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Ideas of Stewart Shapiro, by Text
[American, b.1951, Professor at Ohio State University; visiting Professor at St Andrew's University.]
1989

Structure and Ontology

146

p.60

9626

A structure is an abstraction, focussing on relationships, and ignoring other features

1991

Foundations without Foundationalism


p.225

15944

Secondorder logic is better than set theory, since it only adds relations and operations, and nothing else [Lavine]

Pref

p.17

13624

The 'triumph' of firstorder logic may be related to logicism and the Hilbert programme, which failed

Pref

p.17

13625

Mathematics and logic have no border, and logic must involve mathematics and its ontology

Pref

p.16

13626

Semantic consequence is ineffective in secondorder logic

Pref

p.15

13627

There is no 'correct' logic for natural languages

Pref

p.14

13628

We can live well without completeness in logic

Pref

p.13

13629

Broad standard semantics, or Henkin semantics with a subclass, or manysorted firstorder semantics?

Pref

p.12

13630

Noncompactness is a strength of secondorder logic, enabling characterisation of infinite structures

Pref

p.9

13631

Are sets part of logic, or part of mathematics?

1.1

p.3

13632

Finding the logical form of a sentence is difficult, and there are no criteria of correctness

1.1

p.5

13633

'Satisfaction' is a function from models, assignments, and formulas to {true,false}

1.1

p.6

13634

Satisfaction is 'truth in a model', which is a model of 'truth'

1.1

p.8

13635

'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence

1.2.1

p.12

13636

An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation

1.2.1

p.12

13637

If a logic is incomplete, its semantic consequence relation is not effective

1.3

p.16

13638

Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects

1.3

p.19

13640

Russell's paradox shows that there are classes which are not iterative sets

2.1

p.26

13641

Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals

2.3.1

p.36

13642

Logic is the ideal for learning new propositions on the basis of others

2.5

p.43

13643

Aristotelian logic is complete

2.5.1

p.44

13644

Semantics for models uses settheory

3.3

p.73

13650

Henkin semantics has separate variables ranging over the relations and over the functions

3.3

p.73

13645

In standard semantics for secondorder logic, a single domain fixes the ranges for the variables

4.1

p.79

13646

Compactness is derived from soundness and completeness

4.1

p.80

13648

The LöwenheimSkolem theorems show an explosion of infinite models, so 1storder is useless for infinity

4.1

p.80

13649

Completeness, Compactness and LöwenheimSkolem fail in secondorder standard semantics

4.1

p.80

13647

Choice is essential for proving downward LöwenheimSkolem

4.2

p.85

13651

A set is 'transitive' if contains every member of each of its members

5 n28

p.132

13657

Firstorder arithmetic can't even represent basic number theory

5.1.2

p.105

13652

The 'continuum' is the cardinality of the powerset of a denumerably infinite set

5.1.3

p.106

13653

'Wellordering' of a set is an irreflexive, transitive, and binary relation with a least element

5.1.4

p.109

13654

It is central to the iterative conception that membership is wellfounded, with no infinite descending chains

5.3.3

p.123

13656

Some sets of natural numbers are definable in settheory but not in arithmetic

6.5

p.158

13658

Downward LöwenheimSkolem: each satisfiable countable set always has countable models

6.5

p.158

13659

Upward LöwenheimSkolem: each infinite model has infinite models of all sizes

6.5

p.158

13661

A language is 'semantically effective' if its logical truths are recursively enumerable

6.5

p.159

13660

Maybe compactness, semantic effectiveness, and the LöwenheimSkolem properties are desirable

7.1

p.173

13662

Firstorder logic was an afterthought in the development of modern logic

7.1

p.174

13663

Some reject formal properties if they are not defined, or defined impredicatively

7.1

p.176

13664

Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions

7.1

p.177

13666

Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets

7.2

p.178

13667

Skolem and Gödel championed firstorder, and Zermelo, Hilbert, and Bernays championed higherorder

7.2.1

p.180

13668

Bernays (1918) formulated and proved the completeness of propositional logic

7.2.2

p.182

13669

Can one develop set theory first, then derive numbers, or are numbers more basic?

