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El. knyga: Thin Objects: An Abstractionist Account

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(Proefssor of Philosophy, University of Oslo)
  • Formatas: 288 pages
  • Išleidimo metai: 24-May-2018
  • Leidėjas: Oxford University Press
  • Kalba: eng
  • ISBN-13: 9780192558770
Kitos knygos pagal šią temą:
Thin Objects: An Abstractionist AccountDidesnis paveikslėlis
  • Formatas: 288 pages
  • Išleidimo metai: 24-May-2018
  • Leidėjas: Oxford University Press
  • Kalba: eng
  • ISBN-13: 9780192558770
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Are there objects that are "thin" in the sense that not very much is required for their existence? Frege famously thought so. He claimed that the equinumerosity of the knives and the forks suffices for there to be objects such as the number of knives and the number of forks, and for these objects to be identical. The idea of thin objects holds great philosophical promise but has proved hard to explicate. Oystein Linnebo aims to do so by drawing on some Fregean ideas. First, to be an object is to be a possible referent of a singular term. Second, singular reference can be achieved by providing a criterion of identity for the would-be referent. The second idea enables a form of easy reference and thus, via the first idea, also a form of easy being. Paradox is avoided by imposing a predicativity restriction on the criteria of identity. But the abstraction based on a criterion of identity may result in an expanded domain. By iterating such expansions, a powerful account of dynamic abstraction is developed.

The result is a distinctive approach to ontology. Abstract objects such as numbers and sets are demystified and allowed to exist alongside more familiar physical objects. And Linnebo also offers a novel approach to set theory which takes seriously the idea that sets are "formed" successively.

Recenzijos

Thin Objects is therefore warmly recommended as a novel contribution to the philosophy of mathematics. It gives a thorough understanding of the abstractionist approach, clearly showing its Fregean roots, but at the same time diverging from Frege and the neo-Fregeans in substantial and innovative ways. * Inger Bakken Pedersen, The Mathematical Intelligencer * Recommended. * D. A. Forbes, CHOICE *

