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II; Scientific Laws and Theories

Scientific Laws and Theories

A general survey of the world from the ordinary point of view which we call that of common sense makes apparent to us the existence of two great complexes or domains; that which we call the physical or material and that which we call the psychical or mental. This is the ordinary dualism of matter and mind; a dualism which we all accept for the common purposes of everyday life when we are not philosophizing and which was erected into a philosophical system by Descartes and his followers. We regard ourselves as sentient beings endowed with consciousness with the powers of feeling thinking willing and remembering as psychically active. We appear to receive from without during our waking hours a continuous stream of sense-impressions leading to what we call the perception of external objects and events or grouped sense-impressions. These perceptions succeed one another in time and change with greater or less rapidity. Although our percepts are in a continual state of flux the changes in them are not entirely haphazard but exhibit a considerable degree of regularity of sequence of such a kind that we possess a certain amount of power of prediction of their future course. This power of prediction is an essential condition of the possibility of any actions on our part because in its absence we should have no knowledge of what the effects of any of our actions might be. We find moreover that we possess certain powers of classifying and sorting out our percepts of forming by abstraction from them permanent concepts under which whole classes of our percepts are subsumed. Among the perceptual objects of which we are aware are the bodies of other individuals and we believe on indirect evidence (leaving aside such debatable matters as the existence of telepathy) that these individuals like ourselves have perceptions and powers similar to our own of classifying these perceptions of abstraction of forming concepts and of remembering. We also believe as a matter of inference from experience that the actions of other individuals are influenced by their perceptions in a manner similar to that which we find to be the case in ourselves. Further we believe also on inferential grounds that the percepts of other individuals have a large measure of resemblance to our own; and that other persons discern regularities in the sequences of their perceptions which are closely related to those which we ourselves discern. We all alike have the apparent power of producing by the exercise of our wills changes in the perceptual domain or physical world. Such changes in the external environment involving the use of our voices or the motion of parts of our bodies apparently produced by our wills become percepts for other individuals and thus form the medium of communication between one individual and another. The means of communication of ideas between individuals thus conditioned lead to the formation of a body of common knowledge of the perceptual world which is regarded as accessible to all normal individuals. If we regard this picture as descriptive of what has been going on during an indefinite past and if we take into consideration the cumulative effect of the process combined with the memories of individuals and the traditions of the race we have from a non-philosophical point of view an account of some plausibility of the way in which common or public knowledge of the physical world may be regarded as having grown up. It must however be observed that the experience of an individual is in its fulness of detail unique and incommunicable; it is only incompletely made intelligible to other individuals by means of language or other forms of symbolism. Thus language in its very nature involves abstraction in which the elements of actual individual experience are replaced by symbols which fail to represent with absolute completeness what they are designed to describe.

It is a part of that view of the nature of Natural Science or the Science of physical percepts which I propose to develop that Science is essentially a purposive continuation of the formation of what I have called common knowledge but carried out in a more systematic manner and to a much higher degree of exactitude and refinement. The earliest stages in the formation of scientific knowledge are of the kind which may be described as classificatory. Physical objects are arranged in classes in accordance with observed similarities in the objects assigned to any one class; physical events or sequences of events are classified in accordance with observed regularities or similarities in those events or sequences. There is however always a certain degree of arbitrariness involved in the selection of the precise similarities or regularities which form the basis of the classification; thus at the very commencement of the process of building up scientific knowledge the analyzing and generalizing powers of the mind find scope for their activity and are necessary factors. The result of this process expresses itself in the formation of abstractions or concepts which are not identical with any of the perceptual objects or happenings which conditioned their formation but which serve as a conceptual symbolization or representation of those aspects of the latter which were regarded as alone relevant for the purpose of classification. A rule or law which affords a conceptual description of a particular kind of sequence of physical events in the perceptual world is set up in the first instance in the manner I have indicated on the basis of actual observation or experiment dealing with a selected class of sequences of physical events. It is then extended hypothetically to the descriptive representation of sequences which may occur under conditions not in all respects identical with those in which those observations were first made which suggested the law. When this hypothetical extension is justified by further experiments or observations and especially by its power of prediction of what will happen or be observed under certain conditions the rule or law becomes what is usually called either a scientific law or principle or a law of nature. The term law of nature has too frequently been taken to imply that it meant a purely objective law as it were inherent in the perceptual world which natural phenomena must of necessity obey whereas a law of nature is in reality a conceptual law set up by the activity of the mind of man but conditioned as regards its validity by the perceptual world which must be taken as a datum. The law is constructed by the mind but not purely “arbitrarily;” it has limitations other than and besides those imposed by the canons of thought. In the physical complex there is the essential element of fact which must be taken as a given and unalterable datum required for the construction of the law. Thus the complex of phenomena or appearances which we call the physical world must be taken to be in fact such that the law must stand the test of applicability for the purpose of resuming certain tracts of uniformity in it. Natural Science need not however go beyond this recognition of the existence of this element of fact; it is unnecessary for its purposes to make the assumption that a single law has a precise correspondence with a single definite set of relations which actually subsist in Nature. Still less is it necessary for the purposes of Natural Science to assume that the law corresponds to a set of relations between real “entities”. The formation of precise views on these matters is the task of Philosophers; the man of Science need not go beyond the superficial view which contents itself with the recognition of the essential element of fact in the world of percepts.
