Branching space-times : theory and applications / Nuel Belnap, Thomas M�uller, and Tomasz Placek.

Format:

Book

Author/Creator:

Belnap, Nuel, 1930- author.

Müller, Thomas, 1969- author.

Placek, Tomasz, 1960- author.

Series:

Oxford studies in philosophy of science

Language:

English

Subjects (All):

Determinism (Philosophy)--Mathematics.

Determinism (Philosophy).

Events (Philosophy)--Mathematical models.

Events (Philosophy).

Mathematical physics.

Space and time--Mathematical models.

Space and time.

Chaotic behavior in systems--Mathematical models.

Chaotic behavior in systems.

Probabilities.

Mathematical models.

Mathematics.

Physical Description:

xiii, 441 pages : illustrations ; 24 cm.

Place of Publication:

New York, NY : Oxford University Press, [2022]

Summary:

"This book develops a rigorous theory of indeterminism as a local and modal concept. Its crucial insight is that our world contains events or processes with alternative, really possible outcomes. The theory aims at clarifying what this assumption involves, and it does it in two ways. First, it provides a mathematically rigorous framework for local and modal indeterminism. Second, we support that theory by spelling out the philosophically relevant consequences of this formulation and by showing its fruitful applications in metaphysics. To this end, we offer a formal analysis of modal correlations and of causation, which is applicable in indeterministic and non-local contexts as well. We also propose a rigorous theory of objective single-case probabilities, intended to represent degrees of possibility. In a third step, we link our theory to current physics, investigating how local and modal indeterminism relates to issues in the foundations of physics, in particular, quantum non-locality and spatio-temporal relativity. The book also ventures into the philosophy of time, showing how the theory's resources can be used to explicate the dynamic concept of the past, present, and future based on local indeterminism"-- Provided by publisher.

Contents:

