Ideas from 'Philosophies of Mathematics' by A.George / D.J.Velleman [2002], by Theme Structure

[found in 'Philosophies of Mathematics' by George,A/Velleman D.J. [Blackwell 2002,0-631-19544-0]].

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2. Reason / D. Definition / 7. Contextual Definition
Contextual definitions replace a complete sentence containing the expression
2. Reason / D. Definition / 8. Impredicative Definition
Impredicative definitions quantify over the thing being defined
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
The 'power set' of A is all the subsets of A
Cartesian Product A x B: the set of all ordered pairs in which a∈A and b∈B
The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}}
4. Formal Logic / F. Set Theory ST / 3. Types of Set / e. Equivalence classes
Grouping by property is common in mathematics, usually using equivalence
'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
Even the elements of sets in ZFC are sets, resting on the pure empty set
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I
Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = y
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / c. Axiom of Pairing II
Axiom of Pairing: for all sets x and y, there is a set z containing just x and y
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / p. Axiom of Reducibility
The Axiom of Reducibility made impredicative definitions possible
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / a. Sets as existing
ZFC can prove that there is no set corresponding to the concept 'set'
4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
As a reduction of arithmetic, set theory is not fully general, and so not logical
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
Asserting Excluded Middle is a hallmark of realism about the natural world
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
A 'model' is a meaning-assignment which makes all the axioms true
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
Differences between isomorphic structures seem unimportant
5. Theory of Logic / K. Features of Logics / 2. Consistency
Consistency is a purely syntactic property, unlike the semantic property of soundness
A 'consistent' theory cannot contain both a sentence and its negation
5. Theory of Logic / K. Features of Logics / 3. Soundness
Soundness is a semantic property, unlike the purely syntactic property of consistency
5. Theory of Logic / K. Features of Logics / 4. Completeness
A 'complete' theory contains either any sentence or its negation
6. Mathematics / A. Nature of Mathematics / 3. Numbers / b. Types of number
Rational numbers give answers to division problems with integers
The integers are answers to subtraction problems involving natural numbers
6. Mathematics / A. Nature of Mathematics / 3. Numbers / g. Real numbers
Real numbers provide answers to square root problems
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / d. Actual infinite
The classical mathematician believes the real numbers form an actual set
6. Mathematics / A. Nature of Mathematics / 7. Application of Mathematics
Logicists say mathematics is applicable because it is totally general
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / e. Peano arithmetic 2nd-order
Second-order induction is stronger as it covers all concepts, not just first-order definable ones
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / g. Incompleteness of Arithmetic
The Incompleteness proofs use arithmetic to talk about formal arithmetic
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / c. Fregean numbers
A successor is the union of a set with its singleton
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / d. Hume's Principle
Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle
6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / a. Mathematics is set theory
Set theory can prove the Peano Postulates
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
Talk of 'abstract entities' is more a label for the problem than a solution to it
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
If mathematics is not about particulars, observing particulars must be irrelevant
6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc.
Type theory prohibits (oddly) a set containing an individual and a set of individuals
Type theory has only finitely many items at each level, which is a problem for mathematics
The theory of types seems to rule out harmless sets as well as paradoxical ones.
6. Mathematics / C. Sources of Mathematics / 8. Finitism
Much infinite mathematics can still be justified finitely
Bounded quantification is originally finitary, as conjunctions and disjunctions
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
Gödel's First Theorem suggests there are truths which are independent of proof
The intuitionists are the idealists of mathematics
18. Thought / D. Concepts / 1. Concepts / a. Nature of concepts
Corresponding to every concept there is a class (some of them sets)