Mathematics is
the science of order

It doesn't seem possible for such a world to exist, because we live in a world rich with sensory, physical and social patterns.

Patterns represent organization and order, and as much as one-third of the human brain is engaged in making sense out of the visual patterns in the world around us.

There are only a few basic shapes that provide all of the visual richness of nature: circle, polygon, spiral, helix, meander, branch, radiant. No one knows how the human brain, the most complexly organized structure that we know of anywhere in the universe, puts these shapes together to recognize objects that are made from various combinations of those shapes.

Equally puzzling is how the brain can use symbols based on the properties of counting numbers to describe the patterns that we find in nature, and which exist in our minds. But through the millennia we have found many ways to describe them that have become so complex as to be beyond the abilities of most of us.

We have a name for the study of patterns, whether they be physical, imaginary or abstract. We call it mathematics, the science of order that uses the logic and rules of operations that are ultimately derived from the simple act of counting to study quantity, shape and arrangement. We also use it as a tool of analysis to study relationships and to perform calculations.

There are few areas of human endeavor that are not in some way characterizable mathematically or enhanced somehow by mathematics. In art, music, linguistics, literature and dance, mathematics augments and helps to add beauty to performance and analysis. In business, economics, politics and population dynamics, it is indispensable. It is used in information theory and signal processing, for security encryption and decryption, optimization, game theory and even for recreation. The list goes on and on.

At first glance it might seem odd that mathematics is so pervasively useful to us, but it is not really so strange. To exist, our universe must have an underlying order and rules of operation since there is order, and nature has patterns of many kinds.

There are visual patterns as well as time-ordered patterns. Patterns of repetition, cycles and discrete countable objects all exist and interact in the physical domain of the universe, and regardless of what is behind it, our brains are part of it. They are highly organized groups of cells that exchange signals in an orderly fashion in order to function in a variety of ways organizing, storing, retrieving and communicating information.

Our brains are "wired" to recognize simple shapes such as circles, triangles and squares, and certain specific brain cells are responsible for recognizing lines that lean certain ways and intersect at different angles. By putting together these shapes and lines, we each construct our own unique map of our world. But it is a giant leap from that basic geometric underpinning to writing equations that describe the interaction of an electron with the fabric of space-time surrounding a rotating black hole.

Researchers have found that the shapes that newborns see in the earliest days of life affect the way the brain develops. This implies not only that mathematics is hard-wired into our brains, but also that these earliest perceptions play a role in our cultural development and help us to learn about our environment.

The mathematical nature of the universe was perplexing to the ancient Greek philosophers, who it is said "discovered the mind," being the first to think seriously about thinking for the sake of thinking.

A cult of numerical mysticism arose in ancient Greece 25 1/2 centuries ago, although people had been counting, keeping records and calculating areas and volumes of geometric shapes in one way or another with increasing sophistication for 20 centuries or so before that.

The leader and founder of the cult was Pythagoras of Samos (582-500 B.C.), who discovered as a youth that pipes produced harmonious tones when their lengths were related by fractions consisting of small integers.

Pythagoras and his followers considered that mathematical order ruled the universe. Theirs was a cult that saw mystical significance in numbers and shapes and their interrelationships, sharing with people of all ages the quest to reveal meaning in their own existence and the world in which they lived.

The Pythagoreans went on to discover many numerical patterns, including the famous Pythagorean Theorem: a-squared plus b-squared equals c-squared.

They studied numerology, finding meaning in sequences and sums of numbers, and studied the relationships between numerical patterns and polygons that would define the geometry that Euclid would formalize three centuries later.

The Pythagorean relationships between geometry and numbers would be synthesized with algebra 20 centuries later by Rene Descartes into analytic geometry and would go on to become the backbone of physics that propelled it from speculative philosophy into the bedrock of science and engineering.

Modern mathematicians and scientists avoid mystical interpretations of numerical relationships, and why the universe is mathematical is less of a concern to us today. That notwithstanding, mathematicians still find fascination in patterns of numbers and study them fervently with an ever-growing repertoire of methods and symbols.

There are many numerical patterns, such as the nonrepeating, infinite digits of pi, that seem to have no significance; while others, such as the wavelengths of light emitted from excited atoms, have led to deeper understanding of the laws of nature.

Yet there are some significant numbers, such as the primes, that have patterns but for which no one has found a general rule describing the pattern despite centuries of analysis.

The kinds of patterns that mathematics describes are as diverse and variable as nature itself, but mathematics goes beyond the physical and connects with the deepest part of the abstract brain. There are many mathematical structures that were conceived in the mind and studied before any examples were found in nature, and many that still have not.