Machine generated contents note: 1. Introduction

1.1. Real possibilities

1.2. Representing possibilities via branching vs. possible worlds

1.3. Some thoughts on our modally rich world

1.4. Branching in the landscape of possible-worlds theories: Some comments on modal metaphysics

1.5. Outline of the book

1.6. Exercises to Chapter 1

2. The Foundations of Branching Space-Times

2.1. The underlying ideas of BST

2.2. Histories

2.3. Historical connection

2.4. Density and continuity

2.5. Weiner's postulate and spatio-temporal locations

2.6. Axioms of the common core of BST

2.7. Exercises to Chapter 2

3. Two Options for the Branching of Histories

3.1. Indeterminism as the branching of histories

3.2. On chains in common BST structures

3.3. Extending common BST: two options

3.4. BST92

3.4.1. BST92 in formal detail

3.4.2. Local possibilities

3.4.3. The pattern of branching of BST92

3.4.4. Transitions

3.5. Introducing BSTNF

3.5.1. The new Prior Choice Principle and BSTNF structures defined

3.5.2. Local possibilities and the pattern of branching in BSTNF

3.5.3. Facts about choice sets

3.6. BST92 or BSTNF: Does it matter?

3.6.1. Topological issues: An overview

3.6.2. Translatability results: An overview

3.7. Exercises to Chapter 3

4. Building upon the Foundations of Branching Space-Times

4.1. A variety of events and their occurrence propositions

4.2. Basic transitions

4.2.1. Basic transitions in BST92

4.2.2. A note on basic transitions in BSTNF

4.3. Sets of basic transitions

4.4. Topological aspects of BST

4.4.1. General idea of the diamond topology

4.4.2. Properties of the diamond topology in BST92

4.4.3. The diamond topology in BSTNF

4.5. A note on branching-style semantics

4.6. Exercises to Chapter 4

5. Modal Funny Business

5.1. Motivation for being interested in modal correlations

5.2. Modal funny business

5.2.1. Expected inconsistencies in sets of basic transitions

5.2.2. Combinatorial funny business

5.2.3. Explanatory funny business

5.2.4. On the interrelation of combinatorial and explanatory funny business

5.3. Some consequences of modal funny business

5.4. On MFB in BSTNF

5.5. Exercises to Chapter 5

6. Causation in Terms of causae causantes

6.1. Causation: Causes and effects as BST transitions

6.2. At least an inus condition

6.3. Causae causantes in BST92 in formal detail

6.3.1. Defining causae causantes in BST92

6.3.2. Causae causantes are at least inus conditions

6.3.2.1. Transitions to outcome chains or scattered outcomes

6.3.2.2. Transitions to disjunctive outcomes

6.4. Causation in the presence of modal funny business

6.5. Causae causantes in BSTNF structures

6.6. Conclusions

6.7. Exercises to Chapter 6

7. Probabilities

7.1. Two conditions of adequacy and two crucial questions

7.1.1. Two conditions of adequacy

7.1.2. Two crucial questions

7.1.3. Propensities [µ] and probability measures p

7.2. Causal probability spaces in BST

7.2.1. Probabilities for transitions: The simplest case

7.2.2. Two BST transitions, one basic transition

7.2.3. Two or more transitions and some complications

7.2.4. General probability spaces in BST

7.2.5. Representing transitions in different causal probability spaces

7.2.6. Probabilistic funny business

7.3. Fending off objections to propensities

7.3.1. Some remarks on propensities

7.3.2. Humphreys's paradox

7.3.3. Our diagnosis of Humphreys's paradox

7.3.4. Salmon's corkscrew story: More on conditional propensities and inversion

7.4. Conclusions

7.5. Exercises to Chapter 7

8. Quantum Correlations

8.1. Introducing quantum correlation experiments

8.2. On the BST analysis of quantum correlations

8.3. Explaining modal correlations via instruction sets

8.3.1. Extensions of a surface structure by generic instruction sets

8.3.1.1. The possibility of superdeterministic extensions

8.3.1.2. Splitting in extended structures: The general case

8.3.2. Non-contextual and contextual instruction sets

8.3.2.1. Non-contextual instruction sets

8.3.2.2. Contextual instruction sets

8.3.2.3. On the interrelation of different types of instruction sets

8.3.3. Instruction sets for GHZ

8.3.3.1. The superdeterministic extension

8.3.3.2. C/E independence

8.3.3.3. Non-contextual instruction sets for GHZ

8.3.3.4. Contextual instruction sets for GHZ

8.3.4. Summary of the BST approach to modal structure extensions

8.4. Probabilistic correlations

8.4.1. Probabilistic hidden variables

8.4.2. Extension of a probabilistic surface structure

8.4.3. Single and multiple cases of PFB, and super-independence

8.4.3.1. A structure with a single case of PFB

8.4.3.2. A structure with multiple cases of PFB - super-independence

8.4.4. The Bell-Aspect experiment

8.4.4.1. The set-up of the Bell-Aspect experiment

8.4.4.2. The surface structure for the Bell-Aspect experiment

8.4.4.3. Probabilistic funny business

8.4.4.4. Derivation of the Bell-CH inequality

8.4.4.5. Analysis of the derivation

8.4.4.6. Consequences from our analysis

8.5. Exercises to Chapter 8

9. Branching in Relativistic Space-Times

9.1. Minkowskian Branching Structures

9.1.1. Basic notions

9.1.2. Defining MBSs

9.1.3. Taking stock

9.2. Differential manifolds and BSTNF

9.2.1. Differential manifolds

9.2.2. Differential manifolds and MBSs

9.2.3. Differential manifolds and BSTNF, generally

9.2.4. Differential manifolds in GR

9.3. GR space-times

9.3.1. The initial value problem in GR

9.3.2. An example of the failure of the IVP: Non-isometric extensions of Taub space-time

9.3.3. Can non-Hausdorff manifolds in GR be interpreted modally?

9.3.4. On bifurcating curves in GR

9.3.5. Global and local determinism and indeterminism

9.3.6. A note on closed causal curves and BST

9.3.7. Summary on General Relativity

9.4. Conclusions

9.5. Exercises to Chapter 9

10. A Branching Space-Times Perspective on Presentism

10.1. Introduction

10.2. The problem of defining the present in special relativity

10.3. Making room for an extended dynamic present

10.4. The dynamic present, past, and future: Two approaches

10.5. Dynamic time via causae causantes

10.6. Dynamic time via the semantics of the open future

10.7. The way to guarantee satisfactory dynamic time in BST: Sticky modal funny business

10.8. What does dynamic time look like in MBSs?

10.9. Conclusions

10.10. Exercises to Chapter 10

A. Selected Proofs and Additional Material

A.1. Dedekind continuity

A.2. Formal details of the interrelation of BST92 and BSTNF

A.2.1. Characterizing the transition structure of a BST92 structure

A.2.2. BST92 transition structures are BSTNF structures

A.2.3. From new foundations BSTNF to BST92

A.2.4. Going full circle

A.2.4.1. From BST92 to BSTNF to BST92

A.2.4.2. From BSTNF to BST92 to BSTNF

A.2.5. The translatability of some notions pertaining to MFB

A.3. Proof of Theorem 5.1

A.4. Additional material for Chapter 8

A.4.1. Extensions by one point or by multiple points?

A.4.2. Proofs for Chapter 8

B. Answers to Selected Exercises

B.1. Answers to selected exercises from Chapter 1

B.2. Answers to selected exercises from Chapter 2

B.3. Answers to selected exercises from Chapter 3

B.4. Answers to selected exercises from Chapter 4

B.5. Answers to selected exercises from Chapter 5

B.6. Answers to selected exercises from Chapter 6

B.7. Answers to selected exercises from Chapter 7

B.8. Answers to selected exercises from Chapter 8

B.9. Answers to selected exercises from Chapter 9

B.10. Answers to selected exercises from Chapter 10.

Notes:

Includes bibliographical references and indexes.

ISBN:

9780190884314

0190884312

OCLC:

1273049914

Publisher Number:

99989829553