When we observe the heavenly bodies we become aware that they all participate in one universal motion—a diurnal revolution round the polar axis.

In the case of fixed stars this is most unqualifiedly true, but in the case of the sun, and the planets also, the single motion of revolution can be discerned, modified, and slightly altered by other and secondary motions.

Hence the universal characteristic of the celestial bodies is that they move in a diurnal circle.

But we know that this one great fact which is true of them all has in reality nothing to do with them. The diurnal revolution which they visibly perform is the result of the condition of the observer. It is because the observer is on a rotating earth that a universal statement can be made about all the celestial bodies.

The universal statement which is valid about every one of the celestial bodies is that which does not concern them at all, and is but a statement of the condition of the observer.

Now there are universal statements of other kinds which we can make. We can say that all objects of experience are in space and subject to the laws of geometry.

Does this mean that space and all that it means is due to a condition of the observer?

If a universal law in one case means nothing affecting the objects themselves, but only a condition of observation, is this true in every case? There is shown us in astronomy a vera causa for the assertion of a universal. Is the same cause to be traced everywhere?

Such is a first approximation to the doctrine of Kant’s critique.

It is the apprehension of a relation into which, on the one side and the other, perfectly definite constituents enter—the human observer and the stars—and a transference of this relation to a region in which the constituents on either side are perfectly unknown.

If spatiality is due to a condition of the observer, the observer cannot be this bodily self of ours—the body, like the objects around it, are equally in space.

This conception Kant applied, not only to the intuitions of sense, but to the concepts of reason—wherever a universal statement is made there is afforded him an opportunity for the application of his principle. He constructed a system in which one hardly knows which the most to admire, the architectonic skill, or the reticence with regard to things in themselves, and the observer in himself.

His system can be compared to a garden, somewhat formal perhaps, but with the charm of a quality more than intellectual, a besonnenheit, an exquisite moderation over all. And from the ground he so carefully prepared with that buried in obscurity, which it is fitting should be obscure, science blossoms and the tree of real knowledge grows.

The critique is a storehouse of ideas of profound interest. The one of which I have given a partial statement leads, as we shall see on studying it in detail, to a theory of mathematics suggestive of enquiries in many directions.

The justification for my treatment will be found amongst other passages in that part of the transcendental analytic, in which Kant speaks of objects of experience subject to the forms of sensibility, not subject to the concepts of reason.

Kant asserts that whenever we think we think of objects in space and time, but he denies that the space and time exist as independent entities. He goes about to explain them, and their universality, not by assuming them, as most other philosophers do, but by postulating their absence. How then does it come to pass that the world is in space and time to us?

Kant takes the same position with regard to what we call nature—a great system subject to law and order. “How do you explain the law and order in nature?” we ask the philosophers. All except Kant reply by assuming law and order somewhere, and then showing how we can recognise it.

In explaining our notions, philosophers from other than the Kantian standpoint, assume the notions as existing outside us, and then it is no difficult task to show how they come to us, either by inspiration or by observation.

We ask “Why do we have an idea of law in nature?” “Because natural processes go according to law,” we are answered, “and experience inherited or acquired, gives us this notion.”

But when we speak about the law in nature we are speaking about a notion of our own. So all that these expositors do is to explain our notion by an assumption of it.

Kant is very different. He supposes nothing. An experience such as ours is very different from experience in the abstract. Imagine just simply experience, succession of states, of consciousness! Why, there would be no connecting any two together, there would be no personal identity, no memory. It is out of a general experience such as this, which, in respect to anything we call real, is less than a dream, that Kant shows the genesis of an experience such as ours.

Kant takes up the problem of the explanation of space, time, order, and so quite logically does not presuppose them.

But how, when every act of thought is of things in space, and time, and ordered, shall we represent to ourselves that perfectly indefinite somewhat which is Kant’s necessary hypothesis—that which is not in space or time and is not ordered. That is our problem, to represent that which Kant assumes not subject to any of our forms of thought, and then show some function which working on that makes it into a “nature” subject to law and order, in space and time. Such a function Kant calls the “Unity of Apperception”; i.e., that which makes our state of consciousness capable of being woven into a system with a self, an outer world, memory, law, cause, and order.

The difficulty that meets us in discussing Kant’s hypothesis is that everything we think of is in space and time—how then shall we represent in space an existence not in space, and in time an existence not in time? This difficulty is still more evident when we come to construct a poiograph, for a poiograph is essentially a space structure. But because more evident the difficulty is nearer a solution. If we always think in space, i.e. using space concepts, the first condition requisite for adapting them to the representation of non-spatial existence, is to be aware of the limitation of our thought, and so be able to take the proper steps to overcome it. The problem before us, then, is to represent in space an existence not in space.

