CHAPTER VII

THE ANATOMY OF SOME SCIENTIFIC IDEAS

_I. Fact_

THE characteristic of physical science is, that it ignores all judgments of value: for example, æsthetic or moral judgments. It is purely matter-of-fact, and this is the sense in which we must interpret the sonorous phrase, "Man, the servant and the minister of Nature."

The sphere of thought which is thus left is even then too wide for physical science. It would include Ontology, namely, the determination of the nature of what truly exists; in other words, Metaphysics. From an abstract point of view this exclusion of metaphysical inquiry is a pity. Such an inquiry is a necessary critique of the worth of science, to tell us what it all comes to. The reasons for its careful separation from scientific thought are purely practical; namely, because we can agree about science--after due debate--whereas in respect to metaphysics debate has hitherto accentuated disagreement. These characteristics of science and metaphysics were unexpected in the early days of civilised thought. The Greeks thought that metaphysics was easier than physics, and tended to deduce scientific principles from _a priori_ conceptions of the nature of things. They were restrained in this disastrous tendency by their vivid naturalism, their delight in first-hand perception. Mediæval Europe shared the tendency without the restraint. It is possible that some distant generations may arrive at unanimous conclusions on ontological questions, whereas scientific progress may have led to ingrained opposing veins of thought which can neither be reconciled nor abandoned. In such times metaphysics and physical science will exchange their rôles. Meanwhile we must take the case as we find it.

But a problem remains. How can mankind agree about science without a preliminary determination of what really is? The answer must be found in an analysis of the facts which form the field of scientific activity. Mankind perceives, and finds itself thinking about its perceptions. It is the thought that matters and not that element of perception which is not thought. When the immediate judgment has been formed--Hullo, red!--it does not matter if we can imagine that in other circumstances--in better circumstances, perhaps--the judgment would have been--Hullo, blue!--or even--Hullo, nothing! For all intents and purposes, at the time it was red. Everything else is hypothetical reconstruction. The field of physical science is composed of these primary thoughts, and of thoughts about these thoughts.

But--to avoid confusion--a false simplicity has been introduced above into the example given of a primary perceptive thought. "Hullo, red!" is not really a primary perceptive thought, though it often is the first thought which finds verbal expression even silently in the mind. Nothing is in isolation. The perception of red is of a red object in its relations to the whole content of the perceiving consciousness.

Among the most easily analysed of such relations are the space relations. Again the red object is in immediate perception nothing else than a red object. It is better termed an "object of redness." Thus a better approximation to an immediate perceptive judgment is, "Hullo, object of redness there!" But, of course, in this formulation other more complex relations are omitted.

This tendency towards a false simplicity in scientific analysis, to an excessive abstraction, to an over-universalising of universals, is derived from the earlier metaphysical stage. It arises from the implicit belief that we are endeavouring to qualify the real with appropriate adjectives. In conformity with this tendency we think, "this real thing is red." Whereas our true goal is to make explicit our perception of the apparent in terms of its relations. What we perceive is redness related to other apparents. Our object is the analysis of the relations.

One aim of science is the harmony of thought, that is, to secure that judgments which are logical contraries should not be thought-expressions of consciousness. Another aim is the extension of such harmonised thought.

Some thoughts arise directly from sense-presentation, and are part of the state of consciousness which is perception. Such a thought is, "An object of redness is there." But in general the thought is not verbal, but is a direct apprehension of qualities and relations within the content of consciousness.

Amid such thoughts there can be no lack of harmony. For direct apprehension is in its essence unique, and it is impossible to apprehend an object as both red and blue. Subsequently it may be judged that if other elements of the consciousness had been different, the apprehension would have been of a blue object. Then--under certain circumstances--the original apprehension will be called an error. But for all that the fact remains, there was an apprehension of a red object.

When we speak of sense-presentation, we mean these primary thoughts essentially involved in its perception. But there are thoughts about thoughts, and thoughts derived from other thoughts. These are secondary thoughts. At this point it is well explicitly to discriminate between an actual thought-expression, namely, a judgment actually made, and a mere proposition which is a hypothetical thought-expression, namely, an imagined possibility of thought-expression. Note that the actual complete thought-content of the consciousness is explicitly neither affirmed or denied. It is just what _is_ thought. Thus, to think "two and two make four" is distinct from affirming that two and two make four. In the first case the proposition is the thought-expression, in the second case the affirmation of the proposition is the thought-expression, and the proposition has been degraded to a mere proposition, namely, to a hypothetical thought-expression which is reflected upon.

A distinction is sometimes made between facts and thoughts. So far as physical science is concerned, the facts are thoughts, and thoughts are facts. Namely, the facts of sense-presentation as they affect science are those elements in the immediate apprehensions which are thoughts. Also, actual thought-expressions, primary or secondary, are the material facts which science interprets.

The distinction that facts are given, but thoughts are free, is not absolute. We can select and modify our sense-presentation, so that facts--in the narrower sense of immediate apprehension of sense-presentation--are to some degree subject to volition. Again, our stream of thought-expression is only partially modified by explicit volition. We can choose our physical experience, and we find ourselves thinking; namely, on the one hand there is selection amid the dominant necessity of sense, and on the other hand, the thought-content of consciousness (so far as secondary thoughts are concerned) is not wholly constituted by the selection of will.

Thus, on the whole there is a large primary region of secondary thought, as well as of the primary thoughts of sense-presentation, which is given in type. That is the way in which we do think of things, not wholly from any abstract necessity, so far as we know, but because we have inherited the method from an environment. It is the way we find ourselves thinking, a way which can only be fundamentally laid aside by an immense effort, and then only for isolated short periods of time. This is what I have called the "whole apparatus of commonsense thought."

It is this body of thought which is assumed in science. It is a way of thinking rather than a set of axioms. It is, in fact, the set of concepts which commonsense has found useful in sorting out human experience. It is modified in detail, but assumed in gross. The explanations of science are directed to finding conceptions and propositions concerning nature which explain the importance of these common sense notions. For example, a chair is a common sense notion, molecules and electrons explain our vision of chairs.

Now science aims at harmonising our reflective and derivative thoughts with the primary thoughts involved in the immediate apprehension of sense-presentation. It also aims at producing such derivative thoughts, logically knit together. This is scientific theory; and the harmony to be achieved is the agreement of theory with observation, which is the apprehension of sense-presentation.

Thus there is a twofold scientific aim: (1) the production of theory which agrees with experience; and (2) the explanation of commonsense concepts of nature, at least in their main outlines. This explanation consists in the preservation of the concepts in a scientific theory of harmonised thought.

It is not asserted that this is what scientists in the past meant to achieve, or thought that they could achieve. It is suggested as the actual result of scientific effort, so far as that effort has had any measure of success. In short, we are here discussing the natural history of ideas and not volitions of scientists.

