## A topos foundation for theories of physics: I. Formal languages for physics (2007)

Citations: | 8 - 3 self |

### BibTeX

@MISC{Döring07atopos,

author = {A. Döring and C. J. Isham},

title = {A topos foundation for theories of physics: I. Formal languages for physics},

year = {2007}

}

### OpenURL

### Abstract

This paper is the first in a series whose goal is to develop a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics arises when the topos is the category of sets. Other types of theory employ a different topos. In this paper we discuss two different types of language that can be attached to a system, S. The first is a propositional language, PL(S); the second is a higher-order, typed language L(S). Both languages provide deductive systems with an intuitionistic logic. The reason for introducing PL(S) is that, as shown in paper II of the series, it is the easiest way of understanding, and expanding on, the earlier work on topos theory and quantum physics. However, the main thrust of our programme utilises the more powerful language L(S) and its representation in an appropriate topos.

### Citations

474 |
Categories for the working mathematician
- MacLane
- 1998
(Show Context)
Citation Context ...l quantity, A, is represented by an arrow Aφ : Σφ → Rφ in the topos; and (ii) propositions about the system are understand. The references that we have found most helpful in this series of papers are =-=[7, 8, 10, 9, 11, 12]-=-. Some of the basic ideas are described briefly in the Appendix to this paper. 6We coin the term ‘neo-realist’ to signify the conceptual structure implied by our topos formulation of theories of physi... |

424 |
Introduction to Higher Order Categorical Logic
- Lambek, Scott
- 1986
(Show Context)
Citation Context ...l quantity, A, is represented by an arrow Aφ : Σφ → Rφ in the topos; and (ii) propositions about the system are understand. The references that we have found most helpful in this series of papers are =-=[7, 8, 10, 9, 11, 12]-=-. Some of the basic ideas are described briefly in the Appendix to this paper. 6We coin the term ‘neo-realist’ to signify the conceptual structure implied by our topos formulation of theories of physi... |

149 | A categorical semantics of quantum protocols
- Abramsky
- 2004
(Show Context)
Citation Context ...l generalisations and changes. A recent example of such an attempt is the work of Abramsky and Coecke who construct a categorical analogue of some of the critical parts of the Hilbert space formalism =-=[5]-=-; see also the work by Vicary [6]. Here, we adopt a different strategy based on the intrinsic logical structure that is associated with any topos. 5 3 Of course, the existence of the long-range, and a... |

149 |
Sketches of an elephant: a topos theory compendium
- Johnstone
- 2002
(Show Context)
Citation Context ...l quantity, A, is represented by an arrow Aφ : Σφ → Rφ in the topos; and (ii) propositions about the system are understand. The references that we have found most helpful in this series of papers are =-=[7, 8, 10, 9, 11, 12]-=-. Some of the basic ideas are described briefly in the Appendix to this paper. 6We coin the term ‘neo-realist’ to signify the conceptual structure implied by our topos formulation of theories of physi... |

76 |
Introduction To Spin Foam Models Of Quantum Gravity And Bf Theory”, gr-qc/0010050
- Baez
(Show Context)
Citation Context ...interpretation which would include a (limited) relative-frequency understanding of probability. In fact, most modern approaches to quantum gravity aspire to a formalism that is background independent =-=[28, 29, 30, 31]-=-. So, if a background space does arise, it will be in one of the restricted senses mentioned above. Indeed, it is often asserted that a proper theory of quantum gravity will not involve any direct spa... |

71 | A topos perspective on the KochenSpecker theorem: III. Von Neumann algebras as the base category
- Hamilton, Isham, et al.
- 1999
(Show Context)
Citation Context ...tum theory [1, 2] which serves as a paradigmatic example for the general theory. These ideas are motivated by earlier work by one of us (CJI) and Butterfield on interpreting quantum theory in a topos =-=[21, 22, 23, 24, 26, 25]-=-; see also [20, 27]. In the present paper, we will make precise the sense in which propositions about a system can be represented by sub-objects of an object in a topos. To this end, we introduce a fo... |

63 |
Topoi: The Categorial Analysis of Logic
- Goldblatt
- 1984
(Show Context)
Citation Context |

60 |
Sheaves in Geometry and Logic: A First Introduction to Topos Theory
- MacLane, Moerdijk
- 1992
(Show Context)
Citation Context ...e identity operator ˆ1 ∈ B(H).) The definition of a topos. From our perspective, the most relevant feature of a topos, τ, is that it is a category which behaves in many ways like the category of sets =-=[8, 11]-=-. Most of the precise details are not necessary for the present series of papers, but here we will list some of the most important ones for our purposes: 1. There is a terminal object 1τ in τ; this me... |

