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MATHEMATICS in 10 LESSONS
The Grand Tour
By JERRY P. KING Prometheus Books
Copyright © 2009 Jerry P. King
All right reserved. ISBN: 978-1-59102-686-0
Chapter One
TRUTH AND BEAUTY Like Macbeth's air-drawn dagger, mathematics lives in the mind. Mathematical objects-things like numbers, equations, matrices-are abstract and imagined. They do not belong to the real world. Certainly, mathematical symbols can be written on pages and printed in books that are indeed real-world objects. But the subject itself comes from somewhere deep in the mind. Mathematics is made of pure thought, as are air-drawn daggers. It is not made of ink. Mathematicians make it out of airy nothing. And, as poets do, they give it form:
... as imagination bodies forth The forms of things unknown, the poet's pen Turns them to shapes and gives to airy nothing A local habitation and a name.
As I write I have an illustration before me-perhaps by N. C. Wyeth-of a unicorn standing with his spiraled horn in the air and pale moonlight on his silver flanks. Beyond him lies water and stone towers that shine above the trees like Camelot. It is a lovely picture and in it the silver creature seems truly fabled. But it is only a picture. Yet it is as close as any of us will come to unicorns.
Go look. Search the dark forest. Stand still asa tree. Wait beside the bright pool until the moon wanes and you turn cold as stone. You will see no unicorns. Unicorns do not live on this earth. Neither does mathematics.
As you search, you may find a ledger left behind in some abandoned campsite. On the first page you may see faded mathematical symbols including, let's say, the number 6. But you cannot conclude from this that "sixes" exist in the real world. To do so would be analogous to concluding that unicorns exist because N. C. Wyeth once drew a picture of one of them. What you have found, on the old ledger, is a picture of "the number six"-a symbol that represents a mathematical idea. All those who know elementary mathematics share this idea. It connotes to them other commonly shared ideas.
For example, they know that "six" is the name of the natural number (natural numbers are the numbers: 1, 2, 3, 4, 5, 6, 7, ...) that follows five and precedes seven. Also, they are aware of certain arithmetical properties of 6 such that 6 = 1 + 2 + 3, and 6 = 2 · 3. (Here the "dot" represents multiplication. This is one of the symbols we will use for this operation. Nowhere in this book do we indicate multiplication with the "X" which is commonly used in elementary school. "X" looks too much like "x," which we will use for something else.)
Thus, the number six is an idea that stands apart from the world of reality, as do unicorns or daggers made of air. Moreover, we can extend this discussion from the number six to any mathematical notion. Each mathematical notion-no matter whether it is as simple as that of a positive integer or as complicated as a topological space-is an abstraction and lives in the world of ideas.
As we proceed, it will become increasingly important to formalize this distinction. Think of the two worlds-the world of reality and the world of mathematics-as existing side by side. They can be represented schematically in various ways. Figure 1 shows each world as a simple rectangle, the real world on the left and the mathematical world on the right. Real-world objects-the things that live there-are just what you think: poems and people, palaces and plasterboard, all those things you can see or touch. Real things.
The other, the mathematical world, contains only ideas. But they are special ideas and they have names like number or function or inequality. These are mathematical ideas and they have properties that they inherit from simpler mathematical notions and they can be manipulated and extended-according to the rules of mathematics-to produce more complicated ideas. Part of our purpose in this book is to come to grips with the mathematical world and to understand some of the fundamental ideas that live in it. This will come as our story unfolds. What we want to understand now is the existence of the mathematical world and its separateness from the world of reality.
It is in this sense of separateness, of abstraction, that mathematics is similar to Wyeth's unicorn or to Macbeth's dagger of the mind. All are mere constructs, composed of nothing but pieces of pure thought. But there is a significant sense in which mathematics differs from the others. The dagger comes from the heat-oppressed brain of a man who would be king and who stands on the bloody edge of murder. The unicorn derives from Dark Age myth and from the tales of magic told on long nights around leaping fires. Each has its place in literature and in the unfolding of civilization. But neither is necessary. The world would be pretty much as it is if neither the dagger nor the unicorn had been conceived-not as rich a world, but still the world. Malory and Shakespeare are important to me and I prefer a world containing all their ideas. I do not like to imagine a world without the concepts of unicorn or that of Macbeth's dagger. But I can imagine such a world. And I know it would be not much changed.
