## 3.2 Introduction

The common objective in scientific research, whatever our particular field of interest, is to discover those exact relationships that exist among the measurable quantities of nature. And typically, when our perceptions of those relationships are confirmed by experiment to have sufficient generality, we look upon our perceptions as natural laws. Although we often accept apparent truths about the natural universe that are difficult to quantify (Darwin’s thesis of evolution, for example), the absolutely necessary condition for “exactness” in a scientific law is that its formulation consist of measurable quantities whose definitions depend, in turn, entirely upon a set of dimensions. In this formal sense, such classical sciences as physics and chemistry–as well as much of modern biology–are said to be exact, but obviously, some topics of natural science (such as Darwin’s thesis of evolution) are characteristically inexact.

Dimensional formalities do not, of course, insure scientific infallibility. In fact, dimensionally perfect formulae sometimes turn out to be irrelevant, inaccurate, or erroneous. However, dimensional completeness fulfills the necessary condition of scientific exactness, and we cannot proceed to the more practical concerns of scientific sufficiency until we satisfy dimensional exactitude in our formulae. A scientifically sufficient formulae that is not scientifically exact is meaningless. We shall be concerned here almost exclusively with the dimensional completeness of scientific statements and only incidentally with sufficiencies. Although dimensional methods provide us with the means for determining the formally necessary relationships among the quantities of a scientific formula, we should clearly understand that experimentation, observation, empirical evidence, and insight really determine just which quantities go into the formula in the first place.

So as to distinguish between differing magnitudes of a particular dimension, we must have access to a scale (or unit) of comparison. In saying “the height of that tree is nine meters” we imply that we have chosen length as the dimension of definition and the meter as the corresponding scale unit of comparison. In making the observation let us also note that “length” might have defined our concept of tallness but not necessarily our whole notion of the tree itself. This is not a trivial acknowledgment; it points up the fact that scientific definitions describe attributes of natural entities or occurrences, not their ontological meanings or their elemental reasons for existence. While we might employ the universal law of gravitation to quantify the attribute of attraction between bodies of mass, the law of gravitation consists wholly of dimensional relationships and tells us nothing of the “essence” of gravity, nor why every corporeal body in the universe should have a gravitational field in the first place. Although we may share an intense curiosity about such questions, their answers are usually reserved to inquiries external to science proper. Should perplexity exist over the scientific description of some natural quantity, its origin should be sought in the quantity’s dimensional definition.

In view of these definitional objectives, we must accord to every kind of natural quantity its defining dimension, but these defining dimensions need not be wholly independent of one another. Since we desire that differing natural quantities be related through natural laws, then for simplicity and clarity of definition we should keep the number of independent dimensions to a minimum. This minimum set of independent dimensions we shall call fundamental, and all other dimensions derived from the fundamental set we shall call derived. Similarly, the magnitudes (the scale units) corresponding to the fundamental dimensions are viewed as fundamental magnitudes, and those corresponding to the derived dimensions as derived magnitudes. An obvious example of a derived quantity would be that of velocity–the velocity, say, of an aircraft whose pilot reads derived units of velocity magnitude directly from the scale of his airspeed indicator. Irrespective of the particular scale system on the airspeed indicator, we perceive the velocity dimension to be derived, in general, from the more fundamental set of dimensions, length and time, and with corresponding symbols $$V$$, $$L$$ ,$$T$$ we can write that general dimensional relationship as

$[V] \equiv [\frac{L}{T}]$

read as “the dimension $$V$$ is defined as the ratio of dimensions $$L$$ and $$T$$.” The particular scale measurement of, say, 500 kilometers per hour would correspond to 500 derived units of velocity magnitudes derived from the kilometer and the hour. We would write the scale relationship for this particular example as

$v = 500 \frac{km}{hr}$

with an obvious correspondence between fundamental unit magnitudes “km” and “hr” and the fundamental dimensions $$L$$ and $$T$$. The unit symbol “$$\frac{km}{hr}$$” is as much a part of the value of quantity $$v$$ as is the numerical value “500”. The conventions (the symbolic styles) employed in this example suggest the convenient fact that we can manipulate dimensional and scale symbols as algebraic entities.

There is a certain arbitrariness of choice in the number and nature of the “fundamental” dimensions, whether we are thinking of our needs in choosing dimensions for actual measurement purposes or for the more formalistic requirements of theoretical models (and there is also considerable latitude in the selection of the scale units we might regard as fundamental; those suited for one problem may not be suited for another). Because of the strong historical precedent of Newtonian mechanics, the three natural concepts of space, time, and mass are often regarded as being fundamental in the absolute sense, but, in fact, more than three fundamental dimensions are sometimes desirable, and we shall see that differing sets of fundamental dimensions can be selected, none of which need be regarded as absolutely fundamental.