You are here

IV; Causation and Deterministic Systems

Causation and Deterministic Systems

When a physical event takes place it is usually regarded by common sense as determined by preceding events or processes which are deemed to have caused it to take place. Very frequently some one preceding event is singled out as the cause of the event in question. It is assumed that the particular event the effect would not have taken place in the absence of the cause; and that cause is regarded as affording an explanation of the occurrence of the effect. A somewhat less summary explanation of the event recognizes a plurality of preceding events or processes the absence of any one of which would have entailed the nonoccurrence of the effect. The ordinary notion of causation attributes a certain contingency to what is regarded as the cause of an event; thus an event or sequence A is in general only considered as the cause of an event B when the absence of A can be easily imagined; when this is not the case the invariable succession of B after A does not give rise to the idea that A is the cause of B. For example we do not consider the night as caused by the preceding day although we regard both day and night as belonging to a fixed sequence; in this case the element of contingency is not regarded as present. In scientific thought the notion of causation is expanded so as to embrace a whole complex of conditions some preceding in time and others simultaneous with the particular event in question. That every event has a cause formulates the conception of the determination of the event by a complex of preceding and present conditions. Taking into account the fact of the existence of approximately isolated systems in the physical world the cause of an event is restricted to the notion that it consists of preceding and present conditions forming a limited complex of relevant circumstances; all other circumstances being regarded as irrelevant in regard to the explanation of the event. When the event under consideration and the relevant circumstances are of such a character that they are measurable or in other words capable of being correlated with numbers the event is frequently said to be a function of the relevant conditions. This notion of functionality may be regarded as a more precise form of that of causation. But in the view of common sense a view which has until comparatively recently been shared by men of Science the notion of causation contains something more than the bare idea that the so-called effect has an invariable relation to the cause in point of fact. It includes the notion that the cause some particular event or process or some sub-complex within that cause can be regarded as an active agent which compels the effect regarded as passive in the transaction. When this idea of compulsion by an active agent is an essential component of the notion of causation the effect being regarded as forced by the compelling action of the cause that cause is said to be an efficient cause. It is not difficult to recognize the essentially anthropomorphic character of this notion of efficient causation. When by an exercise of his will a man moves a body or one of his own limbs he regards himself as an active agent who by the power of his will compels the body or limb to move; the moved body being looked upon as purely passive in the occurrence. The organic feeling of exertion which accompanies the action is to him the sign of his activity. In all our interventions in the physical world this feeling of activity dependent upon the consciousness of motion and contraction of muscles is present in greater or lesser degree.

By ejection of this notion of activity into the physical world in general we tend to attribute to the cause of an event the role of the active agent which we feel ourselves to be when we regard an event as caused by ourselves. We do not however usually suppose that consciousness or any psychical factor is necessarily present in a cause when that cause is not a living organism; in this respect the original animistic conceptions which were probably concerned in the ejection of the notion of efficient causation into the external world have taken a modified form. Hume who in his Enquiry concerning human understanding has treated the subject of efficient causation with great clearness and fullness writes1:
No animal can put external bodies in motion without the sentiment of a nisus or endeavour; and every animal has a sentiment or feeling from the stroke or blow of an external object that is in motion. These sensations which are merely animal and from which we can à priori draw no inference we are apt to transfer to inanimate objects and to suppose that they have some such feelings whenever they transfer or receive motion.
The indirect character of the effect of the causation presumed to exist when a man moves a limb has also been pointed out by Hume who wrote2:
We learn from anatomy that the immediate object of power in voluntary motion is not the member itself which is moved but certain muscles and nerves and animal spirits and perhaps something still more minute and more unknown through which the motion is successively propagated ere it reach the member itself whose motion is the immediate object of volition. Can there be a more certain proof that the power by which the whole operation is performed so far from being directly and fully known by an inward sentiment or consciousness is to the last degree mysterious and unintelligible?
