Fundamentals of Measurement

 

Fundamentals of Measurement and
Representation of Natural Systems

by Robert Rosen


 


Ordering Details

(1978) Elsevier North-Holland; ISBN 0444002618 [Amazon] [search used]

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From the Introduction

“The point of departure is the measurement problem, as it appears in physics; the manner in which measurements allow us to characterize subsystems; the role of such subsystems as tools in system analysis; and the relationships existing between different ways of perceiving or interacting with the same system. Our conclusions are: (1) there exists no universal family of of analytic units appropriate for the treatment of all interactions; (2) there are on the contrary many such families of analytic units, all of which are equally “real” and entitled to be treated on the same footing; (3) the appropriate use of natural interactions can enormously extend the class of physical observables accessible to us; (4) the concept of a model must be formulated, in its most general terms, as the sharing of a subsystem by two otherwise distinct systems, capable of imposing the same dynamic on an appropriate system with which they can both interact. We establish these results through a variety of terminologies which turn out to be equivalent: stability, invariance, symmetry, homeostasis.”

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Chapter Excerpts

Chapter 1:    Mathematical Background

“…we will consider the relation between an equivalence relation R imposed on a set S, and an automorphism T on S; i.e., a one-one mapping of S onto itself. These automorphisms will be very important to us later; they will be variously interpreted as building blocks for dynamics, as symmetry elements, and as perturbations.”

Chapter 2:     The Basic Formalism

    “It is essential to realize at this point that the formalism to be developed, although we cast it initially in the framework of natural systems, is in fact applicable to any situation in which a class of objects is associated with real numbers, or in fact classified or indexed by the elements of any set whatever. It is thus applicable to any situation in which classification, or recognition, or discrimination is involved; indeed, one of the aims of our formalism is to point up the essential equivalence of the measurement problem in physics with all types of of recognition or classification mechanisms based on observable properties of the objects being recognized or classified.”

Chapter 3:    Meters and Dynamics

“Let us consider, for example, the situation in physics. A state description of a mechanical system, such as a wave function, is supposed to contain in it the answer to every question which can meaningfully be asked about the system. Yet, given a pair of systems, each of which is individually completely specified from the point of view of its own state description, we cannot specify the manner in which the two systems will interact and what shall constitute a state description of the composite system. This simple fact makes it clear that the state descriptions which are conventionally used for systems in isolation can not possibly be complete, for they do not allow us to answer physically meaningful questions about the systems in terms of such descriptions alone.”

Chapter 4:    Dynamics and Linkage

    “We shall argue here that the principal reason for the difficulties associated with emergent phenomena lie precisely in the tacit assumption that it is appropriate to describe a natural system by a single set of states and accordingly to constrain all dynamics imposed on that system in such a way that the trajectories lie entirely in that set of states. Stated another way, we shall argue that the dynamical formalism we have developed above, in which interactions result in trajectories which pull states out of the set in which they originally lie, automatically carries with it the capability to discuss emergent phenomena in a natural way.

    The characteristic feature of emergent novelty is the need to pass to a new mode of system description after the emergence has occurred. Such a new mode of description is typically characterized by quite different observables than those appearing in the prior description and/or by new linkage relationships between previously defined observables. In all dynamical theories in which sets of states of a system are posited, there is simply no visible source for such new observables; this has always been the basic difficulty.”

Chapter 5:    The Analysis of Dynamics

“In this chapter, we are going to collect together many of the concepts and ideas developed previously for the purpose of determining a set of admissible procedures in terms of which unknown natural systems which can interact with each other (i.e., impose dynamics) and with which we can interact through a set of meters may be best analyzed. Our thrust will be that there is no one mode of analysis which is universally valid for all systems and for all interactions, but rather that each mode of interaction, and each set of meters chosen for observing it, determines a set of corresponding “natural” analytic units. Results of this type are of great significance in physics and biology, for they directly contradict the prevailing reductionistic ideas that there do exist universal analytic units out of which all natural systems can be analyzed.”

Chapter 6:    Symmetry

    “Thus, symmetry considerations are not absolute, but are contingent upon the choice of description in which they are visible; the absolute character symmetry arguments appear to possess in mathematics and physics depends upon a tacit prior choice of the description in which they are visible. By explicitly recognizing that this description is a matter of choice, we on the one hand relativize the concept of symmetry, but on the other hand we can greatly extend its scope and significance.”

Chapter 7:    Similarity in Physics and Biology

“Algebraic topology, by its very nature, involves the construction of algebraic models of geometric objects; the relation of such an algebraic model (e.g., a homology group) to the space it models is specified by a functor between appropriate categories. Likewise, the assertion that two such models are equivalent to each other, in the sense that every property of one of them can be translated into a property of the other, is the essence of the concept of natural equivalence. The point we wish to stress here is that all concepts of of similarity, modelling, observation and recognition naturally fall into this same category-theoretic framework, and that therefore this framework must ultimately play as significant a role in the natural sciences as it had heretofore played in mathematics..”

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