No
physicist or engineer ever solves a real problem. Instead she creates a
model
of the real problem and solves this model problem. This model must
satisfy two
requirements: it must be simple enough to be solvable, and it must be
realistic
enough to be useful; ie., it must be both conceptually
understandable and
empirically fruitful.
The theories and "laws" of physics are also models. Whether in the solving of a particular engineering problem or in the search for the wide ranging laws of physics, the art of scientific analysis consists in the creation of useful models of reality. The model is the interface between reality and the human mind. As such, the model must be expressed in human terms; it is cast in terms of concepts which we create from the data of our experience. Our models speak as much about us, our experience and our modes of thought as they do about the external reality being modeled.
I prefer to speak of models where others might speak of theories because the word "model" emphasizes the criterion of usefulness. We tend to think of a theory as a candidate for some absolute, objective truth; a model is used to convey useful information without the pretense of being unique, complete or ultimate. As an example of the conception, gestation, birth and growth of a model in physics let us consider the history of the "ideal gas law" (PV=RT), which you undoubtedly have studied in your chemistry classes.
Despite
the voluminous abstractions of ancient philosophers, no useful
understanding of
gas behavior emerged throughout ancient history and the middle ages.
The
possibility of a useful model awaited the creation of the thermometer
and the
manometer. Each of these devices uses a thread of mercury imbedded in
glass in
order to generate a number (the length of the mercury thread) which
varies in
value as the device is subjected to varying conditions. Boyle, Charles
and Gay Lussac
investigated the behavior of these devices when
connected to a gas under controlled conditions.
To collapse a very long story, their experimentation resulted in the creation of the empirical relation PV = RT, the variables P and T representing the readings of the manometer and thermometer respectively; R is a constant for a fixed quantity of gas. If we then define P, V and T to be measurements of properties of the gas, PV = RT becomes a useful model of the gas behavior, even though P and T, at this point, have no deeper meaning other than the numbers generated by the specified devices.
That there should exist any (let alone such a simple) relation among the numbers generated by these (or any other) devices is not at all to be expected; such serendipity can only be gratefully contemplated when it appears. It is an instance of the profound meaning in one of Einstein's most famous quotations: "The most incomprehensible thing about the world is that it is comprehensible."
The creation of the model PV = RT was a giant leap forward; and note that the crucial beginning step consisted in the free creation of a set of concepts in terms of which meaningful questions might be put to nature so that nature might respond in a meaningful way. These concepts are not lying in nature awaiting discovery by some passive act of looking; they must be actively created. This is how the properties of matter come to be. This is how we define into existence those measurable properties of reality which we find to be useful. They are human constructs in terms of which we might ask meaningful questions, read nature's answers and organize our understanding into useful and testable models.
Each of these concepts is quantitative in nature: the number generated by a measuring device. Our empirical gas law is simply a relation (and very useful) among the numbers (P,V,T) generated by our measuring devices; it is an empirical model. The numbers generated by measuring devices have no deeper meaning except within the context of a conceptual model of the system being measured and its effect upon the measuring devices.
Boyle did his experimentation in the 1600's, while the pilgrims were colonizing America. It was not until the mid 1800's, while Americans were fighting over slavery, that Joule brought together the theories (models) of Newtonian mechanics and atomism (then hotly contested) to create a conceptual model of the ideal gas as a system of randomly moving point particles. In this model P is quite naturally associated with the Newtonian force concept and accounts for the behavior of the mercury manometer. However there is no a priori mechanical association for the empirical quantity T, the "temperature" of the gas as generated by the thermometer.
Herein lies a wonderfully simple instance of the incredibly awesome power of an empirically based analytical science: the fruitful interaction of experimental and theoretical physics. Newton's laws drove Joule's conceptual model to a very illuminating result: the numerical value of the product PV for Joule's gas is proportional to the total kinetic energy of the randomly moving gas particles. Thus Joule's conceptual model bestows upon the empirical temperature T, in PV = RT, a deeper meaning as a humanly invented property of the gas; it becomes a measure of the energy of random motion of the gas particles.
Thus
it is that the mathematical model PV = RT has
foundations as both an empirical
model and a conceptual model. I present it as a
paradigm to illustrate
the properties of the model in physics:
1) It is a human construct, the offspring of both our experience and our imagination.
2) It is quantitative and speaks of freely defined, measurable properties of matter.
3) It has both an empirical and a conceptual usefulness: it presents a testable numerical equality involving the numbers generated by specified measuring devices, and it offers a conceptual framework for associating a deeper meaning with these numbers.
4) The empirical usefulness of a model is a matter of experimental verification, and once verified this usefulness will remain; future models of a wider scope will include it as a special case.
5) The conceptual usefulness of a model can be a cultural matter, a matter of institutional and personal taste (more of this later).
Our
conceptual models are of course produced from the data of our
experience. Every
now and then I close my eyes and carefully feel an object such as a
piece of
fruit, a table or my own face, and try to imagine what it might be like
to have
never had the sense of sight. What sort of conceptual models might I
fashion as
I explore reality using only the sense of touch? (Try to form the
concept of
the shape of an object without invoking a visual
image.) How could I
appreciate the language of a sighted person? There is no way that a
sighted
person could convey to me his conscious experience of light vs.
darkness, let
alone red vs. green. Our conceptual models could communicate only
through shaky
analogies and metaphors, but our empirical models could unambiguously
communicate regarding the numbers generated by measuring devices.
