Exploring a new frontier beyond the realm of human calculation
Computer Math

Paul Lutus, lutusp@arachnoid.com

(Popular Computing, March 1985)

I

magine that a person is brought forward from the year 1500, and you are to tell him the advantages of literacy. You have a difficult job ‑-

he relies totally on his memory and has had no exposure to books. You want to explain the world of ideas to be found in books, but all he can see are funny symbols. You know that reading has enriched your life, and you want to say so. But you’re afraid he may laugh.
Now imagine that you are sent forward into the future, a future in which computers open the door to mathematics as books have opened the door to literacy. Your guide tells you there is a rich human experience in mathematics, an experience as rewarding as any provided by books, an experience you have been denied. You are told of a beautiful mathematical landscape, but all you can see are funny symbols. Your guide tries to explain that you will be a better person, and you laugh.
We are on the threshold of a revolution in mathematics ‑- how we think about it, how we practice it, and how we learn it. The revolution centers on the computer as a mathematical tool, a tool that is becoming recognized as fundamental to mathematics just as books are to literacy. The mathematical revolution may, in due time, become as important as the computer revolution that preceded it and brought it into being.
As in every revolution, people are taking sides ‑- some would like to see a complete exploration of the computer's role in human mathematics, while others feel that using a computer in math is cheating.

Early Mass Storage

The easiest way to strike terror into educators' hearts is to tell them their students can't read. Literacy is the cornerstone of modern education and the gateway to other subjects. We tend to take widespread literacy for granted, to the extent, in fact, that we forget how recently it came into being.
Things were different before about 1500. If you wanted something read or written, you consulted a specialist who probably lived in a hilltop monastery. Books were rare and expensive, and people memorized things they needed to know. In fact, memorization played the central role in education that literacy plays now.
Johann Gutenberg changed all that. One day Gutenberg was carving the letters for an entire page of text into a block of wood, as was the practice of his time. On this particular day he got careless ‑- his knife slipped, passing completely through the block. Rather than throw away the spoiled block, Gutenberg realized something: if he cut the block into individual letters, he could arrange them in any order ‑- and reuse them as well.
The idea of movable type transformed the art of printing and greatly reduced its cost. As is true for many innovations, the first inexpensive books were used in schools to do old things better -‑ they contained passages to be memorized. Those who gazed too long at the page were considered dull. But in time people began to argue that books made memorization unnecessary. Try to imagine the debate in the schools of the time: "The young aren't memorizing, they're reading!" -- "Don't rely on books -- they're flammable!"
We first used books to store a greater number of ideas than we could comfortably memorize. More recently we have formed an active partnership with books -‑ we put old, unchanging ideas in books, thereby freeing our brains to develop new ideas. This specialization takes advantage of the brain's great strength: the ability to synthesize a new idea out of several old ones ‑- and compensates for its weakness: imperfect recall.

How We Practice Mathematics

The general view of mathematics today is surprisingly similar to our view of literacy in 1500: we don't really need it, memorization is good enough, and besides, specialists can do it for us. But that view is changing. Because of a development as unforeseen as Gutenberg's mishap, as unexpected as the printing press, we may create a mathematics to rival literature.
The first change in the mathematical revolution now underway is one of practice. In universities across the country, computers help solve formerly intractable problems, and explore areas beyond the reach of human calculation.
In the early part of this change, computers solved problems expressed in standard mathematical notation. Results were generally provided in numeric form. More recently, novel problems and explorations are being provided to the computer in unorthodox forms ‑- as one researcher put it, the notation is becoming "more algorithmic than algebraic" ‑- and the results are more likely to be presented in graphic than numeric form. Because of the computer's speed, the process is becoming more interactive as well.
Computers model the behavior of bacteria and viruses too dangerous for direct experiment. Spacecraft designs too costly to launch orbit the computer screens of their designers. Low‑cost computer architectural designs are tested by benign computer earthquakes. And more important, the operators of these machines are gaining mathematical knowledge in a way that can only be compared to the sudden acquisition of literacy in a world of books.

