Jacques Maritain Center :
Theories of Knowledge / by Leslie J. Walker, S.J.
§ 324. Realism interprets truth as 'correspondence' between our concepts and judgments on the one hand, and objective reality on the other, though in true knowledge the nature of objects is revealed to our minds, not completely, but in part. A true concept represents, or, better, in a true concept we know, some real object in some one or more of its aspects or of its relations. Hence, truth must be determined objectively, not by our needs, but by objects themselves, if it is to be truth at all. In perception, in conception, and in judgment, purposes and needs have their proper function, as psychology shows; but their influence, in the main, is restricted to the intent, as opposed to the content, of thought. Certainly we may frame hypotheses in which we provisionally determine the content of thought by our own constructive powers, but a concept so formed is not true unless it is confirmed or verified by reference to the object.
§ 325. Is this position compatible with the standpoint adopted in Physical Science? In general it seems to me that it is. For the majority of physicists admit that determination of the mind by its object is the only possible means of attaining truth; though they hesitate when asked how far physical concepts and laws have been so determined, and sometimes seem to doubt whether they will ever be more than symbolic and useful. Nevertheless, in her aim and in her practice science is almost always objective. The aim of the physicist is to force nature to reveal her inner workings and 'the correlations of natural agencies.' Facts are regarded as something independent of mind and our knowledge of them as objectively determined. Purpose is allowed to determine, not facts themselves, but only what facts are relevant to the problem in hand. Guided by purpose, the scientist selects his facts, taking note of what is relevant and neglecting the rest. But in the sphere of observation and experiment, never, if he can help it, does he allow purpose to determine content. Abnormal cases occur, cases in which associations or the 'will to believe' is exceptionally strong but the careful observer -- and the scientist is, or should be, a careful observer -- will suspect such influences, if present, and will hesitate to trust his conclusions in such a case until they are confirmed by independent evidence. It is only in the framing of hypotheses that the constructive activity of mind plays a part. But a hypothesis is not a law; it is a question, a postulate which we ask nature to answer or confirm. We apprehend some relation as holding between concrete realities -- for even in its interrogative form a hypothesis is suggested by facts we modify that relation and try whether it still holds in nature; or we postulate that the relation holds in other and different cases, and experiment to discover whether it is so. Should reality persistently answer our question in the affirmative, our hypothesis then becomes a law, true approximately and under the given conditions for all cases. Throughout the experimental stage of science it is the aim of the scientist to get the object itself to determine the answer to his questions, as it determines, in part at least, the questions themselves. He wants to know reality as it is. His whole endeavour is, therefore, to exclude subjective considerations and pre-conceived ideas, so that he may read aright the answer that nature gives. He tries more than anyone else to keep in subjection his 'will to believe,' knowing that, if this is impossible, he may as well relinquish all attempts to discover the laws which govern the universe.
§ 326. The scientist, then, fully admits that if scientific concepts and laws are to be verified, they must be determined by their objects, and by their objects alone; and the aim of experiment is to place conditions precisely in order that such a determination may take place. The problem which we have to solve, then, is how far the concepts and laws of physical science in its present stage of development have been de facto determined by objective reality, for upon this will depend their power to give us true knowledge of objective reality. Science, which as a rule is non-assertive on this point, is apt to appear somewhat sceptical on account of its fondness for such terms as symbolic,{1} conventional, probable, approximate, useful. M. Duhem, for instance, tells us that all concepts in Physical Science are symbolic, and all laws approximate, and that hence, strictly speaking, they cannot be said to be either true or false. For this reason he distinguishes common-sense laws from scientific laws, because of the former we may predicate truth and falsity, notions which are inapplicable in science. The reason is, he says, that common-sense laws "simply extract what is common in each particular case to which the law applies, so that in each of those cases to which we apply the law, we shall find concrete objects in which these abstract ideas are realised;" whereas scientific notions, mass, temperature, pressure, are "not only abstract, but symbolic, and have meaning only thanks to physical theories."{2} And from the approximate character or indetermination of the symbol follows the indeterminate and approximate nature of the law.
Common-sense laws are very true, but on condition that the general terms between which they establish a connection be abstractions arising spontaneously from the concrete, abstractions unanalysed, taken en bloc, like the general idea of a carriage or of a horse.{3} . . . A physical law possesses a certitude much less immediate and much more difficult to appreciate than a law of common-sense, though it surpasses the latter in the minute and detailed precision of its predictions.{4}
§ 327. Let us examine a scientific definition and then a scientific law in order to see for ourselves how far its genesis and its precision affect its truth. "The words hot and cold, or hotness and coldness," says Professor Preston, in sketching the development of the notion of Temperature, "refer to the state of a body as judged by the sense of heat," a hot body being "regarded as the source of an influence which affects the sense of heat."{5} We begin, then, with sense-perception. Hotness is some property of objects in virtue of which they affect our temperature-sense, as psychologists sometimes call it. And this, we may note in passing, is probably at bottom what many who belong to the new school of Energetics mean when they say that the data of science are sensations. Ostwald, for instance, who is more objective than Mach, says that all sensations are due to a difference of energy between the organ of sense and the medium which surrounds it. At any rate, the common-sense notion of hotness, like those of sound, light, pressure, is at first objective. True, in the beginning, such notions are denotative rather than connotative; but their connotation is not zero. We know enough to be able to distinguish hotness from other qualities of the same or of different objects; and are thus able to enquire what is its cause, what its nature, what its relations to other properties of matter.
