The purpose of this dissertation is to construct a radically new type of mathematics whose underlying logic differs from the ordinary classical logic used in standard mathematics, and which we feel may be more natural for applications in quantum mechanics. Specifically, we begin by constructing a first order quantum logic, the development of which closely parallels that of ordinary (classical) first order logic --- the essential differences are in the nature of the logical axioms, which, in our construction, are motivated by quantum theory. After showing that the axiomatic first order logic we develop is sound and complete (with respect to a particular class of models), this logic is then used as a foundation on which to build (axiomatic) mathematical systems --- and we refer to the resulting new mathematics as "quantum mathematics." As noted above, the hope is that this form of mathematics is more natural than classical mathematics for the description of quantum systems, and will enable us to address some foundational aspects of quantum theory which are still troublesome --- e.g. the measurement problem --- as well as possibly even inform our thinking about quantum gravity.\n\nAfter constructing the underlying logic, we investigate properties of several mathematical systems --- e.g. axiom systems for abstract algebras, group theory, linear algebra, etc. --- in the presence of this quantum logic. In the process, we demonstrate that the resulting quantum mathematical systems have some strange, but very interesting features, which indicates a richness in the structure of mathematics that is classically inaccessible. Moreover, some of these features do indeed suggest possible applications to foundational questions in quantum theory. \n\nWe continue our investigation of quantum mathematics by constructing an axiomatic quantum set theory, which we show satisfies certain desirable criteria. Ultimately, we hope that such a set theory will lead to a foundation for quantum mathematics in a sense which parallels the foundational role of classical set theory in classical mathematics. One immediate application of the quantum set theory we develop is to provide a foundation on which to construct quantum natural numbers, which are the quantum analog of the classical counting numbers. It turns out that in a special class of models, there exists a 1-1 correspondence between the quantum natural numbers and bounded observables in quantum theory whose eigenvalues are (ordinary) natural numbers. This 1-1 correspondence is remarkably satisfying, and not only gives us great confidence in our quantum set theory, but indicates the naturalness of such models for quantum theory itself. We go on to develop a Peano-like arithmetic for these new "numbers," as well as consider some of its consequences. Finally, we conclude by summarizing our results, and discussing directions for future work.
The purpose of this dissertation is to construct a radically new type of mathematics whose underlying logic differs from the ordinary classical logic used in standard mathematics, and which we feel may be more natural for applications in quantum mechanics. Specifically, we begin by constructing a first order quantum logic, the development of which closely parallels that of ordinary (classical) first order logic --- the essential differences are in the nature of the logical axioms, which, in our construction, are motivated by quantum theory. After showing that the axiomatic first order logic we develop is sound and complete (with respect to a particular class of models), this logic is then used as a foundation on which to build (axiomatic) mathematical systems --- and we refer to the resulting new mathematics as "quantum mathematics." As noted above, the hope is that this form of mathematics is more natural than classical mathematics for the description of quantum systems, and will enable us to address some foundational aspects of quantum theory which are still troublesome --- e.g. the measurement problem --- as well as possibly even inform our thinking about quantum gravity.\n\nAfter constructing the underlying logic, we investigate properties of several mathematical systems --- e.g. axiom systems for abstract algebras, group theory, linear algebra, etc. --- in the presence of this quantum logic. In the process, we demonstrate that the resulting quantum mathematical systems have some strange, but very interesting features, which indicates a richness in the structure of mathematics that is classically inaccessible. Moreover, some of these features do indeed suggest possible applications to foundational questions in quantum theory. \n\nWe continue our investigation of quantum mathematics by constructing an axiomatic quantum set theory, which we show satisfies certain desirable criteria. Ultimately, we hope that such a set theory will lead to a foundation for quantum mathematics in a sense which parallels the foundational role of classical set theory in classical mathematics. One immediate application of the quantum set theory we develop is to provide a foundation on which to construct quantum natural numbers, which are the quantum analog of the classical counting numbers. It turns out that in a special class of models, there exists a 1-1 correspondence between the quantum natural numbers and bounded observables in quantum theory whose eigenvalues are (ordinary) natural numbers. This 1-1 correspondence is remarkably satisfying, and not only gives us great confidence in our quantum set theory, but indicates the naturalness of such models for quantum theory itself. We go on to develop a Peano-like arithmetic for these new "numbers," as well as consider some of its consequences. Finally, we conclude by summarizing our results, and discussing directions for future work.