_II. Objects_

We perceive things in space. For example, among such things are dogs, chairs, curtains, drops of water, gusts of air, flames, rainbows, chimes of bells, odours, aches and pains. There is a scientific explanation of the origin of these perceptions. This explanation is given in terms of molecules, atoms, electrons, and their mutual relations, in particular of their space-relations, and waves of disturbance of these space-relations which are propagated through space. The primary elements of the scientific explanation--molecules, etc.--are not the things directly perceived. For example, we do not perceive a wave of light; the sensation of sight is the resultant effect of the impact of millions of such waves through a stretch of time. Thus the object directly perceived corresponds to a series of events in the physical world, events which are prolonged through a stretch of time. Nor is it true that a perceived object always corresponds to the same group of molecules. After a few years we recognise the same cat, but we are thereby related to different molecules.

Again, neglecting for a moment the scientific explanation, the perceived object is largely the supposition of our imagination. When we recognised the cat, we also recognised that it was glad to see us. But we merely heard its mewing, saw it arch its back, and felt it rubbing itself against us. We must distinguish, therefore, between the many direct objects of sense, and the single indirect object of thought which is the cat.

Thus, when we say that we perceived the cat and understood its feelings, we mean that we heard a sense-object of sound, that we saw a sense-object of sight, that we felt a sense-object of touch, and that we thought of a cat and imagined its feelings.

Sense-objects are correlated by time-relations and space-relations. Three simultaneous sense-objects which are also spatially coincident, are combined by thought into the perception of one cat. Such combination of sense-objects is an instinctive immediate judgment in general without effort of reasoning. Sometimes only one sense-object is present. For example, we hear mewing and say there must be a cat in the room. The transition from the sense-object to the cat has then been made, by deliberate ratiocination. Even the concurrence of sense-objects may provoke such a self-conscious effort. For example, in the dark we feel something, and hear mewing from the same place, and think, Surely this is a cat. Sight is more bold; when we see a cat, we do not think further. We identify the sight with the cat, whereas the cat and the mew are separate. But such immediate identification of a sight object and an object of thought may lead to error; the birds pecked at the grapes of Apelles.

A single sense-object is a complex entity. The sight-object of a tile on the hearth may remain unchanged as we watch it in a steady light, remaining ourselves unchanged in position. Even then it is prolonged in time, and has parts in space. Also it is somewhat arbitrarily distinguished from a larger whole of which it forms part. But the glancing fire-light and a change in our position alters the sight-object. We judge that the tile thought-object remains unchanged. The sight-object of the coal on the fire gradually modifies, though within short intervals it remains unchanged. We judge that the coal thought-object is changing. The flame is never the same, and its shape is only vaguely distinguishable.

We conclude that a single self-identical sight-object is already a phantasy of thought. Consider the unchanging sight-object of the tile, as we remain still in a steady light. Now a sense-object perceived at one time is a distinct object from a sense-object seen at another time. Thus the sight of the tile at noon is distinct from its sight at 12.30. But there is no such thing as a sense-object at an instant. As we stare at the tile, a minute, or a second, or a tenth of a second, has flown by: essentially there is a duration. There is a stream of sight, and we can distinguish its parts. But the parts also are streams, and it is only in thought that the stream separates into a succession of elements. The stream may be "steady" as in the case of the unchanging sight-tile, or may be "turbulent" as in the case of the glancing sight-flame. In either case a sight-object is some arbitrarily small part of the stream.

Again, the stream which forms the succession of sight-tiles is merely a distinguishable part of the whole stream of sight-presentation.

So, finally, we conceive ourselves each experiencing a complete time-flux (or stream) of sense-presentation. This stream is distinguishable into parts. The grounds of distinction are differences of sense--including within that term, differences of types of sense, and differences of quality and of intensity within the same type of sense--and differences of time-relations, and differences of space-relations. Also the parts are not mutually exclusive and exist in unbounded variety.

The time-relation between the parts raises the questions of memory and recognition, subjects too complex for discussion here. One remark must be made. If it be admitted, as stated above, that we live in durations and not in instants, namely, that the present essentially occupies a stretch of time, the distinction between memory and immediate presentation cannot be quite fundamental; for always we have with us the fading present as it becomes the immediate past. This region of our consciousness is neither pure memory nor pure immediate presentation. Anyhow, memory is also a presentation in consciousness.

Another point is to be noted in connection with memory. There is no directly perceived time-relation between a present event and a past event. The present event is only related to the memory of the past event. But the memory of a past event is itself a present element in consciousness. We assert the principle that directly perceived relations can only exist between elements of consciousness, both in that present during which the perception occurs. All other relations between elements of perception are inferential constructions. It thus becomes necessary to explain how the time stream of events establishes itself in thought, and how the apparent world fails to collapse into one single present. The solution of the difficulty is arrived at by observing that the present is itself a duration, and therefore includes directly perceived time-relations between events contained within it. In other words we put the present on the same footing as the past and the future in respect to the inclusion within it of antecedent and succeeding events, so that past, present, and future are in this respect exactly analogous ideas. Thus there will be two events _a_ and _b_, both in the same present, but the event _a_ will be directly perceived to precede the event _b_. Again time flows on, and the event _a_ fades into the past, and in the new present duration events _b_ and _c_ occur, event _b_ preceding event _c_, also in the same present duration there is the memory of the time-relation between _a_ and _b_. Then by an inferential construction the event _a_ in the past precedes the event _c_ in the present. By proceeding according to this principle the time-relations between elements of consciousness, not in the same present, are established. The method of procedure here explained is a first example of what we will call the Principle of Aggregation. This is one of the fundamental principles of mental construction according to which our conception of the external physical world is constructed. Other examples will later on be met with.

The space-relations between the parts are confused and fluctuating, and in general lack determinate precision. The master-key by which we confine our attention to such parts as possess mutual relations sufficiently simple for our intellects to consider is the principle of convergence to simplicity with diminution of extent. We will call it the "principle of convergence." This principle extends throughout the whole field of sense-presentation.

The first application of the principle occurs in respect to time. The shorter the stretch of time, the simpler are the aspects of the sense-presentation contained within it. The perplexing effects of change are diminished and in many cases can be neglected. Nature has restricted the acts of thought which endeavour to realise the content of the present, to stretches of time sufficiently short to secure this static simplicity over the greater part of the sense-stream.

Spatial relations become simplified within the approximately static sense-world of the short time. A further simplicity is gained by partitioning this static world into parts of restricted space-content. The various parts thus obtained have simpler mutual space-relations, and again the principle of convergence holds.

Finally, the last simplicity is obtained by partitioning the parts, already restricted as to space and time, into further parts characterised by homogeneity in type of sense, and homogeneity in quality and intensity of sense. These three processes of restriction yield, finally, the sense-objects which have been mentioned above. Thus the sense-object is the result of an active process of discrimination made in virtue of the principle of convergence. It is the result of the quest for simplicity of relations within the complete stream of sense-presentation.

The thought-objects of perception are instances of a fundamental law of nature, the law of objective stability. It is the law of the coherence of sense-objects. This law of stability has an application to time and an application to space; also it must be applied in conjunction with that other law, the principle of convergence to simplicity from which sense-objects are derived.

Some composite partial streams of sense-presentation can be distinguished with the following characteristics: (1) the time-succession of sense-objects, belonging to a single sense, involved in any such a composite partial stream, is composed of very similar objects whose modifications increase only gradually, and thus forms a homogeneous component stream within the composite stream; (2) the space-relations of those sense-objects (of various senses) of such a composite stream which are confined within any sufficiently short time are identical so far as they are definitely apprehended, and thus these various component streams, each homogeneous, "cohere" to form the whole composite partial stream; (3) there are other sense-presentations occurring in association with that composite partial stream which can be determined by rules derived from analogous composite partial streams, with other space and time relations, provided that the analogy be sufficiently close. Call these the "associated sense-presentations." A partial stream of this sort, viewed as a whole, is here called a "first crude thought-object of perception."