On the other hand, it is inconceivable to imagine a world without mathematics. (Without mathematics there would exist no science beyond mere description and categorization. The world would be absent the great predictive and explanatory powers provided by mathematics. No part of such a world could be free of opinion, of dogma, or of wishful thinking.) The two worlds of figure 1 are fundamentally entwined like the branches of adjacent vines. They are essential for one another. The real world gives us people who create mathematics. The mathematical world gives us truth.
CREATION
A lot has been slipped into the final two sentences of the above. I've mentioned the two philosophically fundamental concepts of creation and truth. We need to take a closer look at each of them.
As we have seen, the people who pursue research in mathematics are called mathematicians. And when they engage in research they refer to the process as "doing mathematics." To "do mathematics" means to set down on paper new mathematics: mathematics that has heretofore not existed. Ordinarily, this new mathematics becomes the content of a research paper that is published in a specialized journal that exists solely for the purpose of receiving such papers. The research paper then becomes accessible to other mathematicians who may read it and be stimulated to do further research.
Only mathematicians do mathematics. In fact, in certain circles this is taken as the definition of the term "mathematician." "A mathematician," they say, "is a person who does mathematics," "and mathematics is what mathematicians do"-obviously circular definitions, but completely intelligible to those in the business. One of my purposes is to make it intelligible for all of us.
Notice that this "definition" of mathematician says nothing about teaching mathematics. Most mathematicians are employed in universities and they do teach mathematics. But that is different from doing mathematics. A research mathematician is likely also to be a teacher of mathematics. But not always. Moreover, there are many, many people who teach mathematics at some level or another who do not pursue mathematics research, and who may even-like most of the educated public-be unaware that such research exists. Teaching is one thing, research is another.
So, when someone introduces himself as a mathematician, the proper response may not be "what kind of mathematics do you teach?" but rather "what kind of mathematics do you do?" The mathematician-being then properly impressed and gratified with your knowledge of his field-will answer something like: "differential equations," or "topology," or "complex analysis," thereby giving you the name of the particular subfield of mathematics in which he works. (There are, incidentally, more than fifty subfields of mathematics identified by Mathematical Reviews, a major publication of the American Mathematical Society.)
A central question arises quickly from all this: "What exactly does a mathematician do when he sets down new mathematics? Does he discover this mathematics or does he create it?
On the surface these two points of view seem clearly distinct. Does a mathematician find new mathematics or does he simply make it up? The first point of view-that of discovery-is known as the "Platonist," or the "absolutist" viewpoint. The other-that of creation-is referred to as the "constructivist" position. The Platonist view is the more passive, seeing mathematicians more or less as discoverers of an existing mathematical reality. The constructivists see mathematicians as makers, as creators of a mathematical world that without them would not exist. But as the question is pursued at deeper philosophical levels, the two views tend to fuse and the mathematician emerges as creator sometimes and as discoverer at other times. An individual mathematician often feels he is inventing some or other specific piece of mathematics while simultaneously uncovering bits of mathematical reality through his work as a whole. However, the fundamental question remains: "Is mathematics created or discovered?"
The question is philosophically complicated and has yet to be settled to anyone's satisfaction. We are not about to settle it here. Nor should we try. But I want to make two points.
The first is that my own view is the constructivist view: mathematics is created. This is the view that will prevail throughout this book and I shall consistently refer to mathematicians as creators of mathematics. I consider the mathematical world (as figure 1 shows) as a world of ideas that is separate from the world of reality. (As I have already noted, the two worlds are entwined. But the separate branches do not touch.) Mathematicians live in the real world and they create the objects that live in the mathematical world.
The second point is that mathematicians themselves are divided on this issue and my position is the minority view. An informal survey of colleagues at various institutions has shown a ratio of about eight to two in favor of discovery over creation. Most mathematicians-I am convinced-believe they discover, not create, mathematics. And there are some nonmathematicians on the discovery side as well. An example is Martin Gardner, the distinguished mathematical expositor, who wrote in a recent book review:
Not only is the Universe mathematically structured, it is made entirely of mathematics.