Logical foundation for quantum theory is considered. I claim that quantum theory correctly represents Nature when mathematical physics embraces and indeed features, logical anomalies inherent in pure mathematics. This approach links undecidability in arithmetic with the logic of quantum experiments. The undecidablity occupies an algebraic environment which is the missing foundation for the 3-valued logic predicted by Hans Reichenbach, shown by him to resolve `causal anomalies' of quantum mechanics, such as: inconsistency between prepared and measured states, complementarity between pairs of observables, and the EPR-paradox of action at a distance. Arithmetic basic to mathematical physics, is presented formally as a logical system consisting of axioms and propositions. Of special interest are all propositions asserting the existence of particular numbers. All numbers satisfying the axioms permeate the arithmetic indistinguishably, but these logically partition into two distinct sets: numbers whose existence the axioms determine by proof, and numbers whose existence axioms cannot determine, being neither provable nor negatable. Failure of mathematical physics to incorporate this logical distinction is seen as reason for quantum theory being logically at odds with quantum experiments. Nature is interpreted as having rules isomorphic to the abovementioned axioms with these governing arithmetical combinations of necessary and possible values or effects in experiments. Soundness and Completeness theorems from mathematical logic emerge as profoundly fundamental principles for quantum theory, making good intuitive sense of the subject.
While it is widely acknowledged that quantum theory is strange, and that the world view it implies is elusive, there is considerably less consensus as to why this is the case. This article will explore the possibility that the theory may be opaque because the mathematical language it employs is inarticulate. To address this possibility, three interconnected mathematical theories will be developed, each specifically designed to handle aspects of quantum theory's subject matter. The first will describe dynamic systems in terms that are general enough to be applicable to any kind dynamics, including both classical and quantum. That description will then be used to construct a theory of experiments, with the aim of understanding what types of experiments scientific methodology allows. Finally, the theory of experiments will found a theory of experimental probabilities applicable to any collection of experiments, so long as their probabilities are internally consistent. The resulting over-arching mathematical language will then be applied to quantum theory. In the initial application, it will be used to explain the core nature and structure of quantum measurement and probability. Following that, it will be used to explore quantum theory's models. Like any theory written in a mathematical language, quantum theory derives its meaning from its class of models. To understand the class quantum models, two models of central importance will be described. The first will be a conceptually simple construction that follows almost immediately from the above theory of dynamics. It fully satisfies both quantum theory and the above mentioned theory of experimentation; additionally, its macroscopic objects obey classical mechanics. The second will utilize a minimal amount of mathematical structure to satisfy the same conditions.
Standard Boolean logic is based on ideas taken from classical, first-order set theory. The first-order logic of quantum mechanics introduced by von Neumann is not a classical one. It is therefore natural to ask whether it is possible to formulate the corresponding theory of quantum sets, their (quantum) topology and higher-order structures. Once we have quantum sets and quantum topologies we can try to quantize essentially anything: phase space, groups, spacetime, etc. In his book Finkelstein reports results of some 40 years of studies in this direction. The book is not addressed to specialists, but to `thoughtful students of physics, mathematics and philosophy', which may justify the fact that this review is written by a person who is not actively working in the field of quantum logic. From this layman's position I must say that the book does not read very well. First of all, the standard language of physics is not naturally suited for speaking about such problems, so Finkelstein invents a new one. Obviously, it is not easy to read a difficult text which, in addition, is written in a language we do not know. It seems that the author should have tried to depart as little as possible from the standard terminology, but he could not resist coining new terms even for things we already know under different names. When I first looked into the table of contents I felt both curiosity and some intimidation since I did not understand the words. To give an example, we find there spinors, cospinors, anti-spinors, anti-cospinors, ambi-spinors, ambi-cospinors, etc. It turned out that they are simply upper- and lower-index, primed or unprimed ordinary spinors, or their second-rank tensor products. Fortunately higher-rank spinors were not needed. The notation also does not help to understand the text. For example, on p 148 the symbol denotes a spacetime index, but three lines later, in equation (11), it is a gauge group parameter, and no comment is made on this. On the other hand, the same object is often denoted in different ways (see, for instance, equation (1) on p 44 and (7) on p 47) and if we add a number of typographical errors the first reading of the book is really tiring. A general philosophy advocated in the book is to use a kind of two-time description which replaces continuous dynamics with discrete algebraic `activity'. The construction begins with a bi-modal logic of initial and final actions and the