Conceptual models are observer dependent and observer limited. As the physicist probes into the behavior of reality she strives to create meaningful conceptual models of that reality, using as raw materials the concepts fashioned from human experience. As she probes deeper she finds that she has to become ever more creative and imaginative, generating abstractions and cross fertilizations of her ideas in order to conceptually model the behavior of reality in human terms.
There is no reason to expect that this process can be extended indefinitely. It seems reasonable to anticipate that beyond a certain level of analysis the behavior of reality cannot be conceptually modeled in literal human terms, even though we may continue to be clever enough to create numerical equalities involving the readings of our instruments. After all, our instruments operate on the same superficial level as our senses.
We are already on the doorstep of this conceptual barrier. The mathematical models of quantum theory defied even the imagination of Albert Einstein; he was never able to conceive a satisfactory conceptual model of the reality behind these equations. As regards creative "weirdness", modern art and music are poor seconds to modern physics, even though the arts operate completely free of any constraints, whereas physics operates under the severe constraint of empirical usefulness!
Suppose
that you are shipwrecked on a desert island and, with nothing better to
do,
decide to create the science of physics from scratch. You decide that
your
first task will be to choose (or design) standards for your
measurements of
space and time intervals. How should you choose a standard measuring
rod and a
standard clock? This is a "catch 22" question: one would like to have
these standards available a priori, so that one can
perform experiments
(both physical and gedanken)
to ask questions of
nature, read her answers and be guided toward a theory about the
behavior of
matter. Yet one's choices of a standard clock and measuring rod already
presuppose considerable understanding about the behavior of matter! For
example, the choice of a standard clock already presupposes a theory
which will
be committed to the conclusion that this particular mechanism ticks at
a
constant rate. Logical consistency will force the theory to this
conclusion.
Choices among theories and choices among standards are inextricably
intertwined.
The dilemma exposed in the above paragraph is not debilitating; we need only replace the word "theory" (a candidate for an absolute, objective truth) with the word "model" (a useful way of describing reality in human terms). In this view, the choice of a clock simply defines into existence a measurable parameter "t" which will be used as a linear time base for the description of the evolution of phenomena. We will be comparing the course of all other phenomena to the succession of ticks of this clock.
Clearly the choice of standards is a matter of free definition. The criterion is not one of truth; it is simply one of usefulness: which choices lead to the most "desirable" empirical and conceptual models of reality? Put another way: how "weird" does the conceptual model have to get in order to be empirically useful? The words "desirable" and "weird" must be defined by you and/or current scientific culture; they are a matter of taste. Historically, and logically, this is an iterative process, as we see more and more details of where the model is leading.
Let me tease you with a famous example (which hopefully you will study in detail later): Einstein, in his 1905 relativity theory, was the first to capitalize upon this freedom of choice (of rods and clocks) in a radical way. His definitions of "desirable" and "weird" were not mainstream. To him the desirable model must preserve the invariance (sameness) of physical law (in particular Maxwell's equations of electrodynamics) for all non-accelerated observers. But conventional wisdom said that the velocities appearing in Maxwell's equations must be measured from an absolute frame of reference (the "aether" frame). This was "desirable" to many; they found it satisfying that the laws of physics should be simple only to an observer at absolute rest. In fact, any deviation of your experimental results from the laws of physics would then furnish you with sufficient data to measure your own absolute velocity. They had been disappointed that Newton's model of mechanics did not allow us to measure our absolute velocity by mechanical experiments (Newton himself must have been disappointed). They were overjoyed that now Maxwell's model of electrodynamics (which includes light) would allow us to measure our absolute velocity using optical experiments.
Einstein conceived a completely different conceptual model for Maxwell's electrodynamics. He sought a model in which these equations could be used with equal validity by all non-accelerated observers, each using the numerical values of all quantities (eg., velocities) as measured from her frame. He dared to redefine the measurement of space and time intervals to make this so. It is to be expected that such a redefinition would force new and worse weirdities into the model; we surely should expect that we will have to design new clocks and measuring rods, with exotic "relativistic" properties. The remarkable result has been that the new weirdities were only cultural, that ordinary clocks and meter sticks behave relativisticaly, and that a vast scope of phenomena have become more simply describable, even phenomena far removed from Maxwell's equations. Widespread acceptance did not come quickly or easily, but today relativity is not only accepted as empirically and conceptually useful; it has become beautiful!
The search for beauty in our conceptual models has always been a driving force and sometimes, as with Einstein's relativity, it seems to have been the sole motivation. Today many, like Einstein, are disappointed in their search for intuitive beauty in the quantum aspects of modern physics. Unlike relativity, the beauty of quantum theory still eludes visceral human appreciation. Perhaps with time we can acquire a taste, but it must begin with an adjustment of our expectations, toward models rather than "theories". Physics does not offer any quieting and ultimate answers.
Physics
has not been idle; there is much for you to learn. To learn means to
make your
own; it is an active process which only begins with
listening and
reading. You must return often to listening and reading, but meaningful
learning comes only from contemplation. Each person must construct his
own
models and his own philosophy of what physics is; these will grow and
develop;
construction is never complete. What I have said here is subject to
criticism
by scientists, philosophers, students and even myself,
as my appreciation of physics continues to develop. These words should
be taken
as providing only a beginning for discussion and contemplation. I have
tried to
express my current philosophy to you; over the years you will build
your own
unique and personal version. Even more than the appreciation of a
symphony or
painting, the understanding of physics is a unique and personal
encounter of a
consciousness with reality.
-Bob Sciamanda, Edinboro
Univ
of PA treborsci@verizon.net