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Ideas from 'Philosophy of Mathematics' by Stewart Shapiro [1997], by Theme Structure

[found in 'Philosophy of Mathematics:structure and ontology' by Shapiro,Stewart [OUP 1997,0-19-513930-5]].

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2. Reason / A. Nature of Reason / 6. Coherence
10237 Coherence is a primitive, intuitive notion, not reduced to something formal
2. Reason / D. Definition / 7. Contextual Definition
10204 An 'implicit definition' gives a direct description of the relations of an entity
4. Formal Logic / D. Modal Logic ML / 1. Modal Logic
10206 Modal operators are usually treated as quantifiers
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
10208 Axiom of Choice: some function has a value for every set in a given set
10252 The Axiom of Choice seems to license an infinite amount of choosing
4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
10207 Anti-realists reject set theory
5. Theory of Logic / B. Logical Consequence / 2. Types of Consequence
10259 The two standard explanations of consequence are semantic (in models) and deductive
5. Theory of Logic / B. Logical Consequence / 5. Modus Ponens
10257 Intuitionism only sanctions modus ponens if all three components are proved
5. Theory of Logic / C. Ontology of Logic / 1. Ontology of Logic
10253 Either logic determines objects, or objects determine logic, or they are separate
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
10251 The law of excluded middle might be seen as a principle of omniscience
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
10212 Classical connectives differ from their ordinary language counterparts; '∧' is timeless, unlike 'and'
5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
10209 A function is just an arbitrary correspondence between collections
5. Theory of Logic / G. Quantification / 6. Plural Quantification
10268 Maybe plural quantifiers should be understood in terms of classes or sets
5. Theory of Logic / I. Semantics of Logic / 4. Satisfaction
10235 A sentence is 'satisfiable' if it has a model
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
10239 The central notion of model theory is the relation of 'satisfaction'
10240 Model theory deals with relations, reference and extensions
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
10238 The set-theoretical hierarchy contains as many isomorphism types as possible
10214 Theory ontology is never complete, but is only determined 'up to isomorphism'
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
10234 Any theory with an infinite model has a model of every infinite cardinality
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
10201 Virtually all of mathematics can be modeled in set theory
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
10213 Real numbers are thought of as either Cauchy sequences or Dedekind cuts
18243 Understanding the real-number structure is knowing usage of the axiomatic language of analysis
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
18245 Cuts are made by the smallest upper or largest lower number, some of them not rational
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
10236 There is no grounding for mathematics that is more secure than mathematics
6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics
10256 For intuitionists, proof is inherently informal
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
10202 Natural numbers just need an initial object, successors, and an induction principle
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic
10205 Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic)
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
10222 Mathematical foundations may not be sets; categories are a popular rival
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
10218 Baseball positions and chess pieces depend entirely on context
10224 The even numbers have the natural-number structure, with 6 playing the role of 3
10228 Could infinite structures be apprehended by pattern recognition?
10230 The 4-pattern is the structure common to all collections of four objects
10249 The main mathematical structures are algebraic, ordered, and topological
10273 Some structures are exemplified by both abstract and concrete
10276 Mathematical structures are defined by axioms, or in set theory
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
10270 The main versions of structuralism are all definitionally equivalent
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism
10221 Is there is no more to structures than the systems that exemplify them?
10248 Number statements are generalizations about number sequences, and are bound variables
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
10220 Because one structure exemplifies several systems, a structure is a one-over-many
10223 There is no 'structure of all structures', just as there is no set of all sets
8703 Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics [Friend]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
10274 Does someone using small numbers really need to know the infinite structure of arithmetic?
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
10200 We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false)
10210 If mathematical objects are accepted, then a number of standard principles will follow
10215 Platonists claim we can state the essence of a number without reference to the others
10233 Platonism must accept that the Peano Axioms could all be false
6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics
10244 Intuition is an outright hindrance to five-dimensional geometry
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
10280 A stone is a position in some pattern, and can be viewed as an object, or as a location
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
10254 Can the ideal constructor also destroy objects?
10255 Presumably nothing can block a possible dynamic operation?
7. Existence / A. Nature of Existence / 1. Nature of Existence
10279 Can we discover whether a deck is fifty-two cards, or a person is time-slices or molecules?
7. Existence / C. Structure of Existence / 7. Abstract/Concrete / a. Abstract/concrete
10227 The abstract/concrete boundary now seems blurred, and would need a defence
10226 Mathematicians regard arithmetic as concrete, and group theory as abstract
7. Existence / D. Theories of Reality / 7. Fictionalism
10262 Fictionalism eschews the abstract, but it still needs the possible (without model theory)
10277 Structuralism blurs the distinction between mathematical and ordinary objects
9. Objects / A. Existence of Objects / 1. Physical Objects
10272 The notion of 'object' is at least partially structural and mathematical
9. Objects / B. Unity of Objects / 3. Unity Problems / e. Vague objects
10275 A blurry border is still a border
10. Modality / A. Necessity / 6. Logical Necessity
10258 Logical modalities may be acceptable, because they are reducible to satisfaction in models
10. Modality / E. Possible worlds / 1. Possible Worlds / a. Possible worlds
10266 Why does the 'myth' of possible worlds produce correct modal logic?
15. Nature of Minds / C. Capacities of Minds / 3. Abstraction by mind
10203 We apprehend small, finite mathematical structures by abstraction from patterns
18. Thought / E. Abstraction / 2. Abstracta by Selection
10229 Simple types can be apprehended through their tokens, via abstraction
18. Thought / E. Abstraction / 3. Abstracta by Ignoring
10217 We can apprehend structures by focusing on or ignoring features of patterns
9554 We can focus on relations between objects (like baseballers), ignoring their other features
18. Thought / E. Abstraction / 7. Abstracta by Equivalence
10231 Abstract objects might come by abstraction over an equivalence class of base entities