Concept formation in mathematics:
This thesis consists of three overlapping parts, where the first one centers around the possibility of defining a measure of the power of arithmetical theories. In this part a partial measure of the power of arithmetical theories is constructed, where ''power'' is understood as c...
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| Format: | Abschlussarbeit Buch |
| Sprache: | English |
| Veröffentlicht: |
Gothenburg
University of Gothenburg
[2011]
|
| Schriftenreihe: | Acta philosophica Gothoburgensia
27 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | This thesis consists of three overlapping parts, where the first one centers around the possibility of defining a measure of the power of arithmetical theories. In this part a partial measure of the power of arithmetical theories is constructed, where ''power'' is understood as capability to prove theorems. It is also shown that other suggestions in the literature for such a measure do not satisfy natural conditions on a measure. In the second part a theory of concept formation in mathematics is developed. This is inspired by Aristotle's conception of mathematical objects as abstractions, and it uses Carnap's method of explication as a means to formulate these abstractions in an ontologically neutral way. Finally, in the third part some problems of philosophy of mathematics are discussed. In the light of this idea of concept formation it is discussed how the relation between formal and informal proof can be understood, how mathematical theories are tested, how to characterize mathematics, and some questions about realism and indispensability. |
| Beschreibung: | 1 Band (verschiedene Seitenzählungen) Diagramme |
| ISBN: | 9789173467056 |
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| 490 | 1 | |a Acta philosophica Gothoburgensia |v 27 | |
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| 520 | 3 | |a This thesis consists of three overlapping parts, where the first one centers around the possibility of defining a measure of the power of arithmetical theories. In this part a partial measure of the power of arithmetical theories is constructed, where ''power'' is understood as capability to prove theorems. It is also shown that other suggestions in the literature for such a measure do not satisfy natural conditions on a measure. In the second part a theory of concept formation in mathematics is developed. This is inspired by Aristotle's conception of mathematical objects as abstractions, and it uses Carnap's method of explication as a means to formulate these abstractions in an ontologically neutral way. Finally, in the third part some problems of philosophy of mathematics are discussed. In the light of this idea of concept formation it is discussed how the relation between formal and informal proof can be understood, how mathematical theories are tested, how to characterize mathematics, and some questions about realism and indispensability. | |
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Datensatz im Suchindex
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|---|---|
| adam_text | Contents 1 2 Introduction................................................................................ И On Explications.................................................................... 13 2.1 2.2 2.3 2.4 3 4 30 33 34 36 37 Abstract Objects and Idealizations................................ 38 Abstract Objects versus Concrete Objects............................... 38 Abstractions and Idealizations.................................................. 41 Structuralism........................................................................ 43 5.1 5.2 5.3 5.4 5.5 6 зо Philosophy of Mathematics in Aristotle.......................... 3.1 Mathematical Objects as Abstractions...................................... 3.2 On the Existence of Mathematical Objects.............................. 3.3 Questions of Truth................................................................... 3.4 On the Relation between Sciences........................................... 3.5 Concluding Remarks................................................................ 4.1 4.2 5 Carnap and Explications........................................................... 13 Some Problems with Carnap’s Position.................................... 19 On the Use of Explications in the Thesis................................ 24 An Overview of Treated Explications...................................... 26 General Remarks....................................................................... Relativist Structuralism............................................................. Universalist
