What is Math About? Harvey

A. Smith

I once mentioned to my dentist that I did research in mathematics. Surprised, he asked, "How can you do research in mathematics? Isn't it all known?" Repeating this, which I thought a funny story, I found my non-mathematical friends shared his view.

Teaching a course on the history of modern mathematics to math majors, I would ask them, as their first homework, to interpret the following unrhymed poem by a mathematician:

Paradox

By Clarence R. Wylie, Jr.

Not truth, nor certainty. These I foreswore

In my novitiate, as young men called To holy orders must abjure the world

‘If…, then…,’ this only I assert; And my successes are but pretty chains

Linking twin doubts, for it is vain to ask If what I postulate be justified, Or what I prove possess the stamp of fact.

Yet bridges stand, and men no longer crawl In two dimensions. And such triumphs stem

In no small measure from the power of this game, Played with the thrice-attenuated shades Of things, has over their originals. How frail the wand, but how profound the spell!

A few, inspired by literature classes, gave strange, recondite—even religious—interpretations. None of them understood that Wylie was precisely answering the title question of this lecture. Much of my course was devoted to explaining that in great detail. Here, I will give a one-lecture synopsis.

Ancient mathematics developed to meet practical ends: deciding how much to tax the peasants, building temples and pyramids, and settling boundary disputes after floods obliterated established markers. The latter use came to be called "geometry," from the Greek for "earth measurement." Allegedly, Egyptian priests, seeking to resolve disputes, developed a technique for getting the disputants to agree. Starting from a few assertions held in common by the parties, they would then argue logically to arrive at a conclusion which all must accept and so resolve the dispute. Some Greeks—notably Pythagoras—were fascinated by this practice. Pythagoras developed a religious cult devoted to seeking "truth" by this logical method, which he encountered while traveling in Egypt. Geometry flourished in Greek culture; the motto over the door to Plato's Academy in Athens read, "Let no one ignorant of geometry enter here."

Euclid of Alexandria wrote Elements of Geometry in the reign of the first Greek Pharaoh, Ptolemy I, around 300 BCE. It remained the standard approach to plane geometry for over two thousand years. Euclid stated ten assumptions, to which he thought every right-thinking person must agree. Five, like, "Things equal to the same thing are equal to each other," he called "axioms." A second five, such as, "Any two points can be connected by a straight line," he called "postulates" because he considered them less basic. Statements arrived at by arguing logically, assuming only the axioms and postulates, were called “theorems.” Today, we don’t usually distinguish between "axioms" and “postulates.” The terms “point,” “line,” and “plane”—the idealized “objects” supposedly being studied—were never defined; everyone thought they sort of knew what those terms meant. Many scholars thought one of Euclid's postulates for plane geometry was overly complicated. It said, "For any line, and any point not on that line, there is exactly one line containing that point which has no point in common with the first line." Two lines in the plane that don't intersect are called "parallel," so this is called "the parallel postulate." They felt such a complicated statement should be provable as a theorem, rather than being a separate postulate. For two thousand years, they tried in vain to prove the parallel postulate from the others, rather than assuming it. In 1733, Girollamo Saccheri, a Jesuit priest, convinced himself he had succeeded and published Euclides ab Omni Naevo Vindicatus (Euclid Cleared of Every Blemish). His approach was to assume the falsity of Euclid's parallel postulate and try to establish a contradiction with the others, thereby proving it. Assuming there could be more than one parallel line, Saccheri correctly proved many strange-seeming theorems, but he never really arrived at a contradiction. Finally, he decided the theorems he proved were so bizarre they contradicted common sense. That, he declared, constituted a contradiction of the other axioms.

Other mathematicians tried a similar, but different, approach. Instead of assuming there was only one parallel line—as Euclid had postulated—or more than one parallel line, as Saccheri had, they assumed there were none. Like Saccheri, they found no contradictions. Soon they realized the theorems they were proving described matters well-known to artists who were drawing scenes in perspective.

Imagine yourself as an artist painting a scene on a flat canvas. Everything along the same line extended from your eye corresponds to a single point on the canvas—the closest point on that line blocks the others from view. Now, think of your eye as being at the center of a sphere, with you looking along a line in a particular direction. To get a familiar picture of this situation, think of the sphere as being a globe of the Earth and the line of sight as being toward the point where the Greenwich meridian (0° longitude) crosses the equator (0° latitude). The north pole is overhead, and the south pole is down. You can see only things in front, between 90° east and 90° west.

Now, imagine that you are at the center of a transparent spherical shell. All points on any particular vertical line you see in space will lie at the same longitude of the sphere, so you see and draw the line as a straight vertical line of longitude; if the original line were extended to infinity at both ends, the line would appear to run from pole to pole. Two different parallel vertical lines at the same longitude are seen as the same line by the artist since one lies behind the other. If they lie at different longitudes, they correspond to different longitudes on the globe corresponding to circles—which, unlike the original parallel lines—intersect at the poles. The lines representing them on your canvas approach one another as they become more distant from the eye but, since your canvas is finite, you cannot show them intersecting. (Parallel lines that are not vertical behave similarly—just think of the globe being rotated so the plane containing both the circle and the artist’s eye passes through the north and south poles.)

