I know, I know, I'm 2 days early in regards to today's degree, whatever, better ahead than behind...been doing more reading than writing lately, here's an excerpt from one of the titles on my list (The Language of Mathematics: Making the Invisible Visible)...It's more Eurocentric than anything (I took out some of those parts), but those that know, will take the best part...All Emphasis is My Own...Peace...Au
Excerpt from the prolouge of:
The Language of Mathematics: Making the Invisible Visible
ISBN 0-7167-3379-X (hardcover)
ISBN 0-7167-3967-4 (paperback)
1. Mathematics—Popular works. I. Title.
© 1998, 2000
What Is Mathematics?
It's Not Just Numbers
What is mathematics? Ask this question of persons chosen at random, and you are likely to receive the answer "Mathematics is the study of numbers." With a bit of prodding as to what kind of study they mean, you may be able to induce them to come up with the description "the science of numbers." But that is about as far as you will get. And with that, you will have obtained a description of mathematics that ceased to be accurate some two and a half thousand years ago!
Given such a huge misconception, there is scarcely any wonder that your randomly chosen persons are unlikely to realize that research in mathematics is a thriving, worldwide activity, or to accept a suggestion that mathematics permeates, often to a considerable extent, most walks of present-day life and society. In fact, the answer to the question "What is mathematics?" has changed several times during the course of history.
After Newton and Leibniz, mathematics became the study of number, shape, motion, change, and space. By the end of the nineteenth century, mathematics had become the study of number, shape, motion, change, and space, and of the mathematical tools that are used in this study.
It was only within the last thirty years or so that a definition of mathematics emerged on which most mathematicians now agree: mathematics is the science of patterns. What the mathematician does is examine 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, purely utilitarian or of little more than recreational interest. They can arise from the world around us, from the depths of space and time, 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 patterns of number and counting.
• Geometry studies 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.
One aspect of modern mathematics that is obvious to even the casual observer is the use of abstract notation: algebraic expressions, complicated-looking formulas, and geometric diagrams. The mathematician's reliance on abstract notation is a reflection of the abstract nature of the patterns he studies.
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, line drawings in the form of blueprints are the most appropriate way to specify the construction of a building. And musical notation is the most appropriate way to convey music, apart from, perhaps, actually playing the piece.
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 notation, concepts, and procedures. For instance, the symbolic notation of algebra is the most appropriate means of describing and analyzing the general behavioral properties of addition and multiplication. The commutative law for addition, for example, 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
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 notation.
Symbols of Progress
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.
Given the strong similarity between mathematics and music, both of which have their own highly abstract notations and are governed by their own structural rules, it is hardly surprising that many (perhaps most) mathematicians also have some musical talent.
In fact, for most of the two and a half thousand years of Western civilization, starting with the ancient Greeks, mathematics and music were regarded as two sides of the same coin: both were thought to provide insights into the order of the universe. It was only with the rise of the scientific method in the seventeenth century that the two started to go their separate ways.
For all their historical connections, however, there was, until recently, one very obvious difference between mathematics and music. Though only someone well trained in music can read a musical score and hear the music in her head, if that same piece of music is performed by a competent musician, anyone with the sense of hearing can appreciate the result. It requires no musical training to experience and enjoy music when it is performed.
For most of its history, however, the only way to appreciate mathematics was to learn how to 'sight-read' the symbols. Though the structures and patterns of mathematics reflect the structure of, and resonate in, the human mind every bit as much as do the structures and patterns of music, human beings have developed no mathematical equivalent of a pair of ears. Mathematics can be 'seen' only with the 'eyes of the mind'.
It is as if we had no sense of heating, so that only someone able to sight-read musical notation would be able to appreciate the patterns and harmonies of music.
In recent years, however, the development of computer and video technologies has to some extent made mathematics accessible to the untrained. In the hands of a skilled user, the computer can be used to 'perform' mathematics, and the result can be displayed in a visual form on the screen for all to see. Though only a relatively small part of mathematics lends itself to such visual 'performance', it is now possible to convey to the layperson at least something of the beauty and the harmony that the mathematician 'sees' and experiences when he/she does mathematics.