The two opposite tendencies either to regard natural laws as entirely constructible by the mind in accordance with what were regarded as a priori necessities of thought or on the other hand to regard them as something entirely external to and only discerned by the mind have both been prominent in the history of Science. It may perhaps be the case that some of the writers who uphold the “descriptive” view of the character of Natural Science have so expressed themselves that the importance of the element of fact or “givenness” in the realm of percepts appears to be unduly minimized. Both the activity of the mind and the data of perception are factors in the genesis of a so-called law of nature. Natural laws have been characterized by Mach1 as “abridged descriptions” and as “comprehensive and condensed reports about facts.” It may be doubted whether such a characterization is quite adequate as applied to conceptual schemes in general. The laws always contain less than the facts because the concepts which the laws employ in their statements are but symbolical representations which leave out of account those differences of individual detail which are la be found in perceptual objects or events which fall linger a particular class. In the words2 of Boltzmann:
It has never been doubted that our ideas are merely images of the objects (or rather symbols for them) which have a certain relationship with the objects and never completely correspond to them but are related to them as letters to sounds or notes to musical tones. Also on account of the limitation of our intellect they are able only to depict a small part of the objects.
There is no absolute line of demarcation between scientific knowledge and common knowledge. The knowledge which we have that solid bodies when unsupported fall to the ground would hardly be described as scientific knowledge but it is not in essence distinct from other rules of the order of phenomena which we do dignify by the name scientific knowledge on account of the fact that they rest upon experience which is less readily analyzed or separated off from the general complex of physical experience or which requires more refined methods of observation. But the law of falling bodies in accordance with the more exact observations made by Galileo involving numerical determination became a genuine scientific law. By the process of hypothetical generalization it was extended by Newton into the general law of gravitation the power of which to represent to a great degree of accuracy the observed motions of the bodies of the solar system was brilliantly established by Newton himself and a number of eminent researchers in the field of Gravitational Astronomy.
A law or rule which refers to some particular kind of sequence of events under certain conditions and which is regarded as a scientific law or principle is very often constructed by means of artificial production of the set of conditions under which the sequence of events in question is observed to take place; the law is then said to have been discovered by experiment. In other cases in which the conditions are not produced artificially but occur without the intervention of the observer the law is said to be discovered by observation. But in all cases the discovery or rather the construction of a scientific law involves that synthetic activity of thought which manifests itself in a constructive process in which actual percepts are employed only as the raw material and starting point of the mental process. In the attempt to discover a scientific law a selective process is requisite in regard to the percepts some greater or lesser part of what is perceived must be ignored as irrelevant to the purpose on hand; this selective procedure amounts to a process of abstraction in which some elements of our actual percepts are removed and not attended to. The degree of abstraction employed and thus the degree in which the concepts differ from percepts varies greatly in different departments of Science and also varies greatly according to the stage of development which a particular department has reached. A scientific law is accordingly always in some greater or lesser degree abstract in the sense that it represents only a part of what is in any individual case actually perceived; it describes a particular sequence of physical events which in an actual case is accompanied by other percepts prevents in relation to which the law has no application. For example when Kepler discovered the law that the earth and planets describe elliptical orbits with the sun in one focus he abstracted from all the physical properties of the sun and planets even from their sizes; all the infinite details of their physical constitution being irrelevant to his particular purpose of describing the main features of their relative motions.