52 | Unsharp quantum logics
- Chiara, Giuntini
- 1994
(Show Context)
Citation Context ...Borel subsets ∆A, ∆B, ∆C of IR such that19 ( ) Ê[A ∈ ∆A] ∧ Ê[B ∈ ∆B] ∨ Ê[C ∈ ∆C] ̸= (3.15) ( ) ( ) Ê[A ∈ ∆A] ∧ Ê[B ∈ ∆B] ∨ Ê[A ∈ ∆A] ∧ Ê[C ∈ ∆C] (3.16) 18 For an excellent survey of quantum logic see =-=[14]-=-. This includes a discussion of a first-order axiomatisation of quantum logic, and with an associated sequent calculus. It is interesting to compare our work with what the authors of this paper have d... |

46 | Toposes and Local Set Theories - Bell - 1988 |

39 |
Space-time and the philosophical challenge of quantum gravity
- Butterfield, Isham
- 2001
(Show Context)
Citation Context ...roper theory of quantum gravity will not involve any direct spatio-temporal concepts, and that what we commonly call ‘space’ and ‘time’ will ‘emerge’ from the formalism only in some appropriate limit =-=[18]-=-. In this case, any instrumentalist interpretation could only ‘emerge’ in the same limit, as would the associated relative-frequency interpretation of probability. In a theory of this type, there will... |

23 |
Some possible roles for topos theory in quantum theory and quantum gravity Found. Phys
- Butterfield, Isham
- 2000
(Show Context)
Citation Context ...tum theory [1, 2] which serves as a paradigmatic example for the general theory. These ideas are motivated by earlier work by one of us (CJI) and Butterfield on interpreting quantum theory in a topos =-=[21, 22, 23, 24, 26, 25]-=-; see also [20, 27]. In the present paper, we will make precise the sense in which propositions about a system can be represented by sub-objects of an object in a topos. To this end, we introduce a fo... |

18 |
Quantum Theory and the Schism in Physics
- Popper
- 1982
(Show Context)
Citation Context ...rstood in a different way. In the physical sciences, one of the most discussed approaches involves the concept of ‘potentiality’, or ‘latency’, as favoured by Heisenberg, Margenau, and Popper [15][16]=-=[17]-=- (and, for good measure, Aristotle). In this case there is no compelling reason why the probability-value space should be a subset of the real numbers. The minimal requirement is that this value-space... |

17 | Kochen-Specker theorem for von Neumann algebras
- Döring
- 2005
(Show Context)
Citation Context ... paradigmatic example for the general theory. These ideas are motivated by earlier work by one of us (CJI) and Butterfield on interpreting quantum theory in a topos [21, 22, 23, 24, 26, 25]; see also =-=[20, 27]-=-. In the present paper, we will make precise the sense in which propositions about a system can be represented by sub-objects of an object in a topos. To this end, we introduce a formal language for e... |

16 |
Some reflections on the status of conventional quantum theory when applied to quantum gravity
- Isham
- 2003
(Show Context)
Citation Context ...s. In this sense there is a direct link between the space in which physical quantities take their values (what we shall call the ‘quantity-value space’) and the nature of physical space or space-time =-=[19]-=-. If conceded, this claim means that the assumption that physical quantities are real-valued is problematic in a theory in which space, or space-time, is not modelled by a smooth manifold. Admittedly,... |

16 | The case for background independence
- Smolin
- 2006
(Show Context)
Citation Context ...interpretation which would include a (limited) relative-frequency understanding of probability. In fact, most modern approaches to quantum gravity aspire to a formalism that is background independent =-=[28, 29, 30, 31]-=-. So, if a background space does arise, it will be in one of the restricted senses mentioned above. Indeed, it is often asserted that a proper theory of quantum gravity will not involve any direct spa... |

16 | Causal Sets: Discrete Gravity (Notes for the Valdivia Summer School - Sorkin - 2005 |

11 | A topos foundation for theories of physics: II. Daseinisation and the liberation of quantum theory
- Döring, Isham
- 2008
(Show Context)
Citation Context ...ds on both the theory-type and the system; and that physical quantities and propositions are represented in the ways indicated above. Papers II and III in the series are concerned with quantum theory =-=[1, 2]-=- which serves as a paradigmatic example for the general theory. These ideas are motivated by earlier work by one of us (CJI) and Butterfield on interpreting quantum theory in a topos [21, 22, 23, 24, ... |