The solution is an easy one. It is provided by the conception of alternativity.

To get our ideas clear let us go right back behind the distinctions of an inner and an outer world. Both of these, Kant says, are products. Let us take merely states of consciousness, and not ask the question whether they are produced or superinduced—to ask such a question is to have got too far on, to have assumed something of which we have not traced the origin. Of these states let us simply say that they occur. Let us now use the word a “posit” for a phase of consciousness reduced to its last possible stage of evanescence; let a posit be that phase of consciousness of which all that can be said is that it occurs.

Let a, b, c, be three such posits. We cannot represent them in space without placing them in a certain order, as a, b, c. But Kant distinguishes between the forms of sensibility and the concepts of reason. A dream in which everything happens at haphazard would be an experience subject to the form of sensibility and only partially subject to the concepts of reason. It is partially subject to the concepts of reason because, although there is no order of sequence, still at any given time there is order. Perception of a thing as in space is a form of sensibility, the perception of an order is a concept of reason.

We must, therefore, in order to get at that process which Kant supposes to be constitutive of an ordered experience imagine the posits as in space without order.

As we know them they must be in some order, abc, bca, cab, acb, cba, bac, one or another.

To represent them as having no order conceive all these different orders as equally existing. Introduce the conception of alternativity—let us suppose that the order abc, and bac, for example, exist equally, so that we cannot say about a that it comes before or after b. This would correspond to a sudden and arbitrary change of a into b and b into a, so that, to use Kant’s words, it would be possible to call one thing by one name at one time and at another time by another name.

In an experience of this kind we have a kind of chaos, in which no order exists; it is a manifold not subject to the concepts of reason.

Now is there any process by which order can be introduced into such a manifold—is there any function of consciousness in virtue of which an ordered experience could arise?

In the precise condition in which the posits are, as described above, it does not seem to be possible. But if we imagine a duality to exist in the manifold, a function of consciousness can be easily discovered which will produce order out of no order.

Let us imagine each posit, then, as having, a dual aspect. Let a be 1a in which the dual aspect is represented by the combination of symbols. And similarly let b be 2b, c be 3c, in which 2 and b represent the dual aspects of b, 3 and c those of c.

Since a can arbitrarily change into b, or into c, and so on, the particular combinations written above cannot be kept. We have to assume the equally possible occurrence of form such as 2a, 2b, and so on; and in order to get a representation of all those combinations out of which any set is alternatively possible, we must take every aspect with every aspect. We must, that is, have every letter with every number.

Let us now apply the method of space representation.

Take three mutually rectangular axes in space 1, 2, 3 (fig. 59), and on each mark three points, the common meeting point being the first on each axis. Then by means of these three points on each axis we define 27 positions, 27 points in a cubical cluster, shown in fig. 60, the same method of co-ordination being used as has been described before. Each of these positions can be named by means of the axes and the points combined.

Thus, for instance, the one marked by an asterisk can be called 1c, 2b, 3c, because it is opposite to c on 1, to b on 2, to c on 3.

Let us now treat of the states of consciousness corresponding to these positions. Each point represents a composite of posits, and the manifold of consciousness corresponding to them is of a certain complexity.

Suppose now the constituents, the points on the axes, to interchange arbitrarily, any one to become any other, and also the axes 1, 2, and 3, to interchange amongst themselves, any one to become any other, and to be subject to no system or law, that is to say, that order does not exist, and that the points which run abc on each axis may run bac, and so on.

Then any one of the states of consciousness represented by the points in the cluster can become any other. We have a representation of a random consciousness of a certain degree of complexity.

Now let us examine carefully one particular case of arbitrary interchange of the points, a, b, c; as one such case, carefully considered, makes the whole clear.

Consider the points named in the figure 1c, 2a, 3c; 1c, 2c, 3a; 1a, 2c, 3c, and examine the effect on them when a change of order takes place. Let us suppose, for instance, that a changes into b, and let us call the two sets of points we get, the one before and the one after, their change conjugates.

The points surrounded by rings represent the conjugate points.

It is evident that as consciousness, represented first by the first set of points and afterwards by the second set of points, would have nothing in common in its two phases. It would not be capable of giving an account of itself. There would be no identity.