Structuralism......................................................... Pattern Structuralism................................................................ Structuralism in the Thesis........................................................ 43 45 46 47 49 Summaries of the papers............................................................ 6.1 Measuring the Power of an Arithmetical Theory..................... 6.2 On Explicating the Concept The Power of an Arithmetical Theory......................................................................................... 6.3 A Note on the Relation Between Formal and Informal Proof. 50 50 52 53
6.4 6.5 7 Indispensability, The Testing of Mathematical Theories, and Provisional Realism................................................................... 53 Mathematical Concepts as Unique Explications...................... 55 Future Work - Some Ideas........................................................ 55 References.................................................................................... 57
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| adam_txt |
Contents 1 2 Introduction. И On Explications. 13 2.1 2.2 2.3 2.4 3 4 30 33 34 36 37 Abstract Objects and Idealizations. 38 Abstract Objects versus Concrete Objects. 38 Abstractions and Idealizations. 41 Structuralism. 43 5.1 5.2 5.3 5.4 5.5 6 зо Philosophy of Mathematics in Aristotle. 3.1 Mathematical Objects as Abstractions. 3.2 On the Existence of Mathematical Objects. 3.3 Questions of Truth. 3.4 On the Relation between Sciences. 3.5 Concluding Remarks. 4.1 4.2 5 Carnap and Explications. 13 Some Problems with Carnap’s Position. 19 On the Use of Explications in the Thesis. 24 An Overview of Treated Explications. 26 General Remarks. Relativist Structuralism. Universalist
Structuralism. Pattern Structuralism. Structuralism in the Thesis. 43 45 46 47 49 Summaries of the papers. 6.1 Measuring the Power of an Arithmetical Theory. 6.2 On Explicating the Concept The Power of an Arithmetical Theory. 6.3 A Note on the Relation Between Formal and Informal Proof. 50 50 52 53
6.4 6.5 7 Indispensability, The Testing of Mathematical Theories, and Provisional Realism. 53 Mathematical Concepts as Unique Explications. 55 Future Work - Some Ideas. 55 References. 57 |
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| spelling | Sjögren, Jörgen 1948- Verfasser (DE-588)1256803235 aut Concept formation in mathematics Jörgen Sjögren Gothenburg University of Gothenburg [2011] 1 Band (verschiedene Seitenzählungen) Diagramme txt rdacontent n rdamedia nc rdacarrier Acta philosophica Gothoburgensia 27 Dissertation Göteborgs universitet 2011 This thesis consists of three overlapping parts, where the first one centers around the possibility of defining a measure of the power of arithmetical theories. In this part a partial measure of the power of arithmetical theories is constructed, where ''power'' is understood as capability to prove theorems. It is also shown that other suggestions in the literature for such a measure do not satisfy natural conditions on a measure. In the second part a theory of concept formation in mathematics is developed. This is inspired by Aristotle's conception of mathematical objects as abstractions, and it uses Carnap's method of explication as a means to formulate these abstractions in an ontologically neutral way. Finally, in the third part some problems of philosophy of mathematics are discussed. In the light of this idea of concept formation it is discussed how the relation between formal and informal proof can be understood, how mathematical theories are tested, how to characterize mathematics, and some questions about realism and indispensability. Mathematik (DE-588)4037944-9 gnd rswk-swf Begriffsbildung (DE-588)4005249-7 gnd rswk-swf Mathematics / Philosophy Concepts (DE-588)4113937-9 Hochschulschrift gnd-content Mathematik (DE-588)4037944-9 s Begriffsbildung (DE-588)4005249-7 s DE-604 Acta philosophica Gothoburgensia 27 (DE-604)BV000651194 27 Digitalisierung BSB München - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032598997&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Sjögren, Jörgen 1948- Concept formation in mathematics Acta philosophica Gothoburgensia Mathematik (DE-588)4037944-9 gnd Begriffsbildung (DE-588)4005249-7 gnd |
| subject_GND | (DE-588)4037944-9 (DE-588)4005249-7 (DE-588)4113937-9 |
| title | Concept formation in mathematics |
| title_auth | Concept formation in mathematics |
| title_exact_search | Concept formation in mathematics |
| title_exact_search_txtP | Concept formation in mathematics |
| title_full | Concept formation in mathematics Jörgen Sjögren |
| title_fullStr | Concept formation in mathematics Jörgen Sjögren |
| title_full_unstemmed | Concept formation in mathematics Jörgen Sjögren |
| title_short | Concept formation in mathematics |
| title_sort | concept formation in mathematics |
| topic | Mathematik (DE-588)4037944-9 gnd Begriffsbildung (DE-588)4005249-7 gnd |
| topic_facet | Mathematik Begriffsbildung Hochschulschrift |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032598997&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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