7.3

p.196

13670

Categoricity can't be reached in a firstorder language

9.1

p.238

13673

The notion of finitude is actually built into firstorder languages

9.1.4

p.243

13674

We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models

9.1.4

p.245

13675

Substitutional semantics only has countably many terms, so Upward LöwenheimSkolem trivially fails

9.1.4

p.246

13676

Only higherorder languages can specify that 0,1,2,... are all the natural numbers that there are

9.3

p.251

13677

Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals

1997

Philosophy of Mathematics


p.258

10279

Can we discover whether a deck is fiftytwo cards, or a person is timeslices or molecules?

Intro

p.4

10200

We distinguish realism 'in ontology' (for objects), and 'in truthvalue' (for being either true or false)

Intro

p.5

10202

Natural numbers just need an initial object, successors, and an induction principle

Intro

p.5

10201

Virtually all of mathematics can be modeled in set theory

Intro

p.11

10203

We apprehend small, finite mathematical structures by abstraction from patterns

Intro

p.13

10205

Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic)

Intro

p.13

10204

An 'implicit definition' gives a direct description of the relations of an entity

Intro

p.16

10206

Modal operators are usually treated as quantifiers

Intro

p.17

10207

Antirealists reject set theory

1

p.24

10208

Axiom of Choice: some function has a value for every set in a given set

1

p.24

10209

A function is just an arbitrary correspondence between collections

1

p.25

10210

If mathematical objects are accepted, then a number of standard principles will follow

2.5

p.53

10213

Real numbers are thought of as either Cauchy sequences or Dedekind cuts

2.5

p.55

10214

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

3

p.42

10212

Classical connectives differ from their ordinary language counterparts; '∧' is timeless, unlike 'and'

3.1

p.72

10215

Platonists claim we can state the essence of a number without reference to the others

3.1

p.74

10217

We can apprehend structures by focusing on or ignoring features of patterns

3.1

p.76

10218

Baseball positions and chess pieces depend entirely on context

3.3

p.84

10220

Because one structure exemplifies several systems, a structure is a oneovermany

3.3

p.85

10221

Is there is no more to structures than the systems that exemplify them?

3.3

p.87

10222

Mathematical foundations may not be sets; categories are a popular rival

3.4

p.95

10223

There is no 'structure of all structures', just as there is no set of all sets

3.5

p.100

10224

The even numbers have the naturalnumber structure, with 6 playing the role of 3

4.1

p.111

10227

The abstract/concrete boundary now seems blurred, and would need a defence

4.1

p.112

10228

Could infinite structures be apprehended by pattern recognition?

4.1 n1

p.109

10226

Mathematicians regard arithmetic as concrete, and group theory as abstract

4.2

p.113

10229

Simple types can be apprehended through their tokens, via abstraction

4.2

p.115

10230

The 4pattern is the structure common to all collections of four objects

4.4

p.96

8703

Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics [Friend]

4.5

p.124

10231

Abstract objects might come by abstraction over an equivalence class of base entities

4.7

p.131

10233

Platonism must accept that the Peano Axioms could all be false

4.8

p.132

10234

Any theory with an infinite model has a model of every infinite cardinality

4.8

p.135

10235

A sentence is 'satisfiable' if it has a model

4.8

p.135

10237

Coherence is a primitive, intuitive notion, not reduced to something formal

4.8

p.135

10236

There is no grounding for mathematics that is more secure than mathematics

4.8

p.136

10238

The settheoretical hierarchy contains as many isomorphism types as possible

4.9

p.138

18243

Understanding the realnumber structure is knowing usage of the axiomatic language of analysis

4.9

p.139

10240

Model theory deals with relations, reference and extensions

4.9

p.139

10239

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

5.2

p.150

10244

Intuition is an outright hindrance to fivedimensional geometry

5.3.4

p.166

10248

Number statements are generalizations about number sequences, and are bound variables

5.4

p.171

18245

Cuts are made by the smallest upper or largest lower number, some of them not rational

5.5

p.176

10249

The main mathematical structures are algebraic, ordered, and topological

6.3

p.187

10251

The law of excluded middle might be seen as a principle of omniscience

6.3

p.188

10252

The Axiom of Choice seems to license an infinite amount of choosing

6.4

p.192

10253

Either logic determines objects, or objects determine logic, or they are separate

6.5

p.194

10254

Can the ideal constructor also destroy objects?

