We shall encounter in the course of our discussion of astronomy a number of concepts, such as the "geocentric" concept or the concept of "a spherical Earth", the concept of "the electron," the concept of "the photon," the concept of "a force," or the concept of "energy." Let us see if we can determine what is implied by such phrases. In doing so, our discussion of concepts, models, laws, theories-elements in the processes of science-in this and following discussion has been heavily shaped by Gerald Holton's and Stephan Brush's book (Introduction to Concepts and Theories in Physical Science, 2nd ed., Addison-Wesley, 1973). Their book, which we encourage you to consult for a more detail discussion, contains many references for additional reading on the conceptual foundations of physical science.
Meaning of Physical Concepts
It is clear historically that scientists limit themselves to certain types of observations and thought processes. Their use of concepts as a reference frame for understanding, their rules for fashioning conceptual schemes, and their type of argument for seeking agreement are unlike those of nonscience. Science has accumulated since the time of the ancient Greeks a set of internationally accepted and reasonably enduring concepts, which are its medium of exchange. To a large degree the secret of science's successful harmony and continuity lies in the nature of these concepts, their definitions, and their ready acceptance as the "coin of the realm."
To make concepts more definite, let us adopt for our discussion the following definition of a physical concept taken in part from Holton:
Physical Concept: A physical concept is a general idea, or notion, or understanding as derived from specific instances or occurrences in the natural world. In general, physical concepts can be defined operationally and most possess a quantitative nature.
The characteristics of concepts-possessing operational definitions and quantitative natures-actually do more to define the notion than does the first sentence. Each of these characteristics aid in the unambiguous communication of problems and results, make possible unambiguous agreement on facts and their interpretations, and knit together the efforts of many independent scientists, even those who are widely separated in subject, time, and place. The renown physicist Albert Einstein (1879-1955) noted that, "The only justification for concepts is that they serve to represent the complex of our experiences." Let us take each characteristic in turn and amplify its meaning.
Concepts from an Operational Point of View
Some physical concepts are neither intuitive nor unambiguously understandable. In such cases, scientists believe that they can still work with such concepts by being able to operationally define them. The first characteristic mentioned in the definition is then the operational meaning of a concept. By an operational definition we signify that:
Operational Definition: One can prescribe an operational procedure, that is, develop a prescribed series of actions, whereby the concept or physical quantity can be measured.
Ideally, each concept used in physical science can be clarified by some such operational definition, and that perhaps is the most important mechanism by which mutual understanding among scientists is possible. For clearly it is more difficult to misinterpret rigorously performed actions than it is words.
Several points need to be considered regarding operational definitions. The first is that to most people everyday notions seem clear and scientific terms mysterious. Such a belief is more a product of our greater familiarity with everyday notions than with scientific terms. A little thought, however, should show that the opposite is actually true. The words of daily life are usually so flexible in definition, so open to emotional coloring, that they lend themselves to a variety of interpretations and contextual meanings. For science, this will not do. Nonscientists generally find it difficult to get used to a highly specific vocabulary and to the apparently picayune insistence by scientists on its rigorous use. However with out a rigorously defined set of terms, meaningful communication in science is impossible.
Second, operational definitions must seem to be just human convention and they do not necessarily tell us what concepts like "length," or "force," really are. Perfectly true, but therein lies their great utility. For we may make laboratory measurements suffice for discussion purposes until such time that questions concerning the deeper significance or reality of a concept can profitably be addressed.
Third, errors associated with measurements in an operational definition do not have the connotation to scientists of wrong, mistaken, or sinful acts, which exist in everyday usage of the word. Errors of measurement are merely departures from exactness. But in time new technology may reduce these departures, so that scientists can at least strife for greater, if not achieve, exactness in their measurements.
Finally, all physical concepts are not of equal importance. Galileo Galilei (1564-1642) pointed out that some concepts are more directly observable, and therefore can play a primary role in science. Other concepts which do not readily lend themselves to being defined operationally must play secondary roles.
Quantitative Nature of Concepts
The second characteristic of a meaningful concept is that most are quantitative in nature. The demand for quantitative concepts rests on an article of faith that is as ancient as it is revered. This article of faith is the belief that nature works according to mathematical laws and that observations are explained when we find the mathematical law relating observable quantities. Quoting Galileo, "Philosophy [i.e., science] is written in that great book which ever lies before our eyes-I mean the Universe-but we cannot understand it if we do not learn the language and grasp the symbols in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles, and other geometrical figures [i.e., mathematical schemes] without whose help it is impossible to comprehend a single word of it, without which one wanders in vain through a dark labyrinth."
To Galileo and his successors, mathematics provides the technique par excellence for comprehending and seeing the unity that exists in nature first perceived by the Greeks through geometry. On a practical level, understanding and manipulating concepts is made possible by the fact that mathematically formulated ideas can be expressed symbolically in equations. Thus mathematics is both a tool for science and a universal language through which scientists communicate.
Because of the quantitative nature of concepts, the important thing for scientists about various experimental apparatus, such as weights, springs, or meter sticks, is no longer their compositions, their colors, or their histories outside of the laboratory. Since all that really matters to scientists are those mathematical relationships involving experimental objects, then the objects cease to be individual entities and they become in the minds of the investigator mathematically-ideal bodies. Their real weight is totally forgotten, and in its place we envision "a point mass." In this sense, science does not deal directly with "real bodies" but with abstractions that exist and move in a hypothetical space of precise mathematical properties. In this mathematical world of the mind, scientists can manipulate the experiment at will, eliminating in their minds all air resistance, regarding the inclined plane as perfectly smooth, or changing one aspect and leaving all other factors untouched. As seen from the outside, this idealized world, rendered in the esoteric language of mathematics and filled with simplifications and exaggerations, is quite analogous to that of the modern painter, poet, or composer. To further our understanding, let us define those concepts which are mathematical ones as:
Mathematical Concepts: Mathematical concepts are those concepts taken from a mathematicised world of ideal and precise objects, and it is a realm in which actions exist as calculations. Such concepts are often justifiable by empirical evidence, but some can not be and they find justification as coherent parts of a fruitful theory. Mathematical concepts are taken seriously only insofar as they yield new understanding of the world of our experiences.
To many scholars mathematics has moved beyond being a tool and a language in science. Mathematical equations have become a way of knowing the world that is unavailable to us in any other way. Equations possess a reality that defies language and visual images. And, they can be just as real, if not more so, in the mind of the scientist as sensual experience.
Especially useful concepts recur in a great many descriptions and laws, often in areas far removed from the context in which they were initially formulated. As you progress through the book, keep in mind our comments on the characteristics of physical concepts, for they will help you to appreciate the outlook astronomers bring to their effort in trying to understand the cosmos.