[Paper Review] A topos perspective on the Kochen-Specker theorem: II. Conceptual Aspects, and Classical Analogues:

C. J. Isham, Jeremy Butterfield|ArXiv.org|Aug 31, 1998

Quantum Mechanics and Applications3 references21 citations

TL;DR

This paper proposes a topos-theoretic framework for valuations in quantum and classical physics, using sieve-valued assignments to physical quantities that generalize the FUNC (functional) principle. It shows that such valuations naturally arise from partial truth conditions and presheaf structures, offering a solution to the Kochen-Specker contextuality problem by defining truth-values as sieves closed under coarse-graining, with key results demonstrating their consistency and generality across both quantum and classical settings.

ABSTRACT

In a previous paper, we have proposed assigning as the value of a physical quantity in quantum theory, a certain kind of set (a sieve) of quantities that are functions of the given quantity. The motivation was in part physical---such a valuation illuminates the Kochen-Specker theorem; and in part mathematical---the valuation arises naturally in the topos theory of presheaves. This paper discusses the conceptual aspects of this proposal. We also undertake two other tasks. First, we explain how the proposed valuations could arise much more generally than just in quantum physics; in particular, they arise as naturally in classical physics. Second, we give another motivation for such valuations (that applies equally to classical and quantum physics). This arises from applying to propositions about the values of physical quantities some general axioms governing partial truth for any kind of proposition.

Motivation & Objective

• To generalize the functional calculus (FUNC) principle in quantum theory beyond single-valued functions, using sieve-valued valuations.
• To show that such valuations are not exclusive to quantum physics but also arise naturally in classical physics through macrostate-based coarse-graining.
• To establish that sieve-valued valuations are a natural consequence of general principles of partial truth in any presheaf of propositions over a category.
• To unify the conceptual and mathematical foundations of contextuality in quantum mechanics via topos-theoretic structures.

Proposed method

• Assign to each physical quantity A a sieve on A, defined as a set of morphisms f:B→A closed under composition with further morphisms, representing possible coarse-grainings.
• Define truth-values for propositions 'A∈Δ' as sieves on A, where the sieve captures all contexts in which the proposition is true.
• Use the presheaf structure on the category of physical quantities (with morphisms as functions/coarse-grainings) to model how truth-values propagate across contexts.
• Apply generalized FUNC: the sieve value of f(A) is the pullback of the sieve value of A along the morphism f, ensuring consistency across contexts.
• Model partial truth via a logical algebra of truth-values, where a proposition is more true if more of its consequences are totally true.
• Use the category-theoretic notion of subobjects and generalized entailment to formalize consequence relations in the presheaf setting.

Experimental results

Research questions

• RQ1 How can the FUNC principle be generalized in a way that avoids the Kochen-Specker contextuality obstruction in quantum theory?
• RQ2 In what sense do sieve-valued valuations naturally arise in classical physics, even in the absence of quantum contextuality?
• RQ3 Can the concept of partial truth in propositions be formalized in a way that leads naturally to sieve-valued valuations?
• RQ4 What is the role of the category of contexts (via coarse-graining morphisms) in defining a consistent valuation structure?
• RQ5 How do the structural properties of presheaves (e.g., global elements, subobjects) relate to the existence or non-existence of global valuations?

Key findings

• Sieve-valued valuations provide a consistent generalization of the FUNC principle in quantum theory, avoiding the Kochen-Specker theorem's obstruction by replacing global real-valued functions with set-valued assignments.
• In classical physics, such valuations naturally emerge from macrostate information, where the set of all functions of a given quantity forms a sieve under coarse-graining.
• The proposed valuation scheme satisfies key physical and logical constraints: monotonicity, null proposition, and the generalized FUNC via pullbacks of sieves.
• The framework generalizes beyond physics: any presheaf of propositions over a small category admits a natural sieve-valued valuation satisfying partial truth principles.
• The truth-value of a proposition is fully determined by the set of all contexts in which its weakenings are totally true, formalizing a natural notion of partial truth.
• The absence of global elements in the spectrum presheaf (a key result of the Kochen-Specker theorem) is reconciled by replacing global functions with sieve-valued valuations that are globally defined.