What is mathematics?

For all the time schools devote to the teaching of mathematics, very little (if any) is spent trying to convey just what the subject is about. Instead, the focus is on learning and applying various procedures to solve math problems. That’s a bit like explaining soccer by saying it is executing a series of maneuvers to get the ball into the goal. Both accurately describe various key features, but they miss the what and the why of the big picture.

Given the demands of the curriculum, I can understand how this happens, but I think it is a mistake. Particularly in today’s world, a general understanding of the nature, extent, power, and limitations of mathematics is valuable to any citizen. Over the years, I’ve met many people who graduated with degrees in such mathematically rich subjects as engineering, physics, computer science, and even mathematics itself, who have told me that they went through their entire school and college-level education without ever gaining a good overview of what constitutes modern mathematics. Only later in life do they sometimes catch a glimpse of the true nature of the subject and come to appreciate its pervasive role in modern life.

1. More than arithmetic

Most of the mathematics used in present-day science and engineering is no more than three- or four-hundred years old, much of it less than a century old. Yet the typical high school curriculum comprises mathematics at least that old — some of it over two-thousand years old!

Now, there is nothing wrong with teaching something so old. As the saying goes, if it ain’t broke, don’t fix it. The algebra that the Arabic-speaking traders developed in the eighth and ninth centuries (the word algebra comes from the Arabic term al-jabr, meaning “restoration” or “reunion of broken parts”) to increase efficiency in their business transactions remains as useful and important today as it was then, even though today we may now implement it in a spreadsheet macro rather than by medieval finger calculation. But time moves on and society advances. In the process, the need for new mathematics arises and, in due course, is met. Education needs to keep pace.

Mathematics can be said to have begun with the invention of numbers and arithmetic, which is believed to have occurred around ten thousand years ago, with the introduction of money. (Yes, apparently it began with money!)

Over the ensuing centuries, the ancient Egyptians and Babylonians expanded the subject to include geometry and trigonometry.¹ In those civilizations, mathematics was largely utilitarian, and very much of a “cookbook” variety. (“Do such and such to a number or a geometric figure, and you will get the answer.”)

The period from around 500 BCE to 300 CE was the era of Greek mathematics. The mathematicians of ancient Greece had a particularly high regard for geometry. Indeed, they treated numbers geometrically, as measurements of length, and when they discovered that there were lengths to which their numbers did not correspond (essentially the discovery of irrational numbers), their study of number largely came to a halt.²

In fact, it was the Greeks who made mathematics into an area of study, not merely a collection of techniques for measuring, counting, and accounting. Around 500 BCE, Thales of Miletus (Miletus is now part of Turkey) introduced the idea that the precisely stated assertions of mathematics could be logically proved by formal arguments. This innovation marked the birth of the theorem, now the bedrock of mathematics. This formal approach by the Greeks culminated in the publication of Euclid’s Elements, reputedly the most widely circulated book of all time after the Bible.³

By and large, school mathematics is based on all the developments I listed above, together with just two further advances, both from the seventeenth century: calculus and probability theory. Virtually nothing from the last three hundred years has found its way into the classroom. Yet most of the mathematics used in today’s world was developed in the last two hundred years, let alone the last three hundred!

As a result, anyone whose view of mathematics is confined to what is typically taught in schools is unlikely to appreciate that research in mathematics is a thriving, worldwide activity, or to accept that mathematics permeates, often to a considerable extent, most walks of present-day life and society. For example, they are unlikely to know which organization in the United States employs the greatest number of Ph.D.s in mathematics. (The answer is almost certainly the National Security Agency, though the exact number is an official secret. Most of those mathematicians work on code breaking, to enable the agency to read encrypted messages that are intercepted by monitoring systems — at least, that is what is generally assumed, though again the Agency won’t say. Though most Americans probably know that the NSA engages in code breaking, many do not realize it requires mathematics, and hence do not think of the NSA as an organization that employs a large number of advanced mathematicians.)

The explosion of mathematical activity that has taken place over the past hundred years or so in particular has been dramatic. At the start of the twentieth century, mathematics could reasonably be regarded as consisting of about twelve distinct subjects: arithmetic, geometry, calculus, and several more. Today, the number of distinct categories is somewhere between sixty and seventy, depending how you count them. Some subjects, like algebra or topology, have split into various subfields; others, such as complexity theory or dynamical systems theory, are completely new areas of study.

