Whole and parts as categorial.

Every existent is a whole of parts, because Space and Time, in different senses, disintegrate each other. Time breaks up Space into spaces, and Space enables Time to consist of times. Each of them, as we have seen, secures the continuity of the other; Space by supplying connection to the fleeting instants of Time, Time by providing elements within the blank identity of Space. It is but repeating the same thing in other words, when we say that, besides sustaining each other's continuity, they break each other up. Time disintegrates Space directly by distinguishing it into successive spaces; Space disintegrates Time indirectly by making it a whole of times, without which whole there would be no separate times either. Considered by themselves they have no parts; they owe their partition to one another in their mutual involvement, and they divide each other in correspondence. In this division Time plays the directer rôle and takes the lead.

What applies to Space and Time as such applies to any space or time as they exist in any empirical being. Everything is in the end, in its simplest terms, a piece of Space-Time and breaks up therefore into parts, of which it is the whole. It is purely an empirical matter, that is a matter arising not from the fundamental character of Space-Time but from the empirical grouping of parts within it, what the whole may be. It may be a line or a volume in which parts are united continuously. It may be an aggregate of things with definite qualities, a pile of shot or a company of soldiers, or a library of books, or a collection of quite heterogeneous things, like the contents of an antiquarian shop or the different members of our bodies. The things thus aggregated are not themselves continuous but discontinuous; but they are continuously related by the space and time which intervenes between them. There would not be aggregate wholes composed of individuals but for the connecting space-time. But the individuals, owing to their specific qualities, form an isolated object of interest apart from their connection within Space-Time, and it is the space-time which they themselves occupy which is resolved by their separation from one another into parts. Moreover it is these aggregates of ‘qualified’ individuals which being nearer to our senses are our first experience of wholes; and it is later and by some effort of reflection that we first dissect individual aggregates like bodies into their constituent parts and later still observe that a bare extension is itself composed of parts. But it remains that the intrinsic resolution of Space-Time through the internal relation of Space and Time is the basis of all distinction of parts, no matter how loosely the whole is united of them.

Number.

Number is the constitution of a whole in relation to its parts; and it is generated in the concurrent or correspondent distinction of parts in space and time within a spatio-temporal whole. It may be described indifferently as a plan of resolution of a whole into parts or of composition of parts into a whole. All existents are numerable or possess number, because in occupying a space-time they occupy parts of space in correspondence with parts of time. It matters not whether the parts be equal or unequal, homogeneous or heterogeneous in their qualities; or whether the wholes are of the same extent of space-time or not. A group consisting of a man and a dog is as much a two as a group of two men or two shillings; though its parts are unequal in quantity and different in quality; and as much two as a group of two elephants or mice which occupy as wholes very different quantities of space-time. To arrive at the number of, a whole of individuals we have to abstract from the quality or magnitude of the individuals. Their number concerns only the constitution of the whole out of its parts or resolution of the whole into them. In itself number is the correspondence of the space and time parts which is involved in this resolution; but it is a consequence of this that the number of a group establishes a correspondence between the members of the group and those of any other group which has the same number-constitution. All twos correspond to one another in virtue of their twoness, that is of the plan of constitution of the whole from its parts. Number is therefore the plan of a whole of parts.

Number is a different category from extensive quantity, though closely connected with it: quantity communicates with number. They are different because it is a different relation of Space to Time which lies at the basis of them. Quantity expresses the fact that Space is a duration, or that Time sweeps out Space in its flight. Number is the concurrent resolution of either into parts. But since this is so, quantity is directly numerable, for in the generation of a quantity there is the making of a whole of parts by successive addition of the parts. Kant in making number the ‘schema’ of quantity noted the connection of the two, but mistakenly overlooked the more important difference. The category whose schema is number, if any such distinction of schema and category could possibly be recognised, as it cannot, would be not quantity but that of part and whole. Intensive quantity does not communicate directly with number, for it is not a whole of parts. The connection is only possible indirectly through correlation of intensities with extensive quantities.

Number a universal.