The notion that causation in the physical world is similar to the efficient causation which we feel to be present when we act affords an explanation of the ancient belief one which has for ages been persistent in scientific thought that all physical action involves the contact of material bodies because our own mode of directly intervening in the physical world involves the placing of some part of our bodies in contact with external matter. The principle known as the “principle of causation” has frequently been stated in the concise form that every natural event has a cause; and this has usually been regarded as a necessary or axiomatic principle of thought to which Nature must conform. It should be observed that the principle leads unavoidably to an endless regress. For if an event have necessarily a cause that cause being considered to be a preceding event or a plurality of such preceding events must itself have a cause and consequently we have a sequence of events in which there can be no initial or first cause.
A result of modern criticism of the notions which underlie scientific thought has been that the conception of efficient causation has been discarded as unnecessary and useless for the purposes of Natural Science. Hume's analysis of the nature of perception made it abundantly clear that efficient causation is not to be discovered in natural phenomena. He writes1:
When we look about us towards external objects and consider the operation of causes we are never able in a single instance to discover any power or necessary connexion; any quality which binds the effect to the cause and renders the one an infallible consequence of the other. We only find that the one does actually in fact follow the other.
It should be observed that the acceptance of Hume's position in this matter does not imply an acceptance of his views in this connection as regards the association of ideas. It is at the present time often thought desirable no longer to use the terms cause or causation in connection with physical phenomena on account of the fact that these terms have usually in the past been taken to imply the notion of efficiency. When the term cause is retained it is taken to be synonymous with the totality of antecedent conditions. The principle of causation taken in the only sense in which it can now be retained in Natural Science is not a logically necessary principle but merely the postulation or rather working hypothesis that it is possible to predict the happening of particular events when certain complexes of antecedent conditions are known. The mode in which this prediction is made in any particular case is by the employment of conceptual laws or schemes which have reference to events of a class to which the particular predicted event belongs. Thus Helmholtz has written1: “The principle of causality is nothing else than the hypothesis that all the phenomena of nature are submitted to law.” I shall however presently show that Helmholtz's statement is capable of a too general interpretation; and that in fact the principle of causality as employed in Natural Science is of a more stringent character than would appear from Helmholtz's formulation of it. The extent to which the postulate of causation actually holds good when applied to natural phenomena we can only find out by experience.
In the more superficial notion of causation of which I spoke at the beginning of the lecture an event A antecedent to an event B is under certain circumstances regarded as the cause of B some interval of time elapsing between A and B. But our sense-impressions are no longer regarded by Psychologists as atomic in the sense of consisting of a set of separate sense-impressions with intervals of time however small between consecutive ones but rather as forming a continuous stream; so that we now have to contemplate instead of two separate events A and B a continuous process to which both A and B belong. An advanced scientific theory does not take account only of the bare fact of the invariability with which B follows A but should give a conceptual account of the process to which A and B belong. In some departments of Science especially those which have become amenable to representation by a Mechanical theory this has in many cases been accomplished. In other departments the efforts to trace continuous processes to which both the so-called cause and the so-called effect belong have met only with partial and fragmentary success. Sometimes what has been accomplished after minute scrutiny amounts to the insertion between A and B of other intermediate observed events between each two consecutive ones of which a hiatus exists of essentially the same character as the original hiatus between A and B themselves; and thus the nexus between A and B has not been successfully represented as a continuous process. Moreover it must be observed that an event at an instant of time can only be regarded as a conceptual limit not as an actual percept because an actually perceived event always has some duration however small that duration may be; and thus the actual events A and B can themselves be regarded only as processes; finite parts of a single process which includes them both.
Although as we have seen the search for efficient causes in natural phenomena is chimerical the belief in the existence and possible discovery of such efficient causes has been of great importance in the whole history of Science. That belief has often been advantageous in stimulating investigators to discover a few stages in the endless regress to which the a priori principle of causation leads; and this has often led to the establishment of the connection between different phenomena although such connections have never given any lasting satisfaction to the desire to realize the notion of efficient causation.
Just as efficient causation cannot be discovered within physical phenomena so also the search for any logical necessity binding together the phenomena of a sequence is doomed to inevitable failure. Logical necessity has its habitat in the domain of thought alone. It binds together the parts of a fully worked out conceptual scheme which represents a physical sequence but no such logical necessity connects the successive stages of the physical sequence itself.