If Gardner is correct then not only is mathematics discovered, but the two worlds of figure 1 actually merge. Everything in the universe turns into mathematics. I do not believe him correct but we must take Mr. Gardner seriously. He has thought deeply and written beautifully about mathematics for many years and he stands squarely in the Platonist camp.
I remain unconvinced. If the Platonists are to prevail, they must come to grips with a fundamental, and yet unanswered, question:
If mathematics is discovered, then it must already exist. Who created it?
Obviously, you can discover only that which already exists. The Platonists must then provide for mathematics a creator other than a mathematician. Or else, they are obliged to provide a process by which mathematics came to be. Those who believe the world to be made of mathematics should first convert the cosmologists. For the cosmologists-who worry about the creation of the universe-clearly are looking at the wrong problem. What the Platonist position requires is not a theory for the origin of the physical universe, but rather one for the origin of mathematics.
We could, of course, put our faith in a revised version of Genesis 1: In the beginning God created mathematics. Otherwise, we need a new cosmology. We need a big bang theory for mathematics.
I have none to offer. Neither have the Platonists. I'll settle for the constructivist viewpoint. Mathematicians create mathematics.
TRUTH
At first, it seems paradoxical to assert, as did the mathematician Alfred Renyi, "One can know more about things which do not exist than about things which do exist." Yet this is the view that must prevail if we are to find truth in the mathematical world rather than in the real world. For the mathematical world consists only of ideas and abstraction, things that certainly do not exist in the sense that a chair or a tree exists.
Renyi is correct. The truth or falsity of any statement about the real world ultimately depends on its confirmation or denial by examination in the light of observation. Ultimately, we must go to the real world and look.
Consider, for example, this statement: all crows are black. Here we have a simple declarative sentence that asserts the existence of some property of the real world, namely, each fowl that belongs to the collection of birds called crows is black in color. It seems obvious that the assertion is true or else it is false. Either there exists a nonblack crow or there does not. If such a bird exists, the statement is false. Otherwise, it is true.
However, there seems to be little chance of determining-with complete certainty-which of these alternatives holds true. A process might have to be developed whereby every single crow on the face of the earth could be examined for the presence or absence of nonblack feathers. Of course, you might be lucky and find early on a nonblack crow. If you do, the process ends, and you know the statement is false.
But you might not locate such a bird. (I've never seen one.) If not, then you continue. You go on observing and recording the blackness of the world's crows one by one until, somehow or other, you have examined them all. The process seems endless and unpractical. How will you know when you are done?
Moreover, there are serious verification problems that go beyond the technical difficulties associated with the examination process. One is the philosophical question of the reliability of observation itself.
All of us-at some time or another-have encountered a situation in which one or another of our senses has failed us-as did Macbeth's eyesight when he saw the floating dagger with its handle toward his hand. Coffee smells like tea when I have a cold. Glass feels like marble underwater. And once from my window, I saw walking in the rain with his old familiar stride a colleague who had been dead for two years. I know-we all know-that our senses are not always completely reliable. How then can we be certain about black crows? We cannot.
Of course, we can come close. If we do the observation carefully and scientifically, we can eliminate most of the sources of potential error. We can be pretty sure. But being pretty sure is not being certain. And it is not what we mean by truth. You do not find truth by looking.
Moreover, our discussion about crows extends to any assertion regarding the real world. In order to test the hypothesis that a given real-world object has some particular property, a scientist observes the object. The observation may be complicated and it may require the use of sophisticated laboratory equipment, but it is observation nevertheless. You cannot be certain of what you see. When a friend claims that the house across the lake is white, your proper response is: "It seems white. On this side."
Because real-world validity depends ultimately on observation, we must look somewhere else for truth. The real world is the place for wonderful things-sunsets and flowers and poetry. It surely is the place for love. But truth lives somewhere else. Truth lives in the mathematical world.
(Continues...)
Excerpted from MATHEMATICS in 10 LESSONS by JERRY P. KING Copyright © 2009 by Jerry P. King. Excerpted by permission.
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