Metatheory

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Contents

1 Contextualism 11.1 Epistemology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.3 Criticisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.4 Experimental research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 References and further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Correspondence principle 62.1 Quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Other scientic theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.1 Bohr model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.2 One-dimensional potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3.3 Multiperiodic motionBohrSommerfeld quantization . . . . . . . . . . . . . . . . . . . 92.3.4 The quantum harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.5 Relativistic kinetic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Denitionism 113.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Eclecticism 124.1 Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 Architecture and art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.3 Psychology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.4 Martial arts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.5 Philology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.6 Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

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4.7 Religion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5 Emic and etic 175.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.3 Examples of etic case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.4 Examples of emic case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.5 Importance as regards personality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.6 Secondary sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

6 Explanatory power 236.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.3 Relation to other criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

7 Metamathematics 257.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.2 Milestones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

7.2.1 Principia Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.2.2 Gdels completeness theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.2.3 Gdels incompleteness theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.2.4 Tarskis denition of model-theoretic satisfaction . . . . . . . . . . . . . . . . . . . . . . . 277.2.5 The impossibility of the Entscheidungsproblem . . . . . . . . . . . . . . . . . . . . . . . 27

7.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

8 Metatheory 308.1 Taxonomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308.2 In mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

9 Neopragmatism 32

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9.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329.1.1 Anglo-analytic inuences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329.1.2 Continental inuences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339.1.3 Wittgenstein and language games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

9.2 Richard Rorty and anti-representationalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349.3 Neopragmatism as distinguished from idealism and epistemic relativism . . . . . . . . . . . . . . . 349.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

10 Proof (truth) 3610.1 On proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

11 Psychologism 3811.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3811.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3811.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

12 Received view 40

13 Relativism 4113.1 Forms of relativism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

13.1.1 Anthropological versus philosophical relativism . . . . . . . . . . . . . . . . . . . . . . . 4113.1.2 Descriptive versus normative relativism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

13.2 Postmodernism and relativism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4213.3 Related and contrasting positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4213.4 Theatre and relativism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4313.5 Catholic Church and relativism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

13.5.1 Leo XIII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4313.5.2 John Paul II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4313.5.3 Benedict XVI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

13.6 Criticisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4413.7 Advocates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

13.7.1 Indian religions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4513.7.2 Sophists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4513.7.3 Bernard Crick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4513.7.4 Paul Feyerabend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4613.7.5 Thomas Kuhn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4613.7.6 George Lako and Mark Johnson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

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13.7.7 Robert Nozick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4713.7.8 Joseph Margolis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4713.7.9 Richard Rorty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4813.7.10 Isaiah Berlin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

13.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4813.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4913.10Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5113.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

14 Semantic holism 5214.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5214.2 Problems with semantic holism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5214.3 Semantic holism and holism of mental content . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5314.4 Semantic holism and conrmational holism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5314.5 Moderate holism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

14.5.1 Arguments against molecularism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5414.6 Holism and compositionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5514.7 Holism and externalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5614.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5714.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

15 Structuralism 5815.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5815.2 Structuralism in linguistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5915.3 Structuralism in anthropology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6015.4 Structuralism in literary theory and criticism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6115.5 History and background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6115.6 Interpretations and general criticisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6215.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6215.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6215.9 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6315.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

16 Succinctness 6516.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

17 Theoretical denition 6617.1 In Dierent Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

17.1.1 Sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6617.1.2 Interdisciplinary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

17.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6717.2.1 In Natural Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6717.2.2 In Medicine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

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17.2.3 In Social Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6817.2.4 In Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

17.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6817.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

18 Theoreticism 7018.1 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

19 Theory of justication 7119.1 Subjects of justication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7119.2 Justications and explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7119.3 Justication is a normative activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7119.4 Theories of justication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7219.5 Justiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

19.5.1 Commonly used justiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7319.6 Criticisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7319.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7419.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7419.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7419.10Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 75

19.10.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7519.10.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7719.10.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Chapter 1

Contextualism

This article is about epistemological and ethical contextualism. For information about semantic contextu-alism, see Context principle.

Contextualism describes a collection of views in philosophy which emphasize the context in which an action, utter-ance, or expression occurs, and argues that, in some important respect, the action, utterance, or expression can onlybe understood relative to that context.[1] Contextualist views hold that philosophically controversial concepts, suchas meaning P, knowing that P, having a reason to A, and possibly even being true or being right only havemeaning relative to a specied context. Some philosophers[2] hold that context-dependence may lead to relativism;[3]nevertheless, contextualist views are increasingly popular within philosophy.[4]

In ethics, contextualist views are often closely associated with situational ethics, or with moral relativism.[5]

In architectural theory, contextualism is a theory of design wherein modern (not be confused with modernism) build-ing types are harmonized with urban forms usual to a traditional city.[6]

1.1 Epistemology

1.1.1 IntroductionIn epistemology, contextualism is the treatment of the word 'knows as context-sensitive. Context-sensitive expressionsare ones that express dierent propositions relative to dierent contexts of use.[7] For example, some terms that arerelatively uncontroversially considered context-sensitive are indexicals, such 'I', 'here', and 'now'. While the word 'I'has a constant linguistic meaning in all contexts of use, whom it refers to varies with context. Similarly, epistemiccontextualists argue that the word 'knows is context sensitive, expressing dierent relations in some dierent contexts.What varies with context is how well-positioned a subject must be with respect to a proposition to count as knowingit. Contextualism in epistemology then is a semantic thesis about how 'knows works in English, not a theory ofwhat knowledge, justication, or strength of epistemic position consists in.[8] However, epistemologists combinecontextualism with views about what knowledge is to address epistemological puzzles and issues, such as skepticism,the Gettier problem, and the Lottery paradox.Contextualist accounts of knowledge became increasingly popular toward the end of the 20th century, particularlyas responses to the problem of skepticism. Contemporary contextualists include Michael Blome-Tillmann, MichaelWilliams, Stewart Cohen, Keith DeRose, David Lewis, Gail Stine, and George Mattey.The main tenet of contextualist epistemology, no matter what account of knowledge it is wedded to, is that knowledgeattributions are context-sensitive. That is, when we attribute knowledge to someone, the context in which we use theterm 'knowledge' determines the standards relative to which knowledge is being attributed (or denied). If we use itin everyday conversational contexts, the contextualist maintains, most of our claims to know things are true, despiteskeptics attempts to show we know little or nothing. But if the term 'knowledge' is used when skeptical hypothesesare being discussed, we count as knowing very little, if anything. Contextualists use this to explain why skepticalarguments can be persuasive, while at the same time protecting the correctness of our ordinary claims to knowthings. It is important to note that this theory does not allow that someone can have knowledge at one moment andnot the other, for this would hardly be a satisfying epistemological answer. What contextualism entails is that in one

1

2 CHAPTER 1. CONTEXTUALISM

context an utterance of a knowledge attribution can be true, and in a context with higher standards for knowledge,the same statement can be false. This happens in the same way that 'I' can correctly be used (by dierent people) torefer to dierent people at the same time.Thus, the standards for attributing knowledge to someone, the contexualist claims, vary from one users context to thenext. Thus, if I say John knows that his car is in front of him, the utterance is true if and only if (1) John believesthat his car is in front of him, (2) the car is in fact in front of him, and (3) John meets the epistemic standards that my(the speakers) context selects. This is a loose contextualist account of knowledge, and there are many signicantlydierent theories of knowledge that can t this contextualist template and thereby come in a contextualist form.For instance, an evidentialist account of knowledge can be an instance of contextualism if its held that strength ofjustication is a contextually varying matter. And one who accepts a relevant alternatives account of knowledge canbe a contextualist by holding that what range of alternatives are relevant is sensitive to conversational context. DeRoseadopts a type of modal or safety (as it has since come to be known) account on which knowledge is a matter ofones belief as to whether or not p is the case matching the fact of the matter, not only in the actual world, but also inthe suciently close possible worlds: Knowledge amounts to there being no nearby worlds in which one goes wrongwith respect to p. But how close is suciently close? Its here that DeRose takes the modal account of knowledge in acontextualist direction, for the range of epistemically relevant worlds is what varies with context: In high standardscontexts ones belief must match the fact of the matter through a much wider range of worlds than is relevant to lowstandards contexts.