In a scientific theory in which conceptual elements are linked together by a scheme of postulations which fixes the relations between the different elements the consequences which are deducible from the structure of the scheme follow by logical necessity from the postulations and definitions contained in the theory. That logical necessity is of the character in accordance with which the conclusion in a syllogism is a necessary inference from the major and minor premisses. But when the scientific theory is employed for the purpose prescribing a complex of physical phenomena there is no justification for a transference of the logical necessity from the conceptual theory to the perceptual phenomena. For there exists no logical necessity that the theory should be applicable to the description of the phenomena; such applicability when it exists is a fact of experience only not a necessity of thought. For example we possess a conceptual scheme which assigns in accordance with the Newtonian dynamical scheme and the law of gravitation the relative accelerations of the bodies of the solar system represented conceptually by points with definite mass-coefficients; and this scheme is employed to deduce the orbits of the points which represent the earth and planets about the point which represents the sun. The forms of the orbits follow by logical necessity from the postulations of the scheme. Yet there is no purely logical necessity that the earth should continue next year to describe even approximately the actual orbit round the sun which it has described during the past year in approximate accordance with the orbit as deduced from the conceptual scheme. For there is no logical necessity that in the future year the conceptual scheme will continue to serve its descriptive purpose. The expectation which we have that it will do so is of overwhelming strength based upon our past experience of the adequacy of the conceptual scheme. This expectation amounts to a subjective certainty the existence of which in our minds is a fact for an explanation of which reference must be made to the Psychologist and the Epistemologist not to the Logician. Thus there is no purely logical necessity that the seasons will recur as they have done in the past; there is only an expectation of such strength that it may be called a practical certainty but not a theoretical or absolute certainty. Without such practical certainty of this character human action would be an impossibility; the great survival value which this subjective feeling of certainty as regards the continuance of a certain uniformity in Nature must have possessed during the stages of evolution may be an important factor in the explanation of its existence. The fact that this kind of expectation which Natural Science accepts as a working hypothesis is found by general experience to be satisfied is one of which Metaphysicians will take account in their theories of the nature and structure of reality; but for the purposes of Natural Science no such inferences of a philosophical kind are necessary.
The very common idea that it is the function of Natural Science to explain physical phenomena cannon be accepted as true unless the word “explain” is used in a very limited sense. The notions of efficient causation and of logical necessity not being applicable to the world of physical phenomena the function of Natural Science is to describe conceptually the sequences of events which are to be observed in Nature; but Natural Science cannot account for the existence of such sequences and therefore cannot explain the phenomena in the physical world in the strictest sense in which the term explanation can be used. Thus Natural Science describes so far as it can how or in accordance with what rules phenomena happen but it is wholly incompetent to answer the question why they happen. When a sequence of phenomena can be imitatively represented by means of some other sequence of a more familiar type this latter is frequently said to afford an explanation of the former. Thus Lord Kelvin said “when I have made a mechanical model I understand a process.” This dictum might conceivably be understood in either of two senses. In the first that in which as the context shows Lord Kelvin understood it himself the model is taken to be a concrete model in which actual material bodies are employed to constitute its parts. Such a model affords an explanation of the phenomena which it pictures in the more limited sense to which I have alluded; it makes plain by means of a reduction of the relatively unfamiliar to the more familiar. Such employment of a model is really only a preliminary but often a very important stage in the process of gradually arriving at a genuine scientific theory of the class of phenomena in question. The next stage commences with an attempt to explain the model that is to proceed further with the model as starting point; unless indeed the working of the model has already been shown to be representable by a conceptual scheme in which case the phenomena in question have by an indirect process been subsumed under a scientific theory. The second sense in which Lord Kelvin's dictum might be interpreted is that the model is of a conceptual character of which the parts are conceptual objects of defined characteristics and related in assigned modes with one another. In this case the subsumption of the particular phenomena under a scientific scheme has been accomplished; to achieve this result is of the very essence of scientific method. The predilection which has been shown during centuries for mechanistic modes of representing all physical phenomena that is for schemes based upon the Mechanics of molar bodies is to be accounted for in large measure at least by the fact that our most familiar experiences of natural phenomena are related to the motions and interaction by contact of gross bodies.