For example, we look at an orange for half a minute, handle it, and smell it, note its position in the fruit-basket, and then turn away. The stream of sense-presentation of the orange during that half-minute is a first crude thought-object of perception. Among the associated sense presentations are those of the fruit-basket which we conceive as supporting the orange.

The essential ground of the association of sense-objects of various types, perceived within one short duration, into a first crude thought-object of perception is the coincidence of their space-relations, that is, in general an approximate coincidence of such relations perhaps only vaguely apprehended. Thus coincident space-relations associate sense-objects into a first crude thought-object, and diverse space-relations dissociate sense-objects from aggregation into a first crude thought-object. In respect to some groups of sense-objects the association may be an immediate judgment devoid of all inference, so that the primary perceptual thought is that of the first crude thought-object, and the separate sense-objects are the result of reflective analysis acting on memory. For example sense-objects of sight and sense-objects of touch are often thus primarily associated and only secondarily dissociated in thought. But sometimes the association is wavering and indeterminate, for example, that between the sound-object of the mew of the cat and the sight-object of the cat. Thus to sum up, the partial stream of sense-perceptions coalesces into that first crude thought-object of perception which is the momentary cat because the sense-perceptions belonging to this stream are in the same place, but equally it would be true to say that they are in the same place because they belong to the same momentary cat. This analysis of the complete stream of sense-presentation in any small present duration into a variety of first crude thought-objects only partially fits the facts; for one reason because many sense-objects, such as sound for instance, have vague and indeterminate space-relations, for example vaguely those space-relations which we associate with our organs of sense and also vaguely those of the origin from which (in the scientific explanation) they proceed.

The procedure by which the orange of half a minute is elaborated into the orange in the ordinary sense of the term involves in addition the two principles of aggregation and of hypothetical sense-presentation.

The principle of aggregation, as here employed, takes the form that many distinct first crude thought-objects of perception are conceived as one thought-object of perception, if the many partial streams forming these objects are sufficiently analogous, if their times of occurrence are distinct, and if the associated sense-presentations are sufficiently analogous.

For example, after leaving the orange, in five minutes we return. A new first crude thought-object of perception presents itself to us, indistinguishable from the half-minute orange we previously experienced; it is in the same fruit-basket. We aggregate the two presentations of an orange into the same orange. By such aggregations we obtain "second crude thought-objects of perception." But however far we can proceed with aggregation of this type, the orange is more than that. For example, what do we mean when we say, The orange is in the cupboard, if Tom has not eaten it?

The world of present fact is more than a stream of sense-presentation. We find ourselves with emotions, volitions, imaginations, conceptions, and judgments. No factor which enters into consciousness is by itself or even can exist in isolation. We are analysing certain relations between sense-presentation and other factors of consciousness. Hitherto we have taken into account merely the factors of concept and judgment. Imagination is necessary to complete the orange, namely, the imagination of hypothetical sense-presentations. It is beside the point to argue whether we ought to have such imaginations, or to discuss what are the metaphysical truths concerning reality to which they correspond. We are here only concerned with the fact that such imaginations exist and essentially enter into the formation of the concepts of the thought-objects of perception which are the first data of science. We conceive the orange as a permanent collection of sense-presentations existing as if they were an actual element in our consciousness, which they are not. The orange is thus conceived as in the cupboard with its shape, odour, colour, and other qualities. Namely, we imagine hypothetical possibilities of sense-presentation, and conceive their want of actuality in our consciousness as immaterial to their existence in fact. The fact which is essential for science is our conception; its meaning in regard to the metaphysics of reality is of no scientific importance, so far as physical science is concerned.

The orange completed in this way is the thought-object of perception.

It must be remembered that the judgments and concepts arising in the formation of thought-objects of perception are in the main instinctive judgments, and instinctive concepts, and are not concepts and judgments consciously sought for and consciously criticised before adoption. Their adoption is facilitated by and interwoven with the expectation of the future in which the hypothetical passes into the actual, and also with the further judgment of the existence of other consciousnesses, so that much that is hypothetical to one consciousness is judged to be actual to others.

The thought-object of perception is, in fact, a device to make plain to our reflective consciousness relations which hold within the complete stream of sense-presentation. Concerning the utility of this weapon there can be no question; it is the rock upon which the whole structure of commonsense thought is erected. But when we consider the limits of its application the evidence is confused. A great part of our sense-presentation can be construed as perception of various persistent thought-objects. But hardly at any time can the sense-presentations be construed wholly in that way. Sights lend themselves easily to this construction, but sight can be baffled: for example, consider reflections in looking-glasses, apparently bent sticks half in and half out of water, rainbows, brilliant patches of light which conceal the object from which they emanate, and many analogous phenomena. Sound is more difficult; it tends largely to disengage itself from any such object. For example, we see the bell, but we hear the sound which comes from the bell; yet we also say that we hear the bell. Again, a toothache is largely by itself, and is only indirectly a perception of the nerve of the tooth. Illustrations to the same effect can be accumulated from every type of sensation.

Another difficulty arises from the fact of change. The thought-object is conceived as one thing, wholly actual at each instant. But since the meat has been bought it has been cooked, the grass grows and then withers, the coal burns in the fire, the pyramids of Egypt remain unchanged for ages, but even the pyramids are not wholly unchanged. The difficulty of change is merely evaded by affixing a technical Latin name to a supposed logical fallacy. A slight cooking leaves the meat the same object, but two days in the oven burns it to a cinder. When does the meat cease to be? Now the chief use of the thought-object is the concept of it as one thing, here and now, which later can be recognised, there and then. This concept applies sufficiently well to most things for short times, and to many things for long times. But sense-presentation as a whole entirely refuses to be patient of the concept.

We have now come to the reflective region of explanation, which is science.

A great part of the difficulty is at once removed by applying the principle of convergence to simplicity. We habitually make our thought-objects too large; we should think in smaller parts. For example, the Sphinx has changed by its nose becoming chipped, but by proper inquiry we could find the missing part in some private house of Western Europe or Northern America. Thus, either part, the rest of the Sphinx or the chip, regains its permanence. Furthermore, we enlarge this explanation by conceiving parts so small that they can only be observed under the most favourable circumstances. This is a wide extension of the principle of convergence in its application to nature; but it is a principle amply supported by the history of exact observation.

Thus, change in thought-objects of perception is largely explained as a disintegration into smaller parts, themselves thought-objects of perception. The thought-objects of perception which are presupposed in the common thought of civilised beings are almost wholly hypothetical. The material universe is largely a concept of the imagination which rests on a slender basis of direct sense-presentation. But none the less it is a fact; for it is a fact that actually we imagine it. Thus it is actual in our consciousness just as sense-presentation also is actual there. The effort of reflective criticism is to make these two factors in our consciousness agree where they are related, namely, to construe our sense-presentation as actual realisation of the hypothetical thought-objects of perception.