1.1.2 Example

It is claimed that Neurophilosophy has the goal of contextualizing. Wemust contextualize questions usually dealt within the physical and epistemological domains into the context of the empirical domain, the domain of observation inthird-person perspective. Rather than approaching themetaphysical and epistemological issues [about the nature andfeatures of brain and mind] from a mind-based (as in traditional philosophy) or brain-reductive (as in neuroscience)perspective, we therefore pursue a brain-based strategy and thus a non-reductive neurophilosophy. The variousarguments against the material or physicalistic view of consciousness...are directly compared with the empirical dataand are thus put into the empirical, that is, neuroscientic context of consciousness.[9]

1.1.3 Criticisms

However, contextualist epistemology has been criticized by several philosophers. Contextualism is opposed to anygeneral form of Invariantism, which claims that knowledge is not context-sensitive (i.e. it is invariant). More recentcriticism has been in the form of rival theories, including Subject-Sensitive Invariantism (SSI), mainly due to the workof John Hawthorne (2004), and Interest-Relative Invariantism (IRI), due to Jason Stanley (2005). SSI claims that itis the context of the subject of the knowledge attribution that determines the epistemic standards, whereas Contex-tualism maintains it is the attributor. IRI, on the other hand, argues that it is the context of the practical interestsof the subject of the knowledge attribution that determines the epistemic standards. Stanley writes that bare IRI issimply the claim that whether or not someone knows that p may be determined in part by practical facts about thesubjects environment.[10] (Contextualism is a misnomer for either form of Invariantism, since Contextualismamong epistemologists is considered to be restricted to a claim about the context-sensitivity of knowledge attribu-tions (or the word knows). Thus, any view which maintains that something other than knowledge attributions arecontext-sensitive is not, strictly speaking, a form of Contextualism.) DeRose (2009) responds to recent attacks oncontextualism, and argues that contextualism is superior to these recent rivals.An alternative to contextualism called contrastivism has been proposed by Jonathan Schaer. Contrastivism, likecontextualism, uses semantic approaches to tackle the problem of skepticism.[11]

1.1.4 Experimental research

Recent work in the new eld of experimental philosophy has taken an empirical approach to testing the claims ofcontextualism and related views. This research has proceeded by conducting experiments in which ordinary non-philosophers are presented with vignettes which involve a knowledge ascription. Participants are then asked to reporton the status of that knowledge ascription. The studies address contextualism by varying the context of the knowledgeascription, e.g., how important it is that the agent in the vignette has accurate knowledge.

1.2. SEE ALSO 3

In the studies completed up to this point, no support for contextualism has been found.[12] This critique of contextual-ism can be summed up as: stakes have no impact on evidence. More specically, non-philosophical intuitions aboutknowledge attributions are not aected by the importance to the potential knower of the accuracy of that knowledge.Some may argue that these empirical studies for the most part have not been well designed for testing contextual-ism, which claims that the context of the attributor of knowledge aects the epistemic standards that govern theirclaims. Because most of the empirical studies don't vary the stakes for the attributor, but for the subject being de-scribed, these studies are more relevant to the evaluation of John Hawthornes Subject-Sensitive Invariantism orJason Stanleys Interest-Relative Invariantismviews on which the stakes for the putative subject of knowledgecan aect whether that subject knowsthan they are of contextualism. However, Feltz & Zarpentine (forthcoming)have tested the stakes for both the subject and the attributor, and the results are not in keeping with contextualism.Experimental work continues to be done on this topic.[13]

1.2 See also Perspectivism Anekantavada Multi-valued logic False dilemma Principle of Bivalence Exclusive disjunction Degrees of truth Fuzzy logic Logical disjunction Logical value Propositional logic Relativism Rhizome (philosophy) Semiotic anthropology Truth Epistemology at Wikiversity

1.3 Footnotes[1] Price (2008).

[2] Feldman (1999).

[3] Mackie (1977)

[4] Price (2008).

[5] Timmons (1998).

[6] Jencks, p. 78-79

[7] Stanley (2005), p. 16.

[8] Stanley (2005), p. 17.

4 CHAPTER 1. CONTEXTUALISM

[9] Northo, Georg (2014) p. 351

[10] Stanley (2005), p. 85.

[11] Schaer (2004).

[12] See Feltz and Zarpentine (2010), May, Sinnott-Armstrong, Hull, and Zimmerman (2010), and Buckwatler (2010).

[13] See, for example, Schaer and Knobe (2011).

1.4 References and further reading Annis, David. 1978. A Contextualist Theory of Epistemic Justication, inAmerican Philosophical Quarterly,15: 213-219.

Buckwalter, Wesley (2010). Knowledge Isn't Closed on Saturday: A Study in Ordinary Language. Reviewof Philosophy and Psychology 1 (3): 395406. doi:10.1007/s13164-010-0030-3.

Cappelen, H. & Lepore, E. 2005. Insensitive Semantics: A Defense of Semantic Minimalism and Speech ActPluralism, Blackwell Publishing.

Cohen, Stuart. 1998. Contextualist Solutions to Epistemological Problems: Scepticism, Gettier, and theLottery. Australasian Journal of Philosophy, 76: 289-306.

Cohen Stuart. 1999. Contextualism, Skepticism, and Reasons, in Tomberlin 1999. DeRose, Keith. 1992. Contextualism and Knowledge Attributions, Philosophy and Phenomenological Re-

search 52: 913-929.

DeRose, Keith. 1995. Solving the Skeptical Problem, Philosophical Review 104: 1-52. DeRose, Keith. 1999. Contextualism: An Explanation and Defense, in Greco and Sosa 1999. DeRose, Keith. 2002. Assertion, Knowledge, and Context, Philosophical Review 111: 167-203. DeRose, Keith. 2009. The Case for Contextualism: Knowledge, Skepticism and Context, Vol. 1, Oxford: OxfordUniversity Press.

Feldman, Richard. 1999. Contextualism and Skepticism, in Tomberlin 1999. Feltz, Adam; Zarpentine, Chris (2010). Do You KnowMore When It Matters Less?". Philosophical Psychol-

ogy 23 (5): 683706. doi:10.1080/09515089.2010.514572.

Greco, J. & Sosa, E. 1999. Blackwell Guide to Epistemology, Blackwell Publishing. Hawthorne, John. 2004. Knowledge and Lotteries, Oxford: Oxford University Press. Jencks, Charles (2002). New Paradigm In Architecture (7th ed.). Yale University Press. ISBN 0-300-09512-0. Mackie, J.L. 1977, Ethics: Inventing Right and Wrong, Viking Press, ISBN 0-14-013558-8. May, Joshua, Sinnott-Armstrong, Walter, Hull, Jay G. & Zimmerman, Aaron. 2010. "Practical Interests, Rel-evant Alternatives, and Knowledge Attributions: An Empirical Study", Review of Philosophy and Psychology(formerly European Review of Philosophy), special issue on Psychology and Experimental Philosophy ed. byEdouard Machery, Tania Lombrozo, & Joshua Knobe, Vol. 1, No. 2, pp. 265273.

Northo, Georg (2014). Neurophilosophy of consciousness: frommind to consciousness. Minding the Brain:A Guide to Philosophy and Neuroscience. Palgrave Macmillan. p. 351. ISBN 9781137406057.

Price, A. W. 2008. ' 'Contextuality in Practical Reason' ', Oxford University Press. Schaer, Jonathan; Knobe, Joshua (2011). Contrastive Knowledge Surveyed. Nous 46 (4): 675708.doi:10.1111/j.1468-0068.2010.00795.x.

Schaer, Jonathan. 2004. From Contextualism to Contrastivism, Philosophical Studies 119: 73-103.

1.5. EXTERNAL LINKS 5

Schier, Stephen. 1996. Contextualist Solutions to Scepticism, Proceedings of the Aristotelian Society,96:317-33.

Stanley, Jason. 2005. Knowledge and Practical Interests. New York: Oxford University Press. TimmonsMark, 1998 Morality Without Foundations: A Defense of Ethical Contextualism Oxford UniversityPress US.

Tomberlin, James (ed.). 1999. Philosophical Perspectives 13, Epistemology, Blackwell Publishing.