Being a plan of constitution of a whole of parts, number is universal or communicates with universality. It is a non-empirical or a priori universal, arising out of Space-Time as such. The various cardinal numbers, 2, 4, 7, etc., are empirical universals which are special plans of whole and parts and are species of the category number. These special numbers have for their particulars the groups of things (or even of parts of areas or lines or hours) which are apprehended empirical embodiments of these universals. Thus the number two is embodied in two pebbles, or two men, or two inches, but is never to be identified with them. The King's gift to mothers with triplets is given for the triplets, not for the number three. Nor is the number two a mere abstraction from concrete groups of two things but is the plan (itself something concrete) on which this group is constructed. Hence however much the observation of collections of things may provoke us to attend to numbers and their combinations, we no more derive arithmetical truths from the things in which they are embodied than we derive geometrical truths, such as that the two sides of the triangle are greater than the third side, from actual measurement of brass triangles or three-cornered fields. These are not the foundations of arithmetic or geometry but only the devices by which kind nature or our teachers cajole us into the exercise of our attention to or reflection upon numbers and figures themselves. Figures in geometry and numbers in arithmetic are the empirical objects so described which we observe for themselves; and numbers are empirical universals in the same way as triangle and sphere and dog are empirical universals. Thus the special numbers are the variable and shifting material in which number as such, the category number, is embodied. This rarefied, but still concrete, material is what Plato described under the name of the “indeterminate dyad,” indicating by the name dyad its capacity of multiform realisation of number as such, and pointing by this superb conception to the way in which we are to understand the real relation between a universal and its sensible particulars.1 It is therefore by no accident but in virtue of the intrinsic character of number and numbers that universality was represented by him as number and the particular universals or forms as particular numbers. It is only elaborating still further the appositeness of this conception when we try, as I have tried before to do, to explain all universals as spatio-temporal plans that are realised in the sensible particulars, which are in themselves spatio-temporal existents constructed on those plans.

Unity.

Arithmetic then is the empirical science whose object is the special or particular numbers (themselves universals) and the relations of them. One conception remains difficult, that of the number 1, itself.2 It has sometimes been thought that I or unity depends on the act of thought {e.g. in counting) which constitutes an object one. But clearly this could only be true if the act of thought were itself enjoyed as one, and thus the explanation would be circular. Now it is safe to say that unity is a notion posterior in development to multiplicity. That 2 is equal to 1 + 1 is not the definition of 2 but something we learn about it, and Kant was perfectly justified in calling such a proposition synthetic. Probably the greatest step ever made in arithmetic was the elementary discovery that the numbers could be obtained from one another by addition of units, or before that stage was reached, that 6 could be got by adding 2 to 4. The numbers are to begin with distinctive individuals, as distinct from one another as a triangle from a square.3 Enumeration was a reduction of this distinctive difference in the empirical material to a comprehensive law of genesis. Bearing this in mind, that numbers have different numerical quality, we may see that unity is the whole which is the same as its parts; or to put the matter otherwise, any object compared with a whole of two parts or of three parts, could be arranged in a series with it, in so far as in the single object the whole and every part coincided. That is, unity is a limiting case of the distinction of whole into parts in which the distinction has vanished, or it is a piece of Space-Time before its division into parts. Thus unity is rather that which is left when 2 is removed from 3 than what is added to 2 in order to make 3. In any case it is a discovery that given a series of numbers, 2, 3, 4, etc., there is a number, unity, belonging to the series and based on the same concurrence of Space and Time as the numbers, from which the other numbers may be derived by addition, when addition is suitably defined. I say, suitably defined; for it is clear that though we may add together things, we do not add together numbers in the same sense; but the sum of two numbers is the plan of a whole whose parts correspond to the parts of each of the two numbers when they are taken together. We discover in this way empirically that 12 is 7 + 5 and 1 + 1 is 2.

Unity it may be observed in passing is different from a unit. It has sometimes been thought that a number is a multiplicity of equal units; but, as we have seen, number has nothing whatever to do with equality of parts in a whole. A unit is in fact a thing (or even a piece of Space or Time) which is used for purposes of measurement. Measurement is effected by securing correspondence with the series of numbers. The simplest and most convenient method of doing this in dealing with things is the adoption of a unit of the same stuff as the thing and taking wholes whose parts are each the unit thing.