The wholesale employment of purely hypothetical thought-objects of perception enables science to explain some of the stray sense-objects which cannot be construed as perceptions of a thought-object of perception: for example, sounds. But the phenomena as a whole defy explanation on these lines until a further fundamental step is taken, which transforms the whole concept of the material universe. Namely, the thought-object of perception is superseded by the thought-object of science.

The thought-objects of science are molecules, atoms, and electrons. The peculiarity of these objects is that they have shed all the qualities which are capable of direct sense-representation in consciousness. They are known to us only by their associated phenomena, namely, series of events in which they are implicated are represented in our consciousness by sense-presentations. In this way, the thought-objects of science are conceived as the causes of sense-representation. The transition from thought-objects of perception to thought-objects of science is decently veiled by an elaborate theory concerning primary and secondary qualities of bodies.

This device, by which sense-presentations are represented in thought as our perception of events in which thought-objects of science are implicated, is the fundamental means by which a bridge is formed between the fluid vagueness of sense and the exact definition of thought. In thought a proposition is either true or false, an entity is exactly what it is, and relations between entities are expressible (in idea) by definite propositions about distinctly conceived entities. Sense-perception knows none of these things, except by courtesy. Accuracy essentially collapses at some stage of inquiry.

_III. Time and Space_

_Recapitulation._--Relations of time and relations of space hold between sense-objects of perception. These sense-objects are distinguished as separate objects by the recognition of either (1) differences of sense-content, or (2) time-relations between them other than simultaneity, or (3) space-relations between them other than coincidence. Thus sense-objects arise from the recognition of contrast within the complete stream of sense-presentation, namely, from the recognition of the objects as related terms, by relations which contrast them. Differences of sense-content are infinitely complex in their variety. Their analysis under the heading of general ideas is the unending task of physical science. Time-relations and space-relations are comparatively simple, and the general ideas according to which their analysis should proceed are obvious.

This simplicity of time and space is perhaps the reason why thought chooses them as the permanent ground for objectival distinction, throwing the various sense-objects thus obtainable into one heap, as a first crude thought-object of perception, and thence, as described above, obtaining a thought-object of perception. Thus a thought-object of perception conceived as in the present of a short duration is a first crude thought-object of perception either actual or hypothetical. Such a thought-object of perception, confined within a short duration, takes on the space-relations of its component sense-objects within that same duration. Accordingly thought-objects of perception, conceived in their whole extents, have to each other the time-relationships of their complete existences, and within any small duration have to each other the space-relationships of their component sense-objects which lie within that duration.

Relations bind together: thus thought-objects of perception are connected in time and in space. The genesis of the objectival analysis of sense-presentation is the recognition of sense-objects as distinct terms in time-relations and space-relations: thus thought-objects of perception are separated by time and by space.

_Whole and Part._--A sense-object is part of the complete stream of presentation. This concept of being a part is merely the statement of the relation of the sense-object to the complete sense-presentation for that consciousness. Also a sense-object can be part of another sense-object. It can be a part in two ways, namely, a part in time and a part in space. It seems probable that both these concepts of time-part and space-part are fundamental; that is, are concepts expressing relations which are directly presented to us, and are not concepts about concepts. In that case no further definition of the actual presentation is possible. It may even then be possible to define an adequate criterion of the occurrence of such a presentation. For example, adopting for the moment a realist metaphysic as to the existence of the physical world of molecules and electrons, the vision of a chair as occurring for some definite person at some definite time is essentially indefinable. It is his vision, though each of us guesses that it must be uncommonly like our vision under analogous circumstances. But the existence of the definable molecules and waves of light in certain definable relations to his bodily organs of sense, his body also being in a certain definable state, forms an adequate criterion of the occurrence of the vision, a criterion which is accepted in Courts of Law and for physical science is tacitly substituted for the vision.

The connection between the relations "whole and part" and "all and some" is intimate. It can be explained thus so far as concerns directly presented sense-objects. Call two sense-objects "separated" if there is no third sense-object which is a part of both of them. Then an object A is composed of the two objects B and C, if (1) B and C are both parts of A, (2) B and C are separated, and (3) there is no part of A which is separated both from B and from C. In such a case the class α which is composed of the two objects B and C is often substituted in thought for the sense-object A. But this process presupposes the fundamental relation "whole and part." Conversely the objects B and C may be actual sense-objects, but the sense-object A which corresponds to the class α may remain hypothetical. For example, the round world on which we live remains a conception corresponding to no single sense-object at any time presented in any human being's consciousness.

It is possible, however, that some mode of conceiving the whole-and-part relation between extended objects as the all-and-some relation of logical classes can be found. But in this case the extended objects as here conceived cannot be the true sense-objects which are present to consciousness. For as here conceived a part of a sense-object is another sense-object of the same type; and therefore one sense-object cannot be a class of other sense-objects, just as a tea-spoon cannot be a class of other tea-spoons. The ordinary way in thought by which whole-and-part is reduced to all-and-some is by the device of points, namely, the part of an object occupies some of the points occupied by the whole object. If any one holds that in his consciousness the sense-presentation is a presentation of point-objects, and that an extended object is merely a class of such point-objects collected together in thought, then this ordinary method is completely satisfactory. We shall proceed on the assumption that this conception of directly perceived point-objects has no relation to the facts.

In the preceding address on "The Organisation of Thought," another mode is suggested. But this method would apply only to the thought-object of perception, and has no reference to the primary sense-objects here considered. Accordingly it must reckon as a subordinate device for a later stage of thought.

Thus the point-object in time and the point-object in space, and the double point-object both in time and space, must be conceived as intellectual constructions. The fundamental fact is the sense-object, extended both in time and space, with the fundamental relation of whole-to-part to other such objects, and subject to the law of convergence to simplicity as we proceed in thought through a series of successively contained parts.

The relation whole-to-part is a temporal or spatial relation, and is therefore primarily a relation holding between sense-objects of perception, and it is only derivatively ascribed to the thought-objects of perception of which they are components. More generally, space and time relations hold primarily between sense-objects of perception and derivatively between thought-objects of perception.

_Definition of Points._--The genesis of points of time and of space can now be studied. We must distinguish (1) sense-time and sense-space, and (2) thought-time of perception and thought-space of perception.

Sense-time and sense-space are the actually observed time-relations and space-relations between sense-objects. Sense-time and sense-space have no points except, perhaps, a few sparse instances, sufficient to suggest the logical idea; also, sense-time and sense-space are discontinuous and fragmentary.

Thought-time of perception and thought-space of perception are the time and space relations which hold between thought-objects of perception. Thought-time of perception and thought-space of perception are each continuous. By "continuous" is here meant that all thought-objects of perception have to each other a time (or space) relation.

The origin of points is the effort to take full advantage of the principle of convergence to simplicity. In so far as this principle does not apply, a point is merely a cumbrous way of directing attention to a set of relations between a certain set of thought-objects of perception, which set of relations, though actual so far as a thought-object is actual, is (under this supposition) of no particular importance. Thus the proved importance in physical science of the concepts of points in time and points in space is a tribute to the wide applicability of this principle of convergence.

