The usual analytic approach to differential geometry was explored in the early 18th century using the defining notion of limit.With the advent of topos theory in the late 20th century, it has been possible to rediscover without the use of limits the differential and integral calculus, and its development into Synthetic Differential Geometry (SDG), in mathematical universes other than that represented by set theory.These universes obey a subtle but well understood generalisation of classical logic: intuitionistic logic.This thesis presents an exploration of the construction of quantum mechanics in the context of SDG.Chapter 1 introduces SDG, intuitionistic logic, topos theory and a particular topos used in the thesis.Chapter 2 explores the new logical perspective afforded by topos theory and SDG in describing the quantum world from an intuitionistic viewpoint.We build a working quantum-mechanical model using topos theory and analyse the physical and philosophical relevance of the logical steps taken, and account for the consistency of the model with its set-theoretic predecessor by suggesting a certain physical interpretation for the mathematics.In chapter 3, we analyse the extension of this quantum mechanics to state spaces with an infinite number of dimensions, and the associated use of various mathematical constructions that are taken for granted in set theory.Chapter 4 reaches an important proof of the lack of validity of the Spectral 4 The Spectral Theorem 108 4.1 Proof of Failure 4.2 The Internal Logic of the Spectral Theorem 4.3 Measurement-Enforced Validity 5 Conclusions 118 5.
We strengthen the case that the new logical perspective afforded by topos theory is suitable to the task of describing the physical world around us. In exploring some of the aspects of construction of a simple quantum-mechanical system in a mathematical universe different from that represented by set theory, we show that more thought and a better appreciation of the assumptions going into any mathematical model of the physical world are needed. We reflect on some of the mathematical consequences of this wider perspective of physics, explaining one interpretation of probabilistic values and numerical calculations in two mathematical universes governed by the less restrictive intuitionistic logic, and known to support the theory of synthetic differential geometry.
The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a C*-algebra of observables A induces a topos T (A) in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra A. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum Σ(A) in T (A), which in our approach plays the role of a quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which is the noncommutative notion of a space). In this setting, states on A become probability
The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a noncommutative C*-algebra of observables A induces a topos
T(A) in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra
A . According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum
Σ(A) in
T(A) , which in our approach plays the role of the quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on
Σ , and self-adjoint elements of A define continuous functions (more precisely, locale maps) from
Σ to Scott’s interval domain. Noting that open subsets of
Σ(A) correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos
T(A) .These results were inspired by the topos-theoretic approach to quantum physics proposed by Butterfield and Isham, as recently generalized by Döring and Isham.
Abstract Quantum set theory (QST) and topos quantum theory (TQT) are two long running projects in the mathematical foundations of quantum mechanics (QM) that share a great deal of conceptual and technical affinity. Most pertinently, both approaches attempt to resolve some of the conceptual difficulties surrounding QM by reformulating parts of the theory inside of nonclassical mathematical universes, albeit with very different internal logics. We call such mathematical universes, together with those mathematical and logical structures within them that are pertinent to the physical interpretation, ‘Q-worlds’. Here, we provide a unifying framework that allows us to (i) better understand the relationship between different Q-worlds, and (ii) define a general method for transferring concepts and results between TQT and QST, thereby significantly increasing the expressive power of both approaches. Along the way, we develop a novel connection to paraconsistent logic and introduce a new class of structures that have significant implications for recent work on paraconsistent set theory.
Abstract Topos theory provides a fundamentally new mathematical language for describing physical systems, replacing set‑theoretic and Boolean foundations with intuitionistic logic, presheaves, and morphisms. In quantum mechanics this eliminates conceptual paradoxes such as superposition, wave‑function collapse, contextuality, and nonlocality. This work demonstrates that the ZEBTS‑SUPER framework (Zero‑Entropy Background Topological Structure — SUPER 4.0) implements the topos approach at the level of operator geometry: χ‑domains, χ‑morphisms, and χ‑spectral logic form a physical realization of the topos of observables. The subobject classifier Ω is realized as the spectrum of the master operator Dχ, and quantum effects become consequences of the topological structure of the χ‑Boundary and its morphisms. The article establishes a strict equivalence between the Isham–Döring topos approach and the χ‑spectral geometry of ZEBTS‑SUPER, and derives experimentally testable predictions for strong‑field physics, quantum gravity, and high‑energy processes.
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 program utilizes the more powerful language L(S) and its representation in an appropriate topos.