It will have been noticed that not only in the more popular notion of causation but also in that more refined and clarified form of the conception which we call scientific causation the cause which conditions the effect precedes in time or is at least not subsequent to that effect; and this even when the notion of efficiency has been completely extruded. Thus in the sequences with which Natural Science deals an earlier part of a sequence is regarded as determining a part which is subsequent in time. The relation between what we call the cause and what we call the effect is not a symmetric relation; we do not in Natural Science regard an effect as conditioned or determined by a cause in the future. For the practical purposes of Science those of prediction it is essential that such an asymmetric determination should be the object of search. But this mode of determination is not the most general which we can take into account.
A deterministic scheme of the most general type is one for which we possess or at least are assured of the existence of a principle or set of rules which is sufficient to determine any single part of the sequence; although a knowledge of that part of the whole sequence which precedes in time the part to be determined may not suffice for such determination. In such a case the whole sequence is subject to a law or formula which makes it possible at least in theory to calculate the particular characteristics of any assigned part of the sequence. Thus all parts of the sequence are mutually related with one another and with the sequence as a whole in such a way that a knowledge of the principle of the whole sequence is requisite for the purpose of calculating the detailed nature of any part of it. In such a sequence there is no asymmetric relationship of cause and effect. The conceptual sequences with which Natural Science has to do and the search for which is essential for the particular purposes of Natural Science are deterministic but not of the most general type. When I said that Helmholtz's definition of causation in Science is too general I meant that it appears to refer to a deterministic scheme of the general type and not to the particular species with which Natural Science is concerned. The methodological assumption of Natural Science is that there are to be found in natural phenomena not merely physical sequences which are describable by means of deterministic schemes but sequences capable of description by that particular species of such schemes as permits of the determination of the present from a knowledge of the past without taking into account the future. In other words those deterministic schemes which Natural Science employs are such that they involve the asymmetric relation of causation.
If is possible to imagine that to an unlimited extent natural phenomena could be representable by deterministic schemes of the general type and yet that only to a more limited extent were the phenomena representable by such deterministic schemes as Natural Science taken in the restricted sense is capable of employing. Determinism in this general sense does not necessarily imply predictability of the future by means of a knowledge of the past or the present. To make such prediction a certain knowledge about the whole complex past present and future would be necessary. The principle of sufficient reason or of Ground and Consequence may be considered as more general than the principle of Causation. The Ground may be embodied in the principle which underlies the whole scheme; whereas when the principle of Causation is applicable there must be a restriction of or specialization in the nature of the Ground of such a character that asymmetric causation is realized in the scheme. A deterministic sequence of the general type does not necessarily. Involve same kind of uniformity that exists in a scheme for which the causal relation holds. This is a fact which admits of a simple illustration. Let us suppose that a particle had been moving during the indefinite long past uniformly in a straight line; let us further suppose that at a certain time it suddenly proceeds to describe with uniform velocity a semi-circle returning to the original straight line at the further extremity of the diameter of the semi-circle and that then it proceeds to move for ever afterwards in the continuation of the original straight line in which it was moving before the semi-circular deviation took place. This is representable as a deterministic system in which a single formula can actually be given by means of which the position of the particle at any assigned time can be calculated. The law or principle embodied in this formula makes provision for the single deviation from uniformity of the motion of the particle which occurs during the semi-circular motion. Yet no observation of the earlier part of the motion would have given any means of predicting the excursion of the particle along the semi-circle; the asymmetric law of causation would be inapplicable in such a case.
It has sometimes been suggested that teleological conceptions may be taken into account in deterministic schemes of the general type. A process in which the teleological factor is involved whether it be completely deterministic or not is one in which the present cannot be regarded as independent of the future parts of the process. Whether this suggestion is of value in general Philosophy is a difficult question which lies outside the boundaries of the present discussion and which I consequently must refrain from discussing. It is quite clear that in that particular type of deterministic schemes which Natural Science employs there is no possibility of any direct representation of the effects of purposive-ness regarded as introducing the element of contingency.
