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

10240

Model theory deals with relations, reference and extensions

10239

The central notion of model theory is the relation of 'satisfaction'

5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms

10214

Theory ontology is never complete, but is only determined 'up to isomorphism'

10238

The set-theoretical hierarchy contains as many isomorphism types as possible

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