CHAPTER I - FUNDAMENTALS OF DISCRETE MODEL THEORY
To deny the concept of infinity is as un-mathematical as it is un-American. Yet, it is precisely a form of such mathematical heresy upon which discrete model theory is built. There are a variety of compelling reasons why this basic concept of both classical and modern mathematical analysis should be considered suspect. Geometrically, the neophyte may be amused by the surface generated by rotating y = 1/x, 1 < x , around the x-axis, since the resulting ģinfinite hornī has a finite volume, but an infinite surface area. It is often said, and, indeed, rather loosely, that one can fill the inside of this surface with a finite amount of paint, yet this quantity of paint is not sufficient to paint the outside. But much more disturbing examples than this exist, for it is possible to describe three-dimensional solids whose volumes are arbitrarily large, but whose surface areas are arbitrarily small (Besicovitch) .
Physically, the concept of infinity has no known counterpart,
since, for example, all solids and particles are believed to be finite
from both the quantitative and qualitative points of view. It is unfortunate
that so many scientists have been conditioned to believe that, say, 10^30
particles can always be approximated well by an infinite number
of points. For, indeed, to approximate a physical particle by a mathematical
point is to neglect the rich structures of the atom and the molecule, while
to approximate 10^30 objects of any type by an infinite number of
such objects is to have enlarged the given set by so great an amount that
the given objects are entirely negligible in the enlarged set, or,
more precisely, in any nondegenerate interval of real numbers 10^30 points
form a set of measure zero (Halmos). Also, from the physical point of view,
as models become more and more sophisticated, the study of dynamical behavior
by means of continuous mathematical models invariably requires the solution
of nonlinear differential equations, and despite the unbelievable volume
of both classical and modern mathematics which exists, no general methods
have ever emerged for solving such equations. Finally, from the physical
point of view, it is worth noting that the concept of infinity has led
to paradoxical results like those related by Zeno as early as 300
B.C. These paradoxes have never been resolved in the sense that the fundamental
problems have merely been shifted to more subtle, less obvious problems
related to the convergence of infinite series (Whitehead).
From the modern computer point of view, the concept of infinity
is completely foreign. Digital computers have largest numbers, smallest
numbers, a finite number of numbers, and only numbers with a finite number
of decimal places. The output from these computers is finite in every sense.
But, further, if one examines the application of computers to approximating
solutions of nonlinear differential equations by numerical methods, which
is one of the most successful areas of computer applications, it is the
concept of infinity, manifested in continuity, which demands the
usually impossible task of proving that a numerical solution converges
to an analytical solution as a grid size converges to zero (see, e.g.,
Greenspan (6) , Henrici, Moore, Richtmyer and Morton, and Urabe).
From a philosophical point of view, the importance of nonlinear
behavior and the application of digital computers to the study of such
behavior have also created reasons for denying the concept of infinity.
As shown in Figure 1.1, it is usual, first, in the development of scientific
Discrete Data From
Discrete Data From
Scientific Experiment ------> Model -------> Numerical Methods
to have experimentation, which results in discrete sets of data. Theoreticians
then analyze these data and, in the classical spirit, infer continuous
models. Should the equations of these models be nonlinear, these would
be solved today on computers by numerical methods, which results again
in discrete data. Philosophically, the middle step of the activity sequence
in Figure 1.1 is inconsistent with the other two steps. Indeed, it would
be simpler and more consistent to replace the continuous model inference
by a discrete model inference, as shown in Figure 1.2, and this can be
accomplished by denying the concept of infinity
Discrete Data From
Discrete Data from
Scientific Experiments -------> Model ------> Numerical Methods
Motivated by the reasons listed above, we will proceed under the following
mathematical and physical assumptions. The concept of infinity and the
consequential concepts of limit, derivative, and integral are reasonable
for the pure mathematical study of real numbers and real functions, but
are not reasonable for the modeling of physical concepts and
Our primary aim will be the study of nonlinear physical behavior.
Classical continuous mathematics will be used only in the study of stability,
where properties of sets of rational numbers can be derived most easily
by considering these numbers as subsets of the real number system. Dynamical
behavior will be studied entirely in terms of arithmetic, or, more precisely,
in terms of high-speed arithmetic, for it is the availability of
the modern digital computer which will make our approach both reasonable
and practical. The dynarnical equations of our models will be difference
equations which, whether linear or nonlinear, will easily be solvable.
Thereby, it is hoped that if an applied scientist is willing to learn the
simple language of a computer, then he need be equipped only with the rudimentary
mathematical knowledge of arithmetic and algebra in order to study highly
complex physical phenomena.
Other related books by Donald Greenspan:
PARTICLE MODELING, Birkhauser, Boston, 1997.
COMPUTER-ORIENTED MATHEMATICAL PHYSICS, Pergamon, N.Y., 1981.
ARITHMETIC APPLIED MATHEMATICS, Pergamon, N.Y., 1980.