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Elizabeth Hodes, Ph.D. 1988 - 1989

Lecture Perspective

EVERYTHING THAT can be known has number . . . For it is not possible to conceive of or to learn anything that has not.

-Philolaus of Croton         

We can discover by purely mathematical constructions the concepts and the laws . . . which furnish the key to the understanding of natural phenomena . . . the creative principle resides in mathematics.

-Albert Einstein        

. . . the pursuit of mathematics is a divine madness of the human spirit.

-Alfred North Whitehead        

MAN COUNTS because he has a deep urge to understand the world he is part of. We seek order as the prerequisite to our understanding. The quest for order involves a fundamental human ability-the knowledge of number-that is, mathematical thinking. Mathematics reflects what human beings are and what they think. It has been called both an art and a science and the language of science. It is one of the oldest and most continuously pursued branches of human thought. Why?

In Greek mythology, three goddesses each coveted a golden apple marked "For the Fairest." The Trojan prince, Paris, was asked to judge which of the three deserved the apple-Hera, queen of the gods, who promised him power; Aphrodite, goddess of love and beauty, who promised him the most beautiful woman in the world; or Athena, goddess of reason and truth, who promised him great wisdom in battle. In this lecture, I hope to convince you that mathematics offers all three goddesses' gifts- Power, Beauty, and Knowledge-along with Mystery.

WITHOUT MATHEMATICS, one will never penetrate to the depths of philosophy. Without philosophy, one will never penetrate to the depths of mathematics. Without both, one will never penetrate to the depths of anything.

-Gottfried Wilhelm Leibniz

The Animal That Counts

Elizabeth Hodes, Ph.D.
Professor of Mathematics

TODAY IS INDEED A high point for me. When a group of faculty members came to my office last spring to announce I had been chosen to be the Faculty Lecturer this year, I felt surprised, proud, appreciated, humble, apprehensive and challenged-and I still do! I am proud and warmed by having been selected by my colleagues and students for this honor- and I want to thank the many of them who provided assistance and support, especially Professors George Frakes and Janice Peterson, the Santa Barbara City College Geology Department, and the technical people behind the scenes, Messrs. Tom Zeiher, Steve Reese and Jeff Barnes.

I feel humble because so many others of our faculty are fine instructors. I feel apprehensive because I want to live up to the high caliber of previous lecturers. Last, I feel challenged. I say challenged because I am fully aware that many of you have a vast unfamiliarity, and possibly even the opposite of appreciation, about the nature of mathematics. Indeed, when I tried my first title-"In Praise of Mathematics"-on a few friends, they suggested that, if I mentioned "mathematics" in the title, nobody would come. I am glad, though, that so many of you were interested in hearing about "The Animal that Counts."

We are the animal that counts because we are an animal with a reasoning mind. We have a deep urge to understand the world of which we are a part, and so we try to find order in that world. We search for patterns. Whether they exist and are discovered or whether we create conceptual order and impose it onto an unordered universe does not matter to me. Suffice it, that we seek order as the prerequisite to understanding. And the ideal form of that quest for order is our ability to count-our knowledge of number, our power not just to find the patterns, but to abstract and establish relationships using mathematics.

For mathematics is not simply in the sums or equations on paper, but "in the mind that gives them meaning." I believe that the ability to conceptualize with numbers is a major part of what separates us from other animals. I find support for this in the extraordinary permanence and universality of mathematics. I maintain that this is because mathematics reflects some fundamental aspect of what human beings are and how they think.

So I want to introduce you to the landscape of the mathematician. Mathematics is a very large country, so I have had to select only a little of what is available to see there. I have regretfully had to pass over many mathematical giants, names like Descartes, Newton, Cauchy, Euler and Gauss, and a wealth of profound mathematical discoveries. I have chosen four examples from the history of mathematics which are linked by a common thread and will, I hope, prove interesting even if you've never had much math. They are brief illustrations of the mathematician's search for order.

Pythagoreans and Square Numbers

We are the animal that counts, I said. We alone have invented numbers and are able to realize many levels of abstraction through them. Though we begin with counting concrete objects, soon we leave the objects behind. I have noticed that even children have a rudimentary sense of abstraction in their counting rhymes.