§ 328. It is in the course of this enquiry that the notion of temperature comes to be scientifically defined; for the scientific definition of temperature presupposes the common-sense notion of hotness. Finding that several pieces of the same substance can be arranged in a series by the sense of heat alone according as one is hotter or colder than another, "we are hence led," says Professor Preston, "to the idea of a scale of hotness, and to enquire how much one body is hotter than another." The estimation of hotness implies some scale or standard of measurement, and when this is chosen
we may speak scientifically of the hotness of a body, and for this purpose the word temperature is employed. The word temperature thus means simply the degree of hotness of a body measured according to some arbitrarily chosen scale. It is a scientific term, and contains all the meaning of the primitive word hotness, as well as the idea of a measure of hotness.{6}
Observing that "one of the most general effects of change of temperature or hotness in a body is change of bulk, or expansion by heat," and that, in the case of certain bodies, their bulk increases continuously in proportion to their hotness,{7} science has selected this change as the basis of a method of indicating temperature. But "the mode by which the change of temperature is indicated by the change of volume remains, of course, a matter of choice, as well as the particular substance employed."{8} For instance, we can measure change of volume directly through expansion indicated by the rise of a liquid, mercury or spirits of wine, in a tube divided into parts of equal or known capacities; or we may allow the liquid to overflow and determine the volume of the overflow, and hence the expansion of that which remains in the tube, by weighing.{9} Again, we may take a gas, air, hydrogen, or nitrogen, as our thermometric substance, in which case temperature may be measured either by "change of volume while the pressure is kept constant, or by change of pressure while the volume is kept constant."{10} Thus, not only is the scale itself arbitrary, but so also is our choice of a thermometric substance, of the property of that substance which we take as a basis of measurement, and of the means by which that measurement is effected.
Is, then, our definition of temperature arbitrary? Strictly speaking, it is not. It is in deciding what particular fact or class of facts the term 'temperature' is to denote that we exercise choice. We find that degrees of hotness may in general be indicated by changes of volume, and we therefore decide to use the term 'temperature' to signify degrees of hotness as indicated in that particular way. It is the choice of the thing to be defined that is arbitrary, not the definition of it; for the definition merely expresses a fact or relation which has been found to hold in nature itself.
§ 329. The question, then, really is whether our definition of temperature expresses an objective fact or not. Do changes of volume and changes of pressure really indicate changes in degree of hotness? If so, our definition is objectively valid; if not, it is merely a creation of the mind.
Now, theoretically, two phenomena which vary concomitantly must be in some way 'causally connected,' in which case changes in the one phenomenon will correspond to changes in the other, in so far as that phenomenon which we may call the effect depends upon the other phenomenon as upon its cause. Hence we may say that in so far as change of volume 'varies continuously' with change of hotness, there is a causal connection between the two, so that change of volume indicates and may be used in order to measure change of hotness. But the statement that change of volume varies continuously with change of hotness applies only to certain cases and even then is not exact. Changes of volume may be due also to changes of pressure, surface-tension, etc., and, in addition to this, there are apparent changes arising from variation in the solid envelope in which the thermometric substance, whether a liquid or a gas, is enclosed. These other causes of variation or of apparent variation the scientist must eliminate or allow for, if his measurements are to be exact; and since he can do neither completely, there will always be some slight error in his results. The personal equation, too, is a source of error, for it varies somewhat; and the lines which indicate the sub-divisions of the thermometric scale have always a certain thickness. Hence the degree of hotness of a body as measured by some chosen scale is not the degree of hotness as it objectively exists. There is always some error; and though of errors in general we may take an average or mean, the result will never be more than approximate. Hence the scientific definition of temperature itself is approximate it does not correspond exactly with the object defined; or rather it implies an impossibility, viz., that changes of hotness can be measured with absolute accuracy by an arbitrary scale; whereas measurement, since it implies quantity or degree, owing to the 'faiblesse de notre esprit' can never be precisely determined.
