Theories of Meaningfulness

Theories of Meaningfulness

Theories of Meaningfulness

Theories of Meaningfulness

Synopsis

Written by one of the masters of the foundation of measurement, Louis Narens' new book thoroughly examines the basis for the measurement-theoretic concept of meaningfulness and presents a new theory about the role of numbers and invariance in science. The book associates with each portion of mathematical science a subject matter that the portion of science is intended to investigate or describe. It considers those quantitative or empirical assertions and relationships that belong to the subject matter to be meaningful (for that portion of science) and those that do not belong to be meaningless. The first two chapters of the Theories of Meaningfulness introduce meaningfulness concepts, their place in the history of science, and some of their traditional applications. The idea that meaningfulness will have different, but interrelated uses is then introduced. To provide formal descriptions of these, the author employs a powerful framework that incorporates pure mathematics, provides for qualitative objects and relations, and addresses the relationships between qualitative objects and pure mathematics. The framework is then applied to produce axiomatic theories of meaningfulness, including generalizations and a new foundation for the famous Erlanger Program of mathematics. The meaningfulness concept is further specialized with the introduction of intrinsicness, which deals with meaningful concepts and relations that are lawful and qualitativeness, which is concerned with qualitative concepts. The concept of empiricalness is then introduced to distinguish it from meaningfulness and qualitativeness. The failure to distinguish empiricalness from meaningfulness and qualitativeness has produced much confusion in the foundations of science literature and has generated many pseudo-controversies. This book suggests that many of these disappear when empiricalness is intersected with the other concepts to produce "meaningful and empirical relations," "empirical laws," and "qualitative and empirical concepts." A primary goal of this book is to show that the new theories of meaningfulness and intrinsicness developed in this book are not only descriptive but are also potent. Asserting that they do more than codify already existing concepts the book: *works out logical relationships between meaningfulness concepts that were previously unrecognized; *clarifies certain well-known and important debates by providing rich languages with new concepts and technical results (theorems) that yield insights into the debated issues and positions taken on them; and *provides new techniques and results in substantive scientific areas of inquiry. This book is about the role of mathematics in science. It will be useful to those concerned with the foundations of science in their respective fields. Various substantive examples from the behavioral sciences are presented.

Excerpt

Since ancient times, the usefulness, power, and certainty of mathematics have aroused wonder. Many of the best minds in western civilization have speculated about the distinctive character of mathematics and its relationship to science. But today many issues remain unresolved. Throughout this book, I explore theories about the relationship of a qualitative situation to mathematical models of the situation, with special emphasis on the qualitative significance of quantitative concepts and statements about the mathematical models. Such theories are called “theories of meaningfulness, ” and as is shown, related theories under various guises have appeared since the beginnings of mathematics and science.

The psychologist S. S. Stevens in the mid-twentieth century was probably the first person to systematically employ the term “meaningfulness” in the scientific literature. He was concerned with issues involving the matching of statistical procedures used to analyze empirical phenomena with the procedures of measurement used to quantify them. He called statistics that matched appropriately with the underlying scaling processes “meaningful” and those that did not “meaningless.” His concept was later generalized in several directions by other measurement theorists. These generalizations are discussed in detail in Chapters 2 and 5.

The meaningfulness theories developed in this book are related to the ones advanced by the measurement theorists. They are, however, motivated by a much broader set of concerns, and correspondingly are applicable to much wider sets of scientific and philosophical issues. They focus on a fundamental problem that is pervasive throughout mathematics and science: If certain concepts are “meaningful” (i.e., are “real, ” “intrinsic, ” “qualitative, ” or “empirical, ” etc.), then what other concepts are “meaningful” (“real, ” etc.)? Instances of this general schema of problems have appeared repeatedly throughout the development of mathematics and science, and the resolutions of some important ones have been the impetus for major intellectual revolutions. the following are examples of issues that gave rise to meaningfulness problems that can be cast in terms of the general schema: Pythagorism, aphilosophical tradition with ancient roots that maintains that reality is reducible to number; the longstanding problem of deciding what curves are geometrical; the . . .

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