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Ideas of Michèle Friend, by Text
[American, fl. 2007, Professor at George Washington University, Washington D.C.]
2007

Introducing the Philosophy of Mathematics


p.128

8713

In classical/realist logic the connectives are defined by truthtables

1.4

p.12

8661

The natural numbers are primitive, and the ordinals are up one level of abstraction

1.4

p.13

8663

Raising omega to successive powers of omega reveal an infinity of infinities

1.4

p.13

8662

The first limit ordinal is omega (greater, but without predecessor), and the second is twiceomega

1.5

p.14

8664

Cardinal numbers answer 'how many?', with the order being irrelevant

1.5

p.15

8665

A 'proper subset' of A contains only members of A, but not all of them

1.5

p.15

8666

Infinite sets correspond onetoone with a subset

1.5

p.16

8667

The 'integers' are the positive and negative natural numbers, plus zero

1.5

p.17

8668

The 'rational' numbers are those representable as fractions

1.5

p.17

8669

Between any two rational numbers there is an infinite number of rational numbers

1.5

p.19

8671

The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps

1.5

p.19

8670

A number is 'irrational' if it cannot be represented as a fraction

1.5

p.21

8672

A 'powerset' is all the subsets of a set

2.3

p.26

8674

The BuraliForti paradox asks whether the set of all ordinals is itself an ordinal

2.3

p.27

8675

Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets'

2.3

p.29

8676

Is mathematics based on sets, types, categories, models or topology?

2.3

p.32

8677

Set theory makes a minimum ontological claim, that the empty set exists

2.3

p.33

8678

Most mathematical theories can be translated into the language of set theory

2.3

p.34

3678

Reductio ad absurdum proves an idea by showing that its denial produces contradiction

2.4

p.36

8680

Classical definitions attempt to refer, but intuitionist/constructivist definitions actually create objects

2.5

p.36

8681

The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts?

2.6

p.42

8682

Major set theories differ in their axioms, and also over the additional axioms of choice and infinity

3.1

p.51

8685

Studying biology presumes the laws of chemistry, and it could never contradict them

3.4

p.64

8688

Concepts can be presented extensionally (as objects) or intensionally (as a characterization)

3.7

p.77

8694

Free logic was developed for fictional or nonexistent objects

4.1

p.82

8696

Structuralist says maths concerns concepts about base objects, not base objects themselves

4.1

p.82

8695

Structuralism focuses on relations, predicates and functions, with objects being inessential

4.4

p.90

8699

Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)?

4.4

p.91

8700

'In re' structuralism says that the process of abstraction is patternspotting

4.4

p.93

8702

In structuralism the number 8 is not quite the same in different structures, only equivalent

4.4

p.93

8701

The number 8 in isolation from the other numbers is of no interest

4.5

p.97

8704

Structuralists call a mathematical 'object' simply a 'place in a structure'

5.1

p.104

8705

Antirealists see truth as our servant, and epistemically contrained

5.1

p.106

8706

Constructivism rejects too much mathematics

5.2

p.106

8707

Intuitionists typically retain bivalence but reject the law of excluded middle

5.2

p.107

8708

Double negation elimination is not valid in intuitionist logic

5.2

p.108

8709

The law of excluded middle is syntactic; it just says A or notA, not whether they are true or false

5.5

p.122

8711

Intuitionists read the universal quantifier as "we have a procedure for checking every..."

6.1

p.128

8712

Mathematics should be treated as true whenever it is indispensable to our best physical theory

6.6

p.149

8716

Formalism is unconstrained, so cannot indicate importance, or directions for research

Glossary

p.172

8721

An 'impredicative' definition seems circular, because it uses the term being defined