§ 330. Ought we, then, to refuse to apply the predicates 'true' and 'false' to scientific definitions on account of their approximate character? By no means. A scientific definition does not claim be exact, and it is only about its 'claim to truth' that the logician has to judge. The scientist, indeed, does not introduce the term 'approximate' into his definitions, but it is always understood. Approxiation is all we can get when, as in science, quantity is involved. But although it implies that knowledge is as yet incomplete, it does not destroy its truth. We do not say that a statistical statement -- v.g., a statement with regard to the population of a country -- is untrue because it is approximate, nor yet do we say that it is neither true nor false. Such statements, like scientific laws and definitions, do not claim to correspond exactly with objective fact, nor is such exactitude required in order for them to be true. That is true which corresponds with fact in so far as correspondence is claimed. We know that all quantitative laws imply a 'margin of error;' but provided we know also that this margin of error is relatively small in comparison with the quantities involved, our statement is what it claims to be, approximate, and in this sense may correspond with reality, and be true. A 'margin of error' is not strictly a margin of error, but rather a margin within which the quantities involved in a statement are known to be invariable. Moreover, not only does the scientist know, when he says that his law is approximate, that such variations are relatively small with respect to the quantities concerned, but he can also determine the limits of these variations. He can tell us, for instance, that his quantities are certainly exact, say, to the fourth decimal place, and probably to the fifth; so that he can also determine the degree of approximation within which any further deduction he may make from the original law will be true.
§ 331. Strictly speaking, then, we cannot say that a scientific definition as such is either arbitrary, symbolic or invalid. The most serious charge we can bring against it is that it is only approximate, and this does not destroy its objective validity. To call a scientific definition such as that of temperature, arbitrary or symbolic, seems to me to be, to say the least, inaccurate and misleading. For a symbol is either a purely arbitrary sign chosen to denote some object, or it is a sign chosen by convention to represent some object on account of a supposed resemblance between that object and the symbol used; and in neither of these senses is the definition of temperature which we have been considering a symbol. What we have defined is not 'degree of hotness' simply, but 'the degree of hotness of a substance considered in relation to the concomitant variations in magnitude of certain other substances.' This is the particular fact or class of facts which the term 'temperature' denotes. True, we choose this particular fact and agree to call it 'temperature;' but this done, we cannot define 'temperature' as we will: its definition is determined by the facts. The denotation of 'temperature,' like that of other terms, is chosen arbitrarily or by convention; but once the denotation is fixed, the connotation will depend upon the facts to which the term is applied. When we say that temperature is 'degree of hotness as indicated by an arbitrarily chosen scale,' we are describing something objective and real. The 'degree of hotness' is objective and real, and so is the 'chosen scale,' for by it we mean the thermometric substances whose variations in magnitude are causally connected with and therefore indicate degrees of hotness. The introduction of the term 'arbitrary' into our definition is a little inaccurate, perhaps; for, though the selection of a thermometric substance is to some extent arbitrary, choice is restricted to those substances which really indicate changes in degrees of hotness. But what is meant is that, provided this condition is fulfilled, it is matter of indifference what substance we choose, or what units we take as a basis of our scale and this also is an objective fact, since there are many substances whose magnitude varies concomitantly with their degree of hotness, or with the degree of hotness of some other substance. What we choose is not the definition, but the thing to be defined; and when the thing defined determines the definition, as it should do and usually does, there is between the two, not merely a symbolic resemblance, but something more ; for the definition -- so far as it goes -expresses true knowledge of a certain class of objective facts.
§ 332. Our definition of 'temperature' is not symbolic, then, if by temperature we mean 'degree of hotness' considered in relation to changes of volume or weight.
Yet there is a sense in which that definition may be said to be symbolic. For 'temperature' may be regarded as a quality of an object, as 'degree of hotness' simpliciter. In fact, it is hotness which the scientist really wishes to define; but instead of defining it, he finds that all he can do is to indicate its changes in degree by certain other changes of a different nature. What the nature of hotness or the nature of temperature is he does not know, but he does know that variations in magnitude correspond to degrees of hotness. Hence he uses variations in magnitude (volume or weight) to signify or symbolise degrees of hotness. The definition of Temperature, therefore, though not itself symbolic, contains a symbolic element, since variations in magnitude in no way reveal to us the nature of temperature, if by temperature we mean degrees of hotness. Yet even this symbolism is symbolism in the less rigid sense of that term. For the variations in magnitude by which we symbolise 'temperature' are not arbitrarily chosen, nor are they chosen by mere convention, but on account of a certain 'resemblance' between the symbol and that which it symbolises. And that 'resemblance' is not merely supposed to exist; but is known to exist and to consist in concomitant variations which imply either a direct causal connection or, at least, a common cause. Our definition, then, though it does not tell us what the nature of 'temperature' is, nevertheless expresses real knowledge about temperature. To call such a definition 'symbolic,' therefore, though true in a certain sense, is somewhat misleading, for a symbol usually means a purely arbitrary sign, and does not necessarily imply any kind of resemblance. Moreover, if we use the term 'temperature' in the strict sense of degree of hotness, quâ measured, our definition is not symbolic at all, as we have seen.