1.5 External links A Brief History of Contextualism - DeRose on the history of contextualism in epistemology. Contextualism in Epistemology - an article by Tim Black on the Internet Encyclopedia of Philosophy. Epistemic contextualism at PhilPapers Epistemic Contextualism entry by Patrick Rysiew in the Stanford Encyclopedia of Philosophy. Epistemic contextualism at the Indiana Philosophy Ontology Project

Chapter 2

Correspondence principle

This article discusses quantum theory and relativity. For other uses, see Correspondence principle (disam-biguation).

In physics, the correspondence principle states that the behavior of systems described by the theory of quantummechanics (or by the old quantum theory) reproduces classical physics in the limit of large quantum numbers. In otherwords, it says that for large orbits and for large energies, quantum calculations must agree with classical calculations.[1]

The principle was formulated by Niels Bohr in 1920,[2] though he had previously made use of it as early as 1913 indeveloping his model of the atom.[3]

The term is also used more generally, to represent the idea that a new theory should reproduce the results of olderwell-established theories (which become limiting cases) in those domains where the old theories work.Classical quantities appear in quantum mechanics in the form of expected values of observables, and as such theEhrenfest theorem (which predicts the time evolution of the expected values) lends support to the correspondenceprinciple.

2.1 Quantum mechanicsThe rules of quantummechanics are highly successful in describing microscopic objects, atoms and elementary parti-cles. But macroscopic systems, like springs and capacitors, are accurately described by classical theories like classicalmechanics and classical electrodynamics. If quantum mechanics were to be applicable to macroscopic objects, theremust be some limit in which quantum mechanics reduces to classical mechanics. Bohrs correspondence principledemands that classical physics and quantum physics give the same answer when the systems become large.[4]

The conditions under which quantum and classical physics agree are referred to as the correspondence limit, or theclassical limit. Bohr provided a rough prescription for the correspondence limit: it occurswhen the quantum numbersdescribing the system are large. Amore elaborated analysis of quantum-classical correspondence (QCC) in wavepacketspreading leads to the distinction between robust restricted QCC and fragile detailed QCC.[5] Restricted QCCrefers to the rst two moments of the probability distribution and is true even when the wave packets diract, whiledetailed QCC requires smooth potentials which vary over scales much larger than the wavelength, which is whatBohr considered.The post-1925 new quantum theory came in two dierent formulations. In matrix mechanics, the correspondenceprinciple was built in and was used to construct the theory. In the Schrdinger approach classical behavior is not clearbecause the waves spread out as they move. Once the Schrdinger equation was given a probabilistic interpretation,Ehrenfest showed that Newtons laws hold on average: the quantum statistical expectation value of the position andmomentum obey Newtons laws.The correspondence principle is one of the tools available to physicists for selecting quantum theories correspondingto reality. The principles of quantum mechanics are broad: states of a physical system form a complex vector spaceand physical observables are identied with Hermitian operators that act on this Hilbert space. The correspondenceprinciple limits the choices to those that reproduce classical mechanics in the correspondence limit.Because quantummechanics only reproduces classical mechanics in a statistical interpretation, and because the statis-

6

2.2. OTHER SCIENTIFIC THEORIES 7

tical interpretation only gives the probabilities of dierent classical outcomes, Bohr has argued that classical physicsdoes not emerge from quantum physics in the same way that classical mechanics emerges as an approximation ofspecial relativity at small velocities. He argued that classical physics exists independently of quantum theory andcannot be derived from it. His position is that it is inappropriate to understand the experiences of observers usingpurely quantum mechanical notions such as wavefunctions because the dierent states of experience of an observerare dened classically, and do not have a quantum mechanical analog. The relative state interpretation of quantummechanics is an attempt to understand the experience of observers using only quantum mechanical notions. NielsBohr was an early opponent of such interpretations.Many of these conceptual problems, however, resolve in the phase-space formulation of quantum mechanics, wherethe same variables with the same interpretation are utilized to describe both quantum and classical mechanics.

2.2 Other scientic theoriesThe term correspondence principle is used in amore general sense tomean the reduction of a new scientic theory toan earlier scientic theory in appropriate circumstances. This requires that the new theory explain all the phenomenaunder circumstances for which the preceding theory was known to be valid, the correspondence limit.For example, Einsteins special relativity satises the correspondence principle, because it reduces to classical me-chanics in the limit of velocities small compared to the speed of light (example below). General relativity reduces toNewtonian gravity in the limit of weak gravitational elds. Laplaces theory of celestial mechanics reduces to Keplerswhen interplanetary interactions are ignored, and Keplers reproduces Ptolemys equant in a coordinate system wherethe Earth is stationary. Statistical mechanics reproduces thermodynamics when the number of particles is large. Inbiology, chromosome inheritance theory reproduces Mendels laws of inheritance, in the domain that the inheritedfactors are protein coding genes.In order for there to be a correspondence, the earlier theory has to have a domain of validityit must work undersome conditions. Not all theories have a domain of validity. For example, there is no limit where Newtons mechanicsreduces to Aristotles mechanics because Aristotles mechanics, although academically dominant for 18 centuries, donot have any domain of validity.

2.3 Examples

2.3.1 Bohr model

Main article: Bohr model

If an electron in an atom is moving on an orbit with period T, classically the electromagnetic radiation will repeatitself every orbital period. If the coupling to the electromagnetic eld is weak, so that the orbit doesn't decay verymuch in one cycle, the radiation will be emitted in a pattern which repeats every period, so that the Fourier transformwill have frequencies which are only multiples of 1/T. This is the classical radiation law: the frequencies emitted areinteger multiples of 1/T.In quantum mechanics, this emission must be in quanta of light, of frequencies consisting of integer multiples of 1/T,so that classical mechanics is an approximate description at large quantum numbers. This means that the energy levelcorresponding to a classical orbit of period 1/T must have nearby energy levels which dier in energy by h/T, andthey should be equally spaced near that level,

En =h

T (En):

Bohr worried whether the energy spacing 1/T should be best calculated with the period of the energy state En , orEn+1 , or some averagein hindsight, this model is only the leading semiclassical approximation.Bohr considered circular orbits. Classically, these orbits must decay to smaller circles when photons are emitted. Thelevel spacing between circular orbits can be calculated with the correspondence formula. For a Hydrogen atom, the

8 CHAPTER 2. CORRESPONDENCE PRINCIPLE

classical orbits have a period T determined by Keplers third law to scale as r3/2. The energy scales as 1/r, so the levelspacing formula amounts to

E / 1r32

/ E 32 :It is possible to determine the energy levels by recursively stepping down orbit by orbit, but there is a shortcut.The angular momentum L of the circular orbit scales as r . The energy in terms of the angular momentum is then

E / 1r/ 1

L2

Assuming, with Bohr, that quantized values of L are equally spaced, the spacing between neighboring energies is

E / 1(L+ ~)2

1L2

2~L3

/ E 32 :

This is as desired for equally spaced angular momenta. If one kept track of the constants, the spacing would be , sothe angular momentum should be an integer multiple of ,

L =nh

2= n~ :

This is how Bohr arrived at his model. Since only the level spacing is determined heuristically by the correspondenceprinciple, one could always add a small xed oset to the quantum numberL could just as well have been (n+.338).Bohr used his physical intuition to decide which quantities were best to quantize. It is a testimony to his skill that hewas able to get so much from what is only the leading order approximation. A less heuristic treatment accounts forneeded osets in the ground state L2, cf. WignerWeyl transform.

2.3.2 One-dimensional potentialBohrs correspondence condition can be solved for the level energies in a general one-dimensional potential. Denea quantity J(E) which is a function only of the energy, and has the property that:

dJ

dE= T

This is the analog of the angular momentum in the case of the circular orbits. The orbits selected by the correspon-dence principle are the ones that obey J=nh for n integer, since

E = En+1 En = dEdJ

(Jn+1 Jn) = 1T

J

This quantity J is canonically conjugate to a variable which, by the Hamilton equations of motion changes withtime as the gradient of energy with J. Since this is equal to the inverse period at all times, the variable increasessteadily from 0 to 1 over one period.The angle variable comes back to itself after 1 unit of increase, so the geometry of phase space in J, coordinatesis that of a half-cylinder, capped o at J = 0, which is the motionless orbit at the lowest value of the energy. Thesecoordinates are just as canonical as x,p, but the orbits are now lines of constant J instead of nested ovoids in x-p space.The area enclosed by an orbit is invariant under canonical transformations, so it is the same in x-p space as in J-.But in the J- coordinates this area is the area of a cylinder of unit circumference between 0 and J, or just J. So J isequal to the area enclosed by the orbit in x-p coordinates too:

J =

Z T0

pdx

dtdt

The quantization rule is that the action variable J is an integer multiple of h.