We juggle the numbers themselves, without constraining images of physical objects. As we do this, we may start to notice patterns that were obscured before. The patterns are in the numbers themselves and are often both surprising and pleasing. Let me illustrate: let's take a look at the odd numbers. (You may be familiar with odd and even numbers from counting off for sports teams or, if you're as old as I am, from the gas crunch of the 1970s.)

If I take the first two odd numbers, I and 3, and add them, I get 4. Now let me add the first three odd numbers I + 3 + 5, giving 9. Keep on, with the first four odds; the sum is now 16. The first five odds give us 25. Can you predict the sum of the first six odds? If you predicted 36, you've caught the pattern! For some of you who can remember your school mathematics, you may recall that 4, 9, 16, 25 and so on are called perfect squares. The pattern we just noticed is that the sums of the first "n" odd numbers will be equal to "n-squared." You can get some insight why this must be so from this diagram (Fig 1).

If you became just a bit excited as you began to see the pattern emerging, you are not alone. Around 500 B.C., there was a cult of number-worshippers, followers of a legendary pre-Hellenic Greek, named Pythagoras. According to the fragmentary writings we have from Plato and other Greeks and Romans, Pythagoras was a mysterious teacher, a guru if you will, who had travelled to Egypt and to Persia and studied the priestly lore of those countries. He taught that there were three attributes of the soul: emotion, intelligence and reason, but that only man possessed reason. He also taught that "number rules the universe," which is why we mathematicians like him.

It was the observation of patterns like the one we have just seen that confirmed his followers, the Pythagoreans, in their conviction that numbers were mysterious and powerful objects. They believed that adepts in the study of numbers could share some of the power, so they studied diligently. Their motives were magical, so they frequently went chasing off onto non-mathematical tracks like numerology. Along the way, however, they made some remarkable discoveries. We shall never be far from the Pythagoreans and their influence as we continue our tour.

The Bequest of the Greeks

If we jump forward 200 years now, we find the Hellenic Greeks building on the number discoveries of the Pythagoreans and adding many geometric discoveries. Geometry began as a very practical study. A concern for measurement and distances was probably natural for a people who were great sea traders and architects. After all, you don't want your temples to the divinities to be lopsided!

But Greek geometry set itself apart from pragmatic geometry to become a study of ideals and an exemplar of deductive reasoning. The Pythagoreans had been convinced that the study of mathematics could reveal to man hidden properties of the world. The Greek geometers conceived of mathematics as an exercise which trained the mind and so they set themselves geometrical tasks not so much for magical power or practical results as for the mental challenge.

One such challenge was the problem of construction. Working with just two tools, a straight edge and a compass, the one to draw lines, the other to draw circles, but neither one enabling you actually to measure lengths or angles, the Greek geometers asked what else can we draw?

Among other things, one can construct certain regular polygons, closed figures, each side the same length, each angle exactly the same size. The construction of these regular polygons was a bit abstract, but had a practical application. By starting with a regular triangle inside a circle, for example, then going half-way between the vertices or corners, we double the number of sides, but still get a regular figure, in this case a hexagon or six-sided polygon. We can get 12 sides, 24 sides, and so on by repeating the process. As we keep going, notice that our polygons have become almost indistinguishable from the circle. The mathematical phrase for this is that "in the limit, the polygons approach the circle." If we measure the length of one side and multiply that by the number of sides we will get the total distance around the entire figure, called the perimeter. Since our polygon is practically coincident with the circle, we can measure the curved distance around the circle by using successively closer "polygonal" approximations.

The construction of regular polygons had other surprising aspects. Triangles, squares and hexagons were relatively easy, but constructing a regular five-sided figure, known as a pentagon, was a real puzzle. You may have noticed that what is difficult to do often gains an aura of mystery? Think of the nickname "Magic" Johnson, for instance. So the pentagon, or the pentagram as it became known in occult lore, was invested with many mysterious powers-rather like our own Pentagon in Washington today.