§ We may, of course, substitute in place of the above definition of temperature the formula V - Vo = vθ,, or V = Vo (1+aθ) where a = v / Vo. Obviously a definition of this kind is in form strictly symbolic, since its terms are now purely arbitrary conventional signs. In reality, however, it is as objective as the real definition for which it is substituted. Its symbols refer to names, and through names to concepts which correspond to objective fact. We might re-write the definition in the form θ = (V - Vo)v, which would mean that the number of degrees or scale-units of temperature in a given body or in the thermometric substance is equal to the ratio of V - Vo, the difference between the actual volume of the thermometric substance at and the volume which the same substance had at the zero of the scale, to V, the degree-measure, or increase in volume of that substance for one degree.{11} Now v is by deftnition the same all along the scale, hence a [ = V / Vo] in the formula V = Vo(1 + a θ), which is called the co-efficient of expansion, or 'expansion per unit volume of the thermometric substance in changing its temperature from 0o to 1o,' should also be the same all along the scale. Yet this, in fact, is not the case. Not only does 'a' vary for different substances, but for most substances it is greater when we measure it between (say) 70o and 71o than it is between 0o and 1o. 'v' and 'a,' then, as soon as we come to deal with an actual case, are approximate, and so also, for that matter, are V and Vo, which are supposed to be constant. Yet our definition is not thereby invalidated, for though the statement V = V0 (1 +a θ) in form is exact, it does not claim to be really exact. We are quite aware that in reality it is approximate, and that we must allow for a margin of error in any particular case to which we may apply it.
§ 334. Hitherto we have been discussing the definition of temperature only in so far as it is based on experimental data; and have found that, though only approximate and in a sense symbolic, it nevertheless gives us true knowledge about objective reality. Physical science, however, is not content with this. It wishes to get at the real nature of temperature, and for this purpose frames hypotheses which, taken together, constitute the theory of Heat. Heat is regarded as 'the molecular energy of matter,' and temperature thus becomes 'the molecular energy of matter as measured by some chosen scale.' Here, then, we have a definition which, as M. Duhem says, is not only 'due to a slow and complex process of elaboration,' but which 'gets its whole meaning from physical theory.' And that theory is not established. It involves numerous hypotheses about the constitution of matter, and the nature of different forms of energy, none of which can as yet be treated as objectively valid and certain. Hence the definition of temperature which is based on such a theory, though useful, cannot be said to be 'true.' Again, it is a symbolic definition that is pictured in terms of mechanical imagery. A 'hot' substance is symbolised as an indefinite number of minute particles of matter all moving about with greater or less rapidity, and behaving according to known mechanical laws. A symbol of this kind corresponds to some extent with objective fact, for particles of matter certainly have mechanical properties. But how far it corresponds, or what other properties of matter are also manifested in the phenomenon of heat we cannot say. By treating heat as if it were the molecular energy of matter, many phenomena may be explained and many experimental laws co-ordinated; but there are phenomena which the theory fails to explain, and cases in which it breaks down completely. At present, then, the molecular theory of heat is neither true nor false. It contains an element of truth, as other theories have done before it; but indefinite modifications will have to be introduced before it can be established, and what precise effects those modifications will have upon the various hypotheses and definitions of which the theory is made up, it is impossible to say. The theoretical definition is symbolic, because though it 'corresponds' with the facts, we cannot say how far it corresponds, or how far it fails so to do. It embraces both an element of truth and an element of error, and between the two it is impossible for us at present to draw the dividing line.
§ 335. Definitions which belong to the theory of physics must be distinguished, therefore, from those which are based directly on experimental facts; and a similar distinction must be made between experimental laws and theoretical hypotheses. The latter are symbolic, provisional, and, strictly speaking, neither true nor false; the former are approximate, but none the less objectively valid. Boyle's or Mariotte's law may be taken as an example of an experimental or empirical law. It is a generalisation based directly on experiment and observation, and states that "at the same temperature, the volumes occupied by the same mass of gas are in inverse proportion to the pressure to which it is subject."{12} This law, M. Duhem tells us, is a "symbolic relation whose application to concrete reality implies that one knows quite a system of theories." Nevertheless, I think that we are justified in regarding it as objectively valid. But let us examine its significance more closely and see for ourselves whether this is the case.
That Boyle's law merely expresses a relation between temperature, volume, pressure and mass, yet does not tell us anything of the intrinsic nature of those entities is clear; and the same may be said of almost all experimental laws. They express relations, sequences, causal connections, not essences. Nevertheless, they may give us true knowledge, and if 'actio sequitur esse,' they tell us something about the nature of the entities concerned (in this particular case, the properties of a gas), even if they do not tell us what that nature is. If we wish to explain the law or to discover why the relation arises, we must have recourse to theory. But considered merely from the experimental point of view, Boyle's law and any other empirical relation of this kind can be established independently of theory, unless, indeed, we apply that term to the complexus of experimental laws themselves. General statements of facts, however, are certainly not what we mean by theory but, on the contrary, are the data by means of which theory is ultimately to be established or condemned.