2.3. EXAMPLES 9

2.3.3 Multiperiodic motionBohrSommerfeld quantizationBohrs correspondence principle provided a way to nd the semiclassical quantization rule for a one degree of freedomsystem. It was an argument for the old quantum condition mostly independent from the one developed by Wien andEinstein, which focused on adiabatic invariance. But both pointed to the same quantity, the action.Bohrwas reluctant to generalize the rule to systemswithmany degrees of freedom. This stepwas taken by Sommerfeld,who proposed the general quantization rule for an integrable system:

Jk = hnk:

Each action variable is a separate integer, a separate quantum number.This condition reproduces the circular orbit condition for two dimensional motion: let r; be polar coordinates fora central potential. Then is already an angle variable, and the canonical momentum conjugate is L, the angularmomentum. So the quantum condition for L reproduces Bohrs rule:

Z 20

Ld = 2L = nh:

This allowed Sommerfeld to generalize Bohrs theory of circular orbits to elliptical orbits, showing that the energylevels are the same. He also found some general properties of quantum angular momentumwhich seemed paradoxicalat the time. One of these results was the that the z-component of the angular momentum, the classical inclinationof an orbit relative to the z-axis, could only take on discrete values, a result which seemed to contradict rotationalinvariance. This was called space quantization for a while, but this term fell out of favor with the new quantummechanics since no quantization of space is involved.In modern quantum mechanics, the principle of superposition makes it clear that rotational invariance is not lost. It ispossible to rotate objects with discrete orientations to produce superpositions of other discrete orientations, and thisresolves the intuitive paradoxes of the Sommerfeld model.

2.3.4 The quantum harmonic oscillatorHere is a demonstration[6] of how large quantum numbers can give rise to classical (continuous) behavior.Consider the one-dimensional quantum harmonic oscillator. Quantum mechanics tells us that the total (kinetic andpotential) energy of the oscillator, E, has a set of discrete values:

E = (n+ 1/2)~!; n = 0; 1; 2; 3; : : : ;

where is the angular frequency of the oscillator. However, in a classical harmonic oscillator such as a lead ballattached to the end of a spring, we do not perceive any discreteness. Instead, the energy of such a macroscopic systemappears to vary over a continuum of values.We can verify that our idea of macroscopic systems fall within the correspondence limit. The energy of the classicalharmonic oscillator with amplitude A, is

E =m!2A2

2:

Thus, the quantum number has the value

n =E

~ ! 1

2=

m!A2

2~ 1

2

If we apply typical human-scale values m = 1kg, = 1 rad/s, and A = 1 m, then n 4.741033. This is a very largenumber, so the system is indeed in the correspondence limit.It is simple to see why we perceive a continuum of energy in this limit. With = 1 rad/s, the dierence betweeneach energy level is 1.05 1034J, well below what we normally resolve for macroscopic systems. One thendescribes this system through an emergent classical limit.

10 CHAPTER 2. CORRESPONDENCE PRINCIPLE

2.3.5 Relativistic kinetic energyHere we show that the expression of kinetic energy from special relativity becomes arbitrarily close to the classicalexpression, for speeds that are much slower than the speed of light.Einstein's mass-energy equation

E =m0p

1 v2/c2 c2 ;

where the velocity, v is the velocity of the body relative to the observer,m0 is the rest mass (the observed mass ofthe body at zero velocity relative to the observer), and c is the speed of light.When the velocity v is zero, the energy expressed above is not zero, and represents the rest energy:

E0 = m0c2:

When the body is in motion relative to the observer, the total energy exceeds the rest energy by an amount that is, bydenition, the kinetic energy:

T = E E0 = m0c2p

1 v2/c2 m0c2

Using the approximation

(1 + x)n 1 + nx

jxj 1

we get when speeds are much slower than that of light or v cwhich is the Newtonian expression for kinetic energy.

2.4 See also Quantum decoherence Classical limit

2.5 References[1] Tipler, Paul; Llewellyn, Ralph (2008). Modern Physics (5 ed.). W. H. Freeman and Company. pp. 160161. ISBN

978-0-7167-7550-8.[2] Bohr, N. (1920), "ber die Serienspektra der Element, Zeitschrift fr Physik 2 (5): 423478, Bibcode:1920ZPhy....2..423B,

doi:10.1007/BF01329978 (English translation in (Bohr 1976, pp. 241282))[3] Jammer, Max (1989), The conceptual development of quantummechanics, Los Angeles, CA: Tomash Publishers, American

Institute of Physics, ISBN 0-88318-617-9, Section 3.2[4] Bohr, Niels (1976), Rosenfeld, L.; Nielsen, J. Rud, eds., Niels Bohr, Collected Works, Volume 3, The Correspondence

Principle (19181923) 3, Amsterdam: North-Holland, ISBN 0-444-10784-3[5] Stotland, A.; Cohen, D. (2006), Diractive energy spreading and its semiclassical limit, Journal of Physics A 39 (10703),

arXiv:cond-mat/0605591, Bibcode:2006JPhA...3910703S, doi:10.1088/0305-4470/39/34/008, ISSN 0305-4470[6] Sells, Robert L.; Weidner, Richard T. (1980), Elementary modern physics, Boston: Allyn and Bacon, ISBN 978-0-205-

06559-2

Chapter 3

Denitionism

Denitionism (also called the classical theory of concepts[1]) is the school of thought in which it is believed that aproper explanation of a theory consists of all the concepts used by that theory being well-dened.[2] This approachhas been criticized for its dismissal of the importance of ostensive denitions.[3]

3.1 References[1] Jack S. Crumley (2006). A Brief Introduction to the Philosophy of Mind. Lanham, Maryland: Rowman & Littleeld. p.

150. ISBN 0-7425-4496-6.

[2] Mario Augusto Bunge (1973). Philosophy of physics. Boston: Springer Science & Business. p. 135. ISBN 90-277-0253-5.

[3] Elwood D. Heiss (2007). Modern Methods and Materials for Teaching Science. Read Books. p. 23. ISBN 1-4067-3830-1.

11

Chapter 4

Eclecticism

Eclecticism in architecture at the intersection of Rkczi Avenue and the Grand Boulevard in Budapest. The Hungarian capital is auniquely unied image of the world.[1]

Eclecticism is a conceptual approach that does not hold rigidly to a single paradigm or set of assumptions, but insteaddraws uponmultiple theories, styles, or ideas to gain complementary insights into a subject, or applies dierent theoriesin particular cases.It can sometimes seem inelegant or lacking in simplicity, and eclectics are sometimes criticized for lack of consistencyin their thinking. It is, however, common in many elds of study. For example, most psychologists accept certainaspects of behaviorism, but do not attempt to use the theory to explain all aspects of human behavior.

12

4.1. ORIGIN 13

4.1 OriginEclecticism was rst recorded to have been practiced by a group of ancient Greek and Roman philosophers whoattached themselves to no real system, but selected from existing philosophical beliefs those doctrines that seemedmost reasonable to them. Out of this collected material they constructed their new system of philosophy. The termcomes from the Greek (eklektikos), literally choosing the best,[2][3] and that from (eklektos),picked out, select.[4] Well known eclectics in Greek philosophy were the Stoics Panaetius and Posidonius, and theNew Academics Carneades and Philo of Larissa. Among the Romans, Cicero was thoroughly eclectic, as he unitedthe Peripatetic, Stoic, and New Academic doctrines. Other eclectics included Varro and Seneca.

