The Theory of Fundamental Measurement

Formalization of Measurement

In this chapter, we introduce the theory of fundamental measurement and the theory of derived measurement, and study the uniqueness of fundamental and derived measures. Fundamental measurement deals with the measurement process that takes place at an early stage of scientific development, when some fundamental measures are first defined. Derived measurement takes place later, when new measures are defined in terms of others previously developed. In this section, we shall' begin with fundamental measurement. Derived measurement will be treated in Section 2.5. Our approach to measurement follows those of Scott and Suppes [1958], Suppes and Zinnes [1963], Pfanzagl [1968], and Krantz et al. [1971].

Russell [1938, p. 176] defines measurement as follows: “Measurement of magnitudes is, in its most general sense, any method by which a unique and reciprocal correspondence is established between all or some of the magnitudes of a kind and all or some of the numbers, integral, rational, or real as the case may be.” Campbell [1938, p. 126] says that measurement is “the assignment of numerals to represent properties of material systems other than number, in virtue of the laws governing these properties.” To Stevens [1951, p. 22], “measurement is the assignment of numerals to objects or events according to rules.” Torgerson [1958, p. 14] says that “measurement of a property … involves the assignment of numbers to systems to represent that property.”

Introduction

The results in Chapter 7 indicate that a general study of partition identities such as the Rogers–Ramanujan identities or the Göllnitz–Gordon identities should help to illuminate these appealing but seemingly unmotivated theorems. In this chapter we shall undertake the foundations of this study. As will become abundantly clear, there are very few truly satisfactory answers to the questions that we shall examine. We shall instead have to settle for partial answers. After presenting the fundamental structure of such problems in the next section, we devote Section 8.3 to “partition ideals of order 1,” a topic which we can handle adequately and which suggests the type of answers we would like for our general questions of Section 8.2. The final section of the chapter describes a large class of partition problems wherein the related generating function satisfies a linear homogeneous q-difference equation with polynomial coefficients. In some ways this final section is unsatisfactory, in that the theory of q-difference equations has not been adequately developed to provide answers generally to questions about partition identities and partition asymptotics; however, the theorems of Section 8.4 do suggest that q-difference equations are indeed worthy of future research.

Foundations

We begin with a simple intuitive observation which forms the basis of our work here. In all the partition identities considered in Chapter 7 (Theorem 7.5, Corollaries 7.6 and 7.7, Theorem 7.11) partition functions were considered that enumerated partitions lying in some subset C of the set of all partitions.

Incidence Matrices

Matrices all of whose entries are either 0 or 1—that is, (0, 1)-matrices—play an important part in linear algebra, combinatorics, and graph theory. In some of these applications it is at times preferable to consider 1 as the “all” element in a Boolean algebra, or the identity element in a field of two elements. In what follows, however, the symbol 1 will represent the positive integer 1, since we shall be mainly concerned with enumerations of systems of distinct representatives and with related problems in the theory of permanents.

Many problems in the theory of nonnegative matrices depend only on the distribution of zero entries. In such cases the relevant property of each entry is whether it is zero or nonzero, and the problem can be often simplified by substituting for the given matrix the (0, 1)-matrix with exactly the same zero pattern.

Definition 1.1. Two m×n matrices A=(aij) and B = (bij) are said to have the same zero pattern if aij = 0 implies bij = 0, and vice versa.

Suppose that A, B, C, and D are nonnegative n-square matrices, and that A has the same zero pattern as B, and C has the same zero pattern as D. Then clearly A + C has the same zero pattern as B + D, and AC has the same zero pattern as BC.

Permanents made their first appearance in 1812 in the famous memoirs of Binet and Cauchy. Since then 155 other mathematicians contributed 301 publications to the subject, more than three-quarters of which appeared in the last 19 years. The present monograph is an outcome of this remarkable re-awakening of interest in the permanent function.

The purpose of the book is to give a complete account of the theory of permanents, their history and applications, in a form accessible not only to mathematicians but also to workers in various applied fields, and to students of pure and applied mathematics. Here is the first complete account of the theory of permanents. It is a survey in the style of MacDuffy The Theory of Matrices and of A Survey of Matrix Theory and Matrix Inequalities, by Marcus and Minc. However, it differs from both works in several respects: the style is more leisurely, the proportion of theorems proved in the text is higher, and the scope is wider—the volume covers virtually the whole of the subject, a feature that no survey of the theory of matrices can even attempt. Apart from many theorems proved in detail, there are numerous results stated without proof. Due to limitation of space, not every known result could be mentioned in the text. The choice of the theorems included in the book reflects, of course, the author's predilections.

A large body of mathematics consists of facts that can be presented and described much like any other natural phenomenon. These facts, at times explicitly brought out as theorems, at other times concealed within a proof, make up most of the applications of mathematics, and are the most likely to survive changes of style and of interest.

This ENCYCLOPEDIA will attempt to present the factual body of all mathematics. Clarity of exposition, accessibility to the non-specialist, and a thorough bibliography are required of each author. Volumes will appear in no particular order, but will be organized into sections, each one comprising a recognizable branch of present-day mathematics. Numbers of volumes and sections will be reconsidered as times and needs change.

It is hoped that this enterprise will make mathematics more widely used where it is needed, and more accessible in fields in which it can be applied but where it has not yet penetrated because of insufficient information.

A large body of mathematics consists of facts that can be presented and described much like any other natural phenomenon. These facts, at times explicitly brought out as theorems, at other times concealed within a proof, make up most of the applications of mathematics, and are the most likely to survive changes of style and of interest.

This ENCYCLOPEDIA will attempt to present the factual body of all mathematics. Clarity of exposition, accessibility to the non-specialist, and a thorough bibliography are required of each author. Volumes will appear in no particular order, but will be organized into sections, each one comprising a recognizable branch of present-day mathematics. Numbers of volumes and sections will be reconsidered as times and needs change.

It is hoped that this enterprise will make mathematics more widely used where it is needed, and more accessible in fields in which it can be applied but where it has not yet penetrated because of insufficient information.

The theory of partitions is one of the very few branches of mathematics that can be appreciated by anyone who is endowed with little more than a lively interest in the subject. Its applications are found wherever discrete objects are to be counted or classified, whether in the molecular and the atomic studies of matter, in the theory of numbers, or in combinatorial problems from all sources.

Professor Andrews has written the first thorough survey of this many sided field. The specialist will consult it for the more recondite results, the student will be challenged by many a deceptively simple fact, and the applied scientist may locate in it the missing identity he needs to organize his data.