calculus book - Mathematics and Computer Science

1.1. THE NATURE OF MATHEMATICS 5
Mathematics and Science
Mathematics was once called “the most exact science”. In the last two
centuries, it has become fairly clear that mathematics is fundamentally
not a science at all. In mathematics, the standard of acceptance of an
idea is logical, deductive proof, about which we say more below. While
mathematicians sometimes perform “experiments” , either with pencil
and paper, or with an electronic computer, the results of a mathematical
experiment are never regarded as definitive. In physics or chemistry,
by contrast, experiment is the sole criterion for validity of an idea. 1
A few minutes’ reflection should reveal the reasons for these radically
differing criteria. Mathematical concepts have no relevant attributes
other than those we ascribe to them, so in principle a mathematician
has complete access to all properties possessed by an object.
In the physical sciences, however, the objects of study are phenomena,
about which information can only be obtained by experiment. No
matter how many experiments are performed, scientists can never be
certain that their knowledge is complete; a more refined experiment
may conflict with existing results, indicating (at best) that an accepted
Law of Nature needs to be modified, or (at worst) that someone has collected
data carelessly. Experimental results are never mathematically
exact, but are subject to “uncertainty” or “experimental error”. Thus,
in the sciences we do not have the same access to our object of study
that we do in mathematics. Laws of Nature—mathematical models of
some aspect of reality—are virtually assured of being approximate.
Despite the differing aims and standards of acceptance, mathematics
and the physical sciences enrich each other considerably. The most
obvious direction of influence is from mathematics to the sciences: The
best available descriptions of natural phenomena are mathematical, and
are astoundingly accurate. For example, total eclipses of the sun can
be predicted hundreds of years in advance, down to the time and locations
at which totality will occur. Less apparent but no less important
is the beneficial influence that physics, biology, and economics have
had on mathematics, particularly in the 20th Century. For whatever
reason, mathematics that describes natural phenomena is deeply interconnected
and full of beautiful, unexpected results. Without the
guiding influence of science, mathematics tends to become ingrown,
specialized, and merely technical.
1 This characterization of science is due to the physicist, R. P. Feynmann.