Popular Science Monthly/Volume 21/August 1882/Transcendental Geometry
TRANSCENDENTAL geometry is the geometry of solids and surfaces in n-dimensional or in curved space. Exactly what surfaces and what solids is a hard question to answer, and the answer is still harder to understand. Let us, then, first find out in what way or ways the science of transcendental geometry arose.
Descartes invented a method of applying algebra to geometry by the well-known Cartesian co-ordinates. As you remember, a point in a plane is determined by two co-ordinates, x and y, for example; a point in space by three, x, y, and z. Now, the question is not unnatural, "What would x, y, z, v, determine?" The natural answer is, "A point in space of four dimensions."
Moreover, we see that, although we have no experience of space of four dimensions, we could form equations between four variables, and transform and combine them as we do in analytic geometry of three dimensions. By adopting a code of interpretation as like to our ordinary code as circumstances would permit, we could interpret the relations of our equations as geometrical relations.
But, as the idea of a fourth dimension to space is almost if not quite inconceivable, let us endeavor to render it less so if possible. Imagine a man deprived of everything but vision, in the way of sensible experience. The world to him would be two dimensional. If, then, he were taken out to drive, he would see continual changes in his plane of vision, but he would ascribe them merely to the effects of time. For example, were he to go through a covered bridge, his sensations might be as follows: A small dark spot, gradually enlarging till it covers the field of vision; then a small bright spot in the middle of it, which would similarly enlarge.
Now, suppose our universe sliced in two by a plane which moved along through it. Suppose sentient beings inhabited this plane. They would perceive at once two dimensions of our universe and the third as a succession in time. So we might suppose ourselves conscious of three dimensions of our universe, and of the fourth as the succession of things in time. Thus we might consider time as a dimension. It is so considered in the mechanical curves of position. Yet we should then have to bring in time relative to time. We will illustrate still further by considering the theory of knots. It is evident that, so long as the line represented in the adjoined figure is kept in the plane, the knot or kink can not be got out of it. But, by turning the loop up, it can be removed at once.
The annexed knot—the type of all knots in ordinary space—can not be undone without severing the ends. In four-dimensional space it could. By this means Zöllner interpreted some of the knot-untying performances of Slade, the American spiritualist. Let us, for example, interpret these facts, using time as fourth dimension, bringing in, of course, time relative to time. If time were a fourth dimension, parts of the state of things at different instants might be visible together. Thus we could have A, C, B, after being tied, joined to A, D, B, and then A, C, B, before being tied.
But we must remember, in passing on, that algebraic equations are capable of other than geometrical interpretations, and that their relations by themselves prove nothing in regard to real or possible relations between external facts. Moreover, the algebraic theory of dimensionality will be interpreted fully by nothing less than a space of infinite dimensionality.
We come now to the most difficult branch of the subject, that of curved surfaces and of curved space. The curvature of a plane curve at any point is the limit of the ratio of the length of the curve to the difference in direction of the initial and terminal tangents. Its differential expression is or . To get the curvature of a curved surface at any point, we slice it up by planes normal to it at that point. On each of these planes it will describe a curve. These curves will have different curvatures at the original point. The reciprocal of the product of the greatest and least of these is called by Gauss the measure of curvature. This name he also applied to an analogous function of the co-ordinates of a point in space. The expression, for a plane curve, of the curvature is the reciprocal of the radius of the circle of closest possible contact at the point investigated. Hence, some have argued that transcendental geometry was inconsistent, in that it talked about the curvature of a space where there were not Euclidean straight lines, hence no radii, and nothing to refer the curvature to. This argument is open to other answers, but it is enough to say that the measure of curvature has no necessary connection with radii.
To return, the condition that a rigid figure can be moved about on a surface without changing its shape, or that a rigid body can be similarly moved in space, is that the measure of curvature of the surface or space is constant in value. Some one might say that, if a body is rigid, no motion can change its shape. This, however, is not true of the mathematically rigid body except under the above conditions, taking the most general definition of a rigid body.