§ 336. Boyle's law, then, expresses a relation which can experimentally be shown to hold between certain entities, of the intrinsic nature of which we know little or nothing for certain. And if on this account you choose to call the relation symbolic, well and good; though the expression is liable to mislead, because it suggests that the law does not express real knowledge. And this is not true, for all the terms denote objective facts. Temperature and mass cannot be defined in their essential nature, but only by means of their relations to other properties, v.g., to volume or to motion and weight. Yet mass, like temperature, signifies an objective fact, and our concept of it is true so far as it goes. For even if mass be ultimately reducible to electromagnetic inertia, it is still true that material bodies have some property in virtue of which, in varying but measurable degree, they tend to keep their state of motion and resist all influences tending to change it, whether in quantity or direction. Doubtless this correspondence can be established in a concrete case only by the use of instruments and by means of measurements and calculations which are often long and complicated. But, as we have seen in regard to temperature, the approximate nature of measurement does not detract from its truth, nor does a long and complicated calculation lead to inaccurate results, for when once our margins of error are known, it is possible to determine with mathematical precision the accuracy of our conclusion.
Boyle's law is inaccurate when stated, as generally it is, in the form which M. Duhem has given, but it is inaccurate because it was proved to hold only for certain gases when subject to moderate pressures, and not because its terms are symbolic or its measurements inexact. For moderate pressures and for the so-called permanent gases, oxygen, hydrogen, and air, the formula (PV = constant) will hold with a margin of error that is comparatively small; but for most gases PV gradually diminishes up to a certain point as the pressure is increased, and after this point is passed it begins to increase; and again its variations are different according to the particular gas which is being examined, and according to the temperature at which the experiment is performed. Yet, while still keeping clear of molecular and other hypotheses in regard to the nature of matter, it is possible to devise more complicated formulae which shall take account of these facts, and which will therefore be applicable in cases where the simple formula failed. These more complicated formulae will, in a certain sense, be more true than those which they supersede, since they will contain a smaller margin of error. Nor does there seem to be any reason for supposing that certitude diminishes as a law becomes more precise, for it will still be possible to construct formulae such that all possible variations from any of the quantities involved will be small in comparison with the quantities themselves.
§ 337. There is, however, a limit to the precision with which experimental science can at present determine the magnitudes it wishes to measure, and if this limit is passed certitude gives place to probability. In proportion as the scientist endeavours to make his measurements more exact, errors due to observation, to the instruments he uses, and to the impurity of the substances upon which he experiments, become of greater importance, till finally the margin of error he is forced to allow, counterbalances the degree of accuracy which he seeks to obtain. It is then that he is driven to invoke the aid of theory. Assuming general principles which apply unreservedly to all material objects, he constructs hypotheses, assigning a definite structure to minute particles of matter upon which it is impossible to experiment individually, and of whose existence he has no immediate experience. From these hypotheses he deduces conclusions whose precision can be as minute and detailed as he wishes to make it, and whose accuracy is unquestionable, granting that his premises are true. But when he comes to apply such conclusions to experimental data he finds that in some cases they are confirmed and in others they are not, and he discovers also that to any individual case there are an indefinite number of theoretical formulae, each of which will apply equally well. Consequently, he is in doubt as to which to choose, and in such a dilemma other considerations being of equal value, the simplest is usually selected. But now it is necessary to explain why the formula selected should not always be confirmed by the facts. In theory, however, this is impossible, for we do not know how far its definitions have an objective counterpart or how far they are merely symbolic; and when a particular formula fails to apply in a given case, we do not know where the error lies, since all formulae and all definitions in theory presuppose an indefinite number of hypotheses of whose validity we are equally uncertain.
§ 338. Clearly, then, the region of theory in Physics differs essentially from that of experimental law. Experimental laws are true so far as they go, or they are false. If qualitative merely, they are true in so far as they establish relations between real properties of matter; false, if supposed adequately to express its whole nature. If quantitative, they are true, provided the limits within which the quantities may vary are known to be relatively small; false, if the quantities involved pretend to be other than approximate. Of theoretical hypotheses, on the other hand, we cannot say definitely that they are true, nor yet that they are false, until they have been verified; and they can only be verified by comparing them with experimental laws already established. Hence the possibility of ever being able to establish a physical theory presupposes that experimental laws are already true; for " le seul contrôle expérimental de la théorie physique qui ne soit pas illogique consiste comparer le système entier de la théorie physique à tout l'ensemble des lois expérimentales et à apprécier si celui-ci est répresenté par celui-li d'une manière satisfaisante."{13} In the experimental stage of physics the scientist may work with symbols, as when he regards the sun as an ideal sphere, whose matter can be treated as if it were massed at its centre; but he knows in this case that, owing to the distance of the sun from the earth and planets, the error he has deliberately introduced cannot affect the degree of accuracy which he wishes to obtain in the deductions he is going to make. The theorist in physics is in a different case. When experience refuses to verify his symbolic formulae, he cannot say where the error lies. It may be in some symbol he is using, or it may be in some hypothesis which is presupposed; he cannot tell. He has no means of comparing his symbols with the object to which they are supposed to correspond. All he can say for certain is that something is wrong somewhere. Such being the case, when the theorist has no means of locating the error, his only alternative is, as M. Duhem says,
