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Ideas of Penelope Maddy, by Text
[American, b.1950, Professor of Logic and Philosophy of Science at the University of California, Irvine.]
I

p.345

17823

If mathematical objects exist, how can we know them, and which objects are they?

II

p.347

17825

Set theory (unlike the Peano postulates) can explain why multiplication is commutative

II

p.347

17824

The master science is physical objects divided into sets

III

p.347

17826

Standardly, numbers are said to be sets, which is neat ontology and epistemology

III

p.349

17828

Numbers are properties of sets, just as lengths are properties of physical objects

III

p.349

17827

Sets exist where their elements are, but numbers are more like universals

IV

p.350

17829

Number words are unusual as adjectives; we don't say 'is five', and numbers always come first

V

p.353

17830

Number theory doesn't 'reduce' to set theory, because sets have number properties

1988

Believing the Axioms I

§0

p.482

13011

New axioms are being sought, to determine the size of the continuum

§1.1

p.484

13014

Extensional sets are clearer, simpler, unique and expressive

§1.1

p.484

13013

The Axiom of Extensionality seems to be analytic

§1.3

p.485

13019

The Iterative Conception says everything appears at a stage, derived from the preceding appearances

§1.3

p.485

13018

Limitation of Size is a vague intuition that overlarge sets may generate paradoxes

§1.5

p.486

13021

The Axiom of Infinity states Cantor's breakthrough that launched modern mathematics

§1.5

p.486

13022

Infinite sets are essential for giving an account of the real numbers

§1.6

p.486

13023

The Power Set Axiom is needed for, and supported by, accounts of the continuum

§1.7

p.487

13024

Efforts to prove the Axiom of Choice have failed

§1.7

p.488

13025

Modern views say the Choice set exists, even if it can't be constructed

§1.7

p.488

13026

A large array of theorems depend on the Axiom of Choice

1990

Realism in Mathematics


p.191

17733

We know mindindependent mathematical truths through sets, which rest on experience [Jenkins]


p.223

8755

Maddy replaces pure sets with just objects and perceived sets of objects [Shapiro]


p.224

8756

Intuition doesn't support much mathematics, and we should question its reliability [Shapiro]

3 §2

p.19

10718

A natural number is a property of sets [Oliver]

1997

Naturalism in Mathematics

§107

p.117

18182

The extension of concepts is not important to me

I Intro

p.1

18163

Mathematics rests on the logic of proofs, and on the set theoretic axioms

I.1

p.5

18164

Frege solves the Caesar problem by explicitly defining each number

I.1

p.7

18167

We can get arithmetic directly from HP; Law V was used to get HP from the definition of number

I.1

p.9

18168

'Propositional functions' are propositions with a variable as subject or predicate

I.1

p.11

18169

Axiom of Reducibility: propositional functions are extensionally predicative

I.1

p.16

18172

Infinity has degrees, and large cardinals are the heart of set theory

I.1

p.17

18175

For any cardinal there is always a larger one (so there is no set of all sets)

I.1 n39

p.15

18171

Cantor and Dedekind brought completed infinities into mathematics

I.2

p.23

18177

In the ZFC hierarchy it is impossible to form Frege's set of all threeelement sets

I.2

p.26

18184

Making set theory foundational to mathematics leads to very fruitful axioms

I.2

p.26

18185

Unified set theory gives a final court of appeal for mathematics

I.2

p.26

18183

Set theory brings mathematics into one arena, where interrelations become clearer

I.2

p.26

18186

Identifying geometric points with real numbers revealed the power of set theory

I.2

p.27

18188

The line of rationals has gaps, but set theory provided an ordered continuum

I.2

p.27

18187

Theorems about limits could only be proved once the real numbers were understood

I.3

p.51

18190

Completed infinities resulted from giving foundations to calculus

I.3

p.52

18191

Axiom of Infinity: completed infinite collections can be treated mathematically

I.3

p.60

18193

The Axiom of Foundation says every set exists at a level in the set hierarchy

I.4

p.66

18194

'Forcing' can produce new models of ZFC from old models

I.5

p.73

18195

A Large Cardinal Axiom would assert everincreasing stages in the hierarchy

I.5

p.74

18196

An 'inaccessible' cardinal cannot be reached by union sets or power sets

II.5

p.131

18204

Scientists posit as few entities as possible, but set theorist posit as many as possible

II.6

p.143

18205

The theoretical indispensability of atoms did not at first convince scientists that they were real

II.6

p.143

18206

Science idealises the earth's surface, the oceans, continuities, and liquids

II.6

p.152

18207

Maybe applications of continuum mathematics are all idealisations

III.8 n1

p.299

10594

Henkin semantics is more plausible for plural logic than for secondorder logic

2011

Defending the Axioms

1.3

p.31

17605

Hilbert's geometry and Dedekind's real numbers were role models for axiomatization

1.3

p.35

17610

The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres

2.3

p.53

17614

The connection of arithmetic to perception has been idealised away in modern infinitary mathematics

2.4 n40

p.56

17615

Every infinite set of reals is either countable or of the same size as the full set of reals

3.3

p.99

17620

Critics of ifthenism say that not all starting points, even consistent ones, are worth studying

3.4

p.82

17618

Settheory tracks the contours of mathematical depth and fruitfulness

5.3ii

p.129

17625

If two mathematical themes coincide, that suggest a single deep truth