Euclid defines a point as without parts and without magnitude. In modern language a point is often described as an ideal limit by indefinitely continuing the process of diminishing a volume (or area). Points as thus conceived are often called convenient fictions. This language is ambiguous. What is meant by a fiction? If it means a conception which does not correspond to any fact, there is some difficulty in understanding how it can be of any use in physical science. For example, the fiction of a red man in a green coat inhabiting the moon can never be of the slightest scientific service, simply because--as we may presume--it corresponds to no fact. By calling the concept of points a convenient fiction, it must be meant that the concept does correspond to some important facts. It is, then, requisite, in the place of such vague allusiveness, to explain exactly what are the facts to which the concept corresponds.

We are not much helped by explaining that a point is an ideal limit. What is a limit? The idea of a limit has a precise meaning in the theory of series, and in the theory of the values of functions; but neither of these meanings apply here. It may be observed that, before the ordinary mathematical meanings of limit had received a precise explanation, the idea of a point as a limit might be considered as one among other examples of an idea only to be apprehended by direct intuition. This view is not now open to us. Thus, again, we are confronted with the question: What are the precise properties meant when a point is described as an ideal limit? The discussion which now follows is an attempt to express the concept of a point in terms of thought-objects of perception related together by the whole-and-part relation, considered either as a time-relation or as a space-relation. If it is so preferred, it may be considered that the discussion is directed towards a precise elucidation of the term "ideal limit" as often used in this connection.

The subsequent explanations can be made easier to follow by a small piece of symbolism: Let _aEb_ mean that "_b_ is part of _a_." We need not decide whether we are talking of time-parts or space-parts, but whichever choice is supposed to be made must be conceived as adhered to throughout any connected discussion. The symbol _E_ may be considered as the initial letter of "encloses," so we read "_aEb_" as "_a_ encloses _b_." Again the "field of _E_" is the set of things which either enclose or are enclosed, _i. e._ everything "_a_," which is such that _x_ can be found so that either _aEx_ or _xEa_. A member of the field of _E_ is called "an enclosure-object."

Now, we assume that this relation of whole-to-part, which in the future we will call "enclosure," always satisfies the conditions in that the relation _E_ is (1) transitive, (2) asymmetrical, and (3) with its domain including its converse domain.

These four conditions deserve some slight consideration; only the first two of them embody hypotheses which enter vitally into the reasoning.

Condition (1) may be stated as the condition that _aEb_ and _bEc_ always implies _aEc_. The fact that an entity _b_ can be found such that _aEb_ and _bEc_ may be conceived as a relation between _a_ and _c_. It is natural to write _E_^2 for this relation. Thus the condition is now written: If _aE_^2_c_, then _aEc_. This can be still otherwise expressed by saying that the relation _E_^2 implies, whenever it holds, that the relation _E_ also holds.

Condition (2) is partly a mere question of trivial definition, and partly a substantial assumption. The asymmetrical relation (_E_) is such that _aEb_ and _bEa_ can never hold simultaneously. This property splits up into two parts: (1) that no instance of _aEb_ and _bEa_ and "_a_ diverse from _b_," can occur, and (2) that _aEa_ cannot occur. The first part is a substantial assumption, the second part (so far as we are concerned) reduces to the trivial convention that we shall not consider an object as part of itself, but will confine attention to "proper parts."

Condition (3) means that _aEb_ always implies that _c_ can be found such that _bEc_. This condition, taken in conjunction with the fact that we are only considering proper parts, is the assertion of the principle of the indefinite divisibility of extended objects, both in space and in time.

An indivisible part will lack duration in time, and extension in space, and is thus an entity of essentially a different character to a divisible part. If we admit such indivisibles as the only true sense-objects, our subsequent procedure is an unnecessary elaboration.

It will be found that a fourth condition is necessary owing to logical difficulties connected with the theory of an infinite number of choices. It will not be necessary for us to enter further on this question, which involves difficult considerations of abstract logic. The outcome is, that apart from hypothesis we cannot prove the existence of the sets, each containing an infinite number of objects, which are here called points, as will be explained immediately.

Now consider a set of enclosure objects which is such that (1) of any two of its members one encloses the other, and (2) there is no member which is enclosed by all the others, and (3) there is no enclosure-object, not a member of the set which is enclosed by every member of the set. Call such a set a "convergent set of enclosure-objects." As we pass along the series from larger to smaller members, evidently we converge towards an ideal simplicity to any degree of approximation to which we like to proceed, and the series as a whole embodies the complete ideal along that route of approximation. In fact, to repeat, the series is a _route of approximation_.

We have now to inquire if the principle of convergence to simplicity may be expected to yield the same type of simplicity for every such convergent route. The answer is, as we might expect, namely, that this depends upon the nature of the properties which are to be simplified.

For example, consider the application to time. Now, time is one-dimensional; so when this property of one-dimensionality has been expressed by the proper conditions, not here stated, a convergent set of enclosure-objects must, considered as a route of approximation, exhibit the properties of one unique instant of time, as ordinarily conceived by the euclidean definition. Accordingly, whatever simplicity is to be achieved by the application to time of the principle of convergence to simplicity must be exhibited among the properties of any such route of approximation.

For space, different considerations arise. Owing to its multiple dimensions, we can show that different convergent sets of enclosure-objects, indicating different routes of approximation, may exhibit convergence to different types of simplicity, some more complex than others.

For example, consider a rectangular box of height _h_ ft., breadth _b_ ft., and thickness _c_ ft. Now, keep _h_ and _b_ constant, and let the central plane (height _h_, breadth _b_) perpendicular to the thickness be fixed, then make _c_ diminish indefinitely. We thus obtain a convergent series of an indefinitely large number of boxes, and there is no smallest box. Thus this convergent series exhibits the route of approximation towards the type of simplicity expressed as being a plane area of height _h_, breadth _b_, and no thickness.

Again, by keeping the central line of height _h_ fixed, and by making _b_ and _c_ diminish indefinitely, the series converges to the segment of a straight line of length _h_.

Finally, by keeping only the central point fixed, and by making _h_, _b_, and _c_ diminish indefinitely, the series converges to a point.

Furthermore, we have introduced as yet no concept which would prevent an enclosure-object being formed of detached fragments in space. Thus we can easily imagine a convergent set which converges to a number of points in space. For example, each object of the set might be formed of two not overlapping spheres of radius _r_, with centres _A_ and _B_. Then by diminishing _r_ indefinitely, and keeping _A_ and _B_ fixed, we have convergence to the pair of points _A_ and _B_.

It remains now to consider how those convergent sets which converge to a single point can be discriminated from all the other types of such sets, merely by utilising concepts founded on the relation of enclosure.

Let us name convergent sets by Greek letters; by proceeding "forward" along any such set let us understand the process of continually passing from the larger to the smaller enclosure-objects which form the set.

The convergent set α will be said to "cover" the convergent set β, if every member of α encloses some members of β. We notice that if an enclosure-object _x_ encloses any member (_y_) of β, then every member of the "tail-end" of β, found by proceeding forward along β from _y_, must be enclosed by _x_. Thus if α covers β, every member of α encloses every member of the tail-end of β, starting from the largest member of β which is enclosed by that member of α.