9 |
Reality in quantum mechanics
- Margenau
- 1949
(Show Context)
Citation Context ...understood in a different way. In the physical sciences, one of the most discussed approaches involves the concept of ‘potentiality’, or ‘latency’, as favoured by Heisenberg, Margenau, and Popper [15]=-=[16]-=-[17] (and, for good measure, Aristotle). In this case there is no compelling reason why the probability-value space should be a subset of the real numbers. The minimal requirement is that this value-s... |

9 | Background independent geometry and Hopf cyclic cohomology
- Connes, Moscovici
- 2005
(Show Context)
Citation Context ...interpretation which would include a (limited) relative-frequency understanding of probability. In fact, most modern approaches to quantum gravity aspire to a formalism that is background independent =-=[28, 29, 30, 31]-=-. So, if a background space does arise, it will be in one of the restricted senses mentioned above. Indeed, it is often asserted that a proper theory of quantum gravity will not involve any direct spa... |

8 | Generic predictions of quantum theories of gravity. arXiv:hep-th/0605.052, to appear in Approaches to Quantum Gravity— Toward a New Understanding of
- Smolin
- 2006
(Show Context)
Citation Context |

6 |
A topos foundation for theories of physics: IV. Categories of systems. arXiv:quant-ph/0703.066
- Döring, Isham
- 2008
(Show Context)
Citation Context ...languages are deductive systems employing intuitionistic logic; as such, they can be used to make, and manipulate, statements about the world as it is revealed in the system under study. In paper IV (=-=[3]-=-) we return once more to the overall formalism and consider what happens to the languages and their representations when the system ranges over the objects in a ‘category of systems’. This category in... |

5 |
Causal sets and the deep structure of space-time. arXiv.org/abs/grqc/0508109
- Dowker
- 2005
(Show Context)
Citation Context ...ace-time ground type symbol M, but then add the axioms for a partial ordering. In that case, Mφ would be a poset-object in τφ, which could be interpreted physically as the τφ-analogue of a causal set =-=[32]-=-. Yet another possibility is to develop a language for history theories, and use it study the topos version of the consistent-histories approach to quantum theory. We will return to some of these idea... |

4 |
A topos foundation for theories of physics: III. Quantum theory and the representation of physical quantities with arrows
- Döring, Isham
- 2008
(Show Context)
Citation Context ...epresentation of the language in a topos τ, the symbol R is mapped to the real-number object in the topos (if there is one). However, the example of quantum theory suggests that this is inappropriate =-=[2]-=-. 202. Then add axioms like ‘: ∀˜r ( + 〈˜r, 0(∗)〉 = ˜r ) ’ where ˜r is a variable of type R, and so on. For another example, consider a point particle moving in three dimensions, with the function sy... |

4 | Is it true; or is it false; or somewhere in between? The logic of quantum theory. Contempory Phys
- Isham
- 2005
(Show Context)
Citation Context ...tum theory [1, 2] which serves as a paradigmatic example for the general theory. These ideas are motivated by earlier work by one of us (CJI) and Butterfield on interpreting quantum theory in a topos =-=[21, 22, 23, 24, 26, 25]-=-; see also [20, 27]. In the present paper, we will make precise the sense in which propositions about a system can be represented by sub-objects of an object in a topos. To this end, we introduce a fo... |

3 |
Philosophic Problems of Nuclear Science
- Heisenberg
- 1952
(Show Context)
Citation Context ... be understood in a different way. In the physical sciences, one of the most discussed approaches involves the concept of ‘potentiality’, or ‘latency’, as favoured by Heisenberg, Margenau, and Popper =-=[15]-=-[16][17] (and, for good measure, Aristotle). In this case there is no compelling reason why the probability-value space should be a subset of the real numbers. The minimal requirement is that this val... |

2 |
Synthetic Differential Geometry (LMS lecture note series: 51
- Kock
- 1981
(Show Context)
Citation Context ...nd Mφ would be a standard differentiable manifold. On the other hand, if the topos τφ admits ‘infinitesimals’, then Mφ could be a manifold according to the language of synthetic differential geometry =-=[13]-=-. A fortiori, the same type of argument applies to the status of ‘time’ in a canonical theory. In particular, it is possible to add a ground type symbol, T , so that, in any representation, φ, the obj... |

1 |
The quantum harmonic oscillator as an adjunction
- Vicary
- 2007
(Show Context)
Citation Context ...recent example of such an attempt is the work of Abramsky and Coecke who construct a categorical analogue of some of the critical parts of the Hilbert space formalism [5]; see also the work by Vicary =-=[6]-=-. Here, we adopt a different strategy based on the intrinsic logical structure that is associated with any topos. 5 3 Of course, the existence of the long-range, and all penetrating, gravitational for... |