Without its algebraic symbols, large parts of mathematics simply would not exist. Indeed, the issue is a deep one, having to do with human cognitive abilities. The recognition of abstract concepts and the development of an appropriate language to represent them are really two sides of the same coin.
The use of a symbol such as a letter, a word, or a picture to denote an abstract entity goes hand in hand with the recognition of that entity as an entity. The use of the numeral '7' to denote the number 7 requires that the number 7 be recognized as an entity; the use of the letter m to denote an arbitrary whole number requires that the concept of a whole number be recognized. Having the symbol makes it possible to think about and manipulate the concept.
This linguistic aspect of mathematics is often overlooked, especially in our modern culture, with its emphasis on the procedural, computational aspects of mathematics. Indeed, one often hears the complaint that mathematics would be much easier if it weren't for all that abstract notation, which is rather like saying that Shakespeare would be much easier to understand if it were written in simpler language.
Sadly, the level of abstraction in mathematics, and the consequent need for notation that can cope with that abstraction, means that many, perhaps most, parts of mathematics will remain forever hidden from the nonmathematician; and even the more accessible parts—the parts described in books such as this one—maybe at best dimly perceived, with much of their inner beauty locked away from view. Still, that does not excuse those of us who do seem to have been blessed with an ability to appreciate that inner beauty from trying to communicate to others some sense of what it is we experience—some sense of the simplicity, the precision, the purity, and the elegance that give the patterns of mathematics their aesthetic value.
The Hidden Beauty in the Symbols
In his 1940 book A Mathematician's Apology, the accomplished English mathematician G. H. Hardy wrote:
The mathematician's patterns, like the painter's or the poet's, must be beautiful, the ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test; there is no permanent place in the world for ugly mathematics. . . . It may be very hard to define mathematical beauty, but that is just as true of beauty of any kind—we
may not know quite what we mean by a beautiful poem, but that does not prevent us from recognizing one when we read it.
The beauty to which Hardy was referring is, in many cases, a highly abstract, inner beauty, a beauty of abstract form and logical structure, a beauty that can be observed, and appreciated, only by those sufficiently well trained in the discipline. It is a beauty ''cold and austere," according to Bertrand Russell, the famous English mathematician and philosopher, who wrote, in his 1918 book Mysticism and Logic:
Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.
Mathematics, the science of patterns, is a way of looking at the world, both the physical, biological, and sociological world we inhabit and the inner world of our minds and thoughts. Mathematics' greatest success has undoubtedly been in the physical domain, where the subject is rightly referred to as both the queen and the servant of the (natural) sciences. Yet, as an entirely human creation, the study of mathematics is ultimately a study of humanity itself.
For none of the entities that form the substrate of mathematics exist in the physical world; the numbers, the points, the lines and planes, the surfaces, the geometric figures, the functions, and so forth are pure abstractions that exist only in humanity's collective mind. The absolute certainty of a mathematical proof and the indefinitely enduring nature of mathematical truth are reflections of the deep and fundamental status of the mathematician's patterns in both the human mind and the physical world.
In an age when the study of the heavens dominated scientific thought, Galileo said, "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."
Striking a similar note in a much later era, when the study of the inner workings of the atom had occupied the minds of many scientists for a generation, the Cambridge physicist John Polkinhorne wrote, in 1986,
Mathematics is the abstract key which tums the lock of the physical universe. In today's age, dominated by information, communication, and computation, mathematics is finding new locks to turn. There is scarcely any aspect of our lives that is not affected, to a greater or lesser extent, by mathematics, for abstract patterns are the very essence of thought, of communication, of computation, of society, and of life itself.
Making the Invisible Visible
We have answered the question "What is mathematics?" with the slogan "Mathematics is the science of patterns." There is another fundamental question about mathematics that can also be answered by a catchy phrase: "What does mathematics do?" By this I mean, what exactly does mathematics give you when you apply it to the study of some phenomenon? The answer is "Mathematics makes the invisible visible.''
Let me give you some examples of what I mean by this answer.
Without mathematics, there is no way you can understand what keeps a jumbo jet in the air. As we all know, large metal objects don't stay above the ground without something to support them. But when you look at a jet aircraft flying overhead, you can't see anything holding it up. It takes mathematics to 'see' what keeps an airplane aloft. In this case, what lets you 'see' the invisible is an equation discovered by the mathematician Daniel Bernoulli early in the eighteenth century.