It is assumed in Euclid that motion of a figure does not alter it. That is, if an angle, A B C, is equal to an angle B C D, it will be equal to it however it is as a whole moved or rotated. This is an assumption that the measure of curvature of the plane or space is constant. Moreover, if we assume it constantly equal to naught, the so-called geometrical axioms that two straight lines can not inclose a surface, etc., are true. For example, a spherical surface has a constant measure of curvature not equal to zero, and positive. Since the shortest distance between two points is a straight line, let us, extending the analogy, call the shortest distance between two points of a spherical surface, lying wholly in that surface, a straight line of that surface. Now, as the measure of curvature of a spherical surface is constant, we can slide a figure about over the surface without altering it, as is evident at once. On a sphere, however, more than one perpendicular can be drawn on the surface from a point to a straight line, and two straight lines can inclose a surface.
In a surface whose curvature is negative, an infinite number of straight lines of the surface can be drawn through a given point which will never meet a given straight line. Such a surface would be like a spool. Some of its sections would be concave and others convex to the same point. We have analogous results in what is called curved space. These results were first suggested by Riemann, who was a pupil of Gauss.
For this mathematical treatment all that is needed is, first, algebra and the differential calculus; secondly, a method of interpreting them geometrically. We have found a code of interpretation for some algebraic equations which give geometrical results, and we apply it so far as we can to all.
So far the mathematicians might have gone without let or hindrance, and there some of them, as Boole and Grassman, stopped. But others thought they had settled whether the geometrical axioms were a priori truths or not. We have just worked out a system of geometry, said they, which is not, as we think, impossible, where these axioms do not hold. Therefore these axioms are the results of an experience of things as they are. If we had had a different order of things, as is possible, these axioms would not have been true nor thought of. I shall, however, try to prove that, although not thought of, they are true.
The geometrical axioms express relations; relations between what?
Geometry is a branch of mathematics. Therefore the geometrical axioms express mathematical relations. What, then, is mathematics, and with what does it deal?
Mathematics, in its widest sense, I will define as the science which treats of logical—that is necessary—relations. Between outside things there are no necessary relations. The relation of cause and effect is sometimes called necessary; but, if so, it is not usually handled mathematically. The relations must, then, be of mental things.
They are not relations between images or imaginations of outside things, for two reasons: First, the relations between imaginations can be no more necessary than the things they image; second, the imaginations of men's minds are different. One may imagine a line as a chalk-line on a blackboard; another, as the edge of a knife; I myself, as the boundary between crystal faces.
However, in all our minds there is something the same in each. It is the concept or idea. And it is of concepts and ideas that mathematics treats.
Here Mill seems to make a mistake. He says, "The points, lines, circles, and squares which any one has in his mind are simply copies of the points, lines, circles, and squares which he has known in his experience." To his mind, then, the function of thought, when we think of circles, is to reproduce some original sensation more or less vividly. This, however, is what I call imagination; and we have tried to prove that imaginations were not the objects of mathematical treatment. Helmholtz acknowledges this when he says that the axioms of geometry, taken by themselves out of all connection with mechanical propositions, represent no relations of real things.
We will notice certain other facts about concepts and words, in connection with their mathematical relations. The first is the persistence of concepts. By this I mean that an idea once formed, by whatever means, experimental or otherwise, does not depend upon the continual recurrence of the same experience for its continued existence.
That is, having once formed an idea of a baby hippopotamus, by having seen one in Barnum's Great Show, I have that idea, which is called into use on various other occasions—such as hearing of it in the newspapers. It is not at all necessary that I should renew the experience every time Barnum comes around. It is, of course, true that a concept may be disused, but its use may be made common as well by unlike as by like experiences.
However, on closer inspection of the hippopotamus, my conception may be new. This leads us to our other all-important distinction and division. Every name has a denotation and a connotation. Its denotation is usually of things, its connotation is conceptual. Some words, proper names especially, correspond to things, the ideas attached to which vary according to the varying aspect of the thing. Other words, however, correspond to ideas; these words are applied or not to things according as there are experiences coming under the concept to which they are attached.
This distinction between words with fixed denotation and varying connotation and words of fixed connotation and varying denotation is quite important, as we shall see. Let us first, however, return to our hippopotamus. This is a word for me of at least partially fixed denotation; it must include the animal that I saw; it must not include an ordinary pig. The connotation would be almost indefinite. This word has, then, a fixed denotation varying connotation, approximately. On the other hand, take the name, rigid body. This is a name with a denotation varying down to zero, perhaps, but its connotation is changeless.