If, however, we can find any set of points in the cubical cluster, which, when any arbitrary change takes place in the points on the axes, or in the axes themselves, repeats itself, is reproduced, then a consciousness represented by those points would have a permanence. It would have a principle of identity. Despite the no law, the no order, of the ultimate constituents, it would have an order, it would form a system, the condition of a personal identity would be fulfilled.

The question comes to this, then. Can we find a system of points which is self-conjugate which is such that when any posit on the axes becomes any other, or when any axis becomes any other, such a set is transformed into itself, its identity is not submerged, but rises superior to the chaos of its constituents?

Such a set can be found. Consider the set represented in fig. 62, and written down in the first of the two lines—

If now a change into c and c into a, we get the set in the second line, which has the same members as are in the upper line. Looking at the diagram we see that it would correspond simply to the turning of the figures as a whole.[2] Any arbitrary change of the points on the axes, or of the axes themselves, reproduces the same set.

Thus, a function, by which a random, an unordered, consciousness could give an ordered and systematic one, can be represented. It is noteworthy that it is a system of selection. If out of all the alternative forms that only is attended to which is self-conjugate, an ordered consciousness is formed. A selection gives a feature of permanence.

Can we say that the permanent consciousness is this selection?

An analogy between Kant and Darwin comes into light. That which is swings clear of the fleeting, in virtue of its presenting a feature of permanence. There is no need to suppose any function of “attending to.” A consciousness capable of giving an account of itself is one which is characterised by this combination. All combinations exist—of this kind is the consciousness which can give an account of itself. And the very duality which we have presupposed may be regarded as originated by a process of selection.

Darwin set himself to explain the origin of the fauna and flora of the world. He denied specific tendencies. He assumed an indefinite variability—that is, chance—but a chance confined within narrow limits as regards the magnitude of any consecutive variations. He showed that organisms possessing features of permanence, if they occurred would be preserved. So his account of any structure or organised being was that it possessed features of permanence.

Kant, undertaking not the explanation of any particular phenomena but of that which we call nature as a whole, had an origin of species of his own, an account of the flora and fauna of consciousness. He denied any specific tendency of the elements of consciousness, but taking our own consciousness, pointed out that in which it resembled any consciousness which could survive, which could give an account of itself.

He assumes a chance or random world, and as great and small were not to him any given notions of which he could make use, he did not limit the chance, the randomness, in any way. But any consciousness which is permanent must possess certain features—those attributes namely which give it permanence. Any consciousness like our own is simply a consciousness which possesses those attributes. The main thing is that which he calls the unity of apperception, which we have seen above is simply the statement that a particular set of phases of consciousness on the basis of complete randomness will be self-conjugate, and so permanent.

As with Darwin so with Kant, the reason for existence of any feature comes to this—show that it tends to the permanence of that which possesses it.

We can thus regard Kant as the creator of the first of the modern evolution theories. And, as is so often the case, the first effort was the most stupendous in its scope. Kant does not investigate the origin of any special part of the world, such as its organisms, its chemical elements, its social communities of men. He simply investigates the origin of the whole—of all that is included in consciousness, the origin of that “thought thing” whose progressive realisation is the knowable universe.

This point of view is very different from the ordinary one, in which a man is supposed to be placed in a world like that which he has come to think of it, and then to learn what he has found out from this model which he himself has placed on the scene.

We all know that there are a number of questions in attempting an answer to which such an assumption is not allowable.

Mill, for instance, explains our notion of “law” by an invariable sequence in nature. But what we call nature is something given in thought. So he explains a thought of law and order by a thought of an invariable sequence. He leaves the problem where he found it.

Kant’s theory is not unique and alone. It is one of a number of evolution theories. A notion of its import and significance can be obtained by a comparison of it with other theories.

Thus in Darwin’s theoretical world of natural selection a certain assumption is made, the assumption of indefinite variability—slight variability it is true, over any appreciable lapse of time, but indefinite in the postulated epochs of transformation—and a whole chain of results is shown to follow.

This element of chance variation is not, however, an ultimate resting place. It is a preliminary stage. This supposing the all is a preliminary step towards finding out what is. If every kind of organism can come into being, those that do survive will present such and such characteristics. This is the necessary beginning for ascertaining what kinds of organisms do come into existence. And so Kant’s hypothesis of a random consciousness is the necessary beginning for the rational investigation of consciousness as it is. His assumption supplies, as it were, the space in which we can observe the phenomena. It gives the general laws constitutive of any experience. If, on the assumption of absolute randomness in the constituents, such and such would be characteristic of the experience, then, whatever the constituents, these characteristics must be universally valid.

