Art. LXVIII.—Transcendental Geometry: Remarks suggested by Mr. Frankland's Paper, “The Non-Euclidian Geometry Vindicated.”*
[Read before the Philosophical Institute of Canterbury, 7th October, 1886.]
In the paper referred to, Mr F. W. Frankland implies that the views he advocates are generally accepted by living mathematicians—e.g., on page 59, paragraph 4: “He [Professor Clifford] says, in common with most living mathematicians who have studied this question, that space may be finite”; and again, on page 60, paragraph 6: “To the expression ‘geometers of the Euclidian school’ I take exception, believing that none such are left, in the sense in which Mr. Skey uses the word. The triumph of the non-Euclidian geometry, or, I will say, the ‘general’ geometry, has been complete. I can safely appeal, on this point, to any distinguished member of any Mathematical Society in Europe or America.”
Now, I am quite aware that, if this were an accurate description of the state of mind of most living mathematicians and distinguished members of Mathematical Societies, it would be an extremely rash proceeding on my part to enter into the controversy. One could only gaze in wonder at those superior beings who roamed at large in space of the (n + 1)th degree, while we poor mortals had to be content with three dimensions.
I cannot think that Mr. Frankland is justified in demanding a greater admission than this: that there are (or have been), distinguished mathematicians holding those views, and that Mathematical Societies have, as in duty bound, allowed the discussion of them in their meetings and in their journals.
[Footnote] * “Trans. N.Z. Inst,” vol. xviii., p. 58
I shall not enter into all the questions raised between Mr. Frankland and his critic, Mr. Skey, but content myself with noticing the three most important propositions laid down in the two papers contributed by the former. These are:—
That the axioms of geometry may be only approximately true:
That the actual properties of space may be somewhat different from those which we are in the habit of ascribing to it:
That the extent of space may be a finite number of cubic miles.
If these propositions are sound, the transcendental geometers may be right; if not, the position of the Euclidian geometers, who maintain that space has three dimensions, and three only, remains unassailed.
The subject, of course, has often been discussed, and the argument on the orthodox side is well represented by Stallo, (“Concepts of Modern Physics,”) and Lotze (“Metaphysic”). While acknowledging my obligations to these great writers, each of whom, however, gives only part of the argument, I shall endeavour to state the case in a somewhat different form:—
(1.) “The axioms of geometry may be only approximately true”; or, again, as Mr. Frankland says in another place, “geometry is a physical and experimental science.”
This idea of geometry, though countenanced by John Stuart Mill, is founded upon a serious misconception as to what the subjects are of which geometry treats. The line of reasoning pursued is shortly as follows:—‘Geometry treats, among other things, of straight lines; but straight lines cannot be conceived apart from objects, and nowhere are we acquainted with lines that are more than approximately straight. Therefore geometry is only an approximate science.’ The argument, as Stallo and others have shown, contains its own answer. How do you know that any given line must be only approximately straight, except by reference to some standard? The very phrase “only approximately straight” implies the existence of such a standard in the mind of the person who makes it. When Mr. Frankland speaks of a line on his supposed manifold as having such feeble curvature as hardly to be distinguishable from a Euclidian straight line, he is really implying this standard. In a similar manner it could be shown that we must admit the concepts of a line, a surface, a plane surface, a right angle, a solid, and so on.
In fact, geometry is the science of such standards as this, or rather of such concepts as this. It has been, I think, rightly defined as the science of the concepts of the limits of the modes of extension. It starts with a limited number of concepts, and
upon them builds up, by a process of deduction, all its propositions. In the physical sciences, on the other hand, whatever concepts we start with, we find that our results have continually to be qualified by bringing in new concepts—so that even in our theories a continual process of approximation is going on.
(2.) We now come to the speculation that the actual properties of space may be different from those generally ascribed to it. This really comes to the same thing as saying that there may be points in space whose position we cannot consider by reference to our Euclidian system of geometry of three dimensions. If we show that by our geometry of three dimensions we could consider the position of all possible points in space, then its methods would suffice for the investigation of any possible form of surface or solid; the so-called geodesics, parallel straight lines which meet, and uniplanar non-parallel straight lines which do not meet—all of which are drawn on this wonderful surface to which Mr. Frankland refers—could be brought to reason by considering the corresponding lines on a similar surface of manageable extent. For it must be possible, on the assumption that three-dimension geometry is sufficient, to obtain a surface of small extent similar to any finite surface whatever.
By many of the transcendental geometers this objection is met by the answer, (which is, I believe, the only possible one), that there may be four or more dimensions in space, not three only, as is usually imagined. Now, as Lotze points out, if there be a fourth dimension in the strict sense of the term, it must be of the same kind as the other three—length, breadth, and thickness: otherwise, our use of the word “dimension” is a misnomer; so also is our use of the word “space.” Time, density, thermal capacity, etc., are all excluded from being regarded as corresponding in any real sense to dimensions of space.
We have three concepts of the methods of extension in space, the three dimensions already referred to: the question is, whether space can be such that we cannot completely examine by reference to our three concepts the form and position of a space which is finite.
Let the position of any point of which we are cognizant in three-dimension space be referred to three co-ordinate axes, O X, O Y, O Z, which are mutually at right angles. All points in our space can be so referred, and every point with any finite and real co-ordinates whatever can have its position assigned to it. Let the fourth dimension be referred to an axis O V: O V must bear the same relation to O X and O Y as O Z does, that is, it must be at right angles to each of them. (This follows from the fact that the fourth dimension must be of the same kind as the other three.)
