July 18, 2006

Algebra and American Civilization

Zernik on Derbyshire’s Unknown Quantity

[VDARE.COM note: Wolfgang Zernik is one of our favorite Today’s Letter writers. We’re pretty fond of John Derbyshire too. This article may seem a bit off-topic for us, but Wolfgang doesn’t think so]

By Wolfgang Zernik

Readers of VDARE.COM will recognize John Derbyshire as a witty and provocative writer of conservative political columns that have occasionally been published here although his regular home is National Review. Derbyshire is a true renaissance man with a variety of interests. He was originally trained as a mathematician and has recently published a well-received book in that field, Prime Obsession.  He is also the author of two novels, Fire from the Sun and Seeing Calvin Coolidge in a Dream, and writes book reviews and literary articles for the New Criterion and other magazines. Now he has published Unknown Quantity: A Real And Imaginary History of Algebra, a book designed for the general reader that meets his usual high standard.

Why is this book being reviewed here, on a webzine best known for dealing with immigration, the National Question and related political matters? One answer is a personal one. Many of us are fans of John Derbyshire and find anything he writes noteworthy.

Another answer is much deeper. It has to do with how we wish the American civilization to develop in the future as a result of the immigration policies we chose now. If we desire that the pursuit of excellence in the arts and sciences should continue, we need to understand the potential for human accomplishment of the immigrant groups we admit. It is helpful in this regard to follow in detail how progress in a particular field, algebra in this case, has been made over the course of history. Whatever else it may be, the history of algebra is after all the story of an intellectual triumph—a splendid victory of the human spirit that we must hope can be often repeated in our great country.

Derbyshire describes his intended audience as "curious non mathematicians unencumbered by fear of formulae". The level of knowledge that is assumed is that covered by a typical high school course. It is of course helpful if the reader has gone beyond that, but it's not necessary. When additional explanations are needed, Derbyshire has thoughtfully inserted "Math Primer" sections where the necessary technical material (but no more than that) is carefully provided.

So if your math education stopped at high school and even if you have forgotten much of that, you should have no problem in reading this book. However, there is no denying the fact that careful attention must be paid. Derbyshire never makes the mistake of underestimating the intelligence of his readers!

The book is divided, like ancient Gaul, into three parts.

Part I "The Unknown Quantity" is a straightforward history of algebra from four thousand years ago through the Middle Ages to the Renaissance. This is the period during which what we can very roughly describe as "high school" algebra was developed. The typical reader will learn more about history than algebra here. However, I certainly leaned some algebra as well—for instance how easy it is to solve cubic or quartic equations.

The time-span involved here is enormous, about three and a half millennia. The story begins where civilization itself began in Babylon and moves on to Egypt, then to the Islamic Empire at its height in the ninth century. Here the modern Hindu-Arab system of numerals was introduced and algebra acquired its name. The story moves on to Italy in the thirteenth century where general methods for solving equations were developed and finally to France where, in the seventeenth century, algebra began to look like the material we learned in high school.

One question that comes to mind: why did it take so long to develop what we now think of as elementary algebra? For thousands of years, only very slow progress was made. As Derbyshire writes in a characteristic aside:

"I fear that at this point the reader may be slipping into the conviction that these ancient and medieval algebraists were not very bright. We stated in 1800 BCE with the Babylonians solving quadratic equations written as word problems and now here we are, 2,600 years later with al-Khwarizmi ... solving quadratic equations written as word problems."

One explanation, as Derbyshire points out, is the very high level at which the subject dwells. Another reason has to be the extraordinarily clumsy notation which was all that was available in the early years. After all, before the introduction of modern numerals, even arithmetic was horrendously difficult. Imagine doing long division with Roman numerals. What a nightmare!

Before the modern notation for a symbolic algebra, equations had to be described in words. This made the finding of solutions very difficult and so it is remarkable that any progress at all was made. Once the notation problem was solved in the seventeenth century, the way was clear for progress into modern algebra.

This brings us to Part 2 of the book entitled "Universal Arithmetic", a name for algebra used by Isaac Newton. At this point the curious nonmathematical reader can look forward to learning something he did not learn in high school.

The seventeenth and eighteenth centuries were a period of consolidation in algebra when little new was invented. What held up progress was a stubborn problem that turned out to be much harder than anticipated. This was the problem of finding a general algebraic solution of the quintic.