Preface xi
Part I Essentials
1 In Search of Thin Objects
3(18)
1.1 Introduction
3(2)
1.2 Coherentist Minimalism
5(2)
1.3 Abstractionist Minimalism
7(2)
1.4 The Appeal of Thin Objects
9(2)
1.5 Sufficiency and Mutual Sufficiency
11(2)
1.6 Philosophical Constraints
13(4)
1.7 Two Metaphysical "Pictures"
17(4)
2 Thin Objects via Criteria of Identity
21(30)
2.1 My Strategy in a Nutshell
21(2)
2.2 A Fregean Concept of Object
23(3)
2.3 Reference to Physical Bodies
26(4)
2.4 Reconceptualization
30(3)
2.5 Reference by Abstraction
33(4)
2.6 Some Objections and Challenges
37(5)
2.6.1 The bad company problem
38(1)
2.6.2 Semantics and metasemantics
38(1)
2.6.3 A vicious regress?
39(1)
2.6.4 A clash with Kripke on reference?
40(1)
2.6.5 Internalism about reference
41(1)
2.7 A Candidate for the Job
42(3)
2.8 Thick versus Thin
45(6)
Appendix 2.A Some Conceptions of Criteria of Identity
46(2)
Appendix 2.B A Negative Free Logic
48(1)
Appendix 2.C Abstraction on a Partial Equivalence
49(2)
3 Dynamic Abstraction
51(26)
3.1 Introduction
51(2)
3.2 Neo-Fregean Abstraction
53(2)
3.3 How to Expand the Domain
55(5)
3.4 Static and Dynamic Abstraction Compared
60(1)
3.5 Iterated Abstraction
61(3)
3.6 Absolute Generality Retrieved
64(2)
3.7 Extensional vs. Intensional Domains
66(11)
Appendix 3.A Further Questions
70(1)
3.A.1 The higher-order needs of semantics
70(1)
3.A.2 Abstraction on intensional entities
70(1)
3.A.3 The need for a bimodal logic
71(2)
3.A.4 The correct prepositional logic
73(1)
Appendix 3.B Proof of the Mirroring Theorem
74(3)
Part II Comparisons
4 Abstraction and the Question of Symmetry
77(10)
4.1 Introduction
77(2)
4.2 Identity of Content
79(2)
4.3 Rayo on "Just is"-Statements
81(2)
4.4 Abstraction and Worldly Asymmetry
83(4)
5 Unbearable Lightness of Being
87(8)
5.1 Ultra-Thin Conceptions of Objecthood
87(2)
5.2 Logically Acceptable Translations
89(1)
5.3 Semantically Idle Singular Terms
90(2)
5.4 Inexplicable Reference
92(3)
Appendix 5.A Proofs and Another Proposition
94(1)
6 Predicative vs. Impredicative Abstraction
95(12)
6.1 The Quest for Innocent Counterparts
95(1)
6.2 Two Forms of Impredicativity
96(2)
6.3 Predicative Abstraction
98(5)
6.3.1 Two-sorted languages
98(2)
6.3.2 Denning the translation
100(1)
6.3.3 The input theory
100(2)
6.3.4 The output theory
102(1)
6.4 Impredicative Abstraction
103(4)
Appendix 6.A Proofs
106(1)
7 The Context Principle
107(28)
7.1 Introduction
107(1)
7.2 How Are the Numbers "Given to Us"?
108(2)
7.3 The Context Principle in the Grundlagen
110(4)
7.4 The "Reproduction" of Meaning
114(3)
7.5 The Context Principle in the Grundgesetze
117(6)
7.6 Developing Frege's Explanatory Strategy
123(6)
7.6.1 An ultra-thin conception of reference
123(1)
7.6.2 Semantically constrained content recarving
124(3)
7.6.3 Towards a metasemantic interpretation
127(2)
7.7 Conclusion
129(6)
Appendix 7.A Hale and Fine on Reference by Recarving
129(6)
Part III Details
8 Reference by Abstraction
135(24)
8.1 Introduction
135(2)
8.2 The Linguistic Data
137(3)
8.3 Two Competing Interpretations
140(3)
8.4 Why the Non-reductionist Interpretation is Preferable
143(5)
8.4.1 The principle of charity
143(1)
8.4.2 The principle of compositionality
144(2)
8.4.3 Cognitive constraints on an interpretation
146(2)
8.5 Why the Non-reductionist Interpretation is Available
148(3)
8.6 Thin Objects
151(8)
Appendix 8.A The Assertibility Conditions
153(2)
Appendix 8.B Comparing the Two Interpretations
155(1)
Appendix 8.C Internally Representable Abstraction
156(1)
Appendix 8.D Defining a Sufficiency Operator
157(2)
9 The Julius Caesar Problem
159(17)
9.1 Introduction
159(1)
9.2 What is the Caesar Problem?
160(2)
9.3 Many-sorted Languages
162(1)
9.4 Sortals and Categories
163(3)
9.5 The Uniqueness Thesis
166(1)
9.6 Hale and Wright's Grundgedanke
167(2)
9.7 Abstraction and the Merging of Sorts
169(7)
Appendix 9.A The Assertibility Conditions
171(2)
Appendix 9.B A Non-reductionist Interpretation
173(1)
Appendix 9.C Defining a Sufficiency Operator
174(2)
10 The Natural Numbers
176(13)
10.1 Introduction
176(1)
10.2 The Individuation of the Natural Numbers
176(2)
10.3 Against the Cardinal Conception
178(4)
10.3.1 The objection from special numbers
179(1)
10.3.2 The objection from the philosophy of language
180(1)
10.3.3 The objection from lack of directness
181(1)
10.4 Alleged Advantages of the Cardinal Conception
182(1)
10.5 Developing the Ordinal Conception
183(2)
10.6 Justifying the Axioms of Arithmetic
185(4)
11 The Question of Platonism
189(16)
11.1 Platonism in Mathematics
189(2)
11.2 Thin Objects and Indefinite Extensibility
191(1)
11.3 Shallow Nature
192(3)
11.4 The Significance of Shallow Nature
195(2)
11.5 How Beliefs are Responsive to Their Truth
197(4)
11.6 The Epistemology of Mathematics
201(4)
12 Dynamic Set Theory
205(18)
12.1 Introduction
205(1)
12.2 Choosing a Modal Logic
206(2)
12.3 Plural Logic with Modality
208(3)
12.4 The Nature of Sets
211(3)
12.4.1 The extensionality of sets
211(1)
12.4.2 The priority of elements to their set
212(1)
12.4.3 The extensional definiteness of subsethood
213(1)
12.5 Recovering the Axioms of ZF
214(9)
12.5.1 From conditions to sets
214(2)
12.5.2 Basic modal set theory
216(1)
12.5.3 Full modal set theory
217(2)
Appendix 12.A Proofs of Formal Results
219(3)
Appendix 12.B A Harmless Restriction
222(1)
Bibliography 223(10)
Index 233
Ųystein Linnebo took up a Professorship at the University of Oslo in 2012, having previously been a Professor at Birkbeck, University of London, and held positions at Bristol and Oxford. He obtained his PhD in Philosophy from Harvard in 2002 and an MA in Mathematics from Oslo in 1995. His main research interests lie in philosophical logic, philosophy of mathematics, metaphysics, early analytic philosophy (especially Frege), as well as parts of philosophy of language and philosophy of science. He has published more than fifty articles and is the author of Philosophy of Mathematics (Princeton University Press 2017).
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