Cybermatics

It may be hard to imagine a mathematical partnership with the computer that will be as powerful as our literate partnership with books. But the first steps are being taken, and the result bears little similarity to old mathematics. Therefore I have coined a new word ‑- "Cybermatics" ‑- to describe this partnership. In cybermatics, a computer provides fast, accurate numeric processing for a human who is thereby freed to specialize in the more creative, intuitive part of mathematical reasoning.
Why has mathematics (as distinguished from cybermatics) been such a hard subject? Most subjects require either linear, stepwise thinking, the specialty of the brain's left hemisphere, or nonlinear, intuitive thinking, the specialty of the right. Math requires both. Mathematicians have to provide rigorous linear processes in one hemisphere while simultaneously visualizing and synthesizing in the other ‑- a very rare skill. This is why there are so few talented mathematicians, and why many individuals can perform arithmetic -- or visualize in four dimensions -- but not do both. Albert Einstein, for example, could easily picture a problem in tensor calculus, but had difficulty with arithmetic.
Cybermatics removes the principal barrier to the world of numbers: the numbers themselves and their low‑level manipulations. This doesn't mean the abandonment of mathematical rigor, in fact the reverse is more likely. In one of the first important achievements of computer mathematics, a group at Columbia University programmed their computer to check textbooks of mathematical formulas used in engineering and science. Of the examined texts, all were found to be in error to some extent, and the worst text had 27 percent incorrect formulas. This wasn't merely an academic exercise -- such formulas are used to build bridges and airplanes.
But not all cybermatic projects are so reasoned, and some are downright silly. In the best examples a clever person makes the computer obedient; in the worst, an obedient person hopes the computer is clever.
There are three categories of cybermatic problems. In the first category, the computer provides complex solutions to trivial problems and the operator might be better off with pencil and paper. The operator relies too much on the computer's speed and power and too little on human cleverness. As an example, we might wish to discover how long an apple takes to fall to the ground. A category‑one solution might simply model the problem in slices of time, applying small increments of gravitational acceleration, letting velocity build up, and measuring the apple's position. The category‑one method will give the operator a reasonably accurate result, but it is a waste of computer power because the falling‑apple problem has an easier solution that can be given in a single formula.
In the second category, the computer increases the speed and accuracy of human calculation but doesn’t replace it. Math problems that could be solved by traditional human methods are solved faster or more accurately by the computer. This "supercalculator" role is reasonable if pedestrian, and is the image most people have of the computer. However, like using books for memorization, it enlists new technology to do old things better.
In the third category, the person‑machine partnership begins producing results that neither could accomplish alone. Standard concepts like problem, equation, and solution begin to break down. Projects are undertaken that are entirely beyond the notation and scope of traditional mathematics. A chemist who once used mathematics to guess at the properties of a molecule now may use cybermatics to create and experiment with new molecules, all before entering the laboratory. A nuclear physicist may save electricity by splitting more atoms in a computer than in an accelerator.
Category three provides a new perspective on mathematics -- a view from the top. For example, rather than trying to verify a system of equations that predicts the growth of a bacterial colony, we can create a graphic image of the colony as it develops and see for ourselves, relying on experience and intuition. In cybermatics, the sequence is more often from experiment to equation than the reverse.

Top-Down Training

At the moment, we teach mathematics from the bottom up. First, we present the linear and logical elements -- clerical skills -- relatively simple operations with absolutely certain outcomes. This part encourages the left‑brain half of the class and discourages the right‑brain half. Then about eight years later, we spring a mathematical surprise and introduce advanced elements -- elements that require visualization, abstract reasoning, and intuition. This part discourages most of those who weren't discouraged by the first part. A very small percentage of the class is encouraged by both parts. These people are called mathematicians and aren't invited to dances.
Cybermatics will be taught in reverse order: from the top down. Children will be first exposed to numbers as they are now first exposed to words, by means of images that encourage further understanding. Colorful displays of computer graphics will demonstrate numeric principles through interactive experiment. Subjects such as calculus and geometry may be presented first, because, although algebraically complex, they are visually simple and have many connections with the world outside the classroom. At present, simply calling something calculus is a sufficient deterrent to further study. In the future, because calculus is the mathematics of change and motion, it is likely to be the centerpiece of introductory cybermatics, offering a visual encouragement that its name is powerless to do.
In the course of training, students will examine these topics in greater depth, element by element, from general to specific. They will have the advantage over today's students of first seeing the whole of which the elements are a part. A typical student will emerge with a grasp of mathematics and practical skills in partnership with the computer that will make the present training system seem like intentional discouragement -‑ which, in light of recent developments, it is.
Average students will use cybermatics to acquire a competence with numbers to equal their competence with words. From literacy training we now expect the ability to dwell comfortably in a world of printed ideas but not necessarily diagram a sentence. From cybermatics training we should expect the ability to explore a new world of numbers, but not necessarily to perform long division.
This training will let us see, for the first time, the beautiful landscape of human mathematics -- until now a wilderness traveled by few. Bertrand Russell said, "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 of art can show."
We can employ cybermatics to correct a historic imbalance that causes us to regard literacy as a necessary skill but dismiss mathematics as the impractical pursuit of specialists. This new skill may change us more profoundly than we realize. Mathematical ideas may come to equal words in richness of expression, providing a new human language that favors no territory and requires no translation.