PART IPRE-RELATIVITY PHYSICS
CHAPTER IMANIFOLDS
IN this chapter we shall be concerned solely with manifolds and their dimensionality; and not with their metrics, which pertains to a different problem entirely.
We all possess a certain instinctive understanding of what is meant by continuity. We notice, for example, that sounds, colours or tactual sensations merge by insensible gradations into other sounds, colours or tactual sensations, without any abrupt transitions. An aggregate of such continuous sensations constitutes what is called asensory continuum or continuous manifold. That continuity is a concept which springs from experience can scarcely be doubted, and it can be accounted for by the inability of our crude senses to differentiate between impressions which are almost alike.
Consider, for example, the succession of musical notes exhibited in the chromatic scale on the piano. Here we are not in the presence of a sensory continuum, for the successive sounds do not merge into one another by insensible degrees. Even an untrained ear can differentiate between aand the-sharp immediately following it. But we can conceive of a piano in which a sufficient number of semitones and intermediary notes have been interposed so that every note would be indistinguishable from its immediate successor and immediate predecessor, although we should still be able to differentiate between non-contiguous notes. It would thus be possible for us to pass through a continuous chain of sounds from any one musical sound to any other without our ear’s ever being able to detect a sudden jump; and this is what we mean by calling our aggregate of soundsa sensory continuum.
Suppose now that we were to remove any one of these notes from our piano (excluding the two extreme ones). The continuity of our chain of sounds would be broken, for when we reached the missing note we should detect a sudden variation in pitch as we passed from the sound immediately preceding the removed note to the one immediately following it. In short, the removal of one of the notes would cut our continuous chain of sounds in two.
The dimensionality of our continuum of sounds is obviously unity, for we can assign successive numbers to the successive notes, starting from some standard note, and by this means determine them without ambiguity.
Let us complicate matters somewhat by assuming that every individual note may be sounded with various intensities, but always in such a way that a note of given intensity can never be distinguished from the one sounded just a little louder or just a little softer. Once again it will be possible for us to pass in a continuous way from a note of feeble intensity to the same note sounded with louder intensity, without our ear’s ever being able to detect a variation in intensity between two successive sounds. More generally we shall be able to pass in a continuous way from a note of given pitch and given intensity to one of some other pitch and some other intensity.
In the case assumed we should be dealing with a two-dimensional continuum or continuous manifold of sounds; for in order to locate a definite sound it would be necessary for us to designate it by two numbers, one specifying its pitch and the other its intensity. We may also notice that whereas in the first case, by removing one of the notes, we were able to cut the manifold of successive pitches into two parts, the removal of one particular note of definite pitch and intensity will now be incapable of effecting this separation.
For instance, if we were to remove the notesounded with a definite intensity it would still be possible to pass in a continuous way from any one note of given intensity to any other note of our continuum by circumscribing the missing note; namely, by choosing some route of transfer which would pass through adiffering in intensity from that of thewe had removed.
In the present case, if we wished to divide our two-dimensional manifold into two parts, it would be necessary to remove some one-dimensional continuum of sounds, for example the one-dimensional continuum formed by all the notes of a given pitch,but varying in intensity, or again of given intensity but varying in pitch. If this were done, it would be impossible for us to pass in a continuous way from a note of definite pitch and intensity of our two-dimensional continuum to a note of any other definite pitch and intensity; for we could never get past the removed line of sounds without our ear’s detecting a sudden change.
We might complicate matters still further by taking into consideration variations in tonality, as for instance the variation which our ear can detect between two given notes of the same pitch and intensity sounded by two different instruments, such as a violin and an organ. Assuming that every one of our notes of given pitch and intensity in our two-dimensional manifold could also vary in tonality by imperceptible degrees, we should be dealing with a three-dimensional sensory continuum in which every note of given pitch, intensity and tonality could be defined unambiguously by the choice of three numbers.