It is possible for each of two convergent sets to cover the other. For example, let one set (α) be a set of concentric spheres converging to their centre _A_, and the other set (β) be a set of concentric cubes, similarly situated, converging to the same centre _A_. Then α and β will each cover the other.

Let two convergent sets which are such that each covers the other be called "equal."

Then it is a sufficient condition to secure that a convergent set α possesses the point type of convergence, if every convergent set covered by it is also equal to it, namely, α is a convergent set with the punctual type of convergence, if "α covers β" always implies that β covers α.

It can easily be seen by simple examples that the other types of convergence to surfaces or lines or sets of points cannot possess this property. Consider, for example, the three convergent sets of boxes in the preceding illustration, which converge respectively to a central plane, a central line in the central plane, and the central point in the central line. The first set covers the second and third sets, and the second set covers the third set, but no two of the sets are equal.

It is a more difficult question to determine whether the condition here indicated as sufficient to secure the punctual type of convergence is also necessary. The question turns on how far thought-objects of perception possess exact boundaries prior to the elaboration of exact mathematical concepts of space. If they are to be conceived as possessing such exact boundaries, then convergent sets converging to points on such boundaries must be allowed for. The procedure necessary for the specification of the complete punctual condition becomes then very elaborate,[3] and will not be considered here.

But such exact determination as is involved in the conception of an exact spatial boundary does not seem to belong to the true thought-object of perception. The ascription of an exact boundary really belongs to the transition stage of thought as it passes from the thought-object of perception to the thought-object of science. The transition from the sense-object immediately presented to the thought-object of perception is historically made in a wavering indeterminate line of thought. The definite stages here marked out simply serve to prove that a logically explicable transition is possible.

We accordingly assume that the condition laid down above to secure the punctual convergence of a convergent set of enclosure-objects is not only sufficient, but necessary.

It can be proved that, if two convergent sets of enclosure-objects are both equal to a third convergent set, they are equal to each other. Consider now any punctual convergent set (α). We want to define the "point" to which α is a route of approximation in a way which is neutral between α and all the convergent sets which are equal to α. Each of these sets is a route of approximation to the same "point" as α. This definition is secured if we define the point as the class formed by all the enclosure-objects which belong either to α or to any convergent set which is equal to α. Let _P_ be this class of enclosure-objects. Then any convergent set (β) which consists of enclosure-objects entirely selected from members of the class _P_ must be a route of approximation to the same "point" as does the original punctual set α; namely, provided that we choose a small enough enclosure-object in β, we can always find a member of α which encloses it; and provided that we choose a small enough enclosure-object in α, we can always find a member of β which encloses it. Thus _P_ only includes convergent sets of the punctual type, and the route of approximation indicated by any two convergent sets selected from _P_ converges to identical results.

_The Uses of Points._--The sole use of points is to facilitate the employment of the principle of Convergence to Simplicity. By this principle some simple relations in appropriate circumstances become true, when objects are considered which are sufficiently restricted in time or in space. The introduction of points enables this principle to be carried through to its ideal limit. For example, suppose _g_ (_a_, _b_, _c_) represents some statement concerning three enclosure-objects, _a_, _b_, _c_, which may be true if the objects are sufficiently restricted in extent. Let _A_, _B_, _C_ be three given points, then we define _g_ (_A_, _B_, _C_) to mean that _whatever_ three enclosure-objects _a_, _b_, _c_ are chosen, such that _a_ is a member of _A_, _b_ of _B_, and _c_ of _C_, it is _always possible_ to find three other members of _A_, _B_, _C_, namely, _x_ a member of _A_, _y_ of _B_, and _z_ of _C_, such that _aEx_, _bEy_, _cEz_, and _g_ (_x_, _y_, _z_). So by going far enough down in the tail-ends of _A_, _B_, _C_ we can always secure three objects _x_, _y_, _z_ for which _g_ (_x_, _y_, _z_) is true.

For example, let _g_ (_A_, _B_, _C_) mean "_A_, _B_, _C_ are three points in a linear row." This must be construed to mean that whatever three objects _a_, _b_, _c_ we choose, members of _A_, _B_, _C_ respectively, we can always find three objects _x_, _y_, _z_, also members of _A_, _B_, _C_ respectively, and such that _a_ encloses _x_, _b_ encloses _y_, _c_ encloses _z_, and also such that _x_, _y_, _z_ are in a linear row.

Sometimes a double convergence is necessary, namely, a convergence of conditions as well as a convergence of objects. For example, consider the statement, "the points _A_ and _B_ are two feet apart." Now, the exact statement "two feet apart" does not apply to objects. For objects _x_ and _y_ we must substitute the statement, "the distance between _x_ and _y_ lies between the limits (2 ± _e_) feet." Here _e_ is some number, less than two, which we have chosen for this statement. Then the points _A_ and _B_ are two feet apart; if, _however we choose the number e_, whatever enclosure-objects _a_ and _b_, members of _A_ and _B_ respectively, we consider, we can always find enclosure-objects _x_ and _y_, members of _A_ and _B_ respectively, such that _a_ encloses _x_ and _b_ encloses _y_, and also such that the distance between _x_ and _y_ lies between the limits (2 ± _e_) feet. It is evident, since _e_ can be chosen as small as we please, that this statement exactly expresses the condition that _A_ and _B_ are two feet apart.

_Straight Lines and Planes._--But the problem of the intellectual construction of straight lines and planes is not yet sufficiently analysed. We have interpreted the meaning of the statement that three or more points are collinear, and can similarly see how to interpret the meaning of the statement that four or more points are coplanar, in either case deriving the exact geometrical statements from vaguer statements respecting extended objects.

This procedure only contemplates groups of finite numbers of points. But straight lines and planes are conceived as containing infinite numbers of points. This completion of lines and planes is obtained by a renewed application of the principle of aggregation, just as a set of first crude thought-objects of perception are aggregated into one complete thought-object of perception. In this way repeated judgments of the collinearity of sets of points are finally, when certain conditions of interlacing are fulfilled, aggregated in the single judgment of all the points of the groups as forming one whole collinear group. Similarly for judgments of coplanarity. This process of logical aggregation can be exhibited in its exact logical analysis. But it is unnecessary here to proceed to such details. Thus we conceive our points as sorted into planes and straight lines, concerning which the various axioms of geometry hold. These axioms, in so far as they essentially require the conception of points, are capable of being exhibited as the outcome of vaguer, less exact judgments respecting the relations of extended objects.

_Empty Space._--It must be observed that the points, hitherto defined, necessarily involve thought-objects of perception, and lie within the space-extension occupied by such objects. It is true that such objects are largely hypothetical, and that we can bring into our hypotheses sufficient objects to complete our lines and planes. But every such hypothesis weakens the connection between our scientific concept of nature and the actual observed facts which are involved in the actual sense-presentations.

Occam's razor, _Entia non multiplicanda præter necessitatem_, is not an arbitrary rule based on mere logical elegancy. Nor is its application purely confined to metaphysical speculation. I am ignorant of the precise reason for its metaphysical validity, but its scientific validity is obvious, namely, every use of hypothetical entities diminishes the claim of scientific reasoning to be the necessary outcome of a harmony between thought and sense-presentation. As hypothesis increases, necessity diminishes.