A complex of physical facts given as percepts appears in the first instance to be confused and irregular on account of its complexity; but the effect of further scrutiny is gradually to reduce it in some degree at least to order when similarities and relatively permanent elements are discovered within the complex; and the ultimate result of this process is that we are enabled mentally to reconstruct groups of facts within the complex. It must however be observed that in Nature there are no precise repetitions in every detail for conditions are never on two occasions absolutely identical. Absolute likeness of sequence exists only in our conceptual schemes of representation not in the world of percepts. With the purely individual Science cannot deal; it operates with the typical and a type is an abstraction. Even in the preliminary work of classification of those groups of sense-impressions which we call physical objects such as is undertaken in the classificatory stage of Zoology or in systematic Botany abstraction is made of individual differences which distinguish one member of a group from another. Thus two plants of the same order genus species and variety are never exactly alike in all respects; they are judged to be of the same variety because they are alike in certain particulars defined in accordance with the arbitrary rules which are employed for the purpose of classification. A plant of a given variety or of a given species is an abstract conception; the successive grades of classification represent different degrees of abstraction in the formation of the concepts that correspond to them. It appears then that even in the earlier stages of scientific thinking what we really work with are not the percepts themselves but concepts which symbolize types of percepts and in the formation of which concepts some of the elements of actual percepts are left out of account.
It is however only in the earlier stages of development of a branch of Science that the procedure consists mainly of the classification of objects by means of the substitution of concepts for the objects themselves or of the construction of the kind of rules which form the simpler laws of sequence of events. As the Science advances such laws are generalized into those more comprehensive laws which are often called scientific principles. A law or principle which has reached a high stage of generality or a group of such laws considered as forming a single body of doctrine is what is known as a scientific theory and this forms a conceptual scheme under which a wide class of physical sequences is subsumed. But whether such a conceptual scheme is called a theory a principle or a law—and that is to some extent a matter of usage or historical accident—its genesis always involves a more or less extensive constructive mental process or synthesis. Such a conceptual scheme involves the employment of conceptual objects the character of which and the relations between which are assigned as postulations of the scheme. These postulations must always be regarded as hypothetical or tentative. The possibility of setting up such a scheme has actual physical experience as its essential condition but the constructive and generalizing work of thought is no less essential. The original function of such a scientific theory or conceptual scheme is to provide an ideal representation of some more or less restricted range of physical phenomena as actually observed that is of certain sequences and regularities in percepts. But the functions of a conceptual scheme are much wider than those of merely describing symbolically what has actually been observed. The scheme is applied hypothetically to predict what will be observed in circumstances which differ in some degree or in some characteristics from those in which the experiments or observations which led up to the theory were made. The value and the range of validity of the particular conceptual scheme have to be estimated by its actual success in the fulfilment of this function of prediction. Thus a scientific theory considered as an hypothesis not as a dogma must be judged by a comparison of its consequences with perceptual fact; and especially by its power of forecasting the occurrence of hitherto unknown or unobserved facts when by more minute observation or by artificial production of experimental tests the occasion for such comparison arises.
The different conceptual theories which are employed in the various branches of Natural Science vary greatly in their degrees of abstraction and in their degrees of precision. In the most abstract and precise theories the language employed is the most precise form of language which we possess that of arithmetic and its generalization in Mathematical Analysis. As any branch of Science progresses to a higher stage of development there are to be found at least portions of it in which it becomes possible to employ theories of this character. There exists a distinction between two species of ideal elements of a conceptual scheme which it is important to recognize. Some concepts have direct perceptual counterparts such concepts having been formed by a direct process abstraction in which what are for the purpose of formation of the scheme irrelevant characteristics of the perceptual objects or processes have been removed by the abstraction. Concepts of the other species have no such direct perceptual counterparts or it is not assumed a priori that they have such; they are formed by an effort of constructive imagination for the purposes of the representative scheme. Although concepts of this latter kind may be regarded as due to the creative activity of thought it must always be remembered that they would never be formed apart from a basis of actual physical experience. In any scientific theories in which both kinds of concepts occur those which have no immediate perceptual counterparts must be regarded at least provisionally as purely ideal elements of the scheme in fact as auxiliaries necessary for the purpose of the formation of a self-contained conceptual scheme which shall serve its purpose of providing a sufficient mode of representation in thought of the particular domain of physical events and objects. In less advanced conceptual schemes as for example in those of the purely classificatory order all the concepts employed may be simply abstract forms or types of perceptual elements. Such concepts are usually however more precise in character than those employed in ordinary intercourse the definition of their characters and relations having sharper outlines. The power which we possess of introducing into a scientific theory as part of the edifice ideal elements which do not directly correspond to clearly defined percepts although essential to the more advanced work of Science is one which has its obvious dangers and these dangers have by no means invariably been avoided in practice. In a workable and satisfactory scientific theory such conceptual elements should be as few in number as possible and of a simple and well-defined character; otherwise the theory has the grave defect of over-complication or even of vagueness and the fundamentally important requisites of a scientific theory which shall serve its purposes of representation and prediction those of clarity and simplicity are not satisfied. Such ideal concepts only serve their purpose provided they are subject to precise definition. In some cases the introduction of such elements into a theory has amounted to little more than the employment of new words which on account of the indistinctness of their denotation have only served the purpose of labelling our ignorance and have consisted of a futile attempt to disguise inability to set up a really adequate representative scheme. If when each new difficulty arises new concepts are invented for the purpose of shelving those difficulties the theory becomes so overloaded that it ultimately perishes by its own weight.