We will now proceed to examine more carefully the poiograph, constructed for the purpose of exhibiting an illustration of Kant’s unity of apperception.

In order to show the derivation order out of non-order it has been necessary to assume a principle of duality—we have had the axes and the posits on the axes—there are two sets of elements, each non-ordered, and it is in the reciprocal relation of them that the order, the definite system, originates.

Is there anything in our experience of the nature of a duality?

There certainly are objects in our experience which have order and those which are incapable of order. The two roots of a quadratic equation have no order. No one can tell which comes first. If a body rises vertically and then goes at right angles to its former course, no one can assign any priority to the direction of the north or to the east. There is no priority in directions of turning. We associate turnings with no order progressions in a line with order. But in the axes and points we have assumed above there is no such distinction. It is the same, whether we assume an order among the turnings, and no order among the points on the axes, or, vice versa, an order in the points and no order in the turnings. A being with an infinite number of axes mutually at right angles, with a definite sequence between them and no sequence between the points on the axes, would be in a condition formally indistinguishable from that of a creature who, according to an assumption more natural to us, had on each axis an infinite number of ordered points and no order of priority amongst the axes. A being in such a constituted world would not be able to tell which was turning and which was length along an axis, in order to distinguish between them. Thus to take a pertinent illustration, we may be in a world of an infinite number of dimensions, with three arbitrary points on each—three points whose order is indifferent, or in a world of three axes of arbitrary sequence with an infinite number of ordered points on each. We can’t tell which is which, to distinguish it from the other.

Thus it appears the mode of illustration which we have used is not an artificial one. There really exists in nature a duality of the kind which is necessary to explain the origin of order out of no order—the duality, namely, of dimension and position. Let us use the term group for that system of points which remains unchanged, whatever arbitrary change of its constituents takes place. We notice that a group involves a duality, is inconceivable without a duality.

Thus, according to Kant, the primary element of experience is the group, and the theory of groups would be the most fundamental branch of science. Owing to an expression in the critique the authority of Kant is sometimes adduced against the assumption of more than three dimensions to space. It seems to me, however, that the whole tendency of his theory lies in the opposite direction, and points to a perfect duality between dimension and position in a dimension.

If the order and the law we see is due to the conditions of conscious experience, we must conceive nature as spontaneous, free, subject to no predication that we can devise, but, however apprehended, subject to our logic.

And our logic is simply spatiality in the general sense—that resultant of a selection of the permanent from the unpermanent, the ordered from the unordered, by the means of the group and its underlying duality.

We can predicate nothing about nature, only about the way in which we can apprehend nature. All that we can say is that all that which experience gives us will be conditioned as spatial, subject to our logic. Thus, in exploring the facts of geometry from the simplest logical relations to the properties of space of any number of dimensions, we are merely observing ourselves, becoming aware of the conditions under which we must perceive. Do any phenomena present themselves incapable of explanation under the assumption of the space we are dealing with, then we must habituate ourselves to the conception of a higher space, in order that our logic may be equal to the task before us.

We gain a repetition of the thought that came before, experimentally suggested. If the laws of the intellectual comprehension of nature are those derived from considering her as absolute chance, subject to no law save that derived from a process of selection, then, perhaps, the order of nature requires different faculties from the intellectual to apprehend it. The source and origin of ideas may have to be sought elsewhere than in reasoning.

The total outcome of the critique is to leave the ordinary man just where he is, justified in his practical attitude towards nature, liberated from the fetters of his own mental representations.

The truth of a picture lies in its total effect. It is vain to seek information about the landscape from an examination of the pigments. And in any method of thought it is the complexity of the whole that brings us to a knowledge of nature. Dimensions are artificial enough, but in the multiplicity of them we catch some breath of nature.

We must therefore, and this seems to me the practical conclusion of the whole matter, proceed to form means of intellectual apprehension of a greater and greater degree of complexity, both dimensionally and in extent in any dimension. Such means of representation must always be artificial, but in the multiplicity of the elements with which we deal, however incipiently arbitrary, lies our chance of apprehending nature.

And as a concluding chapter to this part of the book, I will extend the figures, which have been used to represent Kant’s theory, two steps, so that the reader may have the opportunity of looking at a four-dimensional figure which can be delineated without any of the special apparatus, to the consideration of which I shall subsequently pass on.