The Nature of Mathematics
Mathematics relies on both logic and creativity, and it is pursued
both for a variety of practical purposes and for its basic interest.
The essence of mathematics lies in its beauty and its intellectual
challenge. This essay is divided into three sections, which are
patterns and relationships, mathematics, science and technology and
Firstly, Mathematics is the science of patterns and relationships. As
a theoretical order, mathematics explores the possible relationships
among abstractions without concern for whether those abstractions have
counterparts in the real world. The abstractions can be anything from
strings of numbers to geometric figures to sets of equations. In
deriving, for instance, an expression for the change in the surface
area of any regular solid as its volume approaches zero,
mathematicians have no interest in any correspondence between
geometric solids and physical objects in the real world.
A central line of investigation in theoretical mathematics is
identifying in each field of study a small set of basic ideas and
rules from which all other interesting ideas and rules in that field
can be logically deduced. Mathematicians are particularly pleased when
previously unrelated parts of mathematics are found to be derivable
from one another, or from some more general theory. Part of the sense
of beauty that many people have perceived in mathematics lies not in
finding the greatest richness or complexity but on the contrary, in
finding the greatest economy and simplicity of representation and
proof. As mathematics has progressed, more and more relationships have
been found between parts of it that have been developed separately.
These cross-connections enable insights to be developed into the
various parts; together, they strengthen belief in the correctness and
underlying unity of the whole structure.
Mathematics is also an applied science. Many mathematicians focus
their attention on solving problems that originate in the world of
experience. They too search for patterns and relationships, and in the
process they use techniques that are similar to those used in doing
purely theoretical mathematics. The difference is largely one of
intent. In contrast to theoretical mathematicians, applied
mathematicians, in the examples given above, might study the interval
pattern of prime numbers to develop a new system for coding numerical
information, rather than as an abstract problem. Or they might tackle
the area/volume problem as a step in producing a model for the study
of crystal behavior.
The results of theoretical and applied mathematics often influence
each other. Studies on the mathematical properties of random events,
for example, led to knowledge that later made it possible to improve
the design of...