Commonsense thought also supports this refusal to conceive of all space as essentially depending on hypothetical objects which fill it. We think of material objects as filling space, but we ask whether any objects exist between the Earth and the Sun, between the stars, or beyond the stars. For us, space is there; the only question is whether or not it be full. But this form of question presupposes the meaning of empty space, namely, of space not containing hypothetical objects.

This brings up a wider use of the concept of points, necessitating a wider definition. Hitherto we have conceived points as indicating relations of enclosure between objects. We thus arrive at what now we will term "material points." But the idea of points can now be transformed so as to indicate the possibilities of external relations not those of enclosure. This is effected by an enlargement of the concept of ideal points, already known to geometers.

Define "material lines" to be complete collinear classes of collinear points. Consider now the set of material lines which contain a certain material point. Call such a set of lines an ideal point. This set of lines indicates a possibility of position, which is in fact occupied by that material point common to all the material lines. So this ideal point is an occupied ideal point. Now consider a set of three material lines, such that any two are coplanar, but not the whole three, and further consider the complete set of material lines such that each is coplanar with each of the three material lines first chosen. The axioms which hold for the material lines will enable us to prove that any two lines of this set are coplanar. Then the whole set of lines, including the three original lines, forms an ideal point, according to the definition in its full generality. Such an ideal point may be occupied. In that case there is a material point common to all the lines of the set, but it may be unoccupied. Then the ideal point merely indicates a possibility of spatial relations which has not been realised. It is the point of empty space. Thus the ideal points, which may or may not be occupied, are the points of geometry viewed as an applied science. These points are distributed into straight lines and planes. But any further discussion of this question will lead us into the technical subject of the axioms of geometry and their immediate consequences. Enough has been said to show how geometry arises according to the relational theory of space.

Space as thus conceived is the thought-space of the material world.

_IV. Fields of Force_

The thought-objects of science are conceived as directly related to this thought-space. Their spatial relations are among those indicated by the points of the thought-space. Their emergence in science has been merely a further development of processes already inherent in commonsense thought.

Relations within the complete sense-presentation were represented in thought by the concept of thought-objects of perception. All sense-presentation could not be represented in this way; also the change and disappearance of thought-objects occasioned confusion of thought. A reduction to order of this confusion was attempted by the concepts of permanent matter with primary and secondary qualities. Finally, this has issued in the secondary qualities being traced as perception of events generated by the objects, but--as perceived--entirely disconnected with them. Also the thought-objects of perception have been replaced by molecules and electrons and ether-waves, until at length it is never the thought-object of science which is perceived, but complicated series of events in which they are implicated. If science be right, nobody ever perceived a thing, but only an event. The result is, that the older language of philosophy which still survives in many quarters is now thoroughly confusing when brought into connection with the modern concepts of science. Philosophy--that is, the older philosophy--conceives the thing as directly perceived. According to scientific thought, the ultimate thing is never perceived, perception essentially issuing from a series of events. It is impossible to reconcile the two points of view.

The advantage of the modern scientific concept is that it is enabled to "explain" the fluid vague outlines of sense-presentation. The thought-object of perception is now conceived as a fairly stable state of motion of a huge group of molecules, constantly changing, but preserving a certain identity of characteristics. Also stray sense-objects, not immediately given as part of a thought-object of perception, are now explicable: the dancing light-reflection, the vaguely heard sound, the smell. In fact, the perceived events of the scientific world have the same general definition and lack of definition, and the same general stability and lack of stability, as the sense-objects of the complete sense-presentation or as the thought-objects of perception.

The thought-objects of science, namely, molecules, atoms, and electrons, have gained in permanence. The events are reduced to changes in space-configuration. The laws determining these changes are the ultimate laws of nature.

The laws of change in the physical universe proceed on the assumption that the preceding states of the universe determine the character of the change. Thus, to know the configurations and events of the universe up to and including any instant would involve sufficient data from which to determine the succession of events throughout all time.

But in tracing the antecedents of events, commonsense thought, dealing with the world of thought-objects of perception, habitually assumes that the greater number of antecedent events can be neglected as irrelevant. Consideration of causes is restricted to a few events during a short preceding interval. Finally, in scientific thought it has been assumed that the events in an arbitrarily small preceding duration are sufficient. Thus physical quantities and their successive differential co-efficients up to any order at the instant, but with their limiting values just before that instant, are on this theory sufficient to determine the state of the universe at all times after the instant. More particular laws are assumed. But the search for them is guided by this general principle. Also it is assumed that the greater number of events in the physical universe are irrelevant to the production of any particular effect, which is assumed to issue from relatively few antecedents. These assumptions have grown out of the experience of mankind. The first lesson of life is to concentrate attention on few factors of sense-presentations, and on still fewer of the universe of thought-objects of perception.

The principle by which--consciously or unconsciously--thought has been guided is that in searching for particular causes, remoteness in time and remoteness in space are evidences of comparative disconnection of influence. The extreme form of this principle is the denial of any action at a distance either in time or space. The difficulty in accepting this principle in its crude form is, that since there are no contiguous points, only coincident bodies can act on each other. I can see no answer to this difficulty--namely, either bodies have the same location and are thus coincident, or they have different locations and are thus at a distance and do not act on each other.

This difficulty is not evaded by the hypothesis of an ether, continuously distributed. For two reasons: in the first place, the continuity of the ether does not avoid the dilemma; and secondly, the difficulty applies to time as well as to space, and the dilemma would prove that causation producing change is impossible, namely, no changed condition could be the result of antecedent circumstances.

On the other hand, a direct interaction between two bodies separated in space undoubtedly offends the conception of distance as implying physical disconnection as well as spatial relation. There is no logical difficulty in the assumption of action at a distance as in the case of its denial, but it is contradictory to persistent assumptions of that apparatus of commonsense thought which it is the main business of science to harmonise with sense-presentation, employing only the minimum of modification.

Modern science is really unconcerned with this debate. Its (unacknowledged) conceptions are really quite different, though the verbal explanations retain the form of a previous epoch. The point of the change in conception is that the old thought-object of science was conceived as possessing a simplicity not belonging to the material universe as a whole. It was secluded within a finite region of space, and changes in its circumstances could only arise from forces which formed no essential part of its nature. An ether was called into existence to explain the active relations between these passive thought-objects. The whole conception suffers from the logical difficulties noted above. Also no clear conception can be formed of the sense in which the ether is explanatory. It is to possess a type of activity denied to the original thought-object, namely, it carries potential energy, whereas the atom possessed only kinetic energy, the so-called potential energy of an atom belonging really to the surrounding ether. The truth is, that ether is really excepted from the axiom "no action at a distance," and the axiom thereby is robbed of all its force.

The modern thought-object of science--not yet explicitly acknowledged--has the complexity of the whole material universe. In physics, as elsewhere, the hopeless endeavour to derive complexity from simplicity has been tacitly abandoned. What is aimed at is not simplicity, but persistence and regularity. In a sense regularity is a sort of simplicity. But it is the simplicity of stable mutual relations, and not the simplicity of absence of types of internal structure or of type of relation. This thought-object fills all space. It is a "field"; that is to say, it is a certain distribution of scalar and vector quantities throughout space, these quantities having each its value for each point of space at each point of time, being continuously distributed throughout space and throughout time, possibly with some exceptional discontinuities. The various types of quantity which form the field have fixed relations to each other at each point of time and space. These relations are the ultimate laws of nature.