We can think of our canvas as a plane perpendicular to the direction of vision. If the canvas is infinitely large, every visible point corresponds to a point on the canvas but, as the visible points approach the 90° limit of visibility in any direction, the points on the canvas move infinitely far away toward the edges of the canvas and parallel lines emanating from a distant origin at the center of the canvas would diverge as they approach the edge. This is how artists render a scene in strict perspective. For instance, a road of constant width is shown on the finite canvas as diverging as it gets farther away from its distant source near the center of the painting.

Since it is undefined, we can think of the word “line” as meaning a “great circle” on the visible hemisphere (i.e., a circle centered on the artist’s eye, as the longitudes are.) A great circle arc is the shortest distance between two points on the spherical surface. With the exception of the equator, the latitude circles on a globe are not great circles. If we take “line” to mean the portion of a great circle within the visible hemisphere, including its boundary circle, we can say there are no parallel lines.

Today, mathematicians call the visible hemisphere (together with the infinity circle, its diametrically opposite points being thought of as a single point) the projective plane. (Our artist is "projecting" the hemisphere of his vision onto the plane of the canvas.) A "line" in the projective plane, is defined to mean a great circle on that hemisphere. Similarly, it proved possible to construct a more complicated “hyperbolic” surface on which all Saccheri's theorems hold; as with the spherical surface, except for the parallel postulate, all of Euclid's axioms and postulates hold, but there are many parallels through a point, rather than just one. These examples provide two different "non-Euclidean" geometries. In one there are no parallels, and in the other there are infinitely many. Thus, the parallel postulate is not a theorem that can be proved from the other axioms and postulates.

By the mid-nineteenth century, it became apparent to many mathematicians that, rather than describing "reality," mathematics only studied the logical consequences of a set of axioms. This approach led to the development of wholly new fields of mathematics, including spaces with more than three dimensions—even infinitely many—and "curved" spaces. By the second decade of the twentieth century, when Einstein needed a four-dimensional (three space and one time) curved space for his theory of gravity, which is called general relativity, he found the theory of such spaces was already being studied by some Italian mathematicians and they helped him learn how to use their results. Two decades later, physicists found the mathematics they needed for quantum mechanics in a theory of infinite-dimensional spaces. Today mathematics is seen as being simply the process of studying the logical consequences of a set of axioms about objects that are themselves undefined—just as "line", "point", and “plane” were left undefined by Euclid. If the axioms don't contradict one another, the question of whether they are "true"— in the sense of describing something in the physical universe—doesn't arise. That's a question for physicists and engineers, not mathematicians. Often, when the undefined "objects" (lines, etc.) being discussed in the axioms were given physical interpretations, the results obtained by pursuing the logical consequences of what might seem a bizarre set of axioms proved amazingly useful in unforeseen ways. That is what Wylie's poem is about.

Modern mathematics only says, "Here is a bunch of statements we will call 'axioms’ (The "ifs" of Wylie's poem.) about some undefined objects. We will reason that if they are true, then other statements about those objects, which we will call theorems, must also be true" (the "thens" of the poem.) We make no assertion about the axioms or the theorems themselves being "true" or "certain." Only that if the axioms are assumed to be true then the theorems must logically follow. (Of course, if the axioms contradict one another, some must be discarded, since they can't all be assumed true.)

Let me give a simple example: Suppose there is a (possibly infinite) set of objects which, for convenience, we will sometimes denote by letters. And suppose there is a rule for combining these objects to produce another of the objects. We will denote the combination of the objects a and b (in that order) by a•b. We assume the following axioms:

Axiom 1. For any objects a, b, and c, ((a•b)•c) = (a•(b•c)), where the parentheses denote the order in which the operations are done. That is, if a is combined with b first, (a•b) and the resulting object is combined with c, ((a•b)•c)), it produces the same result as if combine a with (b•c), the result of combining b with c, all in the given order.

Axiom 2. There is a particular object, e, called “the identity,” such that, for any object a, e•a = a•e = a. Combining e with any object, in either order, leaves that object unchanged.

Axiom 3. For any object a, there is an object a-1 (called the inverse of a), such that

A set of objects, with a “combining” operation satisfying 1), 2), and 3) is called a group, and their study is called group theory.

Some easy basic theorems in group theory are:

If e and f are both identities, then e = f. (There is only one identity.)

Proof: f = e•f = e

If c and d are both inverses of a, then c=d. (An object has only one inverse.)