to keep rigidly to the signification of symbols when once they have been fixed, regardless of experimental facts, but to use absolute freedom in the matter of postulation and hypothesis, provided no contradiction is involved and provided, ultimately, all hypotheses be subjected to the test of experience when once a theory is complete.{l4}
§ 339. What function, then, is to be assigned to physical theory? Various answers have been given to this question. It co-ordinates and systematises experimental laws; it synopsises and condenses them in accordance with economy of thought; it enables us to classify what is known and to predict what is unknown; it suggests questions and guides as well as prompts future research. At any rate, then, its function is useful; but is it anything more M. Le Roy would answer in the negative. The symbolic relations which we postulate and express in mathematical language are merely instruments more or less convenient, useful, suitable for the purpose of connecting and systematising experimental laws with a view of bringing them under human control and rendering them more manageable. Like M. Boutroux, he regards science as "a collection of methods for the assimilation of things to our intellect in order to bend them to our will." M. Duhem takes a more objective view, but even he declares that physical theories do not pretend to be explanations of the nature of material things. They are independent of all metaphysical systems.{15}
The question, however, may be raised, whether justifiable thus to separate the useful and the true. Whether, in other words, 'symbolic representation' can render more easy, more rapid, and more sure our reasoning about 'what our senses, aided by instruments, make us know' without in some way corresponding to those same data of experience; or whether, again, it is possible to devise methods for assimilating things to our intellect in order to bend them to our will, unless those methods are based on true principles and are capable of giving knowledge of reality. Can a thing be useful without being in some sense true? I do not see how it can.
§ 340. Theories are not like machines which work automatically, or like physical instruments which may be brought to some degree of perfection by the haphazard process of chopping and changing them at random till they suit the practical purpose we have In view. In science we are dealing with knowledge from first to last. Our 'manipulations of experiences' in their primary purpose are not practical but theoretical, and consist in classification, co-ordination, systematisation and explanation. Verification, too, does not consist in the realisation of a practical end, nor in the successful performance of some physical action; but in a certain identity between the detailed inferences drawn from our theories and given physical facts. With a view to discovering the general principles or laws which underlie these facts, the physicist frames a theory comprising many hypotheses logically connected together, and expressed either in mathematical symbols or in images of particles endowed with some form of energy. These symbols and images, however, are not mere pictures, but concepts. They signify things the existence of which we postulate, and which we suppose to obey known laws, mechanical, dynamical, or electro-magnetic. These laws form the basis of theoretical deduction. From them the physicist infers that under given conditions certain phenomena should occur, and his inferences are often confirmed by fact. What conclusion, then, ought we to draw in regard to the objective significance of these symbols and images, and the laws that govern their relations? Are they, in the strict sense, objectively valid, i.e., do they represent adequately the nature of that for which they stand substitute? No; for in many cases the conclusions deduced from them are not confirmed by experience. Ought we to say, then, that they are neither true nor false? Clearly they are not wholly true, otherwise the conclusions deduced would always be verified; neither, on the other hand, can they be wholly false, for, then, the conclusions ought never to be verified unless per accidens, and in a few cases here and there only. They must contain at least an element of truth, and represent, at any rate, an aspect of reality.
§ 341. Again, the concepts which are involved when we think of facts are of a similar nature to those which we use when we think of hypotheses or laws, which seems to show that facts and hypotheses are not entities of a wholly different order. That on December 28th, 1908, many people were killed in Messina by falling buildings is a fact. That the buildings were shattered owing to an earthquake is also a fact. On the other hand, that the buildings fell to the earth is supposed to have been due to gravity, which is a hypothesis. And that the earth moves round the sun is another hypothesis, itself also accounted for by the further hypothesis of gravitation. Now the concepts which function in our minds when we think of the earth moving round the sun are to a large extent the same as those which we use when we think of buildings falling to the ground, viz., concepts of motion and direction; and we may say the same, I think, of the concepts involved in the Law of Gravitation, viz., distance, mass and force, except that mass is a complex concept involving volume and density, while force can only be conceived as the cause of change from rest or uniform linear motion. Why, then, should we regard hypotheses and facts as entities of wholly different order? If these were really so, it is difficult to see how verification could take place. But if not, why should we make this forced separation between what is useful and what is true? We cannot say that a hypothesis is true until it is verified; yet I can see no reason whatsoever why we should not admit that it may become true. A hypothesis of its very nature seems to me to possess the potentiality of becoming true. And if this is so, it is irrational to deny that hypotheses may contain an element of truth, even though we cannot distinguish what is true in them from what is false, and so cannot tell how far they are true until they have been completely verified.
