If order is a category it might seem eminently to be due to the interference of mind. The mind, it might be thought, compares things in respect of certain characters, e.g. magnitude or shades of colour, and arranges them in a scale in which any one thing precedes another and is in general between that other and some term which precedes itself.1 But a moment's consideration is enough to show that such comparison depends on the characters and relations of the terms themselves, and, what is more pertinent, the acts which the mind performs in arranging terms in an order are themselves in order, only that the order is enjoyed instead of being contemplated. Thus if lines are ordered according to their increasing magnitude, the successive apprehensions of the lines are also ordered in magnitude.
We have order when there are at least three terms of which one is between the other two, that is, when B is between A and C. Order is a category of things because of betweenness of position in Space-Time. This betweenness is, as we have seen, a fundamental feature in Time, and points in Space are between each other in virtue of the Time in which they are generated. What applies to positions in Space-Time applies equally to complexes in Space-Time. We may indeed have things or points which are contemporaneous. But they are between each other in space in virtue of the time in which their positions in space are generated. ‘Between’ is therefore a crude or elementary feature of Space-Time and attaches to the elements of Space-Time themselves and to complexes of those elements.
Betweenness which is the characteristic of order communicates with relation, and order may be resolved into relations. Thus, as Mr. Russell shows, terms x, y, z are in an order when there is a relation R such that x is in the relation R to y and y in the relation R to z, in other words, when there is a transitive relation between the terms, and it is asymmetrical. Thus if the relation is of magnitude, x is greater than y and y than z and x than z, and the relations of y to x and z have a different sense. If the terms are points of time y is before z and after x. This simplest of all orders is at the basis of all order. But though in this way order may be expressed in terms of relation, order is not a mere combination of relations. For the introduction of asymmetry into the transitive relation already implies betweenness. The transitive relation of equality of magnitude would not be sufficient for betweenness of magnitude. Such betweenness can only be generated by a relation which being transitive has direction and is therefore asymmetrical. Betweenness is a crude datum to which the conception of a transitive asymmetrical relation is due. Between is therefore as much a specific datum, though resoluble into two relations of different sense, as a motion along the diagonal is a specific motion though resoluble into components along the sides of the parallelogram. There, too, the mere components are not equivalent to the resultant unless they are really components, that is unless, in the language adopted by Mill, their collocation is also given. Betweenness being thus primordial, order is a category distinct from relation, just as existence is distinct from relation though existence is always in relation to other existence.
Order involves at least three terms, and any three terms may constitute an order, under the conditions in which order is expressed relationally. Each term in the order is ordered according to the nature of that order. But not each term is necessarily between other terms. This is only the case when the series has neither beginning nor end; as in the case of instants or the real numbers. In the order of colours, in the order of precedence of nobility, in the order of species in a genus, and the like, there may be first or last terms or both which are not between in respect of that order; though they will always be between in the fundamental order of Time or Space. Further it is clear that when two terms are said to be in a certain order, as e.g. cause and effect in the order of priority, or two colours in respect of the order of hue or brightness; they are so described in so far as they are selected members from a real order: e.g. in causality the order of time.
A universal character of things.
Order is a difficult conception which I am unequal to the task of treating adequately.2 What has concerned us here has been to indicate that like other categories it is a character of things which is a crude, primordial feature of Space-Time, and can only be indicated as such, or, if described, order is being described like continuity in terms of what is derived from it. It is difficult to discuss the conception at greater length at this stage, for the assurance that all order is in the end spatio-temporal can only be got from considering order in its more special determinations, like the order of numbers or of quantity, which we cannot yet assume to be spatio-temporal. Order has not usually been reckoned among categories at all, and does not form one of the Kantian categories. Yet that all things have order of some sort can readily be seen, if it is only order of position in Time or Space, or quantitative or numerical order.
But the varieties of order are not only these categorial special orders but empirical ones; and some of these may be enumerated. Most important of all, perhaps, is the order of the qualities of a given kind or modality of sensation; for example, that of pitches of sound or hues of colour. Such order has of course no reference to the position of notes on a musical instrument like the piano, or of colours in the solar spectrum. It belongs to sounds or colours as experienced, that is as sounds or colours, which for us are sensa. It is an expression of their ultimate spatio-temporal character. Sounds form an order of pitches, ultimately because their wavelengths are a series in which each is spatio-temporally between two others, and could be known (not heard) as such if the sounds were produced from the one place. Mr. H. J. Watt has even maintained that pitches of sound and hues of colour are not differences of quality, but that there is only one quality, sound, or colour, and pitches and hues are merely terms in an order, determined by the one quality.3 Whether the modalities or classes of sensible qualities themselves, sound, colour, taste, etc., constitute an order, cannot in the present state of our knowledge be asserted.