6.5

p.194

10255

Presumably nothing can block a possible dynamic operation?

6.7

p.205

10256

For intuitionists, proof is inherently informal

6.7

p.207

10257

Intuitionism only sanctions modus ponens if all three components are proved

7.1

p.216

10258

Logical modalities may be acceptable, because they are reducible to satisfaction in models

7.2

p.222

10259

The two standard explanations of consequence are semantic (in models) and deductive

7.2

p.223

10262

Fictionalism eschews the abstract, but it still needs the possible (without model theory)

7.4

p.233

10266

Why does the 'myth' of possible worlds produce correct modal logic?

7.4

p.235

10268

Maybe plural quantifiers should be understood in terms of classes or sets

7.5

p.242

10270

The main versions of structuralism are all definitionally equivalent

8.1

p.245

10272

The notion of 'object' is at least partially structural and mathematical

8.2

p.248

10273

Some structures are exemplified by both abstract and concrete

8.2

p.254

10274

Does someone using small numbers really need to know the infinite structure of arithmetic?

8.3

p.255

10276

Mathematical structures are defined by axioms, or in set theory

8.3

p.255

10275

A blurry border is still a border

8.4

p.256

10277

Structuralism blurs the distinction between mathematical and ordinary objects

8.4

p.259

10280

A stone is a position in some pattern, and can be viewed as an object, or as a location

p.74

p.79

9554

We can focus on relations between objects (like baseballers), ignoring their other features

2000

Thinking About Mathematics

1.1

p.3

8725

Rationalism tries to apply mathematical methodology to all of knowledge

1.2

p.9

8730

'Impredicative' definitions refer to the thing being described

1.2

p.9

8729

Intuitionists deny excluded middle, because it is committed to transcendent truth or objects

2.2.1

p.26

8731

Conceptualist are just realists or idealist or nominalists, depending on their view of concepts

5.1

p.113

8744

Logicism seems to be a nonstarter if (as is widely held) logic has no ontology of its own

6.1.1

p.142

8749

Term Formalism says mathematics is just about symbols  but real numbers have no names

6.1.2

p.144

8750

Game Formalism is just a matter of rules, like chess  but then why is it useful in science?

6.2

p.149

8752

Deductivism says mathematics is logical consequences of uninterpreted axioms

7.1

p.174

8753

Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions

10.1

p.258

8760

Numbers do not exist independently; the essence of a number is its relations to other numbers

10.1

p.259

8761

A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them

10.2

p.265

8762

Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3

10.2

p.267

8763

The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex

10.3 n7

p.272

8764

Categories are the best foundation for mathematics

7.2 n4

p.181

18249

Cauchy gave a formal definition of a converging sequence.

2.1

p.33

10290

Secondorder variables also range over properties, sets, relations or functions

2.1

p.34

10588

Firstorder logic is Complete, and Compact, with the LöwenheimSkolem Theorems

2.1

p.34

10292

Downward LöwenheimSkolem: if there's an infinite model, there is a countable model

2.1

p.34

10590

Up LöwenheimSkolem: if natural numbers satisfy wffs, then an infinite domain satisfies them

2.1

p.34

10294

Secondorder logic has the expressive power for mathematics, but an unworkable model theory

2.2.1

p.36

10591

Logicians use 'property' and 'set' interchangeably, with little hanging on it

2.3.2

p.47

10296

The LöwenheimSkolem Theorems fail for secondorder languages with standard semantics

2.4

p.49

10297

The LöwenheimSkolem theorem seems to be a defect of firstorder logic

2.4

p.50

10298

Some say that secondorder logic is mathematics, not logic

2.4

p.51

10299

If the aim of logic is to codify inferences, secondorder logic is useless

2.4

p.51

10300

Logical consequence can be defined in terms of the logical terminology

n 3

p.52

10301

The axiom of choice is controversial, but it could be replaced