§ 342. Or, again, to put the same argument in another form if the so-called 'symbols' and 'images' of Physical Theory contain no element of truth, how are we to explain the coincidence of experimental fact with theoretical deduction? A precisely similar effect can be produced only by a precisely similar cause; for the nature of the effect proceeds from that of the cause, in which it must be already potentially contained, otherwise it could not get into the effect. The cause may comprise many other properties besides that of producing this particular effect; but, in so far as an effect does proceed from a given cause, the nature of the effect must be already there implicitly in the cause. Now, the particular conclusions that are deduced in physical theory are based upon hypotheses in which things having certain mechanical or electro-magnetic properties are supposed to exist. Hence, in our ideal physical world it is from the nature of these hypothetical entities that certain effects, the phenomena that we anticipate, proceed. Similarly in the real world actual phenomena or events proceed from and are dependent upon the nature of existing entities and their properties. If, then, and in so far as, we find that the phenomena which we anticipate are realised in the concrete, the anticipated and the actual phenomena must proceed from a similar cause; or, in other words, the nature of the entities which we conceive as causes in a verified hypothesis and the nature of real things must be one and the same.
If facts are true, then, the hypothesis which explains them must also be true, and, in so far as our anticipations are realised, the concepts which they presuppose must be objectively valid. Unfortunately, however, our anticipations are never precisely realised, as they are constructed in theory and often enough they are negatived by concrete experience. Since, then, and until we can localise the source of error and rectify our theoretical constructions, it is impossible to say more than that theories contain an element of truth, the degree of which will depend largely upon the precision and the extent of its confirmation by experimental facts.
§ 343. But can we ever distinguish what is true in a theory from what is false? Can we localise error and affirm that it lies in one hypothesis rather than in another? Are there any physical theories or any hypotheses which we know for certain to be true? M. Duhem, as we have seen, holds that the parts of a physical theory are so intimately bound up together that no one hypothesis can be verified apart from the rest. Nothing can be regarded as objectively valid until it is complete, and has been verified as a whole by comparison with the complexus of physical facts. M. Poincaré, on the other hand, takes a different view, and thinks that some hypotheses or laws, if not more certain, are at any rate more probable than others; and this seems the more reasonable view to take, though it by no means follows that there are strictly physical laws, however universal, which can be regarded as certain a priori, or as self-evident. M. Duhem's argument that the law of inertia is not self-evident because it was not recognised as such by the Greeks is hardly conclusive; for a self-evident truth must be understood before its self-evidence can be recognised,{16} and it is quite conceivable that there are truths which would be self-evident if anybody thought about them, but which are not so, 'because no one as yet has thought of them at all.{17} I see no ground, however, for regarding the law of inertia as self-evident. Development is a fact, and so is decay; and, as I have already pointed out, there seems to be no reason a priori why things should not tend constantly to develop or increase in some way or other, or why, on the other hand, they should not tend continuously to diminish and decay. Yet few people would question the truth of the law of inertia now, nor indeed any of Newton's three laws of motion. Why is this? It is precisely because on the three laws of motion, together with that of Gravity, the whole system of Astronomy is based. I know not whether M. Duhem would treat astronomical theory as part of the theory of Physics, or whether he would regard it as a theory sufficiently distinct and sufficiently complete to be verifiable apart from the rest. Astronomers, at any rate, seem to have no doubt as to the validity of the law of Inverse Squares or of the laws of motion, laws which have led to conclusions completely verified by fact, except for a slight margin of error, which can, in most cases, easily be accounted for by the personal equation or by instrumental defects. The law of Gravitation is, at any rate, approximately true if we express it in the form -- between two material bodies there is an attractive force which varies directly as the product of their masses and inversely as the square of their distances; but whether that force is an ultimate property of all material bodies, or whether it is due to a vis a tergo, we do not know. Our knowledge is limited. We know for certain that a relation which we call Gravitation holds between all material bodies, but we do not know whence that relation arises.
§ 344. Let us take another physical theory, the Undulatory Theory of Light. Here, again, we can distinguish what is true in that theory from what is doubtful and possibly false. The theory, taken as a whole, explains nearly all the phenomena of light. In particular, the characteristic hypothesis from which it gets its name is necessary in order satisfactorily to explain refraction and interference, phenomena which seem to show conclusively that the propagation of light is by means of undulatory motion in which there is some form of periodic change transverse to the line of propagation. The subsidiary hypothesis of an 'ether,' which we seem forced to potulate as a medium for the transmission of light, presents certain difficulties. For the properties of this 'ether' approximate at once to those of a solid and to those of a perfect gas. Nor is this difficulty entirely overcome, though it is relieved somewhat, by stating it in a more accurate form as when we say the ratio between the elasticity of the ether and its density must be very large. Another difficulty arises, too, from Michelson's experiment, which apparently shows that no relative motion exists between the ether and the earth, whereas a relative motion between the two must be postulated in order to explain the aberration of light. It has been suggested that this latter difficulty may be obviated by supposing that moving bodies contract in the direction of movement but this hypothesis, though not impossible, cannot be experimentally proved. Further discussion of the difficulties connected with the Undulatory Theory of Light, however, would be beyond the scope of my present purpose. Doubtless, that theory contains an element of error as well as an element of truth. Yet the source of the error can be located, I think, in the postulate of 'ether,' and the essential hypothesis of the theory as stated above does not seem thereby to be affected, and so may be regarded as true.