Besides this important example of empirical order we have such order as that of descent from father to child—the genealogical order, and we have the larger order of descent in animal types determined by distance from a common ancestor; there is the order of greatness from the merest weakling up to the superman, or in moral matters the order of merit which belongs to actions not in virtue of their goodness (for “all goodness ranks the same with God”) but in respect of their largeness or splendour. And the list might be extended to some length. Order is therefore far from being confined to purely categorial orders with which the term is so closely associated in the mind. But it depends ultimately in every case on spatio-temporal betweenness.
Order and the other categories.
Order communicates with existence, as being itself an existent, and as internally constituted of existents. It is relational in itself, and at the same time there may be relation between different orders, as for example in correlation of the order of general intelligence to order of sensibility in some department of sense. Lastly, it communicates with universality; it is a plan. And not only is order universal with regard to its categorial special determinations, like the order of number, but all these categorial orders are universal in respect of their empirical examples; thus we may arrange things in weight or brightness, or even numerical order may assume such particular forms as the order of even or odd or square numbers.
While order thus communicates with universality, its distinctness from universality is a more important matter, and at the same time more difficult to make clear. When points are considered in their order of position, the transitive relation is that of greater (or less) distance from a fixed member of the series (whether distance is taken as equivalent to interval or distance proper, that is unlikeness of position). The relation, distance from a given point, is universal to any of the distances of the points from the fixed point. But then it is not this distance which is the order itself; that order can be resolved into those relations, but is not identical with them. The order is rather that of position in the ordered series, and this is not the universal of the different positions in the series, but is the collective name of all the positions. There is no true universal or plan of construction, to be called position in the series, of which the members of the series are modifications as different dogs are modifications of the plan or law of dog-construction. For position in the series or order implies the order as a whole. To hold that belief would be an instance of the concrete universal over again, like regarding the parabola as the universal of its points, or the State as the universal of its citizens, or the self as the universal of its activities. For a less elementary illustration let us turn to Mr. Watt's conception of pitches as the order of the one quality sound, or colours as the order of colour. It does indeed seem unnatural to hold that there is only one quality sound or colour; rather it would seem that, according to common usage, the pitches and hues are qualities which are ordered in respect of sound or colour. But what is of value in the doctrine is that it recognises order as an intrinsic (I should say categorial) character of the pitches and hues, and sound as such and colour as such are then orders named from the qualities of their members. Now neither sound nor colour is a true universal. There is no quality colour of which the various hues are instances, nor, though this is more difficult to verify, is there probably for experience any universal, sound, of which the various pitches are modifications, certainly no true universal, pitch. In both cases we are considering the colour or sound psychologically as experienced, that is as sound or colour, and not as physical complexes which follow a certain law, in which case both sound and colour are universals. This difference between the relation of colour to the colours, and that of a universal like dog to individual dogs, has long been observed. Mr. Watt's conception enables us to say that colour is the order of colours, and is, I should say, not itself a quality; and the like is probably true of sounds. Thus the order of individuals is not their universal; and individuals regarded as instances of a universal are not considered as ordered in respect of that universal.
Thus order is a characteristic of every existent, distinct from other such pervasive characteristics, and communicating with them; and it appears undoubtedly to be a category and on the same level of rank4 with existence, relation, and universality.
Compare B. Russell, Principles of Mathematics, § 231, p. 242, for the independence of order of any psychological element. “People speak of a series as consisting of certain terms taken in a certain order, and in this idea there is commonly a psychological element. All sets of terms have, apart from psychological considerations, all orders of which they are capable; that is there are serial relations, whose fields are a given set of terms, which arrange those terms in any possible order... Omnipotence itself cannot give terms an order which they do not possess already: all that is psychological is the consideration of such and such an order.”
Mr. Russell has treated it with great fulness, Principles of Mathematics, Pt. iv.
‘The elements of experience and their integration: or modalism,’ Brit. Journ. of Psych., vol. iv., 1911. See also further papers in vols, vi., 1913; vii., 1914; and his later work, The Psychology of Sound (Cambridge, 1917). I do not feel inclined to accept his statement that sound is a quality and pitches merely their order (see below, p. 267), but should regard them as intrinsically qualities forming an order of qualities. On the other hand, when we are dealing not with the sensa but with the sensing of them, we shall see that the corresponding sensings are merely spatio-temporal patterns of response which have no pitch-quality (nor sound quality either), and in respect of them Mr. Watt's doctrine is true.
For rank among the categories see later, chap. ix. pp. 322 ff.