The classification cannot be determined from a collection of individual things alone. For example, to classify apples, oranges, and dogs, we must specify what we are classifying them with respect to. Under the criterion ``Is it a fruit or not?'', they can be sorted into [apples, oranges] and [dogs]. Focusing on the indispensability of such a criterion, this paper formulates the true universals problem as the question: ``Should measurement (sorting, classification) be included among the principles of science?'' What is asked here is not whether measurement is used in practice, but whether measurement is made explicit as a basic structure of the theory. Observable System Theory (OST) is formulated on operator algebras. In the non-commutative case it is as powerful as quantum mechanics, and in the commutative case it is more powerful than statistics.Moreover, OST distinguishes the state of an object, the observable (the criterion of classification), and the measured value, and connects them by the Born principle. Furthermore, it treats changes of the state by the Causality principle. Plato's allegory of the cave and Aristotle's example of medical treatment are presented as examples of a description by a changing state and a fixed observable. Therefore, the universals problem asked in this paper is not about where universals exist, but about what theoretical status should be given to the criterion of classification. We claim that the true universals problem, which has remained open for 2400 years, has been solved by OST, which makes measurement explicit as a principle of science.
The classification cannot be determined from a collection of individual things alone. For example, to classify apples, oranges, and dogs, we must specify what we are classifying them with respect to. Under the criterion Is it a fruit or not?'', they can be sorted into [apples, oranges] and [dogs]. Focusing on the indispensability of such a criterion, this paper formulates the true universals problem as the question: Should measurement (sorting, classification) be included among the principles of science?'' What is asked here is not whether measurement is used in practice, but whether measurement is made explicit as a basic structure of the theory. Observable System Theory (OST) is formulated on operator algebras. In the non-commutative case it is as powerful as quantum mechanics, and in the commutative case it is more powerful than statistics.Moreover, OST distinguishes the state of an object, the observable (the criterion of classification), and the measured value, and connects them by the Born principle. Furthermore, it treats changes of the state by the Causality principle. Plato's allegory of the cave and Aristotle's example of medical treatment are presented as examples of a description by a changing state and a fixed observable. Therefore, the universals problem asked in this paper is not about where universals exist, but about what theoretical status should be given to the criterion of classification. We claim that the true universals problem, which has remained open for 2400 years, has been solved by OST, which makes measurement explicit as a principle of science.
Roughly a quarter of a century ago, the author proposed "Observable System Theory" (OST) as a foundational theory of systems theory. OST describes both statistics and quantum mechanics within a single framework of state–observable–measured value, governed by two explicitly formulated principles: the Born principle and the causal principle. This paper examines Bayesian networks (BNs) from the standpoint of OST, with particular attention to the following fundamental question: what, ultimately, do DAG calculations compute? The answer proposed here is that DAG calculations ultimately compute the probabilistic causality of measured values. Accordingly, OST admits two distinct notions of causality. The first is the causal evolution of states—equivalently, the causal evolution of observables in the Heisenberg picture—governed by OST's causal principle. The second is the probabilistic dependence structure among measured values, represented by a directed acyclic graph (DAG) together with local conditional probability distributions. The main result of this paper is the following fundamental theorem: for every finite Bayesian network specified by a DAG and local conditional probability distributions, there exists a corresponding measurement in OST whose outcome distribution coincides exactly with the joint probability distribution generated by the Bayesian network. We also aim to dispel the notion that theories involving multiple, distinct kinds of causal relation—such as Aristotle's doctrine of the four causes—are inherently untenable. Keywords: Observable System Theory; Bayesian network; directed acyclic graph; measurement; measured value; measurement probability.
Measurement in classical physics Measurements in classical physics are the means by which we come to know the current state of a mechanical system without appreciably disturbing it. The laws of classical mechanics imply that we may interact with the system in such a way that this ideal may be approached arbitrarily closely, at least in principle (i.e., in theory, ignoring practical problems). The aim of this chapter is to examine how far we must revise this programme in the quantum theory of measurement. Did the classical physicist believe in the objective existence of material systems with well-defined position and momentum just because these quantities could be empirically determined with arbitrary precision at the same time? It seems she made a stronger assumption than this; that two successive observations of an apple, first in a tree and then on the ground, are connected to one another by virtue of the fact that the apple actually has a definite position and momentum at each instant, whether it is observed or not. Thus, the programme of classical physics was not merely to predict the results of future experiments by inserting data from previous ones into the laws of motion, but to show how the observations are a function of a causally connected sequence of unique processes undergone by the system of interest, which possesses properties independently of measurements. Measurement was then something of a secondary affair and indeed, prior to quantum mechanics, the theory of measurement as a distinct branch of theoretical physics did not exist.
In classical mechanics, as we know, all observables commute with each other and therefore belong to the same “commuting set.” This means that – in principle – we can measure all observables simultaneously. Then, the state manifests itself as a collection of properties (of values of observables) (see Sec. 1.1), and as such it is observable. On the other hand, quantum-mechanical observables may not commute (see Subsec. 2.1.5). As a consequence, the state is not just a collection of observable values (see also Sec. 2.3) so that we may ask ourselves whether it is observable or not, and therefore raise the following question: What is the nature of the wave function? Is it simply a mathematical tool, i.e. does it only represent the sum of all that we can know about a system? Or does it have an ontological status? Since the early days of quantum mechanics, this question has been very popular. As a first attempt, de Broglie tried to interpret the wave function as describing, in classical terms, a wave field that envelopes a classical particle (see also Subsec. 16.3.2). This theory should predict the existence of empty waves, but several experiments have shown negative results (see also the end of Sec. 9.6). More recently, many physicists have tried to propose some methods in order to measure the wave function. Here we shortly report on some of these proposals and discuss to what extent the quantum state can be measured.
