / C. Sources of Mathematics / 3. Mathematical Nominalism
 18141 Nominalism about mathematics is either reductionist, or fictionalist
 18157 Nominalism as based on application of numbers is no good, because there are too many applications
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
6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
 18158 Ordinals are mainly used adjectively, as in 'the first', 'the second'...
6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
 18127 Simple type theory has 'levels', but ramified type theory has 'orders'
6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
 18147 Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number
 18144 Neo-logicists agree that HP introduces number, but also claim that it suffices for the job
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
 18146 If Hume's Principle is the whole story, that implies structuralism
 18129 Many crucial logicist definitions are in fact impredicative
 18111 Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality
6. Mathematics / C. Sources of Mathematics / 9. Fictional Mathematics
 18159 Higher cardinalities in sets are just fairy stories
 18155 A fairy tale may give predictions, but only a true theory can give explanations
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
 18140 The best version of conceptualism is predicativism
 18138 Conceptualism fails to grasp mathematical properties, infinity, and objective truth values
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
 18132 The predicativity restriction makes a difference with the real numbers
 18133 The usual definitions of identity and of natural numbers are impredicative
 18134 Predicativism makes theories of huge cardinals impossible
 18135 If mathematics rests on science, predicativism may be the best approach
 18136 If we can only think of what we can describe, predicativism may be implied
19. Language / F. Communication / 2. Assertion
 18121 In logic a proposition means the same when it is and when it is not asserted