4.2 Architecture and artThe term eclecticism is used to describe the combination, in a single work, of elements from dierent historical styles,chiey in architecture and, by implication, in the ne and decorative arts. The term is sometimes also loosely appliedto the general stylistic variety of 19th-century architecture after Neo-classicism (c. 1820), although the revivals ofstyles in that period have, since the 1970s, generally been referred to as aspects of historicism.[5]

Eclecticism plays an important role in critical discussions and evaluations but is somehow distant from the actualforms of the artifacts to which it is applied, and its meaning is thus rather indistinct. The simplest denition of thetermthat every work of art represents the combination of a variety of inuencesis so basic as to be of little use.In some ways Eclecticism is reminiscent of Mannerism in that the term was used pejoratively for much of the periodof its currency, although, unlike Mannerism, Eclecticism never amounted to a movement or constituted a specicstyle: it is characterized precisely by the fact that it was not a particular style.

4.3 PsychologyEclecticism is recognized in approaches to psychology that see many factors inuencing behavior and the psyche, andamong those who consider all perspectives in identifying, changing, explaining, and determining behavior.See also: Integrative Psychotherapy

4.4 Martial artsSee also: Hybrid martial arts

Some martial arts can be described as eclectic in the sense that they borrow techniques from a wide variety of othermartial arts. For example, the way of thinking used by Bruce Lee called Jeet Kune Do, is classied as an eclecticsystem. It favors borrowing freely from other systems within a free-oating framework. It does not rigidly hold to asingle paradigm nor set of assumptions or conclusions, but encourages students to learn what is useful for themselves.

4.5 PhilologyIn textual criticism, eclecticism is the practice of examining a wide number of text witnesses and selecting the variantthat seems best. The result of the process is a text with readings drawn from many witnesses. In a purely eclecticapproach, no single witness is theoretically favored. Instead, the critic forms opinions about individual witnesses,relying on both external and internal evidence.Since the mid-19th century, eclecticism, in which there is no a priori bias to a single manuscript, has been thedominant method of editing the Greek text of the New Testament (currently, the United Bible Society, 4th ed. andNestle-Aland, 27th ed.). Even so, the oldest manuscripts, being of the Alexandrian text-type, are the most favored,and the critical text has an Alexandrian disposition.[6]

14 CHAPTER 4. ECLECTICISM

New York Palace, Budapest, Hungary

4.6 Philosophy

In philosophy, Eclectics use elements from multiple philosophies, texts, life experiences and their own philosophicalideas. These ideas include life as connected with existence, knowledge, values, reason, mind, and language. Antiochusof Ascalon (c. 125 c. 68 BC), was the pupil of Philo of Larissa, and the teacher of Cicero. Through his inuence,Platonism made the transition from New Academy Scepticism to Eclecticism.[7] Whereas Philo had still adhered to

4.7. RELIGION 15

Madrid City Council (former Post Head Oce) Madrid, Spain

the doctrine that there is nothing absolutely certain, Antiochus returned to a pronounced dogmatism. Among otherobjections to Scepticism, was the consideration that without rm convictions no rational content of life is possible.[7]He pointed out that it is a contradiction to assert that nothing can be asserted or to prove that nothing can be proved;that we cannot speak of false ideas and at the same time deny the distinction between false and true.[8] He expoundedthe Academic, Peripatetic, and Stoic systems in such a way as to show that these three schools deviate from oneanother only in minor points.[8] He himself was chiey interested in ethics, in which he tried to nd a middle waybetween Zeno, Aristotle and Plato. For instance, he said that virtue suces for happiness, but for the highest gradeof happiness bodily and external goods are necessary as well.[8]

This eclectic tendency was favoured by the lack of dogmatic works by Plato.[9] Middle Platonismwas promoted by thenecessity of considering the main theories of the post-Platonic schools of philosophy, such as the Aristotelian logicand the Stoic psychology and ethics (theory of goods and emotions).[10] On the one hand the Middle Platonists wereengaged like the later Peripatetics in scholarly activities such as the exposition of Platos doctrines and the explanationof his dialogues; on the other hand they attempted to develop the Platonic theories systematically. In so far as it wassubject in this to the inuence of Neopythagoreanism, it was of considerable importance in preparing the way forNeoplatonism.[10]

4.7 ReligionIn religion, Eclectics use elements from multiple religions, applied philosophies, personal experiences or other textsand dogmas to form their own beliefs and ideas, noting the similarities between existing systems and practices, andrecognizing them as valid. These ideas include life, karma, the afterlife, God and Goddess, the Earth, and otherspiritual ideas. This spiritual approach is promoted by Unitarian Universalism. Some use a mix of Abrahamic,Dharmic, Neopagan, Shamanism, Daoic doctrines, New Age, religious pluralism, and Syncretism. Eclectics are mostinterested in what really works, personally and communally.

4.8 See also Eclecticism in architecture Eclecticism in art Eclecticism in music Eclecticism in textual criticism

16 CHAPTER 4. ECLECTICISM

Eclectic medicine Pastiche Perspectivism Polystylism

4.9 References[1] Vroskpe a vilgon egyedlllan egysges, Hungarian Wikipedia, quoted in Eklektikus ptszet: XIX. szzad.

[2] Encyclopdia Britannica in philosophy and theology, the practice of selecting doctrines from dierent systems of thoughtwithout adopting the whole parent system for each doctrine

[3] , Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus Digital Library

[4] , Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus Digital Library

[5] Leonard K. Eaton, The Architecture of Choice: Eclectism in America, 1880-1910, 1975

[6] Aland, B. 1994: 138

[7] Eduard Zeller, Outlines of the History of Greek Philosophy, 13th Edition, page 273




4.10 External links Media related to Eclecticism in art at Wikimedia Commons

Chapter 5

Emic and etic

This article is about the anthropological terms. For emic and etic concepts in linguistics, see emic unit.

Emic and etic, in anthropology, folkloristics, and the social and behavioral sciences, refer to two kinds of eld researchdone and viewpoints obtained;[1] from within the social group (from the perspective of the subject) and from outside(from the perspective of the observer).

5.1 DenitionsThe emic approach investigates how local people think (Kottak, 2006): How they perceive and categorize the world,their rules for behavior, what has meaning for them, and how they imagine and explain things. The etic (scientist-oriented) approach shifts the focus from local observations, categories, explanations, and interpretations to those ofthe anthropologist. The etic approach realizes that members of a culture often are too involved in what they are doingto interpret their cultures impartially. When using the etic approach, the ethnographer emphasizes what he or sheconsiders important.[2]

Although emics and etics are sometimes regarded as inherently in conict and one can be preferred to the exclusion ofthe other, the complementarity of emic and etic approaches to anthropological research has been widely recognized,especially in the areas of interest concerning the characteristics of human nature as well as the form and function ofhuman social systems.[3]

Emic knowledge and interpretations are those existing within a culture, that are determined bylocal custom, meaning, and belief (Ager and Loughry, 2004: n.p.) and best described by a 'native' of theculture. Etic knowledge refers to generalizations about human behavior that are considered universallytrue, and commonly links cultural practices to factors of interest to the researcher, such as economic orecological conditions, that cultural insiders may not consider very relevant (Morris et al., 1999).

Emic and Etic approaches of understanding behavior and personality fall under the study of cultural anthropology.Cultural anthropology states that people are shaped by their cultures and their subcultures, and we must account forthis in the study of personality. One way is looking at things through an emic approach. This approach is culturespecic because it focuses on a single culture and it is understood on its own terms. As explained below, the termemic originated from the specic linguistic term phonemic, from phoneme, which is a language-specic way ofabstracting speech sounds. (Friedman) (Akane) [4][5]

An 'emic' account is a description of behavior or a belief in terms meaningful (consciously or unconsciously)to the actor; that is, an emic account comes from a person within the culture. Almost anything from within aculture can provide an emic account.

An 'etic' account is a description of a behavior or belief by a social analyst or scientic observer (a student orscholar of anthropology or sociology, for example), in terms that can be applied across cultures; that is, an eticaccount attempts to be 'culturally neutral', limiting any ethnocentric, political, and/or cultural bias or alienationby the observer.

17

18 CHAPTER 5. EMIC AND ETIC

When these two approaches are combined, the richest view of a culture or society can be understood. On its own,an emic approach would struggle with applying overarching values to a single culture. The etic approach is helpful inpreventing researchers from seeing only one aspect of one culture and then applying it to cultures around the world.