It turned out that an essential part of the construction of a regular pentagon was to draw two lengths whose ratio to each other is known as the Golden Section or Divine Ratio. Even if you've never heard of it, you have been surrounded by it all your lives. Next time you look at a rectangular table or window or even a deck of cards, check the ratio of the length to the width. Chances are good that you'll find the Divine Ratio In the drawing by Leonardo da Vinci which you have on your program cover, the human figure is based on the Divine Ratio. The nineteenth century psychologist, Gustav Fechner, reported that, in his research, people identified a rectangle whose length and width had this particular ratio, the Divine Ratio, as the most graceful of all shapes.

I'd like to explain how to create two segments that form the Divine Ratio. Relax. You won't be tested on this, but it will give you a nice way to draw perfect five-pointed stars (Fig 2).

1. In a circle whose center is at point O, construct two perpendicular diameters AB and CD.
2. Now find the midpoint of radius OB. Call it M for middle.
3. Then construct a new circle, centered at M with radius MC (the dotted one in the figure), and let this circle intersect the previous diameter AB
   at point E.
4. Point E divides the radius OA into two segments whose ratio is the Divine Ratio.

Moreover, the segment OE can be marked off around the circle exactly 10 times, so that by connecting every other mark, we can get a regular pentagon. This connection to the Divine Ratio imbued the regular pentagon with even greater mystery.

Adding to the attraction of the regular pentagon was yet another fact which fascinated the Pythagoreans, and fascinates me, too. If you try to build solid figures with faces that are identical regular polygons, you might think there are dozens of possibilities. After all, there are hundreds of regular polygons. There are, in fact, only five solids that work. My drawing isn't fantastic, but, in the top row, you see solids composed of four triangles, eight triangles and 20 triangles. Then on the bottom, you see one of six squares, which we know as the cube, and another of 12 pentagons. That's it (Fig 3).

The Greek philosopher, Plato, placed great significance on this. He was much taken with mathematical regularities. In his scheme of the universe, earth was at the center of the universe, because we are most important, and the other five known planets, the sun and stars moved around it at uniform speed in circles, for circles were perfect geometrical shapes. Plato identified four essential elements with four of the regular solids. The four-faced tetrahedron, which pointed upward from a solid base corresponded to fire; the pointy, eight-sided octahedron was linked to air; and the rather smooth 20-sided icosahedron he matched with water. The stable cube he matched with solid earth. But the 12-sided dodecahedron built of that magical figure, the pentagon, Plato identified with the quintessence, the heavenly element. These five Platonic solids and Plato's cosmology figure are in our next stop, but we first need to travel forward in time to the sixteenth century.

Johannes Kepler and 'The Harmony of the Universe'

The animal that counts is an animal able to find abstract relationships. Of course, abstraction is not limited to mathematics. We speak of beauty, of wealth, liberty and justice. But mathematical abstractions sometimes have an astonishing connection to objects of the physical world. Johannes Kepler in his laws of celestial motion found such a connection between a family of ideal curves known as ellipses and the positions and speeds of the planets as they move through space.

Kepler was a fascinating man, whom we know through his naively enthusiastic and candid writings, many of which have been saved. What scientist today would write comments on his work like "Oh, what a mistake!" or "a manifest hallucination"? He was born in 1571 in a Germany of small princedoms, merchant states and religious warfare. In Europe, at that time, the vocabulary and concepts of physics, as we know it, were being born. Chemistry was still alchemy. Artists had only recently introduced perspective into drawing, and architects were experimenting with the domes we see on the cathedrals of Florence and Rome. European traders and explorers were venturing to Africa, Asia and the Americas. New discoveries abounded. Old ideas were changing. The Polish astronomer, Copernicus, had even suggested that the Platonic view of the universe, in which the earth was at the center and the sun and all other heavenly bodies circled around it, should be replaced by a universe in which the earth and five other known planets circled around the sun.