Although as we have seen the notion of efficient causation implying the activity of an agent that compels or necessitates an effect is one which forms no part of the stock of conceptions which Natural Science can utilize it does not in the least follow that the conception of such activity is illusory and must be denied all validity in the more general domain of Thought. The consciousness of activity which we all have whenever we act cannot be shown to be illusory unless it can be conclusively shown that it can be brought under some other category. The mere fact that the notion of efficiency is of anthropomorphic origin is quite insufficient to establish that this conception is illusory or that it is necessarily possible to dispense with it in the account which Thought attempts to give of all the factors in experience. All that has been maintained in the earlier part of this lecture is that Natural Science as distinct from a complete Philosophy of Nature in the method it adopts of describing conceptually that part of our experience which we call physical has no need of the conception of efficient causation and can make no use of it for its own special purposes. It may be a debatable point amongst Psychologists whether the notion of efficient causation is necessary in Psychological Science. In any case the reasons which have led to its extrusion from Natural Science cannot without much further examination be held to be sufficient to justify its exclusion from the categories of Psychology.
I have already referred to the fact that in Biology considerable use has been made of mental factors in relation to Evolution and that teleological conceptions have not always been dispensed with. Unless the theory of psycho-physical parallelism could be raised to something more than a surmise or at most a postulation of which the range of applicability is unknown it is exceedingly difficult to conceive that Natural Science in the narrow sense when applied to living organisms can ever succeed in giving accounts of sequences of physical phenomena in such organisms which shall attain a degree of completeness as great as has been reached in some of the departments of Science which leave living organisms out of account. Neither can it be regarded as certain perhaps not even as probable that the future progress of biological Science will ever enable it in all its departments completely to dispense with the notions of purpose and activity although indefinitely larger tracts of phenomena than at present may be found capable of the descriptive treatment which Natural Science applies. I have already pointed out that in the strictest sense of the term explanation Natural Science explains nothing; it only describes conceptually. It is just the absence of the conceptions of efficient causation and of logical necessity in the kind of account which Natural Science is able to give of portions of the world of physical phenomena that prevents us from regarding such account as explanatory in the strict sense of the term. Philosophy has always occupied itself with attempts to find explanations in the complete sense; and thus to condemn a priori as necessarily illusory the notions of efficient causation activity and purposiveness amounts to a dogmatic denial of all validity and of all possibility of success to metaphysical thought. Some men of Science are apparently prepared to take up this dogmatic attitude towards Metaphysical Philosophy. Their position however admits of no justification by arguments of demonstrative or even very cogent force. It is in opposition to what is at least prima facie the most immediate knowledge we possess that derived from the experience we have when we will and act.
Natural Science deals simply with what is observed to happen in the physical world. By minute examination and comparison of observations it discovers the existence of sequences of sufficient similarity with one another to be capable of being described by rules which are often called Laws of Nature. It synthesizes these rules into conceptual schemes called general theories. It employs these rules and theories tentatively to predict what will happen when a portion of a sequence which appears to fall under one of these laws or theories is observed; the prediction refers to that part of the sequence which has not yet been observed. There is no logical necessity that the prediction should be successful; but in case the rule or theory has been abundantly verified by past observations there is an expectation of success the disappointment of which by age-long habit we do not usually contemplate. This expectation is often so strong that when we are so disappointed in the verification of our predictions we attribute the fact to a mistake in our belief that the sequence concerned is actually sufficiently similar in character to those to which the law or theory is known to be applicable. In fact we search usually with ultimate success for disturbing factors in the conditions which remove the particular sequence in question from that particular type which alone the law or theory is designed to describe. As the result of such further investigations either the law or theory is rejected as insufficiently established; or a wider theory is set up which takes account of what we at first regard as disturbing factors preventing the older theory from being applicable to the new observed sequence so that the wider and amended theory suffices for application to the description of a wider class of sequences than did the older theory.
It is no part of my programme to make a contribution to the interminable discussions with which Philosophers and Theologians have for centuries occupied themselves on the subject of Free Will and Determinism; whether the world can be regarded as a deterministic system or whether it is subject to interventions either continually or sporadically the character and amount of which are even in theory incalculable. There is however one question relevant to these discussions which is in the direct line of thought that concerns itself with the true character of Natural Science. I have already spoken of deterministic schemes as employed in Natural Science; but what precise meaning can be attached to the term deterministic scheme? This question seems to require a somewhat closer examination than I have already given to it especially in view of the fact that it has frequently been assumed without critical examination both by Determinists and by Voluntaryists that the conception of a deterministic scheme is perfectly definite.