5.2 HistoryThe terms were coined in 1954 by linguist Kenneth Pike, who argued that the tools developed for describing linguisticbehaviors could be adapted to the description of any human social behavior. As Pike noted, social scientists have longdebatedwhether their knowledge is objective or subjective. Pikes innovationwas to turn away from an epistemologicaldebate, and turn instead to a methodological solution. Emic and etic are derived from the linguistic terms phonemicand phonetic respectively, which are in turn derived from Greek roots. The possibility of a truly objective descriptionwas discounted by Pike himself in his original work; he proposed the emic/etic dichotomy in anthropology as a wayaround philosophic issues about the very nature of objectivity.The terms were also championed by anthropologists Ward Goodenough and Marvin Harris with slightly dierentconnotations from those used by Pike. Goodenough was primarily interested in understanding the culturally specicmeaning of specic beliefs and practices; Harris was primarily interested in explaining human behavior.Pike, Harris, and others have argued that cultural insiders and outsiders are equally capable of producing emicand etic accounts of their culture. Some researchers use etic to refer to objective or outsider accounts, and emicto refer to subjective or insider accounts.[6]

Margaret Mead was a scientist who studied the patterns of adolescence in Samoa. She discovered that the dicultiesand the transitions that adolescents faced are culturally inuenced. The hormones that are released during pubertycan be dened using an etic framework, because adolescents globally have the same hormones being secreted. How-ever, Mead concluded that how adolescents respond to these hormones is greatly inuenced by their cultural norms.Through her studies, Mead found that simple classications about behaviors and personality could not be used be-cause peoples cultures inuenced their behaviors in such a radical way. Her studies helped create an emic approachof understanding behaviors and personality. Her research deduced that culture has a signicant impact in shaping anindividuals personality. (Friedman) [7] [8]

Carl Jung, a Swiss psychoanalyst, is a researcher who took an etic approach in his studies. Jung studied mythology,religion, ancient rituals, and dreams leading him to believe that there are archetypes used to categorize peoplesbehaviors. Archetypes are universal structures of the collective unconscious that refer to the inherent way peopleare predisposed to perceive and process information. The main archetypes [9] that Jung studied were the persona(how people choose to present themselves to the world), the animus/ anima (part of people experiencing the world inviewing the opposite sex, that guides how they select their romantic partner), and the shadow (dark side of personalitiesbecause people have a concept of evil. Well-adjusted people must integrate both good and bad parts of themselves).Jung looked at the role of the mother and deduced that all people have mothers and see their mothers in a similarway; they oer nurture and comfort. His studies also suggest that infants have evolved to suck milk from the breast,it is also the case that all children have inborn tendencies to react in certain ways. This way of looking at the motheris an etic way of applying a concept cross- culturally and universally.[8]

5.3 Examples of etic case studiesEtic studies, as mentioned, are ones that study one characteristic across various cultures. These studies can giveresearchers an idea of how an idea, like the importance of family, is valued in dierent cultures around the world.This is not to say, however, that emic research cannot be conducted in a cross-cultural context The test of whetherthese are etic or emic concepts resides in their logic-empirical relationship to the cognitive processes. If the veriabilityof an ethnographic statement involves a confrontation with cognitive adequacy or appropriateness, then we are dealingwith emic categories, no matter how many cultures contribute to that confrontation. (See Harris 1968 at 577). Beloware three dierent studies using an etic approach.Study #1: How the Idea of Trust Varies Across Cultures (a book review) [10]

This book found that one of the reasons it is hard to determine whom to trust from another culture is that todetermine trustworthiness, human beings rely on signals and cues from the other person. These signals and cuesare often shaped by ones culture; however, two dierent cultures probably emphasize and respond to dierent

5.3. EXAMPLES OF ETIC CASE STUDIES 19

signals and cues, making it dicult to discern if this person from another culture is trustworthy by your culturesstandards. The authors of this book describe trust as: a psychological state of accepting vulnerability towardthe other party based on positive expectations regarding its behavior (Saunders, Skinner, Dietz, Gillespie &Lewicki, 2012). The authors then dene culture as: a set of basic assumptions manifested in values, beliefs,and norms of social behaviors (Saunders, Skinner, Dietz, Gillespie & Lewicki, 2012).

While studying what characteristics make up trust, Ferrin and Gillespie found that ability, benevolence,and integrity were rated as universal qualities of a trustworthy person (Saunders, Skinner, Dietz, Gillespie &Lewicki, 2012).

One strong example of the dierence in trust building between two cultures is the contrast between tradersin Nigeria and Ghana. In Nigeria, traders will come to trust one another by inviting fellow traders to stayin ones home and get to know his family members. In Ghana, traders develop trust by going to church andfunerals together. This example proves that benevolence and integrity are universal; both are proven throughthe dierent rituals, but it is in how these qualities are established that varies across the cultures.

Study #2: How the Development of a Distinctive Identity Varies Across Cultures [11][12]

This study studied 21 dierent cultural groups in order to gain a sense of how culture aects how people pursueand reach feelings of distinctiveness. The research found that in individualistic cultures, like the United States,distinctiveness is associated with dierence and separateness, while in collectivist cultures it is associatedwith social position (Becker, Vignoles, Owe, Brown, Smith & Easterbrook, 2012). Carefully assessing allresults, researches found that it is in fact a persons surrounding environment (context) and the inherentbeliefs and values of that environment that inuence the dierences between denitions of distinctiveness inindividualistic and collectivist societies.

The study identied distinctiveness as an identity motive (Becker, Vignoles, Owe, Brown, Smith & East-erbrook, 2012). Motivated identity construction theory states that, identity is constructed through a complexinterplay of cognitive, aective, and social interactions processes, all of which occur within particular culturaland local meaning systems (Becker, Vignoles, Owe, Brown, Smith & Easterbrook, 2012). According to thisdenition, identity is completely dened by where an individual is developing that identity.

Overall, a universal constant was that people tended to positively identify characteristics that identied themfrom other individuals. The study proved that, from the cultures studied, all people do strive to attain a levelof distinctiveness from their peers; the dierence among the cultures is in how this separateness is reached.To compare and contrast, the etic study described here found that across cultures, how we distinguish our-selves from others diers. An emic study to follow could see how people within one culture dene themselvesdierently or the in the same ways.

Study #3: How Celebratory Food Preparation Varies Across Cultures [13]

This study studied three dierent cultures to nd the similarities in celebratory food preparation. The re-searchers believed that while the practices may be dierent, the underlying purposes and ideals would becomparative. The study focused on how older women with three dierent cultures: Thailand, New Zealand,and the United States, prepared celebratory food and meals.

In order to keep their study valid and nonbiased, the researchers developed teams to investigate each culturethat were varied in ethnicity themselves so that there would not be a Western bias in the ndings and re-sults (Shordike, Hocking, Pierce, Wright-St. Clair, Vittayakorn, Rattakorn & Bunrayong, 2010). The studyconcluded that there were denitely some general themes spanning the three cultures; for example, all threecultures supported a womans leadership role in the food preparation, all cultures had a complex and diverseway of creating and cooking the food, and each cultures food had a special signicance for the holiday it wasbeing served for (Shordike, Hocking, Pierce, Wright-St. Clair, Vittayakorn, Rattakorn & Bunrayong, 2010).

Etic studies can study any aspect of life that occurs in dierent cultures and note the similarities and dierencesfor research.