preclusion (or `occlusion' ) relation ` ' (essentially iff a transition between and is forbidden). A Hilbert space and a field of coefficients are derived from . Discretization of time goes hand-in-hand with the idea that spacetime should be quantized and, finally, an outline of its construction is given. An intriguing consequence of the Finkelstein formalism is the appearance of internal local gauge groups of the standard model just from the spacetime structure. It seems that some results presented in the book are subject to limitations which are not mentioned by the author. For example, the construction based on will not work correctly if the dynamics between the initial and final acts is nonlinear. This is so because mutually precluding acts do not have to correspond to orthogonal vectors if we use nonlinear eigenvalue problems. So details of evolution between initial and final acts may be more important than one would expect naively. The same concerns the proof that transition probabilities must be of the form . Such subtleties can be ignored in standard quantum mechanics but here we aim at generalization. I find some parts of the text misleading. A canonical example illustrating mathematical and physical meanings of the Finkelstein formalism is a Malusian experiment (i.e. the one involving photons and polarizers), but the Bell theorem given on p 251 cannot distinguish between its single-photon and two-photon versions. Physically the two systems are essentially different and a purely classical Aerts model that gives the Malus law for a single particle is known, even though the theorems and discussion on pp 251 - 3 suggest that this is impossible. Perhaps the original work the theorems were taken from clarifies this point but a reader will not find it for two reasons. First, the name of its author is misprinted. Second, there is no reference to the paper. This is quite typical of the book; the results which are regarded as inessential from the viewpoint of the main argument are oversimplified and quoted without references to original works. The clarity of the presentation would be improved if the author did not try to briefly summarize what he knows about physics (and certainly he knows a lot) but, instead, were more detailed about the material that is really new. In spite of all these minor critical remarks I think the book may be inspiring and opens an entirely new perspective on physics. Many of the problems discussed by Finkelstein remain unsolved but the strategy he proposes looks at least promising and, hopefully, some readers will join the author in his brave attack on the most difficult questions of modern science.
Are there quantum objects? What are they if they are neither given nor resemble anything familiar? To answer these questions we have to abstract from the substantive features of familiar things, delineate the pure logical forms by which we acknowledge objects and show how the forms are fulfilled in quantum theories. We have to explicate, in general terms and without resorting to the given, what we mean by objects. The clear criteria will enable us to affirm the objectivity of quantum theories. Auyang 1995, p. 5 Introduction Cantor, famously, defined a set as ‘… collections into a whole of definite, distinct objects of our intuition or of our thought’ (Cantor 1955, p. 85, our emphasis). On this basis the standard formulations of set theory, and consequently much of contemporary mathematics, are erected. Reflecting upon this definition, and the underlying ontological presuppositions, the question immediately arises, how are we to treat, mathematically, collections of objects which are not distinct individuals? This question becomes particularly acute in the quantum mechanical context, of course. As Yu. I. Manin noted at the 1974 American Mathematical Society Congress on the Hilbert Problems, We should consider possibilities of developing a totally new language to speak about infinity [that is, axioms for set theory]. Classical critics of Cantor (Brouwer et al. ) argue that, say, the general choice is an illicit extrapolation of the finite case.
In this paper I shall outline a possibility for incorporating category theory in an essential way into the foundations of quantum theory. The role for category theory which I shall sketch here is strictly foundational in the sense that it offers no new perspectives for a physicist working within the quantum formalism, but rather provides a way of interpreting the algebraic structure by giving an alternative axiomatization of the underlying logic of the theory. The axiomatization is given without recourse to any classical set theory by invoking methods closely connected with the work of F. W. Lawvere. More precisely, then, a possible approach to the foundations of physics which axiomatizes its algebraic structure in a way which is independent of set theory and hence points, as it were, to a structuralist foundation of physical theories, will be proposed.
It has been argued, most notably by Putnam (1969, 1974) that socalled quantum logic (which we shall often refer to as. orthomodular logic (OML)) is the one true logic,-as has been shown by quantum mechanical experiments. Putnam also has strong universalist tendencies, which would require him to use the same logic in all domains of reasoning, including mathematics, and it has been urged, recently (as one horn of a dilemma) by Hellman (1981), that there might be some problems in working out the classical mathematics of the Hilbertspace foundations of.quantum mechanics in a quantum logic framework. In this paper we show (given one natural framework)(§3) first that if the first-order Peano arithmetic is formulated with quantum logic that it has the same theorems as the classical first-order Peano arithmetic. Distribution (for first-order arithmetical formulas) is a theorem not of quantum logic but rather of arithmetic.