Mathematical thought and its objects:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2008
|
| Edition: | 1. publ. |
| Subjects: | |
| Online Access: | Table of contents only Contributor biographical information Publisher description Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references (p. 343-363) and index |
| Physical Description: | XX, 378 S. 24 cm |
| ISBN: | 9780521452793 0521452791 |
Staff View
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Record in the Search Index
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|---|---|
| adam_text |
Contents
Preface
page
xi
Sources and Copyright Acknowledgments
xix
1
Objects and logic
.1
§1.
Abstract objects
1
§2.
The concept of an object in general: Actuality
3
§3.
Intuitability
8
§4.
Logic and the notion of object
10
§5.
Is whatever is an object?
13
§6.
Being and existence
23
§7.
Abstract objects and their concrete representations
33
2
Structuralism and nominalism
.40
§8.
The structuralist view of mathematical objects
40
§9.
The concept of structure
43
§10.
Dedekind on the natural numbers
45
§11.
Eliminative structuralism and logicism
50
§12.
Nominalism
56
§13.
Nominalism and second-order logic
61
§14.
Structuralism and application
73
3
Modality and structuralism
.80
§15.
Mathematical modality
80
§16.
Modalism
92
§17.
Difficulties of modalism: Rejection of eliminative
structuralism
96
§18.
A noneliminative structuralism
100
vii
Contents
viii
4
A problem about sets
.
§19.
An objection
'
§20.
Ontological conceptions of set 119
§21.
The iterative conception of set 122
§22.
"Intuitive" arguments for axioms of set theory
124
§23.
The Replacement and Power Set axioms
130
5
Intuition
.
138
§24.
Intuition: Basic distinctions
138
§25.
Intuition and perception
143
§26.
Objections to the very idea of mathematical intuition
148
§27.
Toward a viable conception of intuition: Perception and
the abstract
152
§28.
Hilbertian intuition
159
§29.
Intuitive knowledge: A step toward infinity
171
§30.
The objections revisited
179
6
Numbers as objects
.186
§31.
What are the natural numbers?
186
§32.
Cardinality and the genesis of numbers as objects
190
§33.
Finite sets and sequences
199
§34.
Sets and sequences, intuition and number
205
§35.
Difficulties concerning intuition of finite sets
211
§36.
Well, then, what are the numbers? Structuralism put
in its place
218
§37.
Intuition of numbers denied
222
§38.
Appendix
1:
Theories of sets and sequences
225
§39.
Appendix
2:
Relative substitutional semantics for the
language of hereditarily finite sets
231
7
Intuitive arithmetic and its limits
.235
§40.
Arithmetic as about strings: Finitism
235
§41.
The elementary axioms
244
§42.
Logic and intuition
247
§43.
Induction
2ς?
§44.
Primitive recursion
254
§45.
The limits of intuitive knowledge
260
§46.
Appendix 262
Contents ix
8
Mathematical induction
.264
§47.
Induction and the concept of natural number
264
§48.
The problem of the uniqueness of the number structure:
Nonstandard
models
272
§49.
Uniqueness and communication
279
§50.
Induction and impredicativity
293
§51.
Predicativity and inductive definitions
307
9
Reason
.316
§52.
Reason and "rational intuition"
316
§53.
Rational intuition and perception
325
§54.
Arithmetic
328
§55.
Set theory
338
Bibliography
343
Index
365 |
| adam_txt |
Contents
Preface
page
xi
Sources and Copyright Acknowledgments
xix
1
Objects and logic
.1
§1.
Abstract objects
1
§2.
The concept of an object in general: Actuality
3
§3.
Intuitability
8
§4.
Logic and the notion of object
10
§5.
Is whatever is an object?
13
§6.
Being and existence
23
§7.
Abstract objects and their concrete representations
33
2
Structuralism and nominalism
.40
§8.
The structuralist view of mathematical objects
40
§9.
The concept of structure
43
§10.
Dedekind on the natural numbers
45
§11.
Eliminative structuralism and logicism
50
§12.
Nominalism
56
§13.
Nominalism and second-order logic
61
§14.
Structuralism and application
73
3
Modality and structuralism
.80
§15.
Mathematical modality
80
§16.
Modalism
92
§17.
Difficulties of modalism: Rejection of eliminative
structuralism
96
§18.
A noneliminative structuralism
100
vii
Contents
viii
4
A problem about sets
.
§19.
An objection
'
§20.
Ontological conceptions of set 119
§21.
The iterative conception of set 122
§22.
"Intuitive" arguments for axioms of set theory
124
§23.
The Replacement and Power Set axioms
130
5
Intuition
.
138
§24.
Intuition: Basic distinctions
138
§25.
Intuition and perception
143
§26.
Objections to the very idea of mathematical intuition
148
§27.
Toward a viable conception of intuition: Perception and
the abstract
152
§28.
Hilbertian intuition
159
§29.
Intuitive knowledge: A step toward infinity
171
§30.
The objections revisited
179
6
Numbers as objects
.186
§31.
What are the natural numbers?
186
§32.
Cardinality and the genesis of numbers as objects
190
§33.
Finite sets and sequences
199
§34.
Sets and sequences, intuition and number
205
§35.
Difficulties concerning intuition of finite sets
211
§36.
Well, then, what are the numbers? Structuralism put
in its place
218
§37.
Intuition of numbers denied
222
§38.
Appendix
1:
Theories of sets and sequences
225
§39.
Appendix
2:
Relative substitutional semantics for the
language of hereditarily finite sets
231
7
Intuitive arithmetic and its limits
.235
§40.
Arithmetic as about strings: Finitism
235
§41.
The elementary axioms
244
§42.
Logic and intuition
247
§43.
Induction
2ς?
§44.
Primitive recursion
254
§45.
The limits of intuitive knowledge
260
§46.
Appendix 262
Contents ix
8
Mathematical induction
.264
§47.
Induction and the concept of natural number
264
§48.
The problem of the uniqueness of the number structure:
Nonstandard
models
272
§49.
Uniqueness and communication
279
§50.
Induction and impredicativity
293
§51.
Predicativity and inductive definitions
307
9
Reason
.316
§52.
Reason and "rational intuition"
316
§53.
Rational intuition and perception
325
§54.
Arithmetic
328
§55.
Set theory
338
Bibliography
343
Index
365 |
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| spelling | Parsons, Charles 1933-2024 Verfasser (DE-588)118789767 aut Mathematical thought and its objects Charles Parsons 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2008 XX, 378 S. 24 cm txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references (p. 343-363) and index Mathematik Philosophie Mathematics Philosophy Object (Philosophy) Logic Logik (DE-588)4036202-4 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf Mathematik (DE-588)4037944-9 s Logik (DE-588)4036202-4 s b DE-604 http://www.loc.gov/catdir/toc/ecip0716/2007016310.html Table of contents only http://www.loc.gov/catdir/enhancements/fy0803/2007016310-b.html Contributor biographical information http://www.loc.gov/catdir/enhancements/fy0803/2007016310-d.html Publisher description Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016448665&sequence=000008&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
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| title | Mathematical thought and its objects |
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