It has perhaps been in the biological sciences which have to deal with ranges of phenomena of the highest degree of complexity that this kind of danger has been oftenest incurred; we may refer to the various vitalistic hypotheses to illustrate the point. Some such insufficiently defined concepts have often been introduced for the purpose of remedying by means of an enlarged scheme the recognized inadequacy of some older and more restricted descriptive scheme. In Physics for example the temptation to introduce a new ether or a new substance with insufficiently defined properties in order to remove each particular inadequacy of an older mechanical scheme is one to which men of Science have frequently yielded. However in scientific theories of the highest type there may exist such ideal elements Without direct perceptual counterparts the danger of the kind I have indicated being successfully avoided. It is precisely in the proper selection and definition of such conceptual elements of a theory that the highest powers of the great men of Science who are necessarily supreme Artists have been exhibited. Descriptive schemes which employ concepts the smallest in number and the most definite in character are the distinguishing mark of Science at its best.
In some cases it is preferable for expository purposes not to reduce the concepts of a scheme to an absolute minimum but to retain some which are reducible to others as auxiliary concepts. This does not however afford a dispensation from the duty of investigating what is the smallest number of irreducible concepts. A theory is sometimes capable of being stated in more forms than one according to the choice made of the irreducible concepts and thus some latitude arises as to the precise form of the theory. I may refer to the case of molar Dynamics as an illustration of this possibility. Those departments of Natural Science in which the theories or conceptual descriptive schemes are furthest removed from the region of the perceptual and which are thus characterized by a higher degree of abstraction than has as yet been attained by other branches are Mathematics and those parts of Physical Science which have become in a considerable degree amenable to mathematical treatment. It is the relative simplicity of those aspects of the perceptual world with which Mathematics and large parts of those Sciences which concern themselves with non-living matter deal as compared with the Biological Sciences which accounts for their relatively advanced character. Number and Extension with which Arithmetic in the extended sense and Geometry are alone directly concerned can be developed as sciences descriptive of only the most superficial aspects of the perceptual world since they may leave aside almost all the properties and qualities of perceptual things as irrelevant for the purpose of their construction so that in these departments the process of abstraction and idealization can take gigantic strides at a very early period in their genesis. Other branches of Science in which the disentanglement from one another of great complexes of properties qualities and sequences is a much more difficult process are on that account such that the stages of their growth and development are much slower and much more difficult than in the case of Geometry and Mathematical Analysis.
Notwithstanding the very great difference as regards the degrees of abstraction in the conceptual schemes which belong to different branches of Science the difference between them need not I venture to think be regarded as generic. This difference seems rather to be one of degree of advance in the scope and character of the abstract schemes of representation which have at the present time been devised for describing the various regularities of sequence with which the various branches have respectively to deal. On the assumption of the correctness of this view the Science of Geometry which on the perceptual side starts solely with the extensional relations of bodies forms a kind of model to which other branches of Science in their further advance will gradually conform. This conformity with the model may in the cases of most branches of Science not become complete in any time for which we may venture to make a forecast but their advances will consist of progress in the direction of such conformity. Geometry the Science of spatial relations is essentially in its origin a physical science but it has long ago become a deductive Science in the sense that a complete conceptual scheme has been constructed of such a character that all ordinary special spatial properties are logically deducible in their ideal form from the postulations of the scheme itself. These deductions are then applicable for the determination subject to inevitable errors of measurement of the corresponding relations in perceptual space. In our actual Geometry purely experimental determinations of special spatial properties are no longer necessary but only serve for purposes of illustration. It is not inconceivable that the labours of Physicists in a domain on which a flood of light has fallen in the last decades may result in the production of a schematic representation of the composite nature of the atom so precise that the various possibilities as regards the detailed structure of the atom may be worked out. If this were accomplished it might be possible to deduce from such a conceptual scheme the main characteristics of the different chemical elements and the nature and character of their possible combinations. The Science of Chemistry would then have become deductive in the same sense as that in which Geometry is a deductive Science. However far from the attainment of this goal most branches of Science may be the general character of progress in them consists of the discovery of ever larger tracts of phenomena that are capable of conceptual description by means of schemes which succeed in predicting further phenomena which had not been observed until such predictions were tentatively made.