In this world of tumult and change, Kepler was a convert to Copernicanism. He was also committed wholly to the idea that there was a mysterious order to the universe and the key to understanding that order lay in mathematics. At the age of 22, following completion of a degree in theology at the University of Tuebingen, he was offered a post as a teacher of mathematics and astronomy at Gratz. He had studied astronomy, but knew little mathematics. He did not let that stop him-not Kepler. He accepted the appointment and began to study mathematics on his own. Kepler must have had a lot of time for his studies, for, after a grand start with two students his first year, his reputation as a teacher the following year left him with zero students. But he taught himself well. He soon mastered the mathematics of a family of curves known from Greek geometry as conic sections because if you cut through a cone set on a circular base, you get these curves as cross-sections. They are called the ellipse, at the top of the diagram (projected during lecture), the parabola, below it, and the hyperbola, on the right. Kepler became an expert on them, and his knowledge was to be pivotal later on in his astronomical inquiries.

His interest in astronomy revolved around two questions. Why should there be exactly six planets and why were their distances from the sun and their speeds what they were? He may have been pragmatic in asking how fast the planets travel, because, as an astronomer, he was called on to predict solstices and equinoxes. But Kepler's primary motive was not practical. Rather it was a question of faith to him. This world could not be as it was by chance. There had to be some mathematical explanation.

By 1596, the young Kepler thought he had found that explanation in the five Platonic solids. There were six planets, Kepler thought, because there were five solids. Each was separated from its neighbors by one of the regular solids, and, when you ran out of solids, you would run out of planets. The spacing of the orbit of each planet, he felt, was determined by the sizes of the solids. In Kepler's model, the sun was at the center and circling it in a somewhat spherical shell were Mercury and each of the other planets, including earth, nestled up against their corresponding solids. The points of each solid kept the planetary spheres properly spaced apart. You can get some idea of his cosmology from this sketch he made of it (Fig. 4). So taken was he with this scheme that Kepler went on to assign music intervals to the distances between the planets and sun. You may have come across the phrase, "the music of the spheres." He also proposed that marriage laws and other civil matters should be related to these solids. I don't know how effective that would have been, but I do know Kepler had an unhappy marriage.

Kepler's revelation seemed a true vindication of mathematics and its importance. Of course, his wonderful scheme would subsequently fall apart. As we now know, but Kepler did not, there are more than six planets. So the entire significance of the five solids disappears.

Although his findings were flawed, I still consider Kepler a prime example of the mathematician's faith in order. His great pseudo-discovery led him to real discoveries. Trying to defend his ideas to himself, he could not quite get his mathematical results about the distances between the planets to match observational data, and so he sought more precise and complete data.

At that time, the best source for such data was the Danish nobleman and astronomer, Tycho Brahe. Brahe was a colorful character, who, in addition to being a well-known astronomer, was also known for his temper and a false nose he wore after his own had been sliced off in a duel. So Kepler, in 1599, went to work for Brahe. From him, Kepler learned the art of patient and accurate observation of the motions of the moon and planets. Even better, after a long apprenticeship, he was given Brahe's own observations on the motion of the planet Mars and asked to work on this, the most complicated of all the observed orbits. As you can see in this time-lapse photograph (projected during lecture), Mars appears to make fancy curlicues as it moves through space. It was a fortunate assignment for Kepler because Mars' orbit is the least circular planetary orbit. Eighteen months later, when Tycho died, Kepler kept the data in his possession, over the objections of Tycho's relatives and heirs, and continued working.

After eight years of computation, he published in 1609 the Astronomia Nova or New Astronomy; in which he abandoned centuries of earlier astronomical tradition. First of all, he renounced uniform speed. Second, he abandoned circular orbits. Instead, Kepler used his knowledge of the conic sections and his work with the orbit of Mars to argue that the planets, including earth, moved through the heavens in elliptical orbits with the sun at a special geometric point called a focus, and, moreover, that they do not move at a uniform speed, but rather sweep out equal areas in equal times. That is, they go faster at the end of the orbit that is closer to the sun and slower at the opposite extreme
(Fig 5). Kepler's third major insight, that the square of the period was proportional to the cube of the mean distance from the sun, related velocity and location of the planet. It was published in the Harmonice Mundi in 1619, a modest book which sought merely to unite music, geometry, astrology, astronomy-and the ultimate secrets of the universe.

