## Devlin's Angle |

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.