Any attempt to construct a realist interpretation of quantum theory founders on the Kochen-Specker theorem, which asserts the impossibility of assigning values to quantum quantities in a way that preserves functional relations between them. We construct a new type of valuation which is defined on all operators, and which respects an appropriate version of the functional composition principle. The truth-values assigned to propositions are (i) contextual; and (ii) multi-valued, where the space of contexts and the multi-valued logic for each context (...) come naturally from the topos theory of presheaves. The first step in our theory is to demonstrate that the Kochen-Specker theorem is equivalent to the statement that a certain presheaf defined on the category of self-adjoint operators has no global elements. We then show how the use of ideas drawn from the theory of presheaves leads to the definition of a generalised valuation in quantum theory whose values are sieves of operators. In particular, we show how each quantum state leads to such a generalised valuation. A key ingredient throughout is the idea that, in a situation where no normal truth-value can be given to a proposition asserting that the value of a physical quantity A lies in a set D of real numbers , it is nevertheless possible to ascribe a partial truth-value which is determined by the set of all coarse-grained propositions that assert that some function f(A) lies in f(D), and that are true in a normal sense. The set of all such coarse-grainings forms a sieve on the category of self-adjoint operators, and is hence fundamentally related to the theory of presheave. (shrink)

In this paper (a sequel to [4]) I put forward a "local" interpretation of mathematical concepts based on notions derived from category theory. The fundamental idea is to abandon the unique absolute universe of sets central to the orthodox set-theoretic account of the foundations of mathematics, replacing it by a plurality of local mathematical frameworks - elementary toposes - defined in category-theoretic terms.

A general duality connecting the level of a formal theory and of a metatheory is proposed. Because of the role of natural numbers in a metatheory the existence of a dual theory is conjectured, in which the natural numbers become formal in the theory but in formalizing non-formal natural numbers taken from the dual metatheory these numbers become nonstandard. For any formal theory there may be in principle a dual theory. The dual shape of the lattice of projections over separable (...) Hilbert space is exhibited. The crucial ingredient of this dual theory is set-theoretical (Cohen) forcing. Then, some attempts have been made to exhibit the dual shape of the exotic smooth (small) structures on topological ℝ4. There are striking similarities of both pictures. Nonstandard solutions of the equations of motion of 10-dimensional superstring of IIB type are described as a model theoretic extension of standard ones. Based on Brans conjecture one can predict the existence of sources of gravity in 4 dimensions. By their dual origins they carry quantum information as well. The generalized Maldacena conjecture appears naturally, which sheds light on a definition of background independent string theory. (shrink)

As an approach to a Theory of Everything a framework for developing a coherent theory of mathematics and physics together is described. The main characteristic of such a theory is discussed: the theory must be valid and and sufficiently strong, and it must maximally describe its own validity and sufficient strength. The mathematical logical definition of validity is used, and sufficient strength is seen to be a necessary and useful concept. The requirement of maximal description of its own validity and (...) sufficient strength may be useful to reject candidate coherent theories for which the description is less than maximal. Other aspects of a coherent theory discussed include universal applicability, the relation to the anthropic principle, and possible uniqueness. It is suggested that the basic properties of the physical and mathematical universes are entwined with and emerge with a coherent theory. Support for this includes the indirect reality status of properties of very small or very large far away systems compared to moderate sized nearby systems. Discussion of the necessary physical nature of language includes physical models of language and a proof that the meaning content of expressions of any axiomatizable theory seems to be independent of the algorithmic complexity of the theory. Gödel maps seem to be less useful for a coherent theory than for purely mathematical theories because all symbols and words of any language must have representations as states of physical systems already in the domain of a coherent theory. (shrink)

In this paper we construct the ortholattices arising in quantum logic starting from the phenomenologically plausible idea of a collection of ensembles subject to passing or failing various “tests.” A collection of ensembles forms a certain kind of preordered set with extra structure called anorthospace; we show that complete ortholattices arise as canonical completions of orthospaces in much the same way as arbitrary complete lattices arise as canonical completions of partially ordered sets. We also show that the canonical completion of (...) an orthospace of ensembles is naturally identifiable as the complete lattice of properties of the ensembles, thereby revealing exactlywhy ortholattices arise in the analysis of “tests” or experimental propositions. Finally, we axiomatize the hitherto implicit concept of “test” and show how they may be correlated with properties of ensembles. (shrink)

We give an almost explicit presentation of exotic functions corresponding to some exotic smooth structure on topologically trivial R4. The construction relies on the model-theoretic tools from the previous paper. We can formulate unexpected, yet direct connection between ‘‘localized’’ exotic small R4’s and some noncommutative spaces. The formalism of QM can be interpreted in terms of exotic smooth R4’s localized in spacetime. A new way of looking at the problem of decoherence is suggested. The 4-dimensional spacetime itself has built-in means (...) which may enforce a kind of decoherence. (shrink)

We discuss some ways in which topos theory (a branch of category theory) can be applied to interpretative problems in quantum theory and quantum gravity. In Sec.1, we introduce these problems. In Sec.2, we introduce topos theory, especially the idea of a topos of presheaves. In Sec.3, we discuss several possible applications of topos theory to the problems in Sec.1. In Sec.4, we draw some conclusions.

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. (shrink)

Using set theory in the first part of his book, and proof theory in the second, Gaisi Takeuti gives us two examples of how mathematical logic can be used to obtain results previously derived in less elegant fashion by other mathematical techniques, especially analysis. In Part One, he applies Scott- Solovay's Boolean-valued models of set theory to analysis by means of complete Boolean algebras of projections. In Part Two, he develops classical analysis including complex analysis in Peano's arithmetic, showing that (...) any arithmetical theorem proved in analytic number theory is a theorem in Peano's arithmetic. In doing so, the author applies Gentzen's cut elimination theorem. Although the results of Part One may be regarded as straightforward consequences of the spectral theorem in function analysis, the use of Boolean- valued models makes explicit and precise analogies used by analysts to lift results from ordinary analysis to operators on a Hilbert space. Essentially expository in nature, Part Two yields a general method for showing that analytic proofs of theorems in number theory can be replaced by elementary proofs. Originally published in 1978. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905. (shrink)