This view of the nature of a scientific theory the most general outline of which I have traced that it essentially consists of a conceptual scheme designed by the synthetic activity of the mind working with the data of perception for the purpose of representing particular classes of sequences and regularities in our percepts has been powerfully advocated in recent times by Kirchhoff and Mach on the continent and by K. Pearson in this country; some indications of a similar view are to be found in the writings of Auguste Comte and other earlier writers. I propose to follow it out in further detail and to illustrate its application in various special departments of Science which I shall consider in later lectures. It will be found that the adoption of this point of view involves changes in the older traditional conceptions of the relations between scientific thought and general Philosophy. These changes will be seen to have the effect of according to Natural Science a more independent position in relation to Philosophical theories than it has often been supposed to occupy. But they will also have the effect of placing limits to the functions of Natural Science as one of the factors which contribute to the construction of our general view of the world; and also of assimilating in a large degree though not wholly the attitude of Natural Science towards the great speculations of Philosophy with the attitude of what is usually called common sense. The ambiguity of meaning of various expressions employed in their writings by men of Science often makes it difficult to be sure whether or how far they are in agreement with the descriptive view of the nature of Natural Science of which I here give an account.
We have taken as our starting point the fact that we all have streams of percepts which we call our physical experience. The reduction to some kind of order and regularity of particular kinds of such percepts by means of a representation of them by conceptual schemes is what I have maintained to be the function of scientific theories and laws. The important question however arises that as a percept always involves a percipient who is the percipient whose percepts are conceptually described in this manner?
We cannot take Science to deal merely with the private percepts of some one particular person. The very essence of scientific knowledge is that it shall be what we may call public knowledge that is not the partially incommunicable knowledge of some one individual. It is true that at a particular time a particular scientific theory may only be known to a single person its discoverer; but it is essentially capable of being communicated to and understood by other persons whose training and previous knowledge fit them to receive it. In the form in which a scientific theory or law is usually stated it seems to describe facts or occurrences which are independent of any and every observer but the presence of some percipient is always implied. This implied percipient must be regarded as placed in circumstances which enable him to have the perceptions and these circumstances include the use of necessary instruments for extending the scope of his senses such as telescopes microscopes etc.; he is assumed also to be endowed with normal powers of perception. The answer to the question who is the implied percipient in a descriptive scientific theory is then that the theory consists of the conceptual description of what would be observed by any normally constituted observer supposed to be placed in a proper position and under suitable circumstances and provided with the necessary appliances. Such an hypothetical observer is always implied in the statement of a theory or law although he is usually not explicitly referred to. I should however observe that in the latest physical theory that of general relativity certain special circumstances relating to the observer are taken into account as part of the theory itself.
No scientific theory is designed to describe at once the whole of the perceptions which an observer would have under given circumstances but only particular classes of perceptions arbitrarily separated from the rest in accordance with the particular kinds of percepts and their sequences which the theory is designed to describe that is according to the particular range of phenomena with which the particular theory deals. Thus for example a dynamical theory of the motions of molar bodies will be a conceptual scheme designed to describe only the motions of actual bodies as they would appear under assigned circumstances to an observer who perceived those motions and the theory would have no concern with the perceptions which such observer would have at the same time of the colours and other physical characteristics of the moving bodies or of any sounds which they might emit; these would form the subject matter of other theories. Thus the actual perceptions of an observer are described as it were piece-meal by different theories belonging to different branches of Science. A vast part of our percepts has hitherto not been found to be amenable to abstract description of a precise character by any conceptual scheme.
Considerable branches of Science are concerned with attempts to trace back into the remote past complexes of phenomena of certain special kinds that is to give an historical retrospect in a particular domain. Some branches of Science also undertake to perform a similar function as regards the future. Geology for example undertakes the task of tracing the history of the earth's crust; Cosmical Astronomy attempts to depict the history of the evolution of the solar system and of stellar systems. Biological Science has made great efforts to construct a general account of the evolution of living organisms and their various species. It is obvious that no single percipient such as we are acquainted with can be regarded as having followed out the changes even in a single geological period or the evolution of a single race of animals let alone the evolution of the solar system.
I do not however think that the general scientific schemes which purport to describe what has happened in very long but of course strictly finite periods of time need be regarded as falling under a different category from those which describe the short time processes observable by a single percipient. We may regard them all as hypothetical attempts to describe what would actually have been perceived by an observer at any time falling within one of the long periods in contemplation had such an observer been present at that time. These historical accounts have been constructed on the basis of a knowledge of present conditions and of actually observed processes which take place during short periods. Their construction involves the piecing together of a large number of short time processes and of utilizing our conceptual knowledge in such a way as to construct a conceptual scheme of a strictly general character which serves to give a consistent account of what we may hypothetically regard as having led up to the actual present conditions. It is of course clear that these histories of processes through long periods involve a large element of hypothesis and a degree of uncertainty and especially of indefiniteness as regards lengths of time much greater than those which attach to conceptual accounts of short time processes which can be observed by a single percipient.
