Mathematics and the Imaginationby Edward Kasner, James Newman
Anyone who gambles, plays cards, loves puzzles, or simply seeks an intellectual challenge will love this amusing and thought-provoking book. With wit and clarity, the authors deftly progress from simple arithmetic to calculus and non-Euclidean geometry. "Charming and exciting." — Saturday Review of Literature. Includes 169 figures.See more details below
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Anyone who gambles, plays cards, loves puzzles, or simply seeks an intellectual challenge will love this amusing and thought-provoking book. With wit and clarity, the authors deftly progress from simple arithmetic to calculus and non-Euclidean geometry. "Charming and exciting." — Saturday Review of Literature. Includes 169 figures.
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Mathematics and the Imagination
By Edward Kasner, James Newman
Dover Publications, Inc.Copyright © 1967 Ruth G. Newman
All rights reserved.
New Names for Old
For out of oldefeldes, as men seith, Cometh al this newe corn fro yeer to yere; And out of olde bokes, in goodfeith, Cometh al this newe science that men lere.
Every once in a while there is house cleaning in mathematics. Some old names are discarded, some dusted off and refurbished; new theories, new additions to the household are assigned a place and name. So what our title really means is new words in mathematics; not new names, but new words, new terms which have in part come to represent new concepts and a reappraisal of old ones in more or less recent mathematics. There are surely plenty of words already in mathematics as well as in other subjects. Indeed, there are so many words that it is even easier than it used to be to speak a great deal and say nothing. It is mostly through words strung together like beads in a necklace that half the population of the world has been induced to believe mad things and to sanctify mad deeds. Frank Vizetelly, the great lexicographer, estimated that there are 800,000 words in use in the English language. But mathematicians, generally quite modest, are not satisfied with these 800,000; let us give them a few more.
We can get along without new names until, as we advance in science, we acquire new ideas and new forms.
A peculiar thing about mathematics is that it does not use so many long and hard names as the other sciences. Besides, it is more conservative than the other sciences in that it clings tenaciously to old words. The terms used by Euclid in his Elements are current in geometry today. But an Ionian physicist would find the terminology of modern physics, to put it colloquially, pure Greek. In chemistry, substances no more complicated than sugar, starch, or alcohol have names like these: Methylpropenylenedihydroxycinnamenylacrylic acid, or, Oanhydrosulfaminobenzoine, or, protocatechuicaldehydemethylene. It would be inconvenient if we had to use such terms in everyday conversation. Who could imagine even the aristocrat of science at the breakfast table asking, "Please pass the O-anhydrosulfaminobenzoic acid," when all he wanted was sugar for his coffee? Biology also has some tantalizing tongue twisters. The purpose of these long words is not to frighten the exoteric, but to describe with scientific curtness what the literary man would take half a page to express.
In mathematics there are many easy words like "group," "family," "ring," "simple curve," "limit," etc. But these ordinary words are sometimes given a very peculiar and technical meaning. In fact, here is a booby-prize definition of mathematics: Mathematics is the science which uses easy words for hard ideas. In this it differs from any other science. There are 500,000 known species of insects and every one has a long Latin name. In mathematics we are more modest. We talk about "fields," "groups," "families," "spaces," although much more meaning is attached to these words than ordinary conversation implies. As its use becomes more and more technical, nobody can guess the mathematical meaning of a word any more than one could guess that a "drug store" is a place where they sell ice-cream sodas and umbrellas. No one could guess the meaning of the word "group" as it is used in mathematics. Yet it is so important that whole courses are given on the theory of "groups," and hundreds of books are written about it.
Because mathematicians get along with common words, many amusing ambiguities arise. For instance, the word "function" probably expresses the most important idea in the whole history of mathematics. Yet, most people hearing it would think of a "function" as meaning an evening social affair, while others, less socially minded, would think of their livers. The word "function" has at least a dozen meanings, but few people suspect the mathematical one. The mathematical meaning (which we shall elaborate upon later) is expressed most simply by a table. Such a table gives the relation between two variable quantities when the value of one variable quantity is determined by the value of the other. Thus, one variable quantity may express the years from 1800 to 1938, and the other, the number of men in the United States wearing handle-bar mustaches; or one variable may express in decibels the amount of noise made by a political speaker, and the other, the blood pressure units of his listeners. You could probably never guess the meaning of the word "ring" as it has been used in mathematics. It was introduced into the newer algebra within the last twenty years. The theory of rings is much more recent than the theory of groups. It is now found in most of the new books on algebra, and has nothing to do with either matrimony or bells.
Other ordinary words used in mathematics in a peculiar sense are "domain," "integration," "differentiation." The uninitiated would not be able to guess what they represent; only mathematicians would know about them. The word "transcendental" in mathematics has not the meaning it has in philosophy. A mathematician would say: The number ?, equal to 3.14159 ..., is transcendental, because it is not the root of any algebraic equation with integer coefficients.