5.4 Examples of emic case studiesEmic studies, as mentioned, are ones that study the eects an individuals culture has on their personality and theirbehaviors. These studies can give researchers an idea of how culture is central in determining how one acts. Beloware two dierent studies using an emic approach.Study #1: Constructing Maternal Knowledge Frameworks; How Mothers Conceptualize Complementary Feeding

The study, conducted by Eva Monterrossa and her team, focused on using an emic framework to explain ma-ternal knowledge and conceptualizing what behaviors mothers in Morelos, Mexico used to most ecientlynourish their children. The study dened eight dierent concepts which were 1) probaditas (the idea of in-troducing small tastes of foods), (2) preparing separate foods for infants, (3) readiness to eat solid foods, (4)appropriate consistency, (5) the value of variety, (6) child likes and dislikes, (7) money and food costs, and (8)healthiness of foods (positive and negative foods) (Monterrossa). The study constructed an emic frameworkto show how maternal knowledge guides meaningful behavior and the relationships among the concepts, andhow lived social realities inuence complementary feeding. This study found that maternal knowledge wasdierent culturally and it was hard to dene a universal way that mothers interact with their childrens eatinghabits and preferences. For example, most of the women in Morelos cooked all the food for the children fromscratch, while in the United States, many mothers purchase pre-cooked meals or use pre-packaged food fortheir children.[14]

Study #2: Testing the Eect of Risk on Intertemporal Choice in the Chinese Cultural Context

Another study by Yan Sun and Su Li looked at the intertemporal choices that Chinese participants made.According to the study, intertemporal choices were dened as a CHOICE between alternatives that dier insize and time to delivery. Many decisions that individuals and organizations make in the real world depend ona trade-o between immediate pleasure and later benets. (Sun) For example, the study states that people inAmerican and Western cultures do not attach much importance to saving money and that some individualseven use credit card debt to maintain consumption. The results of the study stated that, In contrast, manyChinese are accustomed to saving at least some portion of money. This emic focused study shows what animpact culture has on inuencing certain behaviors in people. People in dierent cultures are brought up indierent ways that control their behaviors and decisions. In the famous classic Aesop fable, the grasshopper,who luxuriated during the warm summer, is often regarded as a bad guy by many Chinese children, whereasthe ant, who stored food for the upcoming winter, is categorized as a good guy. This study shows through anemic perspective that the Chinese culture emphasizes saving behaviors while American and Western culturesdo not emphasize the same behaviors. (Sun) [15]

5.5 Importance as regards personalityEmic and etic approaches are important to understanding personality because problems can arise when concepts,measures, and methods are carelessly transferred to other cultures in attempts to make cross- cultural generalizationsabout personality. It is hard to apply certain generalizations of behavior to people who are so diverse and culturallydierent. One example of this is the F-scale (Macleod).[16] The F-scale, which was created by Theodor Adorno, isused to measure Authoritarian Personality, which can, in turn, be used to predict prejudiced behaviors. This test,when applied to Americans accurately depicts prejudices towards black individuals. However, when a study wasconducted in South Africa using the F-Scale, (Pettigrew and Friedman) [8] results did not predict any anti-Blackprejudices. This study used emic approaches of study by conducting interview with the locals and etic approaches bygiving participants generalized personality tests.

5.6 Secondary sourcesWork and Family: An International Research Perspective

This collection of studies and information tackles the disparity between men and women in the workplace invarious cultures. The research nds that universally, creating equality between men and women in the workplace will lead to better results and better working environments for all.[17]

5.7. SEE ALSO 21

Cross-Cultural Psychiatry:

This piece of work studies mental health care and how it must vary, in order to be eective, among cultures.For example, Chinese families try to handle illness on their own before seeking outside help. Chinese familieswill keep the sick individual in the home as long as possible and see it as a failure if they have to resort tosending the family member to a hospital. On the other hand, Japanese families value health; therefore, a sickindividual would be quarantined and sent away so as not to infect the rest of the family or put a blemish on thename (Gaw, 1982).[18]

Socioemotional Development in Cultural Context:

This piece of work studies childrens emotional attachments and developments across cultures. One ndingshowed that Italian mothers will focus on bringing the child into the family life keeping the child up with thefamily until he or she falls asleep and waking him or her up from a nap to join in with family meals. On theother hand, Parisian mothers main goal is to achieve a proper presentation of the child; i.e., cleanliness,good manners, and emotional self-regulation (Chen & Rubin, 2011). Mothers around the world have dierentprimary goals for their children, but universally, most mothers reach their expectations by loving and protectingthe child.[19]

5.7 See alsoOther explorations of the dierences between reality and humans models of it:

Blind men and an elephant Internalism and externalism Mapterritory relation

5.8 References[1] EE intro, SIL

[2] Kottak, Conrad (2006). Mirror for Humanity, p. 47. McGraw-Hill, New York. ISBN 978-0-07-803490-9.

[3] Jingfeng, Xia (2013). An Anthropological Emic-Etic Perspective On Open Access Practices Academic Search Premier.

[4] Friedman, Howard S; Schustack, MiriamW (2012), Personality: Classic Theories and Modern Research (PRINT), Boston:Pearson Allyn & Bacon.

[5] Akane (Oct 2011), Using one or more examples explain emic & etic concepts, CN: SIS.

[6] http://www.sil.org/~{}headlandt/ee-intro.htm

[7] Papua New Guinea: Sex and Temperament - Margaret Mead: Human Nature and the Power of Culture - Exhibitions -Library of Congress. loc.gov. Retrieved 21 May 2015.

[8] Friedman, Howard S., and Miriam W. Schustack. Personality: Classic Theories and Modern Research. Boston: PearsonAllyn & Bacon, 2012. Print.

[9] Kendra Cherry. What Are Jungs 4 Major Archetypes?". About.com Education. Retrieved 21 May 2015.

[10] Boston College Libraries Login (PDF). bc.edu. Retrieved 21 May 2015.


[12] Becker, M., & et, A. (2012). Culture and the distinctiveness motive: Constructing identity in individualistic and collectivistcontexts. Journal of personality and social psychology, 102(4), 833-855.



[14] Eva C. Monterrosaa, b, , , Gretel H. Peltoa, Edward A. Frongillob, Kathleen M. Rasmussena (2012): How mothers con-ceptualize complementary feeding, Department of Health Promotion, Volume 59, Issue 2, October 2012, Pages 377384

[15] Yan Sun & Shu Li (2011): Testing the Eect of Risk on Intertemporal Choice in the Chinese Cultural Context, The Journalof Social Psychology, 151:4, 517-522

[16] Theories of Personality. simplypsychology.org. Retrieved 21 May 2015.

[17] Poelmans, S. (2005). Work and family: An international research perspective. Taylor & Francis Group.

[18] Herrera , J., Lawson, W., & Sramek, J. (1999). Cross cultural psychiatry. Wiley-Blackwell.

[19] Chen , X., & Rubin, K. (2011). Socioemotional development in cultural context. The Guilford Press.

5.9 Further reading Creswell, J. W. (1998), Qualitative Enquiry and Research Design: Choosing among Five Traditions, London,UK: Sage.

Dundes, Alan (1962), From Etic to Emic Units in the Structural Study of Folktales, Journal of AmericanFolklore, 75, No. 296: 95105, doi:10.2307/538171.

Jingfeng, Xia (2013), An Anthropological Emic-Etic Perspective On Open Access Practices, Academic SearchPremier.

Goodenough, Ward (1970), Describing a Culture, Description and Comparison in Cultural Anthropology(Cambridge, UK: Cambridge University Press): 104119, ISBN 978-0-202-30861-6.

Harris, Marvin (1976), History and Signicance of the Emic/Etic Distinction, Annual Review of Anthropol-ogy 5: 329350, doi:10.1146/annurev.an.05.100176.001553.

Harris, Marvin (1980), Chapter Two: The Epistemology of Cultural Materialism, Cultural Materialism: TheStruggle for a Science of Culture (New York, NY, USA: Random House): 2945, ISBN 978-0-7591-0134-0.

Headland, Thomas; Pike, Kenneth; Harris, Marvin (eds) (1990), Emics and Etics: The Insider/Outsider Debate,Sage.

Jahoda, G. (1977), In Pursuit of the Emic-Etic Distinction: Can We Ever Capture It?", Basic Problems inCross-Cultural Psychology (Y.J. Poortinga, ed.): 5563.

Kitayama, Shinobu; Cohen, Dov (2007), Handbook of Cultural Psychology, New York, NY, USA: GuilfordPress.

Kottak, Conrad (2006), Mirror for Humanity, New York, NY: McGraw Hill, ISBN 978-0-07-803490-9. Nattiez, Jean-Jacques (1987), Music and Discourse: Toward a Semiology of Music (Musicologie gnrale et

smiologue, 1987). Translated by Carolyn Abbate, ISBN 978-0-691-02714-2. Pike, Kenneth Lee (ed.) (1967), Language in Relation to a Unied Theory of Structure of Human Behavior(2nd ed.), The Hague, Netherlands: Mouton.