Let us consider a finite system of ideal objects endowed with a set of properties or qualities and subject to a defined set of relations with one another. We regard this system as subject to change as a certain variable usually taken to represent time takes up a continuous set of values. At any one time we regard the system as being in a certain state. By a known state we mean that we have precise information as regards the positions motions and all other relevant facts relating to the various properties of and relations between the ideal bodies as they are at the time at which the state is known. In what kind of language is it possible to express this knowledge of the state of the system at a particular time? The only language that we possess which would appear to be adequate for this purpose is that of Arithmetic in which numbers are employed. We therefore assume that the state of the ideal system can be completely specified by a finite set of numbers. This amounts to the assumption that all the positions properties qualities and mutual relations which constitute the state of the system can be completely specified by means of a finite set of numbers. This again involves the assumption that all these factors of the state are capable of exact measurement; that is that each one of them can be correlated with one number or with several numbers. It is necessary to consider in some detail what this correlation involves. In the perceptual domain there are two kinds of magnitude known as extensive and intensive respectively. Only extensive magnitudes are capable of direct measurement that is of correlation to a degree of approximation depending on the fineness of our instruments with the system of numbers. The possibility of such correlation depends upon the fact that an extensive magnitude can be regarded as consisting of units all of which are identical in respect of magnitude; this is the equivalent to the property that two extensive magnitudes may be added together their sum forming another magnitude of the same kind. The sizes spatial positions velocities and accelerations of perceptual bodies are all extensive magnitudes. On the other hand such qualities as temperature and colour are not extensive magnitudes; they are not directly additive as are extensive magnitudes although they may be regarded as having greater or less intensity. But nevertheless some qualities which are not extensive magnitudes may be correlated with such magnitudes and in that case they are indirectly measurable and the intensities can then be placed in correspondence with numbers. The correlation of different kinds of intensive magnitude with the number scale has actually been effected as in the case of temperature or in that of colour which has been correlated with several numbers. Unless this correlation has been accomplished for the case of any particular intensive magnitude the quality which it represents cannot be specified numerically as regards amount. All actual measurements direct or indirect are more or less inexact being dependent on the defects of our senses and of the instruments which we may employ to extend the scope of our senses. In a conceptual scheme these practical measurements are replaced by ideally exact measurements not subject to errors of observation; thus the correlation of magnitudes with numbers in a conceptual scheme becomes ideally exact.
We now assume that in our finite conceptual system all the elements which characterize the state of the system are capable of exact correlation each with one number or with several numbers. On this assumption the complete state of the system at a given time is specified by a certain set of numbers. In order to represent changing states of the system we have to consider a set of variables whose values depend on the time-variable; and which for a particular value of the time-variable coincide with the numbers that represent the state of the system at that particular time. If we consider the ideal system during a particular interval of the time-variable we may conceive the variables of the set by the specification of values of which a state of the system is defined to be functions of the time-variable. If the forms of all these functions are known we have the means of calculating to any assigned degree of approximation the state of the system at any time falling in the time-interval for which the functions are defined; thus the state of the system at every point of that time-interval is determinate and theoretically calculable. In the case of a particular conceptual system such as we are here considering the postulations and definitions which we make as regards the relations between the component parts of the system and the laws governing the changes in that system may suffice actually to determine these functions. When these functions are all known to us the conceptual scheme is not only deterministic but is actually determined in the sense that we have such a complete knowledge of the character of the system as a whole that we can by simple calculation determine in detail its state at any assigned time falling in the interval for which it is a deterministic system.