Finally, the reference of number to the corresponding parts of space and time within any space—time may serve to explain why as a matter of history the extension of the idea of number from integers to fractions, irrationals and other numbers has been accomplished in connection with geometrical facts and has arisen out of them.

Number in extension.

In this account of number I have ventured to differ from Messrs. Frege and Russell's often-cited definition of cardinal number as the class of classes similar to a given class; though it is clear that so far as my version of the matter may be taken to be correct it is arrived at by reflection on their doctrine and is suggested by it, and is merely a translation of it into metaphysical language. In fact they define number in extensional terms, which is proper to mathematics, while the account here given is the intensional side of the same subject-matter.4 Moreover, it has been indicated that if number is the constitutive correspondence of Space and Time whereby a whole is a whole of parts, it would follow from this that there is correspondence between the members of all classes which have the same constitutive number. From the side of extension then a cardinal number may be described as a class of such classes. But this description starts with entities belonging to classes; that is it begins with finites, at the very lowest finite spaces or durations, and number is defined by reference to them. Just for that reason the definition tells us something which is true about number, but does not tell us what number is, any more than to describe man as the class of men tells us what man is. It gives us a description of number and not acquaintance with it. It is thus not a metaphysical account of number but something which follows from number; and it would not therefore, so far as I can see, explain why any existent is numerable. Before, it has been suggested that this method of defining number makes it amenable to mathematical treatment, and that it offers a notable instance of the difference between mathematical and metaphysical treatment of the same thing. Consequently it is not in the least pretended that the account here given could be used for making arithmetical discoveries; while on the other hand the extensive definition of number is. It is not the business of metaphysics to make discoveries in arithmetic, which employs such concepts as are most suitable to its own purposes. The metaphysical definition may be useless for mathematical purposes. It is enough that it should be useful for metaphysical purposes. The two accounts refer to the same reality; but while the one, the metaphysical one, points to it with the finger, the other describes it.

Number and counting.

Number is apprehended through counting, but the act of counting does not explain number. Number is a category which belongs to all existents as wholes of parts in Space-Time, and it applies to mind and mental acts in the same sense as to external things. The whole of. enjoyment experienced in counting five is a whole of mental acts and has number like the external thing that is counted. Hence we learn number in counting groups of material things, as in exchanging sheep and oxen for cowrie shells or dollars, or in measuring lengths by our feet, or estimating the height of a horse by our hands. But the counting itself is only compresent with number and is itself numerable.

Since number is constitutive of a whole of parts, we do not count unless we experience a whole as made of parts. Hence it is that as a matter of fact we may find processes performed which simulate counting, but where objects are taken in, we say, as a whole, but not as a whole of parts. A thing may have parts without having its parts recognised as parts and without therefore being in the strict sense a whole. We take in a crowd by its individual look of magnitude or extension. A boy may identify a card used in a musical-box, pierced with a vast number of holes in an intricately complex arrangement, and name its tune; but he clearly is not counting or discriminating parts. Many of the performances of animals which seem like counting may, as Mr. Bradley has pointed out, be explained without reference to counting. Apart from any indications that may be given by human beings to the animal, a group of three things looks or feels different from one of two; and this may be sufficient for the purpose. It cannot be said that the arithmetical powers of the lower animals have been established, and scepticism is not unbecoming in respect of horses and dogs, no less than of pigs. If such capacity of real counting were established our estimate of animals or ourselves would undergo some modification. But it would at most be a chapter added to the story of when and how the mind comes to apprehend number. Metaphysically the interest of counting does not lie here but in the fact which may be verified in all the categories that when the mind is aware of number it also enjoys itself as number.5

• 1.

  See J. Burnet: Greek Philosophy from Thales to Plato, ch. xvi. pp. 320 ff.

• 2.

  I do not attempt the difficult problem of the number zero.

• 3.

  Compare on this matter F. H. Bradley, Logic, pp. 370, 371; and below, p. 319.

• 4.

  Hence it will be observed this account applies directly to all numbers, while the definition by classes applies directly only to integers.

• 5.

  See later Bk. III. ch. vi.