Transcendental is a very exalted name for a small number, but it was coined when it was thought that transcendental numbers were as rare as quintuplets. The work of Georg Cantor in the realm of the infinite has since proved that of all the numbers in mathematics, the transcendental ones are the most common, or, to use the word in a slightly different sense, the least transcendental. We shall talk of this later when we speak of another famous transcendental number, e, the base of the natural logarithms. Immanuel Kant's "transcendental epistemology" is what most educated people might think of when the word transcendental is used, but in that sense it has nothing to do with mathematics. Again, take the word "evolution," used in mathematics to denote the process most of us learned in elementary school, and promptly forgot, of extracting square roots, cube roots, etc. Spencer, in his philosophy, defines evolution as "an integration of matter, and a dissipation of motion from an indefinite, incoherent homogeneity to a definite, coherent heterogeneity," etc. But that, fortunately, has nothing to do with mathematical evolution either. Even in Tennessee, one may extract square roots without running afoul of the law.
As we see, mathematics uses simple words for complicated ideas. An example of a simple word used in a complicated way is the word "simple." "Simple curve" and "simple group" represent important ideas in higher mathematics.
The above is not a simple curve. A simple curve is a closed curve which does not cross itself and may look like Fig. 2. There are many important theorems about such figures that make the word worth while. Later, we are going to talk about a queer kind of mathematics called "rubber-sheet geometry," and will have much more to say about simple curves and nonsimple ones. A French mathematician, Jordan, gave the fundamental theorem: every simple curve has one inside and one outside. That is, every simple curve divides the plane into two regions, one inside the curve, and one outside.
There are some groups in mathematics that are "simple" groups. The definition of "simple group" is really so hard that it cannot be given here. If we wanted to get a clear idea of what a simple group was, we should probably have to spend a long time looking into a great many books, and then, without an extensive mathematical background, we should probably miss the point. First of all, we should have to define the concept "group." Then we should have to give a definition of subgroups, and then of self-conjugate subgroups, and then we should be able to tell what a simple group is. A simple group is simply a group without any self-conjugate subgroups— simple, is it not?
Mathematics is often erroneously referred to as the science of common sense. Actually, it may transcend common sense and go beyond either imagination or intuition. It has become a very strange and perhaps frightening subject from the ordinary point of view, but anyone who penetrates into it will find a veritable fairyland, a fairyland which is strange, but makes sense, if not common sense. From the ordinary point of view mathematics deals with strange things. We shall show you that occasionally it does deal with strange things, but mostly it deals with familiar things in a strange way. If you look at yourself in an ordinary mirror, regardless of your physical attributes, you may find yourself amusing, but not strange; a subway ride to Coney Island, and a glance at yourself in one of the distorting mirrors will convince you that from another point of view you may be strange as well as amusing. It is largely a matter of what you are accustomed to. A Russian peasant came to Moscow for the first time and went to see the sights. He went to the zoo and saw the giraffes. You may find a moral in his reaction as plainly as in the fables of La Fontaine. "Look," he said, "at what the Bolsheviks have done to our horses." That is what modern mathematics has done to simple geometry and to simple arithmetic.
There are other words and expressions, not so familiar, which have been invented even more recently. Take, for instance, the word "turbine." Of course, that is already used in engineering, but it is an entirely new word in geometry. The mathematical name applies to a certain diagram. (Geometry, whatever others may think, is the study of different shapes, many of them very beautiful, having harmony, grace and symmetry. Of course, there are also fat books written on abstract geometry, and abstract space in which neither a diagram nor a shape appears. This is a very important branch of mathematics, but it is not the geometry studied by the Egyptians and the Greeks. Most of us, if we can play chess at all, are content to play it on a board with wooden chess pieces; but there are some who play the game blindfolded and without touching the board. It might be a fair analogy to say that abstract geometry is like blindfold chess—it is a game played without concrete objects.) Above you see a picture of a turbine, in fact, two of them.
A turbine consists of an infinite number of "elements" filled in continuously. An element is not merely a point; it is a point with an associated direction—like an iron filing. A turbine is composed of an infinite number of these elements, arranged in a peculiar way: the points must be arranged on a perfect circle, and the inclination of the iron filings must be at the same angle to the circle throughout. There are thus an infinite number of elements of equal inclination to the various tangents of the circle. In the special case where the angle between the direction of the element and the direction of the tangent is zero, what would happen? The turbine would be a circle. In other words, the theory of turbines is a generalization of the theory of the circle. If the angle is ninety degrees, the elements point toward the center of the circle. In that special case we have a normal turbine (see left-hand diagram).
There is a geometry of turbines, instead of a geometry of circles. It is a rather technical branch of mathematics which concerns itself with working out continuous groups of transformations connected with differential equations and differential geometry. The geometry connected with the turbine bears the rather odd name of "turns and slides."
