March 2003

# The Forgotten Revolution

We mathematicians are used to the fact that our subject is widely misunderstood, perhaps more than any other subject (except perhaps linguistics). Misunderstandings come on several levels.

One misunderstanding is that the subject has little relevance to ordinay life. Many people are simply unaware that many of the trappings of the present-day world depend on mathematics in a fundamental way. When we travel by car, train, or airplane, we enter a world that depends on mathematics. When we pick up a telephone, watch television, or go to a movie; when we listen to music on a CD, log on to the Internet, or cook our meal in a microwave oven, we are using the products of mathematics. When we go into hospital, take out insurance, or check the weather forecast, we are reliant on mathematics. Without advanced mathematics, none of these technologies and conveniences would exist.

Another misunderstanding is that, to most people, mathematics is just numbers and arithmetic. In fact, numbers and arithmetic are only a very small part of the subject. To those of us in the business, the phrase that best describes the subject is "the science of patterns," a definition that only describes the subject properly when accompanied by a discussion of what is meant by "pattern" in this context.

It's not hard to find the reasons for these common misconceptions. Most of the mathematics that underpins present-day science and technology is at most three or four hundred years old, in many cases less than a century old. Yet the typical high school curriculum covers mathematics that is for the most part at least five hundred and in many cases over two thousand years old. It's as if our literature courses gave students Homer and Chaucer but never mentioned Shakespeare, Dickens, or Proust.

Still another common misconception is that mathematics is mainly about performing calculations or manipulating symbolic expressions to solve problems. But this misconception is different. Whereas a scientist or engineer - indeed anyone who has studied any mathematics at the university level - will not harbor the first two misconceptions, possibly only pure mathematicians are likely to be free of this third misconception. The reason is that until 150 years ago, mathematicians themselves viewed the subject the same way. Although they 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.

In the middle of the nineteenth century, however, a revolution took place. One of its epicenters was the small university town of Goettingen in Germany, where the local revolutionary leaders were the mathematicians Lejeune Dirichlet, Richard Dedekind, and Bernhard Riemann. In their new conception of the subject, the primary focus was not performing a calculation or computing an answer, but formulating and understanding abstract concepts and relationships - a shift in emphasis from doing to understanding. Within a generation, this revolution would completely change the way pure mathematicians thought of their subject. Nevertheless, it was an extremely quiet revolution that was recognized only when it was all over. It is not even clear that the leaders knew they were spearheading a major change.

The 1850s revolution did, after a fashion, eventually find its way into school classrooms in the form of the 1960s "New Math" movement. Unfortunately, by the time the message had made its way from the mathematics departments of the leading universities into the schools, it had been badly garbled. To mathematicians before and after 1850, both calculation and understanding had always been important. The 1850 revolution merely shifted the emphasis as to which of the two the subject was really about and which was the supporting skill. Unfortunately, the message that reached the nation's school teachers in the 60s was often, "Forget calculation skill, just concentrate on concepts." This ludicrous and ultimately disastrous strategy led the satirist Tom Lehrer to quip, in his song New Math, "It's the method that's important, never mind if you don't get the right answer." (Lehrer, by the way, is a mathematician, so he knew what the initiators of the change had intended.) After a few sorry years, "New Math" (which was already over a hundred years old) was dropped from the syllabus.

For the Goettingen revolutionaries, mathematics was about "Thinking in concepts" ( Denken in Begriffen). Mathematical objects were no longer thought of as given primarily by formulas, but rather as carriers of conceptual properties. Proving was no longer a matter of transforming terms in accordance with rules, but a process of logical deduction from concepts.

Among the new concepts that the revolution embraced are many that are familiar to today's university mathematics student; function, for instance. Prior to Dirichlet, mathematicians were used to the fact that a formula such as y = x^2 + 3x - 5 specifies a rule that produces a new number (y) from any given number (x). Dirichlet said forget the formula and concentrate on what the function does. A function, according to Dirichlet, is any rule that produces new numbers from old. The rule does not have to be specified by an algebraic formula. In fact, there's no reason to restrict your attention to numbers. A function can be any rule that takes objects of one kind and produces new objects from them.

Mathematicians began to study the properties of abstract functions, specified not by some formula but by their behavior. For example, does the function have the property that when you present it with different starting values it always produces different answers? (The property called bijectivity.)

This approach was particularly fruitful in the development of real analysis, where mathematicians studied the properties of continuity and differentiability of functions as abstract concepts in their own right. In France, Augustin Cauchy developed his famous epsilon-delta definitions of continuity and differentiability - the "epsilontics" that to this day cost each new generation of mathematics students so much effort to master. Cauchy's contributions, in particular, indicated a new willingness of mathematicians to come to grips with the concept of infinity. Riemann spoke of their having reached "a turning point in the conception of the infinite."

In 1829, Dirichlet deduced the representability by Fourier series of a class of functions defined by concepts. In a similar vein, in the 1850s, Riemann defined a complex function by its property of differentiability, rather than a formula, which he regarded as secondary. Karl Friedrich Gauss's residue classes were a forerunner of the approach - now standard - whereby a mathematical structure is defined as a set endowed with certain operations, whose behaviors are specified by axioms. Taking his lead from Gauss, Dedekind examined the new concepts of ring, field, and ideal - each of which was defined as a collection of objects endowed with certain operations.

Like most revolutions, this one had its origins long before the main protagonists came on the scene. The Greeks had certainly shown an interest in mathematics as a conceptual endeavor, not just calculation, and in the seventeenth century, Gottfried Leibniz thought deeply about both approaches. But for the most part, until the Goettingen revolution, mathematics remained primarily a collection of procedures for solving problems. To today's mathematicians, however, brought up entirely with the post-Goettingen conception of mathematics, what in the 19th century was a revolution is simply taken to be what mathematics is. The revolution may have been quiet, and to a large extent forgotten, but it was complete and far reaching. The only remaining question is how long it will take nonmathematicians to catch up.

Devlin's Angle is updated at the beginning of each month.

Mathematician Keith Devlin ( devlin@csli.stanford.edu) is the Executive Director of the Center for the Study of Language and Information at Stanford University and "The Math Guy" on NPR's Weekend Edition. Thid column was adapted from his most recent book The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time, published last fall by Basic Books.