In those advanced scientific theories which are of the quantitative kind that is which describe conceptually the results of actual measurements there is always an element of approximation in the application of the theory because all actual measurement is approximate only. Absolutely exact measurement is conceptual only and resides in the conceptual scheme not in the actual measurements that can be made in the perceptual domain. Thus even in the case of a quantitative theory there is an essential element of approximativeness in the power or scientific theory to represent actual perception; much more is this so in the numerous cases in which the theory is not quantitative but qualitative only. It is in general only in the more advanced branches of Science that quantitative Mathematical representation is available. But in many branches such as Chemistry and Physiology there are in ever increasing numbers particular ranges of phenomena which have been made amenable to such treatment although in some cases such ranges of phenomena form but a very small part of the whole complex of phenomena with which each such particular department of Science deals.
An essential characteristic of every scientific theory is that it only serves to give a conceptual description of a range of phenomena which is of a limited and circumscribed character spatially and temporarily. It only describes what an observer would perceive in some portion of space which however great is limited by his powers of perception even when those powers are increased by the employment of appropriate instruments. Further it only describes what will happen in some limited lapse of time however great that period may be. It also only describes those percepts which are of some more or less limited type. Thus the percepts described by a theory are bounded spatially temporarily and generically. There always remain possible percepts beyond any particular range to describe which the theory is applied. As regards time the theory is wholly inadequate to describe any absolute beginning or end of sequences of phenomena. The period of time to which it may be applied is capable of being extended in both directions at least hypothetically either within the scope of the particular theory or by means of some more comprehensive theory applicable to phenomena which take place in a greater period of time during which we can assign at least tentatively the relevant circumstances. Just as absolute origins and terminations of sequences of phenomena are beyond the reach of scientific theory so also is unbounded perceptual space. Even the perceptions of the Astronomer are bounded by the limitations of the telescope and of the photographic plate; the space of his perceptions is strictly finite however enormous its dimensions may be as estimated in our ordinary systems of measurement. Spatial perceptions have also lower limits imposed by the limitations of the microscope.
Although conceptually we can regard portions of actual space or of time as capable of indefinite increase at any actual stage of the indefinite regress in which we are thus involved the portion attained is strictly finite. Thus we are not entitled to regard Science as dealing with the whole of time or the whole of space or with the whole Universe. It is indeed not clear that we are entitled to assert that either of these is a whole in the sense that to such an hypothetical infinite whole the same categories are applicable as in the case of a finite whole. These considerations would appear to indicate the existence of limits to the application of the methods of Natural Science. The existence of such limits has however frequently been disregarded at least in appearance in statements made by men of Science of high distinction. It has often been said that the whole energy of the Universe or the whole mass of the Universe is constant or that the whole entropy of the Universe tends to increase. To such statements a meaning can only be assigned by assuming them to be elliptical expressions to denote for example that in any finite region which received no accession of mass from without the total mass remains constant.
We have seen that the construction of rules relating to sequences and regularities in Nature the so-called laws of Nature and the more comprehensive schemes which we call scientific theories are the work of the human mind utilizing the raw data of perception. We might imagine that the perceptual world could have been so intricate and irregular in character that it would have been impossible to set up such rules or schemes under which phenomena could be subsumed. Science as we know it would then have been impossible it could never have begun to exist. But life as we experience it would also have been impossible because that common knowledge of sequences in Nature upon which the possibility rests of forecasting the results of our actions would have been absent. The possibility of the existence of Natural Science accordingly depends upon the fact that there is in our percepts that is in what we call Nature a considerable degree of regularity in the sequences of phenomena. This fact is often described as the Uniformity of Nature. What the scope of this uniformity may be and whether it is subject to extraneous interferences are questions which cannot be answered a priori; only experience is relevant in any attempt to answer them. It is not necessary for the purposes of Natural Science to assume that the complex of physical phenomena is such that it is even theoretically capable of description by a single all-embracing conceptual scheme. It is possible that the present state of Science in which different conceptual schemes are employed for resuming different tracts of the perceptual domain is the only possible state. Although the actual progress of Science has involved the gradual welding together of separate conceptual schemes into larger wholes which embrace tracts of phenomena that were earlier treated separately it is not a necessary assumption for Natural Science that this process of unification can even theoretically become complete. In other words it is not necessary to assume that Nature can even theoretically be subsumed under a single interconnected rational scheme.