As before, we should find that the removal of a single note of given pitch, intensity and tonality, or even the removal of a one-dimensional continuum of notes such as all those of given intensity and pitch, but varying in tonality, was quite insufficient to effect a separation in our three-dimensional manifold. In the present case we should haveto remove some two-dimensional continuum—say, all notes of given intensity but varying in pitch and tonality. Only then should we have effected a separation between any given element and any other one, rendering it impossible for a continuity of sound impressions to extend between the two elements. By proceeding in this way indefinitely it is obvious that we can conceive of sensory continua of any number of dimensions; there is no need to limit ourselves to three.
In a general way we may say, therefore, that a sensory continuum is-dimensional when, in order to render a path of continuous passage impossible between any two of its elements, it is necessary to remove a sub-continuum ofdimensions. This sub-continuum itself is known to bedimensional because we can separate it into two parts only by removing from it a sub-continuum ofdimensions, etc., till we finally get a sub-sub-sub ... continuum, which can be separated by removal of a single element. But such an element no longer constitutes a continuum, since passage in it is excluded; its dimensionality is then zero; so that the continuum it separates in two is obviously one-dimensional.
A sensory continuum would also be given by a succession of weights placed on one’s hand, each only slightly heavier than the preceding one. Again our tactual impressions might yield a sensory continuum. In view of the fact that it is possible to account for the rise of the concept of space, even in the consciousness of a blind man, through the sole means of his tactual impressions, it may be of interest to discuss briefly an illustration of a tactual continuum—that obtained by exploring the surface of our skin by means of pinpricks. If these pinpricks are sufficiently close to one another, it will be impossible for us to differentiate between them and we shall always experience the sensation of one solitary pinprick. We can thus consider the sensory continuum obtained by some definite chain of pinpricks extending, let us say, from our elbow to our hand. This particular chain of sensations exhibits all the characteristics of a one-dimensional sensory continuum, since every one of its elements is indistinguishable from its immediate neighbours and since the removal of one of these elements (pinpricks) would create a hiatus rendering it impossible for us to pass in a continuous way from elbow to hand, along the chain.
But if we should now consider all the possible chains of pinpricks extending from a point on our elbow to a point on our hand, the mere removal of one particular pinprick from one particular chain would be insufficient to interfere with the continuous passage. We might always follow one of the other chains, or even, following the same chain up to the missing element, skip round the latter without sensory continuity being interfered with. In the present case the only way to render this continuous passage impossible would be to remove some continuous chain of pinpricks—say, those circling round our wrist. The sensory skin-continuum would now be divided into two parts, and as the continuum removed was one-dimensional, we should conclude that theskin continuum as manifesting its sensitivity to tactual stimuli was two-dimensional.
In a similar way, in crude geometry we recognise a wire as one-dimensional, since by removing a point of the wire our finger cannot pass in a continuous way from one extremity to the other. Likewise, a surface is regarded as two-dimensional because only by cutting it along a line is it possible to interrupt the smooth passage of our finger from any one point to any other. The mere removal of a point on the surface would not interfere with the continuous passage as it did in the case of the wire. It is the same for a volume. Only a surface can divide it in two; hence volume is three-dimensional.[3]
When we seek to determine the dimensionality of perceptual space, itself a sensory continuum produced by the superposition of the visual, the tactual and the motive continua, the problem is more difficult. It would be found, however, that perceptual space has three dimensions; but as the necessary explanations would require several chapters we must refer the reader to Poincaré’s profound writings for more ample information.
Summarising, we may say that our belief in the tri-dimensionality of space can be accounted for on the grounds of sensory experience.
Now the subject of our investigations up to the present point has been the dimensionality of sensory continua and the general characteristics of sensory continuity; considerations relating to measurement, or to the extensional equality of two continuous stretches in our continua, have not been entered upon. Neither has any definition of what is meant by a straight line been introduced at this stage. As a result, metrical geometry, which deals with measurements, and projective geometry, which deals with the projections of points, cannot be discussed. The only type of geometry we can consider at this stage is that purely qualitative non-metrical type calledAnalysis Situs, which deals solely with problems of connectivity.
