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
Coherence is a primitive, intuitive notion, not reduced to something formal
2. Reason / D. Definition / 7. Contextual Definition
An 'implicit definition' gives a direct description of the relations of an entity
4. Formal Logic / D. Modal Logic ML / 1. Modal Logic
Modal operators are usually treated as quantifiers
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
Axiom of Choice: some function has a value for every set in a given set
The Axiom of Choice seems to license an infinite amount of choosing
4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
Anti-realists reject set theory
5. Theory of Logic / B. Logical Consequence / 2. Types of Consequence
The two standard explanations of consequence are semantic (in models) and deductive
5. Theory of Logic / B. Logical Consequence / 5. Modus Ponens
Intuitionism only sanctions modus ponens if all three components are proved
5. Theory of Logic / C. Ontology of Logic / 1. Ontology of Logic
Either logic determines objects, or objects determine logic, or they are separate
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
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
Classical connectives differ from their ordinary language counterparts; '∧' is timeless, unlike 'and'
5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
A function is just an arbitrary correspondence between collections
5. Theory of Logic / G. Quantification / 6. Plural Quantification
Maybe plural quantifiers should be understood in terms of classes or sets
5. Theory of Logic / I. Semantics of Logic / 5. Satisfaction
A sentence is 'satisfiable' if it has a model
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
The central notion of model theory is the relation of 'satisfaction'
Model theory deals with relations, reference and extensions
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
Theory ontology is never complete, but is only determined 'up to isomorphism'
The set-theoretical hierarchy contains as many isomorphism types as possible
5. Theory of Logic / J. Model Theory in Logic / 3. L÷wenheim-Skolem Theorems
Any theory with an infinite model has a model of every infinite cardinality
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
Virtually all of mathematics can be modeled in set theory
6. Mathematics / A. Nature of Mathematics / 3. Numbers / g. Real numbers
Real numbers are thought of as either Cauchy sequences or Dedekind cuts
Understanding the real-number structure is knowing usage of the axiomatic language of analysis
6. Mathematics / A. Nature of Mathematics / 3. Numbers / i. Reals from cuts
Cuts are made by the smallest upper or largest lower number, some of them not rational
6. Mathematics / A. Nature of Mathematics / 6. Proof in Mathematics
For intuitionists, proof is inherently informal
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
There is no grounding for mathematics that is more secure than mathematics
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / d. Peano arithmetic
Natural numbers just need an initial object, successors, and an induction principle
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / b. Greek arithmetic
Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic)
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / e. Caesar problem
Property extensions outstrip objects, so shortage of objects caused the Caesar problem
6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / b. Mathematics is not set theory
Mathematical foundations may not be sets; categories are a popular rival
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / a. Structuralism
Baseball positions and chess pieces depend entirely on context
The even numbers have the natural-number structure, with 6 playing the role of 3
Could infinite structures be apprehended by pattern recognition?
The 4-pattern is the structure common to all collections of four objects
The main mathematical structures are algebraic, ordered, and topological
Some structures are exemplified by both abstract and concrete
Mathematical structures are defined by axioms, or in set theory
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / b. Varieties of structuralism
The main versions of structuralism are all definitionally equivalent
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / c. Nominalist structuralism
Is there is no more to structures than the systems that exemplify them?
Number statements are generalizations about number sequences, and are bound variables
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / d. Platonist structuralism
Because one structure exemplifies several systems, a structure is a one-over-many
There is no 'structure of all structures', just as there is no set of all sets
Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / e. Structuralism critique
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
We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false)
If mathematical objects are accepted, then a number of standard principles will follow
Platonists claim we can state the essence of a number without reference to the others
Platonism must accept that the Peano Axioms could all be false
6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics
Intuition is an outright hindrance to five-dimensional geometry
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
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
Can the ideal constructor also destroy objects?
Presumably nothing can block a possible dynamic operation?
7. Existence / A. Nature of Existence / 1. Nature of Existence
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
The abstract/concrete boundary now seems blurred, and would need a defence
Mathematicians regard arithmetic as concrete, and group theory as abstract
7. Existence / D. Theories of Reality / 6. Fictionalism
Fictionalism eschews the abstract, but it still needs the possible (without model theory)
Structuralism blurs the distinction between mathematical and ordinary objects
9. Objects / A. Existence of Objects / 1. Physical Objects
The notion of 'object' is at least partially structural and mathematical
9. Objects / B. Unity of Objects / 3. Unity Problems / e. Vague objects
A blurry border is still a border
10. Modality / A. Necessity / 6. Logical Necessity
Logical modalities may be acceptable, because they are reducible to satisfaction in models
10. Modality / E. Possible worlds / 1. Possible Worlds / a. Possible worlds
Why does the 'myth' of possible worlds produce correct modal logic?
15. Nature of Minds / C. Capacities of Minds / 3. Abstraction by mind
We apprehend small, finite mathematical structures by abstraction from patterns
18. Thought / D. Concepts / 6. Abstract Concepts / b. Abstracta from selection
Simple types can be apprehended through their tokens, via abstraction
18. Thought / D. Concepts / 6. Abstract Concepts / c. Abstracta by ignoring
We can apprehend structures by focusing on or ignoring features of patterns
We can focus on relations between objects (like baseballers), ignoring their other features
18. Thought / D. Concepts / 6. Abstract Concepts / g. Abstracta by equivalence
Abstract objects might come by abstraction over an equivalence class of base entities