Throughout the history of Science we find that the most potent influence on investigation has been exercised by the notion of efficient causation. The principle of causation or sometimes the more general principle of sufficient reason has been regarded as a necessary principle to which Nature must conform and the aim of Science has been regarded as that of discovering the precise modes in which the principle is realized in natural phenomena. But in Natural Science conceived in the manner I have explained such a priori principles are replaced by the postulate or working hypothesis that sequences can be found in Nature which are capable of being described by laws or sets of laws embodied in determinate ideal schemes and that it is possible to an extent of which we do not know any limits to discover what these schemes are. That these schemes can be applied with absolute precision to concrete cases in Nature is an illusion dependent upon the imperfections of our senses even when their power is extended by using instruments which often prevents us from perceiving all those individual peculiarities of a concrete phenomenon which differentiate it from other phenomena of the same class described by the same conceptual scheme. To construct laws we frequently isolate artificially a part of what happens from the rest; there are always however in a given case accessory phenomena. In a scientific laboratory elaborate precautions are frequently made in order to isolate phenomena and to reduce disturbing influences as much as possible but there remains some residual which cannot always be completely ignored when the experiment is regarded in relation to some simple law of an exact kind. Thus a scientific law states what would happen under ideal conditions more or less imperfectly realized in actual cases.
We have seen that all conceptual schemes are applicable only to the description of what goes on in a portion of the physical domain limited in various ways. This involves the isolation for the purpose of conceptual description of particular domains in the perceptual world. The success of a conceptual scheme in discharging its descriptive function in relation to a particular perceptual complex depends upon the fact that for the purpose on hand that particular perceptual complex may be treated as an isolated system in which all perceptual elements that do not belong to it are ignored. Thus the existence of approximately isolated systems in the world of physical phenomena is a fact upon which the possibility of the existence of Natural Science as we know it depends. For example a descriptive scheme for the motions of the bodies in the solar system would be impossible except for the fact that the solar system is approximately isolated in the sense that the disturbing effects of gravitation due to the stars can be left out
A conceptual scheme which successfully serves its purpose in the particular department of Science to which it belongs must satisfy certain conditions. First it must have logical consistency or coherence as a scheme which is not self-contradictory; the various conceptual elements which it employs and the set of definitions and postulated relations contained in it must form a consistent whole in which there is a complete logical nexus between the parts. Secondly the scheme must satisfy the test of providing an actually adequate description of sequences of phenomena of the class to which it is intended to apply; it must have applicability in relation to physical phenomena. It is quite possible to set up conceptual schemes which satisfy the first of these conditions that of logical coherence but which either do not satisfy or are not known to satisfy the condition of applicability. A conceptual scheme which differs from that of Newton's law of gravitation in taking the law to be that of the inverse cube or any other power of the distance instead of Newton's law of the inverse square could be worked out in detail and would satisfy the condition of logical coherence as fully as does Newton's law; it would however not be applicable for even the approximate representation of the motions of the bodies of the solar system. It would be in conflict with fact in the perceptual domain. A particular system of abstract Geometry may be unimpeachable as a conceptual scheme and yet it may be inapplicable to the description of actual spatial percepts.
A third condition that of relative simplicity should be satisfied by a scientific theory if it is to be used with effectiveness in the building up of Science. It may happen that two or more conceptual theories satisfy the conditions of logical coherence and of applicability. In that case that theory will be chosen amongst all those which are applicable to the same range of phenomena which is of the simplest character.
The most crucial test of applicability of a scientific theory is that it should enable us actually to predict the results of experiments or observations of sequences of phenomena under conditions consistent with the scope of the theory which are made subsequently to the constitution of that theory. If it fails to do this successfully either the theory must be abandoned and a search must be made for a more adequate one by which it can be replaced or else the determination of disturbing elements must be sought in the conditions under which the new experiments were made of such a kind that it can be shown that the complex of conditions in these experiments was not in accord with those postulated by the theory. In other words it must be shown that the observed complex of percepts did not in these experiments possess that feature of approximate isolation which the theory presupposes.
It has frequently happened in the history of Science that a theory which had a very considerable measure of success as a representative scheme failed to satisfy one of the requisites I have described. It has then been abandoned in favour of some other theory which appeared more nearly to satisfy the requisite conditions. In the first place the theory may be abandoned on account of internal defects in the conceptual scheme itself in the matters of logical validity and clarity of definition of the conceptual elements. In some cases a clear statement in a logical form of a conceptual theory has only been given as the result of critical investigation long after the theory was first constructed and applied for its proper descriptive purposes. Also a theory apart from defects in its own nature may be superseded by a new theory of a more comprehensive character one which is capable of representing a wider range of phenomena than the older one. When such supersession takes place the older theory does not lose its logical consistency and whatever degree of applicability it ever possessed.