Proof:

One example of a group is all the integers {…,-3,-2,-1,0,1,2,3,…} with • being the usual + operation of addition, e being the number 0, and the inverse of an integer being its negative. Another example is the set of positive rational number with • being the usual operation of multiplication, e being the rational number 1, and the inverse of a positive rational number being its reciprocal: (2/3) • (3/2) = (3/2) • (2/3) = (1/1) = e upon cancellation of the 2’s and the 3’s.

In these examples the operation • also satisfies the rule a•b = b•a. We might conjecture that this is a theorem that could be proved for all groups. That is not true. If we add it as another axiom, we get a special set of groups, called Abelian groups, named after the early 19th century Norwegian mathematician Niels Henrich Abel. Groups in which a•b = b•a doesn’t always hold are called “non-Abelian.” The group of all possible motions of an object in space, with a•b meaning “motion b, followed by motion a,” the identity being not moving, and the inverse of a motion being the reverse motion is an example of a group which is non-Abelian. (To see this, face north and let a be the motion of lying down and b be that of turning to the east. If you turn before lying down, you end up lying on your back, with your legs pointing east.

If you lie down first and then turn east, you end up on your side facing east with your legs pointing north.) You can also try this exercise with a book or a die.

All of the above examples of groups have infinitely many objects. Again, this does not follow from the axioms. A set with only two elements e and a and the operations e•a = a•e = a, a•a = e, (a is its own inverse,) and e•e = e is a group. (Let e represent the set of all even integers and a represent all the odd integers. and • be the usual addition. The operations then just say that the sum of any even and any odd integer is odd, the sum of any two odd integers is even, and the sum of any two even integers is also even.)

There are mathematicians who spend their lives studying the theory of groups. It was a notable achievement when they were able to show how to construct all the finite groups, a project only completed, in 2004, after decades of effort. It took hundreds of pages to establish their results! Group theory is considered a part of "Abstract Algebra," (sometimes called "Modern Algebra" in older texts.")

There are also "Modern Geometries"—even geometries with only finitely many points and lines. They are sometimes named after the mathematician who first studied them. For instance, "Fano's Geometry" and "Young's Geometry" share the following four axioms, but they differ in a fifth, with Fano using the projective plane axiom of no parallels and Young using Euclid's parallel axiom of exactly one parallel:

Axiom 1. There exists at least one line.

Axiom 2. There are exactly three points on every line.

Axiom 3. Not all points are on the same line.

Axiom 4. There is exactly one line on any two points (two points determine a line.)

Axiom 5. (Fano) There is a common point on any two lines (i.e. there are no parallel lines.)

Axiom 5. (Young) For each line L and point P, not on L, there is exactly one line on P having no point in common with L. (i.e. Euclid’s “parallel postulate.”)

You might be interested in proving (or looking up) that Fano's geometry has exactly seven points and seven lines, while Young's geometry has exactly nine points and thirteen lines.

Young children are taught arithmetic without any axioms—often they start by counting their fingers. Does that mean that elementary arithmetic is not part of what we now call mathematics? In 1889, the Italian mathematician Giuseppe Peano introduced five axioms about “natural numbers”.

Axiom 1. Zero is a natural number.

Axiom 2. Every natural number has a “successor”, which is also a natural number.

Axiom 3. Zero is not the successor of any natural number.

Axiom 4. If the successors of natural numbers are the same, those natural numbers are the same.

Axiom 5. If a set contains zero and also contains the successor of every natural number in the set, then the set contains all the natural numbers.

Starting from those axioms, one can define all of simple arithmetic, indeed all of the classical objects of analysis: the negative integers, the rational numbers, the real numbers, and the complex numbers. This was expounded, brilliantly and completely, in the little book Grundla- gen der Analysis (Foundations of Analysis) that the great number theorist Edmund Landau initially wrote for the education of his daughters°. Published in 1930, it became an international best seller and is still in print and widely available in many languages, including English.

Wylie’s poem tells us that mathematics is not about truth; it’s about logical consistency. There can in fact be a danger in believing that mathematics describes nature. It may lead to overconfidence in making statements about nature based on mathematical assumptions, and a belief the future can be predicted using mathematics. Sometimes it can, and philosophers have wondered why this is so, but mathematical results only reflect what logically follows from the assumptions being made, which may or may not be valid. Many predictions based on mathematical models are highly accurate, but others have proved to be extremely misleading because their underlying assumptions turned out not to be true. The hope is always that in such cases, avé Einstein, a “truer” set of axioms can be found. But it is not guaranteed.

The axioms of plane geometry may be close enough to "truth" for laying out the boundaries of a farmer's field, but for determining the boundaries of a large state or world-wide exploration, the geometry of a sphere is needed. For even higher accuracy, the fact that the earth is not a sphere, but is flattened at the poles to be (approximately) an "oblate spheroid" may be required. For astronomical distances and powerful gravitational fields, the curved space-time of Einstein's relativity is needed. And to understand the physics involved in the basic workings of a microchip, one needs mathematical spaces of infinite dimension.

The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colours or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.

G. H. Hardy