If we have not actually determined the forms of the functions but have convinced ourselves by sufficiently cogent reasoning that these functions exist as implicitly defined by the general laws which hold for the system we still regard the system as deterministic although only theoretically determined. For values of the time-variable outside the range for which the system is known to be deterministic the functions employed may become meaningless or in any case may have no application to the system. Without any essential difference in the definition it is possible to contemplate a system which is deterministic for indefinitely great ranges of value of the time-variable of either sign. A conceptual system which is deterministic in accordance with this definition of the meaning of the term is deterministic in the more general sense; there is no distinction between the relation of past or present with future states different from the relation of future with past or present states. The whole aggregate of states of the system for all the time during which it is deterministic are bound together by one common nexus. But a deterministic system of the particular type which Natural Science can employ must be such that the forms of the functions for the whole time can be inferred from a knowledge of some or all of the states of the system prior to a time falling within the time-interval for which the functions validly represent the system. In this case future states of the system are determined by past states on the assumption of the actual or of the theoretical existence of the junctions. This may he stated in the form that junctions determining the states of the system up to some particular time can be continued beyond that particular time so as to continue to represent the states of the system for some finite or indefinitely great interval of time sub sequential to the particular time considered. In this manner we can represent to ourselves a deterministic scheme in which the asymmetric causal relation (not efficient causation) exists. Having determined the forms of the functions for all the times up to a particular time a knowledge that the system is deterministic in this sense involves the postulation that it is possible to determine by some kind of continuation the forms of the functions for future values of the time-variable and thus by employment of these functions to determine states of the system at future times. This involves the postulation that the laws which regulate the system continue to be valid in the future.
Is it possible to extend the definition of a deterministic system so as to apply to the case in which the states of the system are only representable by means of a non-finite set of functions? It is impossible exhaustively to exhibit in any manner an infinite set of functions or even an infinite set of numbers. If such an infinite set of functions is to be regarded as known or determinate we must possess the means when some finite part of the set is known of determining any one of the others by means of some finite set of rules. Thus the states of the system are virtually determined by a finite set of functions together with a finite set of rules for the determination of any other of the functions when the former have already been determined. This amounts to the virtual reduction of the system to a finite one and it is only on the assumption that this reduction is possible that the conception of a deterministic system whose states are specified by an infinite set of variables has any meaning.
We are undoubtedly able to define particular conceptual schemes of a deterministic type and in particular such as include the causal relation in the sense in which that term is employed without involving the notion of efficient causation. Does there exist in the perceptual world anything that corresponds to such an ideal deterministic scheme? The success of physical Science depends in a considerable degree upon the fact that we are able to mark off in the perceptual world approximately isolated domains in which selected kinds of sequences or events can be described with a very considerable degree of approximation by means of ideal deterministic systems to which such sequences are correlated. It must however be observed that such correlation is confined in each case to a limited set of the relations between the physical objects of the perceptual domain and never embraces the whole of their properties or relations. For one class of properties or relations we may require one ideal deterministic scheme and for another class we may require another such scheme. Science cannot be said to have succeeded even in the case of a strictly limited approximately isolated group of percepts in correlating all their observed relations and all their changes with a single ideal deterministic scheme. The notion of correlation of isolated systems with particular deterministic ideal systems may have its scope widened so as to take account of the detailed differences between one perceptual system and another one of a similar character by supposing that the functional relations in the ideal scheme involve a certain number of variable parameters to which particular values have to be assigned when the ideal scheme is to be applied to describe what happens in one particular perceptual system belonging to the given class.
In those cases in which we are able to correlate the phenomena of a certain class in an approximately isolated group of percepts with an ideal deterministic scheme which serves as a description of those phenomena the correlation can never be taken to be valid beyond some limited period of time of greater or less duration. That this is the case depends at least partially upon the fact that the perceptual system is only approximately isolated; and that outside some limited period of time the relations of the system with other systems external to it may produce an effect upon the system which is no longer negligible; the deviations of the system from the conceptual scheme which represents it may be cumulative in amount and may produce after a sufficiently long time an effect which makes the correlation no longer even approximately valid. It thus appears that a group of phenomena can only be shown to be representable by means of an ideal deterministic scheme to a certain degree of approximation and only for some limited time which may however in some cases be very great according to our ordinary measures of time. The description is never applicable to represent in one scheme all the phenomenal aspects of the perceptual objects which have to do with the phenomena described by the scheme. It must also be remembered that many phenomena have hitherto proved too intractable to admit of complete representation by any kind of conceptual scheme least of all by a scheme which we can describe as deterministic in the precise sense that it is expressible by relations of magnitudes.