5.10 External links Emic and Etic Standpoints for the Description of Behavior, chapter 2 in Language in Relation to a Unied

Theory of the Structure of Human Behavior, vol 2, by Kenneth Pike (published in 1954 by Summer Instituteof Linguistics)

Emics and Etics: The Insider/Outsider Debate, edited by Thomas Headland, Kenneth Pike, and Marvin Harris(published in 1990 by Sage Publications)

Chapter 6

Explanatory power

Explanatory power is the ability of a hypothesis or theory to eectively explain the subject matter it pertains to. Theopposite of explanatory power is explanatory impotence.In the past, various criteria or measures for explanatory power have been proposed. In particular, one hypothesis,theory or explanation can be said to have more explanatory power than another about the same subject matter- if more facts or observations are accounted for;- if it changes more surprising facts into a matter of course (following Peirce);- if more details of causal relations are provided, leading to a high accuracy and precision of the description;- if it oers greater predictive power, i.e., if it oers more details about what we should expect to see, and what weshould not;- if it depends less on authorities and more on observations;- if it makes fewer assumptions;- if it is more falsiable, i.e., more testable by observation or experiment (following Popper).Recently, David Deutsch proposed that the correct hypothesis or theory, the one that stands out among all possibleexplanations, is that specic explanation that- is hard to vary.By this expression he intends to state that the correct theory, i.e., the true explanation, provides specic details whicht together so tightly that it is impossible to change any one detail without aecting the whole theory.

6.1 IntroductionPhilosopher and physicist David Deutsch oers a criterion for a good explanation that he says may be just as importantto scientic progress as learning to reject appeals to authority, and adopting formal empiricism and falsiability. ToDeutsch, these aspects of a good explanation, and more, are contained in any theory that is specic and hard tovary. He believes that this criterion helps eliminate bad explanations which continuously add justications, andcan otherwise avoid ever being truly falsied.[1]

6.2 ExamplesDeutsch takes examples from Greek mythology. He describes how very specic, and even somewhat falsiabletheories were provided to explain how the gods sadness caused the seasons. Alternatively, Deutsch points out, onecould have just as easily explained the seasons as resulting from the gods happiness - making it a bad explanation,because it is so easy to arbitrarily change details.[1] Without Deutschs criterion, the 'Greek gods explanation' couldhave just kept adding justications. This same criterion, of being hard to vary, may be what makes the modernexplanation for the seasons a good one: none of the details - about the earth rotating around the sun at a certain anglein a certain orbit - can be easily modied without changing the theorys coherence.[1]

23

24 CHAPTER 6. EXPLANATORY POWER

Carbon Cycle

Rivers

Storage in GtC

Atmosphere 750

Deep Ocean38.100

Vegetation 610

Sediments 150

Fossil Fuels &Cement Production4,000

Surface Ocean 1,020

Dissolved Organic Carbon

Chapter 7

Metamathematics

Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories,which are mathematical theories about other mathematical theories. Emphasis on metamathematics (and perhapsthe creation of the term itself) is due to David Hilbert's attempt to prove the consistency of mathematical theoriesby proving a proposition about a theory itself, i.e. specically about all possible proofs of theorems in the theory;in particular, both a proposition A and its negation not A should not be theorems (Kleene 1952, p. 55). However,metamathematics provides a rigorousmathematical technique for investigating a great variety of foundation problemsfor mathematics and logic, among which the consistency problem is only one (Kleene 1952, p. 59).

7.1 History

Metamathematical metatheorems about mathematics itself were originally dierentiated from ordinary mathematicaltheorems in the 19th century to focus on what was then called the foundational crisis of mathematics. Richardsparadox (Richard 1905) concerning certain 'denitions of real numbers in the English language is an example ofthe sort of contradictions that can easily occur if one fails to distinguish between mathematics and metamathematics.Something similar can be said around the well-known Russells paradox (Does the set of all those sets that do notcontain themselves contain itself?).Metamathematics was intimately connected to mathematical logic, so that the early histories of the two elds, duringthe late 19th and early 20th centuries, largely overlap. More recently, mathematical logic has often included the studyof new pure mathematics, such as set theory, recursion theory and pure model theory, which is not directly related tometamathematics.Serious metamathematical reection began with the work of Gottlob Frege, especially his Begrisschrift.David Hilbert was the rst to invoke the term metamathematics with regularity (see Hilberts program). In hishands, it meant something akin to contemporary proof theory, in which nitary methods are used to study variousaxiomatized mathematical theorems (Kleene 1952, p. 55).Other prominent gures in the eld include Bertrand Russell, Thoralf Skolem, Emil Post, Alonzo Church, StephenKleene, Willard Quine, Paul Benacerraf, Hilary Putnam, Gregory Chaitin, Alfred Tarski and Kurt Gdel. In par-ticular, arguably the greatest achievement of metamathematics and the philosophy of mathematics to date is Gdelsincompleteness theorem: proof that given any nite number of axioms for Peano arithmetic, there will be true state-ments about that arithmetic that cannot be proved from those axioms.Today, metalogic and metamathematics are largely synonymous with each other, and both have been substantiallysubsumed by mathematical logic in academia.

7.2 Milestones

25

26 CHAPTER 7. METAMATHEMATICS

7.2.1 Principia Mathematica

Main article: Principia Mathematica

Principia Mathematica, or PM as it is often abbreviated, was an attempt to describe a set of axioms and inferencerules in symbolic logic from which all mathematical truths could in principle be proven. As such, this ambitiousproject is of great importance in the history of mathematics and philosophy,[1] being one of the foremost products ofthe belief that such an undertaking may be achievable. However, in 1931, Gdels incompleteness theorem proveddenitively that PM, and in fact any other attempt, could never achieve this lofty goal; that is, for any set of axiomsand inference rules proposed to encapsulate mathematics, there would in fact be some truths of mathematics whichcould not be deduced from them.One of the main inspirations and motivations for PM was the earlier work of Gottlob Frege on logic, which Russelldiscovered allowed for the construction of paradoxical sets. PM sought to avoid this problem by ruling out theunrestricted creation of arbitrary sets. This was achieved by replacing the notion of a general set with notion ofa hierarchy of sets of dierent 'types', a set of a certain type only allowed to contain sets of strictly lower types.Contemporary mathematics, however, avoids paradoxes such as Russells in less unwieldy ways, such as the systemof ZermeloFraenkel set theory.

7.2.2 Gdels completeness theorem

Main article: Gdels completeness theorem

Gdels completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondencebetween semantic truth and syntactic provability in rst-order logic. It makes a close link between model theory thatdeals with what is true in dierent models, and proof theory that studies what can be formally proven in particularformal systems.More formally, the completeness theorem says that if a formula is logically valid then there is a nite deduction (aformal proof) of the formula.It was rst proved by Kurt Gdel in 1929. It was then simplied in 1947, when Leon Henkin observed in his Ph.D.thesis that the hard part of the proof can be presented as the Model Existence Theorem (published in 1949). Henkinsproof was simplied by Gisbert Hasenjaeger in 1953.Gdels completeness theorem says that a deductive system of rst-order predicate calculus is complete in the sensethat no additional inference rules are required to prove all the logically valid formulas. A converse to completeness issoundness, the fact that only logically valid formulas are provable in the deductive system. Together with soundness(whose verication is easy), this theorem implies that a formula is logically valid if and only if it is the conclusion ofa formal deduction.

7.2.3 Gdels incompleteness theorem

Main article: Gdels incompleteness theorem

Gdels incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of allbut the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gdel in 1931,are important both in mathematical logic and in the philosophy of mathematics. The two results are widely, butnot universally, interpreted as showing that Hilberts program to nd a complete and consistent set of axioms for allmathematics is impossible, giving a negative answer to Hilberts second problem.The rst incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an"eective procedure" (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truthsabout the relations of the natural numbers (arithmetic). For any such system, there will always be statements aboutthe natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, anextension of the rst, shows that such a system cannot demonstrate its own consistency.

7.3. SEE ALSO 27

7.2.4 Tarskis denition of model-theoretic satisfactionMain article: T-schema

The T-schema or truth schema (not to be confused wit

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