Philosophy of mathematics for the masses: extending the scope of the philosophy of mathematics
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| Main Author: | |
|---|---|
| Format: | Thesis Book |
| Language: | English |
| Published: |
Stockholm
Department of Philosophy, Stockholm University
2016
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | 256 Seiten |
| ISBN: | 9789176493519 |
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Record in the Search Index
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| adam_text | Contents
Acknowledgements ix
1 Introduction 11
1.1 Interpretations of Benacerraf’s dilemma............. 15
1.1.1 The realist horn of the dilemma............. 15
1.1.2 The anti-realist horn of the dilemma........ 19
1.2 Outline of the thesis.................................. 21
1 Realist Epistemologies 25
Overview of realist theories 27
2 Cognizing numbers versus Number Cognition 37
2.1 Introduction........................................ 38
2.2 Number cognition as cognition of numbers............... 42
2.2.1 Apparent mental arithmetic..................... 43
2.2.2 Cross-modal comparisons........................ 44
2.3 Quantities and Quantity representations............. 45
2.3.1 Two theories of quantity....................... 47
2.3.2 Amounts as quantities.......................... 50
2.3.3 Representing quantities spatially.............. 52
2.4 An alternative: cognition of quantity.................. 54
2.4.1 Empirical support.............................. 55
2.4.2 Small and large quantités...................... 58
2.4.3 Challenges for a quantity based account .... 60
3 Hale and Wright’s neo-logicism 65
3.1 Hale and Wright’s solution............................. 66
3.1.1 Fixing identity and application conditions ... 67
3.2 Evaluating the implicit definition story............ 70
3.2.1 The Caesar Problem.............................. 71
3.2.2 Further problems in getting to the application
conditions...................................... 75
3.3 Epistemic claims from Hale and Wright.................. 76
3.3.1 The first interpretation — an internalist construal 76
3.3.2 The second interpretation - a move to extemalism 79
3.3.3 The third interpretation ֊ dropping knowledge
requirements.................................... 82
3.3.4 The fourth interpretation — justification without
possessing concepts ............................ 83
3.4 Epistemic claims in practice........................... 85
3.4.1 The completeness of the first interpretation . . 86
3.4.2 Completeness of the second and third interpre-
tation ................................................ 88
3.4.3 The fourth interpretation as including too little 91
4 The conceptual strategy and pedagogy 95
4.1 Jenkins’s general account.............................. 97
4.1.1 Jenkins’s view of concepts as representing parts
of the world................................ 97
4.1.2 Problems about extending Jenkins’s account. . 100
4.1.3 A worry about composition of concepts .... 101
4.1.4 An attempt to solve the two worries ........... 102
4.2 Stages of conceptual competence................... 105
4.2.1 Concepts and conceptions................... 106
4.2.2 Action conceptions......................... 107
4.2.3 Process conceptions ........................... 109
4.2.4 Object conceptions......................... 110
4.2.5 Schema conceptions......................... 112
4.3 Problems for the conceptual strategy.................. 114
4.3.1 Acquiring knowledge from concepts.......... 115
4.3.2 Indeterminacy relating to the grasped concept . 117
5 Parsons’s mathematical intuition 123
5.1 Parsons’s account..................................... 124
5.1.1 The underlying ontology.................... 124
5.1.2 The stroke language........................ 127
5.1.3 Intuition of stroke strings.................. 128
5.1.4 Intuitive knowledge of axioms................ 130
5.2 Issues of Completeness................................ 134
5.2.1 Alternatives to strokes........................ 134
5.2.2 The interpretations of Parsons’s account .... 138
5.3 Correctness worries................................... 140
5.3.1 Intuition that and number cognition.......... 141
5.3.2 Consequences................................. 143
5.3.3 Summary........................................ 144
6 Linsky and Zalta’s use of descriptions 147
6.1 Linsky and Zalta’s account............................ 148
6.1.1 A plenitude of abstract objects................ 148
6.1.2 An epistemology based on descriptions .... 151
6.2 (In)Completeness of the account....................... 156
6.2.1 Preliminary knowledge for reference............ 157
6.2.2 Using definite descriptions.................... 160
6.2.3 Summary........................................ 164
II Anti-Realist Semantics 167
Overview of anti-realist theories 169
7 Reformulated mathematical content 179
7.1 Introduction.......................................... 180
7.2 Reformulating in terms of proof....................... 182
7.3 Ordinary people and proof............................. 186
7.4 Are these generally problematic cases?................ 191
7.4.1 Field’s fictionalism........................... 191
7.4.2 Yablo’s nominalism............................. 193
7.4.3 Variations on Field’s and Yablo’s reformulations 196
7.4.4 Balaguer’s reformulation using standard models 199
7.5 Conclusion............................................ 201
8 Semantics in terms of proof 203
8.1 Introduction.......................................... 204
8.2 Giving a proof-conditional semantics.................. 205
8.3 The requirements on speakers ......................... 209
8.4 The limited capacities of speakers..................214
III The general project 221
9 Background assumptions and method 223
9.1 Introduction..........................................224
9.2 Ordinary people doing mathematics.....................226
9.3 Truth-conditional and Proof-conditional semantics . . 237
9.4 Experimental Philosophy.............................. 241
10 Conclusion 247
Svensk Sammanfattning cclvii
Bibliography cclxxi
Index cclxxxvii
|
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| spelling | Buijsman, Stefan 1995- Verfasser (DE-588)1142265447 aut Philosophy of mathematics for the masses extending the scope of the philosophy of mathematics Stefan Buijsman Stockholm Department of Philosophy, Stockholm University 2016 © 2016 256 Seiten txt rdacontent n rdamedia nc rdacarrier Dissertation Stockholms universitet 2016 Philosophie (DE-588)4045791-6 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf Benacerraf, Baruj / 1920-2011 (DE-588)4113937-9 Hochschulschrift gnd-content Mathematik (DE-588)4037944-9 s Philosophie (DE-588)4045791-6 s DE-604 Digitalisierung BSB Muenchen - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029954770&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Buijsman, Stefan 1995- Philosophy of mathematics for the masses extending the scope of the philosophy of mathematics Philosophie (DE-588)4045791-6 gnd Mathematik (DE-588)4037944-9 gnd |
| subject_GND | (DE-588)4045791-6 (DE-588)4037944-9 (DE-588)4113937-9 |
| title | Philosophy of mathematics for the masses extending the scope of the philosophy of mathematics |
| title_auth | Philosophy of mathematics for the masses extending the scope of the philosophy of mathematics |
| title_exact_search | Philosophy of mathematics for the masses extending the scope of the philosophy of mathematics |
| title_full | Philosophy of mathematics for the masses extending the scope of the philosophy of mathematics Stefan Buijsman |
| title_fullStr | Philosophy of mathematics for the masses extending the scope of the philosophy of mathematics Stefan Buijsman |
| title_full_unstemmed | Philosophy of mathematics for the masses extending the scope of the philosophy of mathematics Stefan Buijsman |
| title_short | Philosophy of mathematics for the masses |
| title_sort | philosophy of mathematics for the masses extending the scope of the philosophy of mathematics |
| title_sub | extending the scope of the philosophy of mathematics |
| topic | Philosophie (DE-588)4045791-6 gnd Mathematik (DE-588)4037944-9 gnd |
| topic_facet | Philosophie Mathematik Hochschulschrift |
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