The truth or the falsehood of a scientific theory is often spoken of in a manner which obscures the real nature of such theories. It is the fact of applicability or the degree of applicability of one logically coherent theory as compared with another which is their distinguishing feature. Assuming that both of them are logically coherent conceptual schemes both of which are hypothetical in their nature that one will be preferred for many purposes which has the greater range of applicability and hence the greater power of prediction. If the Einstein theory of gravitation or some modification of it finally supersedes that of Newton as may very likely happen it will be because the former is capable of representing some phenomena which the Newtonian theory fails to represent and also because as part of a wider theory which represents phenomena of a different class from the gravitational it is better fitted than the Newtonian theory of gravitation to form part of a comprehensive scheme relating to a wide class of physical phenomena. But for the purpose of describing nearly all the relative motions of the bodies of the solar system the Newtonian scheme is demonstrably sufficient and it will always be as adequate for that purpose as it has proved itself to be in the past. Thus it is in possessing greater range and accuracy in its power of description and not in what should properly be described as its truth unless truth be understood to mean degree of applicability that the Einstein theory will be regarded as superior to that of Newton. For all we know the Einstein theory itself may one day be superseded by some still more general and comprehensive scheme that is applicable to the description of a still greater range of phenomena.
In general the method of setting up hypothetical theories to be tested by their power of representing and predicting phenomena may be described as methodological pragmatism. That does not of course imply the acceptance of Pragmatism as a general philosophical system.
In some cases there may be diversity in the manner in which a conceptual scheme can be applied to the purpose of describing the phenomena for the representation of which it was constructed. This may especially be the case in a theory expressed in a mathematical form in which some only of the conceptual elements have direct perceptual correspondents the remaining elements being required to ensure the coherence and validity of the theory itself. A principle of correspondence is then required which shall assign the nature of the correspondence between perceptual elements and conceptual elements of the abstract scheme and such principle of correspondence is not necessarily unique. An example of this is found in the theory of the Electromagnetic field as formulated by Maxwell. There has been general agreement that Maxwell's equations do represent what can be observed in the field but different modes have been suggested of placing into correspondence what can be observed with the vectors which appear in the equations.
In the most abstract branch of Science Pure Mathematics the powers which the mind possesses of idealization and generalization have been so extensively employed that conceptual schemes have been erected which are not known to be applicable to the description of any physical phenomena although such schemes could never have come into being without an ultimate starting point in physical perception. This tendency on the part of Pure Mathematics to outrun the exigencies of applications has sometimes been made a subject of reproach to Mathematicians especially on the part of Physicists. Mathematicians have been accused of wasting their intellectual efforts in a region of bare formalism which has cut itself off from all connection with the actual world. It has been urged that Mathematicians have an excessive tendency to occupy themselves with ideas that are too remote from the physical order in which Mathematics had its origin and in which it should still find its proper applications. Experience has however shown in many instances that it is exceedingly hazardous to prophesy that a Mathematical theory however remote it may apparently be from the physical order of things will not sooner or later find an application within that order. It is quite true that although Pure Mathematics is in its origin a part of Natural Science having arisen like every other branch of Science in efforts to deal with certain aspects of the perceptual world it has become by rapid stages a deductive and formal Science in a sense and to a degree which is not the case at present in any other branch of Natural Science. Nevertheless Mathematical thought as its history very clearly indicates is one of the normal forms of mental activity. If it be allowed to develop itself on its own lines free from all fetters imposed upon it by the supposed necessity of keeping itself in close relationship with other departments which can make use of it as a tool it will not prove itself to be in default when new demands are made upon it by other branches of Science as they gradually reach those stages in which they require the help of refined Mathematical methods. It would be easy to establish in detail that many of the most important applications which have been made of Mathematics were wholly unforeseen by those who developed the Mathematical ideas and methods so employed. Had not Mathematicians during more than a century investigated those systems of Geometry that are known as non-Euclidean and which appeared to many Philosophers and Physicists to be an extreme instance of the addiction of Mathematicians to lines of thought which could lead nowhere the recent theory of Einstein could not have been formulated because the way would not have been prepared for the conception of the four dimensional manifold with a non-Euclidean metric which appears in the scheme as space-time and the theory of tensors which Einstein has employed in the elaboration of his scheme would not have existed. The new Physics and Chemistry may at no distant time require weapons forged by Mathematicians; the theories of functions of differential equations of groups; thus providing applications for the most refined methods which Pure Mathematics can devise. It will be seen in later lectures that this time has in fact already arrived.