Introduction: When I first started blogging as an experiment some of the early posts I made were regarding the philosophical question of defining the field of Mathematics. Over time the responses I've received and practical questions about the needs of certain segments of my audience have led me to revise and summarize previous discussions onto one post for a new audience.

What is mathematics?

The earliest so-called mathematics had few symbols and mostly dealt with how to solve common problems that arise in various situations requiring numbers. Today many say math is being able to tell time, make change, or balance a checkbook. While a mathematician will agree that there are mathematical principles involved and those skills are an important part of literacy in our culture the suggestion would probably make him or her grimace. A mathematicians daily pursuit has little resemblance to mundane tasks of accounting. For mathematicians mathematics is a both a pursuit of knowledge and frequently a source of pleasure. Consider for example a mathematician who takes up the art of origami for fun, then chooses to investigate origami for work. The effort results in the ability to push the limits of the art beyond the traditional frog or swan type models, bringing paper folding to a new level. That same mathematician may later be consulted on how to fold airbags into the steering wheel of the latest automobile. Modern origami artists and automobile manufacturers have both indeed benefited from mathematics in this way. In our modern society mathematics has penetrated and affected every aspect of our daily life.

Mathematics has always existed at some level in every society throughout history, but for modern mathematicians, mathematics begins with the Greeks. It is with the Greeks that we first see some of the hallmarks of what the experts call mathematics in the modern sense. Greek philosophers took practical problems that occupied other cultures and turned them into philosophical pursuits in an attempt to understand the fundamental truth of everything. For example, Egyptians had to redraw land boundaries every year after the Nile flooded croplands. This surveying was very important because a person’s wealth depended largely on the amount of land for crops that they owned. Greek philosophers such as Thales studied Egyptian knowledge and began arriving at general principles from which further truths can be deduced. In one account Thales used the concept of ratio, to accurately determine the height of the great pyramid with only his walking stick. Similarly, a Greek librarian and scholar named Eratosthenes generations later accurately estimated the circumference of the earth by noticing the angle of that the sun made with the earth at different locations. He did it by drawing a circles and triangles. Reading his work inspired Columbus centuries later to attempt to sail around the globe. The Greek thinkers learned to explore the abstract world of shapes and numbers and then apply them to solve myriads of practical problems. The use abstraction and logic set them apart from earlier thinkers that extends to the heart of mathematical thinking today.

During the enlightenment the father of modern philosophy Rene Descartes, whose pursuits centered on studying mathematics, saw a fly on the ceiling of his bedroom and realized he had found the key to unifying geometry and algebra -an insight that would be essential in the development of many modern technological aids. He realized that he could describe the location of the fly by its distance from the edges of the walls and corners of his room. From this he produced a coordinate system to represent the location of any object in space by drawing intersecting lines like the edges of the walls in his room. This method of using coordinates is central to the way we make maps, how computers locate information on a hard drive, guide airplanes, and produce computer graphics today. Mathematician Keith Devlin in The Math Gene speculates that the evolution of the unique human ability for language inevitably leads to the capacity for the higher abstraction of mathematics. Devlin believes that man first evolved to be able to comprehend only what we perceive through our senses, then to imagine real objects removed from view, followed by the ability use language and symbols to represent real objects and combinations thereof, and finally the ability to see deep underlying structure or the ability to do math. For example in a recently developed algebra 1 curriculum, the authors of the text shows that the path of a basketball and the sum of the lengths of the bars on the screen of your cell phone indicating signal strength are similar structures mathematically. While abstraction can sometimes intimidate students trying to learn mathematics, fundamentally it is the process of simplifying a problem into its fundamental parts revealing structure that makes otherwise unsolvable or complex problems more workable.

Identifying underlying structure is meaningless without the ability to analyze individual components. The Greek educator Euclid wrote a definitive text on mathematics that was studied as a core text for all educated peoples for hundreds of years called The Elements. The text has been seen as a model of logical analysis. In The Elements Euclid summarized all of Greek learning in math by starting with 10 basic assumptions and a few definitions. Abraham Lincoln, like many others, studied The Elements in his spare time while practicing law to sharpen his ability to make logical arguments in the courtroom. Like Lincoln, for generations educators have seen mathematics as a powerful tool for teaching the principles of clear and precise thinking. As in Euclid’s day, all truths in modern mathematics are established through the use of pure logic.

Interestingly enough, logic itself has received a fairly rigorous mathematical treatment. George Boole (1815-1864) developed an algebra of logic by representing statements with symbols and performing an analysis of what can be said given various relationships between different statements. His work is not only the basis for the instruction of mathematicians in logic, but also for any who wishes to study logic at the university, and it is the foundation of the theory behind computer programming today. Since Boole’s time some have been led to exclaim that mathematics is simply logic in its purest and simplest form. While some describe mathematics as the study of patterns, others can rebut with the observation Rudy Rucker made in the book Mind Tools, the study of logic describes patterns of thought.