Connectivity relates to the types of paths of continuous passage from one part of a continuum to another. Manifolds may possess the same dimensionality and yet differ in connectivity. Thus, the connectivity of a sphere differs from that of a torus or doughnut; since the doughnut, in contrast to the sphere, presents a hole or discontinuity through its centre. Yet both sphere and doughnut are two-dimensional surfaces.
In Analysis Situs, metrical considerations obviously play no part. From a metrical point of view, although a sphere differs in shape from an ellipsoid, yet the connectivity or Analysis Situs of the two surfaces is exactly the same. We may add that there exists an Analysis Situs for every continuous manifold, so that we may conceive of an Analysis Situs ofdimensions corresponding to an-dimensional manifold.
Our next task is to determine how a metrics can be established in a sensory continuum. Consider, for example, a continuous stretch of shades of grey passing from white to black. What do we mean exactly when we say that some definite shade is twice as dark as another? Obviously no definite meaning can be assigned to this statement until we have posited some convention permitting us to establish comparisons.
As another instance, take the case of a continuous stream of sounds varying in pitch. What do we mean by saying that some particular musical note is twice as high in pitch as some other, or that the interval between two notes is equal to the interval between two others? If we were to be guided solely by our ear we might assert that as certain musical notes, though differing in pitch, yet appear to present a certain undefinable similarity (the successive octaves), the intervals between these successive similar notes should be considered equal or congruent. We should thus define as equal the extension of notes subtending the successive octaves of a given musical note.
But if now we had learnt to measure the frequencies of vibration of the various musical sounds, a new type of measurement would immediately suggest itself. Starting from any musical note—say, the middleof the piano, we should find that the octave ofwas vibrating twice as fast, the followingthree times as fast, and the second octave of our originalfour times as fast. It would then appear plausible to define equal intervals between musical notes by the differences in their rates of vibration, and we should infer that the distance betweenand its first octave was equal to the distance between this last note and the following,and equal again to the distance between thisand the following superoctave.
We should thus have obtained a definition of equal stretches of sounds which was at variance with our original definition, in which all octave intervals were regarded as equal or congruent. In view of these conflicting results, we could not well escape the conclusion that a sensory continuum of itself offers us no precise means of defining equal stretches, and that whatever definition we might finally select would be a mere matter of choice, an arbitrarily posited convention.
And yet in the case of space, itself a sensory continuum, men have found no difficulty in agreeing on a common system of measurements. As we shall see in the following chapters, the definition of the equality of different stretches of space to which men were unavoidably led was imposed upon them by the behaviour of certain bodies located in space, bodies which were deemed to remain rigid, hence to occupy equal volumes and equal lengths of space wherever they were displaced. For the present, however, we may leave these metrical considerations aside and confine our attention to the general concept of mathematical or geometrical space, which is the subject of study of the pure mathematician.
The concept of a sensory continuum, hence of perceptual space, as presented to us by crude experience, contains certain contradictions and peculiarities which it was necessary to eliminate before it could be subjected to rigorous mathematical treatment. In the firstplace, this perceptual space is not homogeneous, and the principle of sufficient reason demands that pure empty conceptual space be homogeneous and isotropic, the same everywhere and the same in all directions.
This homogeneity of space permits us to foresee that it must be unbounded, since a boundary would suggest a discontinuity of structure, defining an inside and an outside, hence a lack of homogeneity. Prior to Riemann’s discoveries it was thought that the absence of a boundary would necessitate the infiniteness of space. To-day we know that this belief is unjustified, for a space can be finite and yet unbounded; and two major varieties of such spaces have been discovered by mathematicians.
But the inherent inconsistencies which endure in all sensory continua constituted a still more important reason for compelling mathematicians to idealise perceptual space. In a sensory continuum, as we have seen, a sensationcannot be distinguished from its immediate successor, the sensation;neither canbe differentiated from.Yet no difficulty is experienced in differentiatingfrom.Expressed mathematically, these facts yield the inconsistent series of relations;;.Now, an inconsistency of this sort precludes all mathematical treatment. In mathematics magnitudes cannot be both equal and unequal; they must either be one or the other. The mathematician is therefore compelled to idealise the sensory continuum of experience by assuming that were it not for the crudeness of our senses, the points or sensations,andwould all be distinguishable, and that in place of;and,we should have;;.