An imaginary being might have the same O X and O Y, but might have O V instead of O Z for his third axis.
In choosing our arbitrary axes, let us suppose we begin by fixing the position of O X. In any given plane through O X there is only one axis, O Y, at right angles to O X. By making the plane revolve about O X, we shall make it coincide in succession with all the planes that can be drawn through O X, and O Y will coincide in succession with all the straight lines that can be drawn perpendicular to O X. Let one of these be chosen for the axis O Y; let it be O Y1. But O Z is also at right angles to O X; therefore it is one of the possible O Y's; so is O V. But there is only one series of possible O Y's; therefore O Z and O V must both be in the same series. Now, the particular O Y which is taken as O Z, must be at right angles to O Y1; so must the particular O Y taken as O V. But in the series of O Y's there is only one straight line which is perpendicular to O Y1. Therefore O Z must be that line; so also must O V. Hence O Z and O V must be identical: the imaginary being's space is identical with ours, and he would be cognizant of no points, or of no properties of space, of which we were not also cognizant.
I am aware that this argument is only the reproduction in mathematical form of the argument from common sense; but the only ground, I think, on which it can be overthrown is, that the fourth dimension is not comparable with the other three—length, breadth, and height, to which we refer our notions of the extension of bodies; that is, it is not a dimension of space at all, in our sense of the term. Not being a dimension of space, it cannot aid us in finding any points in space other than those known to us by our three dimensions.
It has been said, I think, by the authors of “The Unseen Universe,” that though space may be of three dimensions with us, yet at some great distance it may have a higher number of dimensions. But space, as space, must be homogenous; to assert anything else is, as Stallo has shown, to confound space with the matter or with the structures which are in it. To explain the use of the word “structure” here, I proceed to distinguish between two of the meanings attached to the word “space.” So far, there has been no danger of ambiguity. But we cannot go further without distinguishing between what is sometimes called structure-space, and absolute space.
Consider the piece of chalk I hold in my hand: it occupies space; outside it there is space not occupied by the chalk. The space occupied by the chalk, the form of which we identify with the form of the piece of chalk, is what is called structure-space. Other bodies besides the piece of chalk in question are said to occupy space. It is possible, indeed, that no space is empty; but the very fact of our being able to think of it as empty or not empty shows that we have formed a concept of space apart from the structures which are in it. This is absolute space, or the
universe of space. Absolute space, in short, is the sum of all structure-spaces and all potential structure-spaces.
In order to be finite, it must, as the new school admit, return into itself, as the circumference of a circle or the surface of a sphere or spheroid returns into itself. Now, since by our three-dimension geometry (the sufficiency of which has, I trust, been made clear) we can always obtain a figure of small extent similar to any corresponding figure of finite extent; we could, if space were finite, obtain a similar small structure-space which likewise returns into itself. But we cannot—the phrase is meaningless; therefore the universe of space cannot be finite.
The only possible meaning that could logically be given to the statement that the Universe is finite, is that the structure-space occupied by all the bodies subject to the physical conditions known to us is a finite number of cubic miles.* But outside this structure-space, again, there must be space, just as there is space outside my piece of chalk.
It is possible that I may be accused of neglecting the argument upon which Mr. Frankland relies—that, namely, which is based upon the assumption that all the axioms of Euclid are true, except the twelfth; and that the twelfth is not true. That axiom is easily shown to be identical with the modern substitute†; the advantage of the latter being, to my mind, the fact that it is at once seen to flow directly from the concept of parallel straight lines; whereas Euclid's 12th needs the 28th proposition before its force can be properly appreciated. I do not know whether it would not be just as easy to approach the subject by assuming as the axiom the second part of Euclid I., 29:—“If a straight line fall upon two parallel straight lines, it shall make the exterior angle equal to the interior and opposite angle on the same side.” This is an immediate consequence, hardly more than a re-statement, of the concept of parallel straight lines (which may be roughly described as straight lines drawn in the same direction).
What Mr. Frankland seems to lose sight of is this: That the notion of parallel straight lines is as truly a concept as is that of a straight line; that the definitions are not and cannot be equivalents for the concepts; they are merely indexes to the nature of the several concepts; and, in like manner, the axioms are indexes of certain concepts so closely related to those pointed to in the definitions as to need no detailed proof.
The inclusion of the twelfth axiom does not make geometry an experimental science. The very question brought as an
[Footnote] *In this, of course, there must be included not merely the space these bodies occupy in a literal sense, but the whole space within the range of which all phenomena connected with them take place.
[Footnote] † Through the same point there cannot be two straight lines, each of which is parallel to a third straight line.
illustration by Mr. Frankland and others will serve to show this. That illustration is as follows:—Vertical lines on the earth's surface were once thought to be parallel; they are now almost universally considered to be inclined to one another. This is a purely physical question, not in pari materiâ with the present. Geometry, as a science of concepts, gives standards to which we may refer physical facts; among its standards are the plane and the sphere; formerly, it was thought that the surface of the earth was nearly a plane; it is now known that it more nearly approaches the sphere, and still more nearly the spheroid. But no standard, no concept of geometry, has been altered by this correction of our physical ideas.
One word more. If the concepts of Euclidian geometry were useless as standards to which to refer actual physical facts, Euclidian geometry would have to go; or if any other geometry gave equally valuable standards, it would have to be admitted by the side of the Euclidian. Otherwise, it must be rejected, however pretty it may be as a playground of the imagination.