The newly revised edition of this well-known book categorizes prevalent views on measurement as described by quantum mechanics, discusses basics agreed to by many who on other issues disagree, and describes some of the problems. Although many paragraphs rest unchanged from the first (1991) edition, results found since 1991 are used to reorganize and sharpen the focus of the book, so that even the old paragraphs are now understood differently and more clearly. One advance in the second edition is a new and more general no-go theorem; another is separation of formal derivations from interpretation. The dozen or so definitions essential to following the book even casually are stated clearly. By this means the authors succeed pretty well in achieving their stated desire to make the book comprehensible to those who do not share their philosophic predispositions. While I differ in philosophy from the authors, attending to their discussion uncovered weaknesses in my own thinking and improved it. The mathematical language of quantum mechanics expresses a system under measurement by states (as rays in a Hilbert space) and a hamiltonian operator, distinct from the apparatus that produces outcomes, which is expressed by a positive operator-valued (POV) measure. In this way quantum language separates state preparation from state measurement. Often the Hilbert space is viewed as a tensor product of subspaces, allowing further conceptual separations. The authors take the stance that quantum mechanics calls for splitting the empirical world into parts, the most essential being (1) objective systems S (to be observed) and (2) apparatus A (preparation and registration devices that produces outcomes (or, as the authors write, definite pointer values) of the observed system). In some interpretations, one or both of an observer and an environment appear as additional parts. What is to be included in the system S versus the apparatus A is not specified by the language, but is open to the theoretician; a probe, for example, can be analysed as either part of the apparatus or as part of the system, as was discussed by von Neumann in work cited by the authors. With what should a theory of measurement be concerned? A broad division of approaches is defined by how one answers the question: should one try to do away with the cut between measuring apparatus and system? The authors phrase this as a choice of REFERENT, a term which in my mind puts them on shaky semantic ground; anyway The Quantum Theory of Measurement is concerned with implications of the `yes'. The book unfolds with the power of a tragedy, in which the logical consequences lead, step-by-step, to the statement and proof of theorem 6.2.1, which, roughly speaking, asserts the non-existence of a solution to the problem of getting definite pointer values from measuring apparatus that exhibits the superposition demanded by its description as a quantum system. Possible responses to this theorem are discussed, among them the many-worlds interpretation and a modification of quantum mechanics that elevates decoherence to a principle. Citations of the literature are thorough, and many points of view are crisply summarized. The book is indispensable to both those who work in the broad direction chosen by the authors and to those who wish to set any other direction in context.
The concept of measurement plays both a central and a problematic role in quantum mechanics. First of all, and unusually, the word ‘measurement’ figures in the fundamental axioms of quantum mechanics - in principle (3) of Chapter 2 for example - and in this quantum physics is quite unlike classical physics as a whole. For though a measurement made on a classical system naturally involves an interaction between the system and a measuring apparatus, there is nothing special according to classical physics about measurement interactions. Describing the measuring process is a straightforward problem of applied classical physics. The classical laws of motion do not break down during the measuring process. For example, suppose you observe a classical particle using light bounced off it. In the classical account you can make the disturbance on the observed system as small as you like and, ‘in principle', you can observe the system with arbitrary accuracy. ‘In principle’ - a favorite expression of the philosopher of physics - means what it always means in physics. The contrast is with ‘in practice'. Of course, in practice there are all sorts of limitations on measurement. There are also some theoretical limitations, like those due to thermodynamics. But these are not the limitations imposed by the theory we are considering, namely classical mechanics. Such limitations as there are are imposed by some other theory. (The expression ‘in principle’ tends to be used when we are discussing the principles of a particular theory and, theoretically but not in practice, physics is a collection of disjoint theories.)
Recently we proposed the linguistic Copenhagen interpretation of quantum mechanics, which is called quantum language or measurement theory. This theory is valid for both quantum and classical systems. Thus, we think that quantum language is one of the most powerful scientific theories, like statistics, and thus, it is the scientific completion (i.e., the destination) of dualistic idealism. If so, we can introduce the concept “progress” in the dualistic idealism. For example, we can assert that [Plato → Descartes → Kant → Wittgenstein → quantum language], where [“X” → “Y”] means that “Y” is more like quantum language than “X”. In this paper, we will study the problem of universals from the perspective of quantum language (i.e., from the scientific perspective of ignoring any religious perspective). And we can be confident of the progress of both of the two time series [Plato → Anselmus → Thomas Aquinas → quantum language] and [Descartes → Thomas Aquinas → quantum language] in dualistic idealism. The reader may find it surprising that Scholastic philosophy is more scientific than Cartesian-Kantian philosophy. However, this is because Descartes gave up the pursuit of “universals” and presented dualism as a visible “mind-matter dualism” so that it could be familiar to the general public. This made the Cartesian-Kantian philosophy socially successful, but unscientific. The problem of universals has not always been clear in the long history of philosophy. The reader should be convinced that the reason is that the problem of universals has been discussed in an incomplete non-scientific dualism to this day.