* * *
The circle is one of the oldest figures in mathematics. The straight line is the simplest line, but the circle is the simplest nonstraight curve. It is often regarded as the limit of a polygon with an infinite number of sides. You can see for yourself that as a series of polygons is inscribed in a circle with each polygon having more sides than its predecessor, each polygon gets to look more and more like a circle.
The Greeks were already familiar with the idea that as a regular polygon increases in the number of its sides, it differs less and less from the circle in which it is inscribed. Indeed, it may well be that in the eyes of an omniscient creature, the circle would look like a polygon with an infinite number of straight sides. However, in the absence of complete omniscience, we shall continue to regard a circle as being a nonstraight curve. There are some interesting generalizations of the circle when it is viewed in this way. There is, for example, the concept denoted by the word "cycle," which was introduced by a French mathematician, Laguerre. A cycle is a circle with an arrow on it, like this:
If you took the same circle and put an arrow on it in the opposite direction, it would become a different cycle.
The Greeks were specialists in the art of posing problems which neither they nor succeeding generations of mathematicians have ever been able to solve. The three most famous of these problems—the squaring of the circle, the duplication of the cube, and the trisection of an angle—we shall discuss later. Many well-meaning, self-appointed, and self-anointed mathematicians, and a motley assortment of lunatics and cranks, knowing neither history nor mathematics, supply an abundant crop of "solutions" of these insoluble problems each year. However, some of the classical problems of antiquity have been solved. For example, the theory of cycles was used by Laguerre in solving the problem of Apollonius: given three fixed circles, to find a circle that touches them all. It turns out to be a matter of elementary high school geometry, although it involves ingenuity, and any brilliant high school student could work it out. It has eight answers, as shown in Fig. 6(a).
They can all be constructed with ruler and compass, and many methods of solution have been found. Given three circles, there will be eight circles touching all of them. Given three cycles, however, there will be only one clockwise cycle that touches them all. (Two cycles are said to touch each other only if their arrows agree in direction at the point of contact.) Thus, by using the idea of cycles, we have one definite answer instead of eight. Laguerre made the idea of cycles the basis of an elegant theory.
Another variation of the circle introduced by the eminent American mathematician, G. J. Keyser, is obtained by taking a circle and removing one point. This creates a serious change in conception. Keyser calls it "a patho-circle," (from pathological circle). He has used it in discussing the logic of axioms.
We have made yet another change in the concept of circle, which introduces another word and a new diagram. Take a circle and instead of leaving one point out, simply emphasize one point as the initial point. This is to be called a "clock." It has been used in the theory of polygenic functions. "Polygenic" is a word recently introduced into the theory of complex functions —about 1927. There was an important word, "monogenic," introduced in the nineteenth century by the famous French mathematician, Augustin Cauchy, and used in the classical theory of functions. It is used to denote functions that have a single derivative at a point, as in the differential calculus. But most functions, in the complex domain, have an infinite number of derivatives at a point. If a function is not monogenic, it can never be bigenic, or trigenic. Either the derivative has one value or an infinite number of values—either monogenic or polygenic, nothing intermediate. Monogenic means one rate of growth. Polygenic means many rates of growth. The complete derivative of a polygenic function is represented by a congruence (a double infinity) of clocks, all with different starting points, but with the same uniform rate of rotation. It would be useless to attempt to give a simplified explanation of these concepts. (The neophyte will have to bear with us over a few intervals like this for the sake of the more experienced mathematical reader.)
The going has been rather hard in the last paragraph, and if a few of the polygenic seas have swept you overboard, we shall throw you a hexagonal life preserver. We may consider a very simple word that has been introduced in elementary geometry to indicate a certain kind of hexagon. The word on which to fix your attention is "parhexagon." An ordinary hexagon has six arbitrary sides. A parhexagon is that kind of hexagon in which any side is both equal and parallel to the side opposite to it (as in Fig. 7).
If the opposite sides of a quadrilateral are equal and parallel, it is called a parallelogram. By the same reasoning that we use for the word parhexagon, a parallelogram might have been called a parquadrilateral.
Here is an example of a theorem about the parhexagon: take any irregular hexagon, not necessarily a parhexagon, ABCDEF. Draw the diagonals AC, BD, CE, DF, EA, and FB, forming the six triangles, ABC, BCD, CDE, DEF, EFA, and FAB. Find the six centers of gravity, A', B', C, D', E', and F' of these triangles. (The center of gravity of a triangle is the point at which the triangle would balance if it were cut out of cardboard and supported only at that point; it coincides with the point of intersection of the medians.) Draw A'B', B'C, CD', D'E', E'F', and F'A'. Then the new inner hexagon A'B'CD'E'F' will always be a parhexagon.
Excerpted from Mathematics and the Imagination by Edward Kasner, James Newman. Copyright © 1967 Ruth G. Newman. Excerpted by permission of Dover Publications, Inc..
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