Ideas from 'Philosophy of Mathematics' by David Bostock [2009], 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
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18137
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Impredicative definitions are wrong, because they change the set that is being defined?
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4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
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18122
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Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism
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4. Formal Logic / F. Set Theory ST / 1. Set Theory
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18114
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There is no single agreed structure for set theory
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4. Formal Logic / F. Set Theory ST / 3. Types of Set / a. Types of set
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18107
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A 'proper class' cannot be a member of anything
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
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18115
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We could add axioms to make sets either as small or as large as possible
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
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18139
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The Axiom of Choice relies on reference to sets that we are unable to describe
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4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
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18105
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Replacement enforces a 'limitation of size' test for the existence of sets
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5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
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18109
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The completeness of first-order logic implies its compactness
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18108
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First-order logic is not decidable: there is no test of whether any formula is valid
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5. Theory of Logic / G. Quantification / 4. Substitutional Quantification
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18123
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Substitutional quantification is just standard if all objects in the domain have a name
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5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
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18120
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The Deduction Theorem is what licenses a system of natural deduction
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5. Theory of Logic / L. Paradox / 4. Paradoxes in Logic / c. Berry's paradox
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18125
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Berry's Paradox considers the meaning of 'The least number not named by this name'
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
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18101
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Each addition changes the ordinality but not the cardinality, prior to aleph-1
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18100
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ω + 1 is a new ordinal, but its cardinality is unchanged
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
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18102
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A cardinal is the earliest ordinal that has that number of predecessors
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
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18106
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Aleph-1 is the first ordinal that exceeds aleph-0
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
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18095
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Instead of by cuts or series convergence, real numbers could be defined by axioms
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18099
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The number of reals is the number of subsets of the natural numbers
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
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18093
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For Eudoxus cuts in rationals are unique, but not every cut makes a real number
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / k. Infinitesimals
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18110
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Infinitesimals are not actually contradictory, because they can be non-standard real numbers
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6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
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18156
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Modern axioms of geometry do not need the real numbers
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
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18097
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The Peano Axioms describe a unique structure
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
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18149
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There are many criteria for the identity of numbers
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18148
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Hume's Principle is a definition with existential claims, and won't explain numbers
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18145
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Many things will satisfy Hume's Principle, so there are many interpretations of it
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
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18143
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Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set!
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
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18116
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Numbers can't be positions, if nothing decides what position a given number has
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18117
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Structuralism falsely assumes relations to other numbers are numbers' only properties
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6. Mathematics / C. Sources of Mathematics / 3. Mathematical Nominalism
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18141
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Nominalism about mathematics is either reductionist, or fictionalist
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18157
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Nominalism as based on application of numbers is no good, because there are too many applications
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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
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18150
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Actual measurement could never require the precision of the real numbers
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6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
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18158
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Ordinals are mainly used adjectively, as in 'the first', 'the second'...
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
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18127
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Simple type theory has 'levels', but ramified type theory has 'orders'
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
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18144
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Neo-logicists agree that HP introduces number, but also claim that it suffices for the job
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18147
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Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
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18111
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Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality
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18146
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If Hume's Principle is the whole story, that implies structuralism
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18129
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Many crucial logicist definitions are in fact impredicative
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6. Mathematics / C. Sources of Mathematics / 9. Fictional Mathematics
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18159
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Higher cardinalities in sets are just fairy stories
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18155
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A fairy tale may give predictions, but only a true theory can give explanations
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
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18140
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The best version of conceptualism is predicativism
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18138
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Conceptualism fails to grasp mathematical properties, infinity, and objective truth values
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
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18131
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If abstracta only exist if they are expressible, there can only be denumerably many of them
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18132
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The predicativity restriction makes a difference with the real numbers
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18133
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The usual definitions of identity and of natural numbers are impredicative
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18134
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Predicativism makes theories of huge cardinals impossible
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18135
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If mathematics rests on science, predicativism may be the best approach
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18136
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If we can only think of what we can describe, predicativism may be implied
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19. Language / F. Communication / 2. Assertion
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18121
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In logic a proposition means the same when it is and when it is not asserted
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