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
Impredicative definitions are wrong, because they change the set that is being defined?
4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism
4. Formal Logic / F. Set Theory ST / 1. Set Theory
There is no single agreed structure for set theory
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / c. Basic theorems of ST
Cantor proved that all sets have more subsets than they have members
4. Formal Logic / F. Set Theory ST / 3. Types of Set / a. Types of set
A 'proper class' cannot be a member of anything
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
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 / f. Axiom of Infinity V
Frege, unlike Russell, has infinite individuals because numbers are individuals
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
The Axiom of Choice relies on reference to sets that we are unable to describe
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / p. Axiom of Reducibility
Axiom of Reducibility: there is always a function of the lowest possible order in a given level
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
Replacement enforces a 'limitation of size' test for the existence of sets
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
The completeness of first-order logic implies its compactness
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
Substitutional quantification is just standard if all objects in the domain have a name
5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
The Deduction Theorem is what licenses a system of natural deduction
5. Theory of Logic / L. Paradox / 4. Paradoxes in Logic / c. Berry's paradox
Berry's Paradox considers the meaning of 'The least number not named by this name'
6. Mathematics / A. Nature of Mathematics / 3. Numbers / b. Types of number
ω + 1 is a new ordinal, but its cardinality is unchanged
Each addition changes the ordinality but not the cardinality, prior to aleph-1
6. Mathematics / A. Nature of Mathematics / 3. Numbers / c. Priority of numbers
A cardinal is the earliest ordinal that has that number of predecessors
6. Mathematics / A. Nature of Mathematics / 3. Numbers / f. Cardinal numbers
Aleph-1 is the first ordinal that exceeds aleph-0
6. Mathematics / A. Nature of Mathematics / 3. Numbers / g. Real numbers
Instead of by cuts or series convergence, real numbers could be defined by axioms
The number of reals is the number of subsets of the natural numbers
6. Mathematics / A. Nature of Mathematics / 3. Numbers / i. Reals from cuts
For Eudoxus cuts in rationals are unique, but not every cut makes a real number
Dedekind says each cut matches a real; logicists say the cuts are the reals
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / l. Infinitesimals
Infinitesimals are not actually contradictory, because they can be non-standard real numbers
6. Mathematics / B. Foundations for Mathematics / 2. Axioms for Geometry
Modern axioms of geometry do not need the real numbers
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / d. Peano arithmetic
Zero is a member, and all successors; numbers are the intersection of sets satisfying this
The Peano Axioms describe a unique structure
PA concerns any entities which satisfy the axioms
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / d. Hume's Principle
Many things will satisfy Hume's Principle, so there are many interpretations of it
Hume's Principle is a definition with existential claims, and won't explain numbers
There are many criteria for the identity of numbers
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / e. Caesar problem
Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set!
One-one correlations imply normal arithmetic, but don't explain our concept of a number
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / e. Structuralism critique
Numbers can't be positions, if nothing decides what position a given number has
Structuralism falsely assumes relations to other numbers are numbers' only properties
6. Mathematics / C. Sources of Mathematics / 3. Mathematical Nominalism
Nominalism about mathematics is either reductionist, or fictionalist
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
Actual measurement could never require the precision of the real numbers
6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
Ordinals are mainly used adjectively, as in 'the first', 'the second'...
6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
Simple type theory has 'levels', but ramified type theory has 'orders'
6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
Neo-logicists agree that HP introduces number, but also claim that it suffices for the job
Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
Many crucial logicist definitions are in fact impredicative
If Hume's Principle is the whole story, that implies structuralism
Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality
6. Mathematics / C. Sources of Mathematics / 9. Fictional Mathematics
Higher cardinalities in sets are just fairy stories
A fairy tale may give predictions, but only a true theory can give explanations
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
The best version of conceptualism is predicativism
Conceptualism fails to grasp mathematical properties, infinity, and objective truth values
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
If abstracta only exist if they are expressible, there can only be denumerably many of them
The predicativity restriction makes a difference with the real numbers
The usual definitions of identity and of natural numbers are impredicative
Predicativism makes theories of huge cardinals impossible
If we can only think of what we can describe, predicativism may be implied
If mathematics rests on science, predicativism may be the best approach
19. Language / H. Pragmatics / 1. Assertion
In logic a proposition means the same when it is and when it is not asserted