While I'm on the subject of flying, what is it that causes objects other than aircraft to fall to the ground when we release them? "Gravity," you answer. But that's just giving it a name; it doesn't help us to understand it.
It's still invisible. We might as well call it 'magic'. To understand gravity, you have to 'see' it. That's exactly what Newton did with his equations of motion and mechanics in the seventeenth century. Newton's mathematics enabled us to 'see' the invisible forces that keep the earth rotating around the sun and cause an apple to fall from a tree onto the ground. Both Bernoulli's equation and Newton's equations use calculus. Calculus works by making visible the infinitesimally small. That's another example of making the invisible visible.
Here's another: Two thousand years before we could send spacecraft into outer space to provide us with pictures of our planet, the Greek mathematician Eratosthenes used mathematics to show that the earth was round. Indeed, he calculated its diameter, and hence its curvature, with 99 percent accuracy. Today, we may be close to repeating Eratosthenes' feat by discovering whether the universe is curved. Using mathematics and powerful telescopes, we can 'see' into the outer reaches of the universe. According to some astronomers, we will soon see far enough to be able to detect any curvature in space, and to measure any curvature that we find.
Knowing the curvature of space, we can then use mathematics to see into the future to the day the universe comes to an end. Using mathematics, we have already been able to see into the distant past, making visible the otherwise invisible moments when the universe was first created in what we call the Big Bang.
Coming back to earth at the present time, how do you 'see' what makes pictures and sounds of a football game miraculously appear on a television screen on the other side of town? One answer is that the pictures and sounds are transmitted by radio waves—a special case of what we call electromagnetic radiation. But, as with gravity, that answer just gives the phenomenon a name; it doesn't help us to 'see' it. In order to 'see' radio waves, you have to use mathematics. Maxwell's equations, discovered in the nineteenth century, make visible to us the otherwise invisible radio waves.
Here are some human patterns we can 'see' through mathematics:
• Aristotle used mathematics to try to 'see' the invisible patterns of sound that we recognize as music.
• He also used mathematics to try to describe the invisible structure of a dramatic performance.
• In the 1950s, the linguist Noam Chomsky used mathematics to 'see' and describe the invisible, abstract
patterns of words that we recognize as grammatical sentences. He thereby turned linguistics from a fairly
obscure branch of anthropology into a thriving mathematical science.
Finally, using mathematics, we are able to look into the future:
• Probability theory and mathematical statistics let us predict the outcomes of elections, often with
• We use calculus to predict tomorrow's weather.
• Market analysts use various mathematical theories to try to predict the behavior of the stock market.
• Insurance companies use statistics and probability theory to predict the likelihood of an accident during the
coming year, and set their premiums accordingly.
When it comes to looking into the future, mathematics allows us to make visible another invisible—that which has not yet happened. In that case, our mathematical vision is not perfect. Our predictions are sometimes wrong. But without mathematics, we cannot see into the future even poorly.
The Invisible Universe
Today, we live in a technological society. There are increasingly few places on the face of the earth where, when we look around us toward the horizon, we do not see products of our technology: tall buildings, bridges, power lines, telephone cables, cars on roads, aircraft in the sky. Where communication once required physical proximity, today much of our communication is mediated by mathematics, transmitted in digitized form along wires or optical fibers, or through the ether. Computers—machines that perform mathematics—are not only on our desktops, they are in everything from microwave ovens to automobiles and from children's toys to pacemakers for those with heart problems. Mathematics—in the form of statistic—is used to decide what food we will eat, what products we will buy, what television programs we will be able to see, and which politicians we will be able to vote for. Just as society burned fossil fuels to drive the engines of the industrial age, in today's information age, the principal fuel we burn is mathematics.
And yet, as the role of mathematics has grown more and more significant over the past half century, it has become more and more hidden from view, forming an invisible universe that supports much of our lives.
Just as our every action is governed by the invisible forces of nature (such as gravity), we now live in the invisible universe created by mathematics, subject to invisible mathematical laws.