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Ideas of Philip Kitcher, by Text
[British, b.1947, Studied at Cambridge, then pupil of Thomas Kuhn. Professor at Columbia.]
1984

The Nature of Mathematical Knowledge


p.64

6298

Kitcher says maths is an idealisation of the world, and our operations in dealing with it [Resnik]

Intro

p.5

12387

Mathematical knowledge arises from basic perception

Intro

p.12

12412

My constructivism is mathematics as an idealization of collecting and ordering objects

01.2

p.17

12413

A 'warrant' is a process which ensures that a true belief is knowledge

01.3

p.22

12389

Knowledge is a priori if the experience giving you the concepts thus gives you the knowledge

01.4

p.24

12390

A priori knowledge comes from available a priori warrants that produce truth

01.6

p.29

12416

We have some selfknowledge a priori, such as knowledge of our own existence

02.2

p.45

12418

In long mathematical proofs we can't remember the original a priori basis

02.3

p.46

12392

Mathematical a priorism is conceptualist, constructivist or realist

03.1

p.50

12420

If mathematics comes through intuition, that is either inexplicable, or too subjective

03.2

p.53

12393

Intuition is no basis for securing a priori knowledge, because it is fallible

03.3

p.61

18061

Mathematical intuition is not the type platonism needs

04.1

p.65

18063

Conceptualists say we know mathematics a priori by possessing mathematical concepts

04.6

p.86

18064

If meaning makes mathematics true, you still need to say what the meanings refer to

04.I

p.68

12423

Analyticity avoids abstract entities, but can there be truth without reference?

05.2

p.92

18065

We derive limited mathematics from ordinary things, and erect powerful theories on their basis

06.1

p.105

18066

The old view is that mathematics is useful in the world because it describes the world

06.1

p.107

18067

Abstract objects were a bad way of explaining the structure in mathematics

06.2

p.108

18068

Arithmetic is made true by the world, but is also made true by our constructions

06.2

p.109

18069

Arithmetic is an idealizing theory

06.2

p.111

18070

We develop a language for correlations, and use it to perform higher level operations

06.3

p.112

18071

A oneoperation is the segregation of a single object

06.4

p.124

12395

Real numbers stand to measurement as natural numbers stand to counting

06.5

p.142

18072

Constructivism is ontological (that it is the work of an agent) and epistemological (knowable a priori)

06.5

p.143

18074

Intuitionists rely on assertability instead of truth, but assertability relies on truth

06.5

p.144

18075

Idealisation trades off accuracy for simplicity, in varying degrees

07.5

p.176

12425

Complex numbers were only accepted when a geometrical model for them was found

07.5

p.176

18077

The defenders of complex numbers had to show that they could be expressed in physical terms

09.3

p.206

18078

The interest or beauty of mathematics is when it uses current knowledge to advance undestanding

09.4

p.212

12426

The 'beauty' or 'interest' of mathematics is just explanatory power

10.2

p.238

18083

With infinitesimals, you divide by the time, then set the time to zero

p.89

p.105

20473

If experiential can defeat a belief, then its justification depends on the defeater's absence [Casullo]

2000

A Priori Knowledge Revisited

§II

p.69

12428

Many necessities are inexpressible, and unknowable a priori

§II

p.69

12429

Knowing our own existence is a priori, but not necessary

§VII

p.86

12430

Classical logic is our preconditions for assessing empirical evidence

§VII

p.87

12431

I believe classical logic because I was taught it and use it, but it could be undermined