§ 345. This view of the applicability of truth to scientific theories is not altogether foreign to the minds of physicists whose expressions -- as far as physical theory is concerned -- are often pragmatic in tone. M. Duhem seems to imply that when physical theory is complete and when it has been completely verified it will become not only useful, but true, and true in the realist sense. M. Poincaré is even more emphatic. Science, he says, is not merely a rule for action -- it gives us knowledge of objects. Its utility lies in its power to enable us to foresee events, but foresight implies sight, and the value of prediction depends upon the accuracy with which events have previously been represented. The symbols or figured concepts, which in mathematical physics form the terms between which relations hold, may vary, but the equations to which they lead are the same. Mass and energy cannot as yet be accurately defined, but conservation of mass and conservation of energy point at least to something which in reality is constant throughout. Even the famous postulate of Simplicity or Thought-economy is, in the opinion of M. Poincaré, not wholly subjective. Simple laws are verified à peu près ; and this cannot be due to chance. There must be some cause for it in the nature of objective reality. Finally, he points out that, though Mechanism is distrusted on account of its tendency to Realism, yet one of the chief conclusions to be drawn from Maxwell's work is that we can always give to the material universe a mechanical explanation if we wish. Nor will such an interpretation be altogether symbolic, for electric oscillations, the movements of a pendulum, and all periodic phenomena manifest "une parenté intime qui correspond à une realité profonde."
§ 346. The significance of statements such as these seems to point to something more than a parallelism between the logical connections of a physical theory and the system of experimental laws upon which it is based. Moreover, this 'parallelism' -- which, in the opinion of Hertz, is the term that best expresses the relation of theory to fact -- itself requires explanation. If the concepts which are used in physical theory, are merely ideal constructions logically connected together, how comes it about that they correspond symbolically, point for point and connection for connection, with the facts to which they are applied? The epistemology of physical theory cannot rest content without some further explanation. Parallelism cannot be ultimate, but, like concomitant variations, seems to postulate some causal connection. The concepts of physical theory are ultimately derived from experience, and the conclusions of physical theory bring us back again to concrete experience in which more or less completely they are realised. In neither case is the coalescence whole and entire yet contact with reality is thereby established, whence it would seem to follow that between logical connections on the one hand, and physical relations on the other, there must also be some form of correspondence, the nature of which the term parallelism is hardly adequate to express.
The possibility of parallelism, of useful symbolism, and of predicating and controlling events by means of calculations based on theoretical assumptions can be explained only on the further assumption that the concepts which physical theory employs have some kind of objective truth. These concepts may not be wholly true, still less adequately express the nature of objective reality; but they are not mere symbols or mere picturesque representations. They give us at any rate some knowledge of our material environment. Most hypotheses, even when verified, tell us little of the inner nature of things. The knowledge which they give concerns for the most part their operations and the relations one to another. Still 'actio sequitur esse' and the nature of a relation depends upon the objects it relates. Mechanism is an attempt to penetrate deeper into the nature of material things, and though it has failed to reduce everything to matter and motion or to show that no properties exist besides the mechanical, it represents a true, if a one-sided and partial, aspect of the material world.
It is not irrational, therefore, to interpret the aim of physical science in a realistic sense, or even to say that some of the hypotheses which belong to physical theory give us real knowledge of the laws which govern the universe in which we live. Realism is not incompatible with physical science. On the contrary, it gives to its speculations a richer meaning and a fuller significance than any that Pragmatism affords; and it raises within us a rational hope that the realistic terminology of which the scientist, whether deliberately or from force of habit makes use, will in the end turn out to be something more than mere metaphor; for, as M. Duhem remarks, "Au fur et à mesure que les methodes expérimentales progressent, l'indétemination du symbole abstract que l'expérience physique fait correspondre au fait concret va en diminuant."{18}
{1} La Théorie Physique, chap. v., § 2.
{2} Ibid., § 1.
{3} Ibid., chap. viii., § 5.
{4} Ibid., chap. v., § 5.
{5} Theory of Heat, p. 12.
{6} Theory of Heat, p. 13. (Italics mine.)
{7} For a thermometric substance the law of the increase of bulk must be one of simple proportion, and this law will hold only within certain limits, exclusive of change of phase.
{8} Op. cit., p. 24.
{9} Ibid., p. 112.
{10} Ibid., p. 116.
{11} Preston, p. 19.
{12} Duhem, La Théorie Physique, chap. v., § I.
{13} La Théorie Physique, chap. vi., § 5.
{14} Ibid., chap. v., §§ 7, 9.
{15} Ibid., chap. vii., § 1.
{16} compare the scholastic distinction between truths which are per se nota quoad se, and per se nota quoad nos.
{17} cf. chap. xv. re Self-Evident Truths.
{18} La Théorie Physique, chap. v., § 3.