You may remember that in high school you learned the simple formula for the solution of the quadratic equation. What you probably did not learn was that, in the sixteenth century, Italian mathematicians had developed similar although of course more complicated solutions for the cubic and quadratic i.e. equations where the highest power of x was three or four. The next logical step was to find a general solution for those equations where the highest power of x was five, the notorious quintics. The trouble was—no one was able to do it. Imagine the frustration.

For two hundred years, algebraists lived and died trying in vain to solve the next obvious problem. Some must have thought that this was the end of algebra. But of course it wasn't.

By the beginning of the nineteenth century it began to dawn on some algebraists that the reason they could not solve the quintic was perhaps that a solution does not exist. Some publications began to appear claiming to prove that the quintic has no general algebraic solution. But the idea was too novel to accept. For some twenty years or so, there were bitter arguments about whether these proofs were valid.

In the end, however, a universally accepted proof that the quintic has no solution was published. The year was 1826. Derbyshire tells us that this year closes the first great epoch in the history of algebra.

What happened in the first decades of the nineteenth century was the gradual discovery of new mathematical objects. Algebraists realized that the "unknown variable" did not necessarily have to represent a number. Instead, it could stand for any object as long as the rules for combining such objects were specified. This is the basis for modern algebra—or rather algebras, for there are now several.

The simplest way of generalizing is to go from a single number to an array of numbers with rules determining how such arrays are to be added or multiplied. Two examples are vectors and matrices. Derbyshire devotes a chapter to each.

If you took a physics course in high school, you may remember that in physics a vector is defined as a quantity that has both a magnitude and a direction. Examples are velocity, force and electric field strength. Since we live in a three-dimensional world, a vector can be represented by its three components (north-south, east-west and up-down). Thus a physics vector is an array of three numbers.

Mathematicians, however, have a more general way of defining vectors. When you read the book, you will easily learn some very modern math about vector spaces and algebras. At the end of the day, though, it is still true that a vector is a one-dimensional array, and so the first modern algebra is vector algebra.

Matrices are also arrays, but in two dimensions rather than just one. For example, an (i x j) matrix has i rows and j columns. Since the rules for manipulating and combining matrices are known, a matrix algebra easily follows.

The history of matrix algebra is however more complicated than my simplistic explanation suggests. It involves detours through Han dynasty China over 2,000 years ago and seventeenth century Japan. The history of algebra, like that of mathematics as a whole, has always been a world-wide phenomenon.

Finally, we come to Part 3 which has the honest but slightly ominous title "Levels of Abstraction".

Here the story is brought up to date. In the nineteenth century, algebra continued to move further into the realm of the abstract. The first and by far the most useful (especially to physicists) of these new objects are groups.

Consider, for example, a child's play cube sitting on your desk. You can rotate that cube in a number of ways so that it will still look the same—for example, turn it through 90 degrees or 180 degrees. Two such rotations applied in sequence will clearly give you just another rotation.

We can express this by saying that all possible rotations of the cube form a set. Combining two members of the set yields just another member of the same set. Sets with this property are, in a natural use of language, called groups.

In this case, we have defined the group of rotations of a cube. If you specify how all the elements of a group are combined, you will have, by the previous definition, specified an algebra but an algebra whose elements represent operations rather than numbers. So here's a simple but not rigorous example of an abstract algebra.

Derbyshire describes more abstract algebras such as rings and fields. As these are more complicated—also less useful—I'm going to take a pass on further explanations. (I do want to express some surprise however at the way algebraists have taken the word "field" which is a basic technical term in physics and have used it in a different and quite unconnected sense—rather like President Bush has hijacked the term “immigration reform”!

And now for something completely different. As Derbyshire says several times, the history of algebra is essentially a story of increasing levels of abstraction. The trouble is that the average reader can take only so much of this.

Look, let's be frank, the eyes may tend to glaze over. The successful author has to find a way to deal with this problem. Derbyshire does deal with it and as readers of Prime Obsession will remember he does so by telling lots of entertaining personal stories.

Mathematicians (as far as I know) are no more odd than members of other professions. But they certainly have their share of strange characters and sometimes, too, they lived in what the Chinese would call interesting times. So the fund of interesting, scandalous, sexual or political anecdotes is very large and Derbyshire makes skillful use of it. The more abstract the material, the more Derbyshire uses the personal to lighten the technical.

For example, the important chapter on group theory is engagingly entitled "Pistols at Dawn" and begins by telling the dramatic life story of Evariste Galois, the brilliant young man who founded the field of group theory and died tragically in a duel at the age of only 20. It is only after several pages of this entertainment that the reader finds himself in technical material where careful attention must again be paid.

Enjoy the book!