These three laws were important statements about universal relations. They were expressed in mathematical terms, for Kepler was a mathematician. But his mathematical work gave physicists a framework for their investigations. Kepler's great successor, the genius, Isaac Newton, was able to establish all three of Kepler's laws from one physical principle, a universal Law of Gravitation. Newton, I have to confess, was the more famous and the greater scientist, for he created the link between celestial motion and terrestrial physics. Nonetheless, Kepler is an embodiment of the animal that counts, and the modern exploration of space owes much to him and his Pythagorean faith.

Matrices and Modern Physics

We have traveled from patterns in numbers to regular polygons, to regular solids, and finally to mathematically expressed laws that connect the motions of the planets with cross-sections of cones. Along the way, we have seen that mathematics can both reveal pleasing patterns and provide essential tools for describing physical reality The esthetic and practical aspects of mathematics seem inseparably intertwined.

Curiously often, in fact, what has appeared in mathematics to be a creation of pure imagination, having no counterpart in the physical world, has later turned out to model the natural world in an unanticipated and unsuspected way. Perhaps that is why mathematics is sometimes called an art and sometimes a science. This quality is well illustrated by the last stop on our tour. Kepler connected mathematics to the motions of the planets through the vast expanse of space. My final illustration connects a mathematical invention, called a matrix, to the motion of the electron in the tiny world of the atom.

Matrices-or matrix if you have only one of them-were developed only a little over a hundred years ago by the independent efforts of an array of mathematicians. The Irishman, Sir William Rowan Hamilton, was guided by an interest in the geometric representation of complex numbers, that is, numbers whose squares may produce negative numbers, a very upsetting idea to mathematicians then and to algebra students now. In pursuit of this problem, he developed in 1843 an entirely new mathematical object called a quaternion because it had four parts to it. The Englishman, Arthur Cayley, was led to matrices around the same time through his work on problems of multi- dimensional geometry. Other important contributions came from the German mathematician, Hermann Grassmann, and the American, Benjamin Pierce, and his son, Charles S. Pierce.

What these men did was not to create new rules, as Kepler had, but rather an entire new class of objects. These objects were "multi-dimensional." They were not single numbers at all, but involved arrangements of many numbers in rows and columns. In a matrix like the one you see here (Fig 6), we speak of the one-one element meaning first row, first column, or the three-five element, meaning third row, fifth column, but the matrix is much more than the sum of its parts.

Matrices are used today to describe many disparate phenomena in which the relationships to be studied are numerous, complicated and interconnected, as in economics, aeronautics, statistics, weather forecasting, tracing the spread of epidemics, or predicting voter behavior. They can be used to predict the results of successively altering mathematical figures by stretching them, rotating them, inverting them, projecting them onto other surfaces, and so on-changes known as mathematical trans-formations. They are also useful in finding solutions to systems of several equations that have all to be true simultaneously-such as equations describing the effects of the planets' mutual gravitational pulls on one another.

Matrices, as you might surmise, are somewhat more complicated than ordinary numbers. A matrix summarizes the information about an entire system. By manipulating the single entity, the matrix, you can work with the entire system. Moreover, a matrix, in one sense which you mustn't take literally, can act either like a noun or a verb-that is, it can behave like a number and have things done to it or as an operator that does things to other matrices.

They have their own special rules of addition and multiplication, too. In order to fit its inventors' needs, multiplication of matrices had to violate a basic property of ordinary arithmetic, the property of commutativity. Commutativity is a property your elementary school teachers counted on when they told you that four multiplied by five is equal to five multiplied by four, that when you want to check your addition on a column of figures, first add from the top down and then add from the bottom up and make sure that the two results agree. Commutativity guarantees that you do not have to worry about which object comes first in a calculation. Now multiplication and addition of ordinary numbers are commutative. Division is an example of something that is not commutative. Six divided by three gives a very different result from three divided by six. Multiplication of two matrices very definitely was not commutative.

This makes them cumbersome to handle, and, besides, the problems they were being applied to in the nineteenth century-geometrical transformations, the representations of complex numbers-were of somewhat limited interest, so for the next 30 years matrices remained the preserve of a small group of mathematicians. Among these mathematicians, however, were two of the premier mathematicians of this century, Richard Courant and David Hilbert, at the University of Goettingen in Germany. Courant was preparing an encyclopedic work on mathematical methods of physics and, in the course of this enormous project, his assistant, the young mathematician, Pascual Jordan, became interested in the esoteric algebra of matrices. He turned to them as Kepler had turned his mind to the conic sections, and became quite knowledgable about them.