The dramatic growth in mathematics led in the 1980s to the emergence of a new definition of mathematics as the science of patterns. According to this description, the mathematician identifies and analyzes abstract patterns — numerical patterns, patterns of shape, patterns of motion, patterns of behavior, voting patterns in a population, patterns of repeating chance events, and so on. Those patterns can be either real or imagined, visual or mental, static or dynamic, qualitative or quantitative, utilitarian or recreational. They can arise from the world around us, from the pursuit of science, or from the inner workings of the human mind. Different kinds of patterns give rise to different branches of mathematics. For example:

• Arithmetic and number theory study the patterns of number and counting.
• Geometry studies the patterns of shape.
• Calculus allows us to handle patterns of motion.
• Logic studies patterns of reasoning.
• Probability theory deals with patterns of chance.
• Topology studies patterns of closeness and position.
• Fractal geometry studies the self-similarity found in the natural world.

2. Mathematical notation

One aspect of modern mathematics that is obvious to even the casual observer is the use of abstract notations: algebraic expressions, complicated-looking formulas, and geometric diagrams. The mathematicians’ reliance on abstract notation is a reflection of the abstract nature of the patterns they study.

Different aspects of reality require different forms of description. For example, the most appropriate way to study the lay of the land or to describe to someone how to find their way around a strange town is to draw a map. Text is far less appropriate. Analogously, annotated line drawings (blueprints) are most appropriate for representing the construction of a building. And musical notation is most appropriate for representing music on paper. In the case of various kinds of abstract, formal patterns and abstract structures, the most appropriate means of description and analysis is mathematics, using mathematical notations, concepts, and procedures.

For example, the commutative law for addition could be written in English as:

When two numbers are added, their order is not important.

However, it is usually written in the symbolic form:

m+n = n+m

While the symbolic form has no significant advantage for a simple example such as this, such is the complexity and the degree of abstraction of the majority of mathematical patterns, that to use anything other than symbolic notation would be prohibitively cumbersome. And so the development of mathematics has involved a steady increase in the use of abstract notations.

Though the introduction of symbolic mathematics in its modern form is generally credited to the French mathematician Françoise Viète in the sixteenth century, the earliest appearance of algebraic notation seems to have been in the work of Diophantus, who lived in Alexandria some time around 250 CE. His thirteen volume treatise Arithmetica (only six volumes have survived) is generally regarded as the first algebra textbook. In particular, Diophantus used special symbols to denote the unknown in an equation and to denote powers of the unknown, and he had symbols for subtraction and for equality.

These days, mathematics books tend to be awash with symbols, but mathematical notation no more is mathematics than musical notation is music. A page of sheet music represents a piece of music; the music itself is what you get when the notes on the page are sung or performed on a musical instrument. It is in its performance that the music comes alive and becomes part of our experience. The music exists not on the printed page but in our minds. The same is true for mathematics. The symbols on a page are just a representation of the mathematics. When read by a competent performer (in this case, someone trained in mathematics), the symbols on the printed page come alive — the mathematics lives and breathes in the mind of the reader like some abstract symphony.

To repeat, the reason for the abstract notation is the abstract nature of the patterns that mathematics helps us identify and study. For example, mathematics is essential to our understanding the invisible patterns of the universe. In 1623, Galileo wrote,

The great book of nature can be read only by those who know the language in which it was written. And this language is mathematics.⁴

In fact, physics can be accurately described as the universe seen through the lens of mathematics.

To take just one example, as a result of applying mathematics to formulate and understand the laws of physics, we now have air travel. When a jet aircraft flies overhead, you can’t see anything holding it up. Only with mathematics can we “see” the invisible forces that keep it aloft. In this case, those forces were identified by Isaac Newton in the seventeenth century, who also developed the mathematics required to study them, though several centuries were to pass before technology had developed to a point where we could actually use Newton’s mathematics (enhanced by a lot of additional mathematics developed in the interim) to build airplanes. This is just one of many illustrations of one of my favorite memes for describing what mathematics does: mathematics makes the invisible visible.

3. Modern college-level mathematics

With that brief overview of the historical development of mathematics under our belts, I can start to explain how modern college math came to differ fundamentally from the math taught in school.

Up to about 150 years ago, although mathematicians had long ago expanded the realm of objects they studied beyond numbers (and algebraic symbols for numbers), they still regarded mathematics as primarily about calculation. That is, proficiency at mathematics essentially meant being able to carry out calculations or manipulate symbolic expressions to solve problems. By and large, high school mathematics is still very much based on that earlier tradition.

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