But it is obvious that a continuum idealised in this way becomes atomic or discrete, since betweenand,as betweenand,no intermediary points have been mentioned. In order to re-establish continuity the mathematician is forced to postulate that between any two pointsandthere exist an indefinite number of intermediary points, such that no one of these points has an immediate neighbour. In other words, the continuum is infinitely divisible.
Thus the magnitudes 1 and 2 are not neighbours, since a number of rational fractions separate them. And no two of these fractional numbers are immediate neighbours, since whichever two such numbers we choose to select, we can always discover an indefinite number of other fractional numbers existing between them. Between any two points on a line in our continuum, however close together they may be, we have thus interposed an indefinite number of rational fractions defining points; yet, despite this fact, we have by no means eliminated gaps between the various points along our line.
The Greek mathematician Pythagoras was the first to draw attention to this deficiency after studying certain geometrical constructions. He remarked, for instance, that if we considered a square whose sides were of unit length, the diagonal of the square (as a result of his famous geometrical theorem of the square of the hypothenuse) would be equalto.Now,is an irrational number and differs from all ordinary fractional or rational numbers. Hence, since all the points of a line would correspond to rational or ordinary fractional numbers, it was obvious that the opposite corner of the square would define a point which did not belong to the diagonal. In other words, the sides of the square meeting at the opposite corner to that whence the diagonal had been drawn, would not intersect the diagonal; and we should be faced with the conclusion that two continuous lines could cross one another in a plane and yet have no point in common.
The only way to remedy this difficulty was to assume that the point corresponding toand in a general way points corresponding to all irrational numbers (such as,and radicals) were after all present on a continuous mathematical line. Accordingly, mathematical continuity along a line was defined by the inclusion of all numbers whether rational or irrational, and a similar procedure was followed for a mathematical continuum of any number of dimensions. In this way mathematicians obtained what is known as theGrand Continuum, orMathematical Continuum.[4]
Now, it is obvious that although the mathematical continuum is still called a continuum, it differs considerably from the popular conception of a continuum, where every element merges into its neighbour. However this may be, the mathematical continuum, and with it mathematical continuity, are as near an approach to the sensory continuum and to sensory continuity as it is possible for mathematicians to obtain. The sensory continuum itself is barred from mathematical treatment owing to its inherent inconsistencies.
And here an important point must be noted. In a sensory continuum considered as a chain of elements, an understanding of nextness orcontiguity, hence an understanding of order, was imposed upon us by judgments of identity in our sensory perceptions. But the same no longer holds in the case of a mathematical aggregate of points, owing to the absence of that merging condition which guided us in the sensory manifolds. Theoretically, we may with equal justification order these points in whatever way we choose, and by varying the order in which we pass from point to point, we should find that the dimensionality of the aggregate varied in consequence.[5]
Dimensionality is thus a property of order, and order must be imposed before dimensionality can be established.
In practice, the geometrician will retain the type of ordering relation imposed by our sensory experience and will conceive, by abstraction, of a mathematical space of points which will manifest itself as three-dimensional when this ordering relation is adhered to. There is nothing to prevent him, however, from conceiving of mathematical spaces of any whole number of dimensions, either by modifying the ordering relation or again by modelling his mathematical manifold on some-dimensional sensory continuum.
Let us suppose, then, that we have conceived of a three-dimensional mathematical space, obtained as an abstraction from the three-dimensional space of common experience. Just as was the case with our sensory continuum, the mathematical continuum will be amorphous; no intrinsic metrics will be inherent in it, hence it will present us with no definite geometry. The definition of congruence, that is, of the equality of two spatial stretches and more generally of two volumes, and the identity of shapes and sizes, will remain as conventional as before; and it will be only after we have introduced measuring conventions into an otherwise indifferent mathematical space that Metrical Geometry as opposed to Analysis Situs will be possible. However, the discussion of these points will be reserved for the next chapters.