For example, consider an electron. There is a scalar distribution of electricity, which is what is ordinarily called the electron. This scalar distribution has a volume-density ρ at the time _t_ at any point (_x_, _y_, _z_). Thus ρ is a function of (_x_, _y_, _z_, _t_), which is zero except within a restricted region. Furthermore, at any time _t_, as an essential adjunct, there is a continuous space distribution at each point of the two vectors (_X_, _Y_, _Z_), which is the electric force, and (α, β, γ), which is the magnetic force. Lastly, individuality is ascribed to the scalar electric distribution, so that in addition to its conservation of quantity--involved in the assumed laws--it is also possible to assign the velocities with which the various individual parts of the distribution are moving. Let (_u_, _v_, _w_) be this velocity at (_x_, _y_, _z_, _t_).

This whole scheme of scalar and vector quantities, namely, ρ, (_X_, _Y_, _Z_), (α, β, γ), (_u_, _v_, _w_) is interconnected by the electromagnetic laws. It follows from these laws that the electron, in the sense of the scalar distribution ρ, is to be conceived as at each instant propagating from itself an emanation which travels outwards with the velocity of light _in vacuo_, and from which (_X_, _Y_, _Z_) and (α, β, γ) can be calculated, so far as they are due to that electron. Thus the field, at any time, due to the electron as a whole depends on the previous history of the electron, the nearer to the electron the more recent being the relevant history. The whole scheme of such a field is one single thought-object of science: the electron and its emanations form one essential whole, namely one thought-object of science, essentially complex and essentially filling all space. The electron proper, namely, the scalar distribution ρ, is the focus of the whole, the essential focal property being that the field at any instant is completely determined by the previous history of the focus and of its space relations through all previous time. But the field and the focus are not independent concepts, they are essentially correlated in one organised unity, namely, they are essentially correlated terms in the field of one relation in virtue of which the entities enter into our thoughts.

The fields of a group of electrons are superposed according to the linear law for aggregation, namely, pure addition for analogous scalar quantities and the parallelogram law for analogous vectors. The changes in motion of each electron depend entirely on the resultant field in the region it occupies. Thus a field can be viewed as a possibility of action, but a possibility which represents an actuality.

It is to be noted that the two alternative views of causation are here both included. The complete field within any region of space depends on the past histories of all the electrons, histories extending backwards in proportion to their distances. Also this dependence can be conceived as a transmission. But viewing the cause which effects changes on the electron within that region, it is solely that field within the region, which field is coincident with that electron both in time and in space.

This process of conceiving the actuality underlying a possibility is the uniform process by which regularity and permanence is introduced into scientific thought, namely, we proceed from the actuality of the fact to the actuality of possibility.

In conformity with this principle, propositions are the outgrowth from actual thought-expressions, thought-objects of perceptions from crude sense-objects, hypothetical thought-objects of perception from actual thought-objects of perception, material points from hypothetical infinite suites of hypothetical thought-objects of perception, ideal points from material points, thought-objects of science from thought-objects of perception, fields of electrons from actual mutual reactions of actual electrons.

The process is a research for permanence, uniformity, and simplicity of logical relation. But it does not issue in simplicity of internal structure. Each ultimate thought-object of science retains every quality attributed to the whole scientific universe, but retains them in a form characterised by permanence and uniformity.

_V. Conclusion_

We commenced by excluding judgments of worth and ontological judgments. We conclude by recalling them. Judgments of worth are no part of the texture of physical science, but they are part of the motive of its production. Mankind have raised the edifice of science, because they have judged it worth while. In other words, the motives involve innumerable judgments of value. Again, there has been conscious selection of the parts of the scientific field to be cultivated, and this conscious selection involves judgments of value. These values may be æsthetic, or moral, or utilitarian, namely, judgments as to the beauty of the structure, or as to the duty of exploring the truth, or as to utility in the satisfaction of physical wants. But whatever the motive, without judgments of value there would have been no science.

Again, ontological judgments were not excluded by reason of any lack of interest. They are in fact presupposed in every act of life: in our affections, in our self-restraints, and in our constructive efforts. They are presupposed in moral judgments. The difficulty about them is the absence of agreement as to the method of harmonising the crude judgments of commonsense.

Science does not diminish the need of a metaphysic. Where this need is most insistent is in connection with what above has been termed "the actuality underlying a possibility." A few words of explanation may render the argument clearer, although they involve a rash approach to metaphysical heights which it is not the purpose of this paper to explore.

The conception of subject and object in careless discussion covers two distinct relations. There is the relation of the whole perceiving consciousness to part of its own content, for example, the relation of a perceiving consciousness to an object of redness apparent to it. There is also the relation of a perceiving consciousness to an entity which does not exist in virtue of being part of the content of that consciousness. Such a relation, so far as known to the perceiving consciousness, must be an inferred relation, the inference being derived from an analysis of the content of the perceiving consciousness.

The bases for such inferences must be elements in consciousness directly known as transcending their immediate presentation in consciousness. Such elements are universal logical truths, moral and æsthetic truths, and truths embodied in hypothetical propositions. These are the immediate objects of perception which are other than the mere affections of the perceiving subject. They have the property of being parts of the immediate presentations for individual subjects and yet more than such parts. All other existence is inferred existence.

In this chapter we are more directly concerned with truths embodied in hypothetical propositions. Such truths must not be confused with any doubtfulness which attaches to our judgments of the future course of natural phenomena. A hypothetical proposition, like a categorical judgment, may or may not be doubtful. Also like a categorial judgment, it expresses a fact. This fact is twofold: as a presentation in consciousness, it is just this hypothetical judgment; as expressing a categorical fact, it states a relation which lies beyond consciousness, holding between entities thereby inferred.

But this metaphysical analysis, short though it be, is probably wrong, and at the best will only command very partial assent. Certainly; and this admission brings out the very point which I wished to make. Physical science is based on elements of thought, such as judgments registering actual perceptions, and judgments registering hypothetical perceptions which under certain circumstances would be realised. These elements form the agreed content of the apparatus of commonsense thought. They require metaphysical analysis; but they are among the data from which metaphysics starts. A metaphysic which rejects them has failed, in the same way as physical science has failed when it is unable to harmonise them into its theory.

Science only renders the metaphysical need more urgent. In itself it contributes little directly to the solution of the metaphysical problem. But it does contribute something, namely, the exposition of the fact that our experience of sensible apparent things is capable of being analysed into a scientific theory, a theory not indeed complete, but giving every promise of indefinite expansion. This achievement emphasises the intimate relation between our logical thought and the facts of sensible apprehension. Also the special form of scientific theory is bound to have some influence. In the past false science has been the parent of bad metaphysics. After all, science embodies a rigorous scrutiny of one part of the whole evidence from which metaphysicians deduce their conclusions.

FOOTNOTES:

[Footnote 3: Cf. _Révue de Métaphysique et de Morale_, May 1916, where this question is dealt with by the author at the end of an article, "La théorie relationniste de l'espace."]