The assertion seems to be certainly true that Natural Science has not succeeded in showing that all that happens in a group of objects however small that group may be is deterministic in the sense that it can be completely represented by an ideal deterministic scheme of the precise kind that I have depicted. Attempts which I shall describe in later lectures have been made to show that all the happenings in a group of perceptual objects or even in the whole perceptual world could be represented by certain kinds of conceptual schemes especially of the kind called an atomic theory in which everything is regarded as dependent upon the interactions of sup positive elementary particles. All such schemes have proved utterly inadequate to give a representation of more than a part of what is observed to happen. Even the theoretical possibility of the existence of a deterministic scheme applicable to the whole physical world and to all that happens in it is open to many and grave objections; partly dependent upon the difficulty to which I have before alluded of extending to the world as a whole conceptions which can only be regarded as significant when applied to a finite part of the world and partly dependent upon our inability to estimate the cumulative effects of those individual peculiarities of perceptual objects which have always to be disregarded by Natural Science. Even Laplace when he contemplated the scheme which I referred to in my first lecture had to postulate the existence of a kind of superhuman calculator; and the complex of electrical and sub-molecular phenomena which have been discovered since Laplace's time has indefinitely increased the difficulty of imagining what even the main outlines of such a scheme would have to be.
Moreover the possibility of regarding even theoretically the physical world or a finite part of it as capable of complete description by means of a deterministic scheme cannot be contemplated without some consideration of the relations of the physical world to the psychical world. If indeed the theory of psycho-physical parallelism could be established completely the psychical world could be completely ignored and no special difficulty would arise from its existence as a system which exercises no influence whatever upon physical phenomena. If there exists complete detailed parallelism between the two domains it would appear to follow as a consequence of the assumption that the physical world is completely describable by a rigidly deterministic scheme that the same assertion could be made as regards the psychical world. On the other hand on an interactionist theory of the relation between the two domains the influence of the psychical world upon the physical must be in some way included in the deterministic scheme which ex hypothesi is applicable to the physical world. If this could not be done the physical world would cease to be representable with absolute completeness by means of the deterministic scheme; the influence of the psychical factor would then be of a character which could only be described as an interference with the physical order of things. If the view to which I have referred in the third lecture were accepted that the physical and the psychical worlds are in reality only parts or aspects of one single fundamental system the assumption of the deterministic character of the physical world would appear to involve the assumption that the whole fundamental system which would embrace both what we call physical and what we call psychical is in some sense deterministic. At all events those processes or events in it which appear to us to belong to the physical world would be regarded as forming a deterministic system and possibly other events or processes in it might be regarded as free or not determined; but these latter would then have to be considered to have no influence on the former otherwise we should be faced with the same difficulty as in the case of the theory of interaction between two disparate domains.
It is difficult if not impossible to attach a precise meaning to the conception of a deterministic psychical scheme of a kind similar to the meaning which we attach to a deterministic physical scheme. For psychical qualities processes and events are not prima facie measurable in a manner similar to that in which physical qualities processes and events may be measurable. What Psychologists measure are not psychical events or processes but their physical concomitants; and these measurements could only be taken to represent psychical measurements on the unproved assumption of some complete and exact correlation between the two domains extending even to the transference of numerical measurements from the physical to the psychical domain. This transference may be and probably is sufficient and of value for certain purposes; but the assumption that all psychical states processes and events are completely and exactly measurable by means of a transference of the measurements of physical concomitants is at least a very large assumption and one extremely difficult to justify.
The conclusion of the whole matter seems to be that the conception that the whole world of physical phenomena or that a finite part of that world is theoretically capable of being represented by a unified deterministic scheme is unproved and unprovable. All that Natural Science has established is that tracts of phenomena can be found which are sufficiently represented for certain purposes by means of deterministic schemes. A very large part of Natural Science has not yet reached the stage in which deterministic schemes of the kind which involve relations of number are applicable; much of it is in the stage in which only classification in abstract types can be employed and to which precise measurement is not yet applicable. But it is a working hypothesis employed in all the more advanced departments and stages of scientific thought that tracts of phenomena can be discovered to which deterministic schemes can be applied for the purposes of precise description and of prediction. The justification for this postulation is to be found in the past successes of such advanced parts of Natural Science and we are not acquainted with barriers which will prevent ever larger tracts of phenomena from being correlated with deterministic descriptive schemes.