Thus we see that mathematics may be defined as the science of the relations of concepts. Its vocabulary, too, must be one of fixed connotation. That is why symbols are so useful; their connotation does not vary unconsciously.
Benjamin Peirce defines mathematics as the science that draws necessary conclusions. Mill says, "The problem is—given a function, what function is it of some other function?" It is obvious that necessary conclusions can be drawn only so far as there are relations fixed whence to draw them; the function must be given before we find its relations with other functions.
Now, I wish to insist, as strongly as I can, that any set of concepts become fit for mathematical handling as soon as their relations are unfolded, and this is what I have so far proved. If you ask, "Whence these concepts?" my answer is, "From experience." From it comes the "element of intuition" that Stallo says is an element in every geometrical axiom. Space itself is but a product of experience. If a man could only hear or taste, would he have our concept of space? I trow not.
Let us now, after this long digression, return to our transcendentalists. Euclidean geometry and non-Euclidean alike are mathematical. Verbally they come to different conclusions, but neither conclusion affects facts. The difference is here, it seems to me. Transcendental geometry is the offspring of analytic, though some have tried to treat it otherwise. The relations that it handles are at first algebraic relations that may apply to anything. Then applying the geometric nomenclature to algebraic expression, calling expressions of the first degree linear, etc., it interprets these results geometrically. Its definitions, thus, are different from those of Euclid; the ideas connoted by its vocabulary are different; its concepts are not the same. It is not wonderful, then, that it gets a broader field of relations.
We decide, then, that from their respective definitions the Euclidean and the transcendental geometry are true. And this is, perhaps, the most important point to settle, for the transcendentalists have said that, although the geometrical definitions were true, the axioms need not be. We, however, say that the axioms, or what you will, of parallelism, etc., are part of the connotation of the words defined, and are simultaneously given. Of course, some experience is necessary to make us form any concepts.
The question now to be answered is, then, Which are the best definitions? But it must be remembered that, as long as we are dealing with mathematics, we are never dealing with real things. Thus Helmholtz is wrong in saying that by adding any mechanical axioms or principles we can obtain an empirical science out of geometry, if the science thus obtained is purely mathematical.
Mathematical concepts can have two virtues in varying degrees, namely, simplicity and resemblance to, or rather correspondence with, external reality. First, they must be simple, that is, their relations to one another must be easily handled; second, their relations must correspond more or less closely with the relations of some set of external things. They do not correspond absolutely. There are no external things which have the properties of mathematical straight lines except approximately. Take the annexed figure.
One would not hesitate to call AB, BC, CB, straight lines, and to say that the triangle ABC has the sum of its angles equal to 180°. The error he would make (they are drawn with compasses) we always make in kind, though not in degree, in applying mathematics to realities. I wish to make clear that the relation between mathematical truths and external facts, is one of resemblance, not identity. What the essence of resemblance is I shall not discuss.
No external facts can do more than change the utility of the two geometries. At present, for simplicity and accuracy of resemblance to external facts, the Euclidean geometry need not fear being swallowed up. If, however, facts should be discovered which could be most simply correlated to transcendental truths, transcendental geometry might become important.
Let us recapitulate. We have tried to show that mathematics deals only with concepts, and that the two geometries are, therefore, also conceptual. Their apparent discrepancy we tried to account for by showing that they used different concepts. We showed that, although concepts might be originated by sensations, they were not, nor were affected by, external facts. The relation between mathematical truths and external facts is one of more or less resemblance, not of identity. Nor can the resemblance be ever proved to be perfect.
The Euclidean geometry has as great facility in accommodating itself to all known facts as the transcendental, and greater simplicity. It is therefore of greater practical utility. The mathematical truth of each is not affected by experience.
Thus transcendental geometry, with its egg-shells turned inside-out without cracking, its knots mysteriously untied, its worlds where the background of everything is a man's own head, is from its conceptual basis, as a creation of man's mind, true. It is a pretty mathematical diversion; it is, as yet, nothing more.