There my story might have ended, except that matrices, which had arisen from the study of mathematical transformations, of setting loose a matrix "operater" on another matrix, had several important features. They were not commutative, as I said before. They assigned numbers to a particular spot described by two indicators or indices-a row and a column. Remember the one-one element and the three-five element? Moreover, matrices have another property, but here I must digress for a moment. We sometimes characterize mathematical entities as being either continuous or discrete. A continuum is an indivisible spread of numbers, like the flow of time. You might think of a discrete variable as something like time on a digital watch-it stays 2:01 for a whole minute and then suddenly changes to 2:02 with no intermediate values. Discrete mathematics involves distinguishable, separate, discontinuous values. Matrices, with their individual entries in each row-column cell, are part of discrete mathematics. And these were exactly the kinds of properties that two physicists, also at Goettingen, Max Born and Werner Heisenberg, were looking for around 1925 to describe the light energies observed to be emitted from atoms.

I am very much reminded of Kepler, who could not reconcile theoretically predicted values with observed values and who, after eight years of work, finally found a solution by discarding old ideas and turning to the ellipse. Born and Heisenberg were dissatisfied with certain "after-the-fact" corrections that had to be made in applying the equations of classical physics to models of the atom that had been developed early in this century. Heisenberg especially felt that attempts to use the concepts of position, time, velocity, momentum and so on to describe atomic states were not suitable. He thought progress required that these concepts be abandoned, just as Kepler had abandoned circular orbits and uniform speed. Instead, thought Heisenberg, the description of the atom must be based on what could be observed: on optical measurements of the frequency, polarity and intensity (or amplitude) of light emitted when electrons moved from one stationary state to another, not on the unobservable things like position or momentum. He used sets of complex numbers to represent these "observable" quantities. Each complex number in the set was linked to two indices, the original state of the electron and its final state. But his sets had a strange "commutation principle." Certain multiplications he needed to perform were not commutative in a very specific way.

He took his ideas to Born, who noticed that some of Heisenberg's rules of computation for his "states" and "observables" were rather like vaguely-recalled rules of computation with-you guessed it-matrices. In one of those "right place at the right time" coincidences, Born was travelling on a train and happened to mention to a friend his need for an assistant who understood matrices. Sitting in the same compartment on the train was Jordan-do you remember him--the mathematician who had mastered matrix algebra. So began a collaboration of Born, Heisenberg and Jordan. By substituting matrices for the continuous position and momentum variables of classical mechanics equations, they made a quantum jump in physicists' ability to describe atomic behavior. And so once again, mathematics showed its extraordinary ability to provide a structure for thinking about reality.

Conclusion

Many people unfortunately associate mathematics with tedious calculating. That's a shame, because one of the most important achievements in mathematics is the correct formulation of a problem, not the calculation of a numerical solution. Mathematics makes the accomplishments of science and engineering possible because it gives them a structure in which to formulate problems and a language for discussing these problems. We have seen the power of this language, ranging from a curiosity about number, to a search for patterns and causal chains, combining an urge toward abstraction and generalization with freedom to explore any possible system the mind can create. Like all science and all art, mathematics brings order out of chaos.

When I was in grade school, I read about something called a geode. This is one (displayed during lecture). On the outside, it is dull, unimposing, possibly even unattractive. The stone is very hard and this one required a diamond saw to cut it open. But the interior-ah, the interior of a geode is a glittering masterpiece! Mathematics is like this geode. Not easy to get at, but, once opened, a beautiful crystalline world, a magical and mighty and mysterious realm of the mind-but different from other mysteries like art, music, or poetry because it appears to be the language of nature.

Plato wrote that Socrates had said, "The understanding of mathematics is necessary for a sound grasp of ethics." I think he had in mind that its study reinforces our rational capacity-which is in some disrepair at this moment. If we are the most important animal, the animal that most counts in this world because of our power, both destructive and constructive, then we cannot ignore the study of mathematics.