While we might talk about how science today uses mathematics as part of its process of analysis such including such diverse studies as linguistics, food science, medicine, criminal investigation, engineering, psychology, geology, astronomy, biology, and so on; or how the layman benefits- many people explore, read, and dabble in mathematics for its own sake, as poets do in their treatment of language. Magazines and books are printed regularly with mathematical diversions and puzzles. Books on mathematics are found on library shelves along with other books of literature. The classicist Mortimer Adler described in his text How to Read a Book the relation to reading mathematics versus other literature. Adler’s description of how mathematics varies from other literature can be summarized by the diagram below.

In his discussion of literature he places mathematics on the extreme left, mathematical terms are well defined and have precise meanings. On the far right the meaning and use of words is often transient and are often chosen for the sound they make or some connotation, thought, or feeling that they conjure. You can change the words and change the experience. The left hand is about constructing thoughts and arguments. The right with constructing feelings and experiences. Often when students don't know a word in a novel, they skip it and may more or less unaffected in their understanding of the story. If you don't understand a term in mathematics however, you can easily be completely lost. Math vocabulary is far more critical for success in mathematics. Words in math and science are generally used in very specific ways and have special meanings that need to be understood in order to fully appreciate what is written.

Like other literature mathematics has its own beauty, although it may be different from what is typically admired in other types of writing. Mathematics seems to contradict much of the conventional wisdom of what makes good reading. In the typical English class students are told be careful about repetition of key words and may be told to include synonyms to add variety. In contrast, almost every math book an upper division text may contain a legend of symbols that can be used to convey meaning preferentially, leading to an increased use of the pictograms versus other syntax. Arguments are typically written in a sort of hieroglyphic to make concise arguments, because it is the experience of the ideas rather than the words that is most important.

To express this idea of beauty in another way, in the typical interaction between people, we find that people fail to say what they mean and don't necessarily mean what they say. This extends to the novel, where a primary aim of the writer is engaging the reader by creating suspense. Suspense generally comes from ambiguity and the language follows suit. Precision can be relatively unimportant. So, we judge fiction beautiful when it contains ambiguities. We revel in the complex metaphors and consider the special rhythm, the pitter-patter the words make as they strike the palate, and we find it beautiful. But, if we were to invert these aesthetics and travel in the opposite direction, replacing ambiguity with certainty and nuance with clarity; if we instead begin with plain assumptions and proceed with unobstructed intention and unambiguous meanings, the result is the language of pure reason, the language of our ideas, the conversation of mathematics. Mathematician Paul Erdos said, “Why are numbers beautiful? It's like asking why is Beethoven's Ninth Symphony beautiful. If you don't see why, someone can't tell you. I know numbers are beautiful. If they aren't beautiful, nothing is.”

Like Mr. Erdos others have compared mathematics to music. Keith Devlin speaking on the aesthetic value of mathematics cites an interesting study in The Math Gene,

In fact, the connection between mathematics and music may go deeper-right to the structure of the very device that creates them both: the human brain; using modern imaging techniques that show which parts of the brain are active while the subject performs various mental or physical tasks; researchers have compared the brain images produced by professional musicians listening to music with those of professional mathematicians working on a mathematical problem. The two images are very similar, showing that the expert musicians and expert mathematicians appear to be using the same circuits.

Unfortunately, that is not how many people feel about mathematics. Mr. Devlin believes that this is due to the level of training required to learn to read math

When a mathematician looks at a page of mathematical symbols she does not 'see' the symbols, any more than the trained musician 'sees' the musical notes on a sheet of music. The trained musicians eyes read straight through the symbols to the sounds they represent. Similarly a trained mathematician reads straight through the mathematical symbols to the patterns they represent

H.E. Huntley in The Divine Proportion, speculates that a golden rectangle whose sides have a particular ratio is pleasing because of the relationship between the times it takes for the eye to scan the lengths of the sides of the rectangle, is identical to the brain in effect as the duration of notes in a musical composition. Although unfamiliar with Huntley's hypothesis to explain why some shapes are more pleasing than others, a friend who has much more training in music than math is always asserts when posed the question, “math is music.”

To students, mathematics is often a gateway or prerequisite to the study other subjects, and unfortunately, it can seem suspiciously like doing a series of push-ups. Students often don’t read the great works of mathematics or learn much beyond surviving their required courses. However for many amateurs and professionals alike math can be a sort of game; a monthly puzzle in Scientific American, or even an engaging art form. For the scientists it is often a language to describe the universe. In industry, math types are often called as consultants to solve complex problems, predict the future, and research logistical solutions. In general, mathematics involves the use of deductive logic applied to the abstraction of ideas, problems, objects, or phenomena to their most fundamental parts. For, many the process is informative, useful, beautiful, and stimulating.

Understanding mathematics can improve your ability to think clearly, see the world from a new perspective, and open the possibility of furthering your ability to do science, pursue careers in today’s technology, improve your literacy, and simply develop your brain. For students enrolling in a new math course this year; I would like to welcome you to a new adventure in learning.