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Part 1. The Mechanical Theory

Lecture 2: Abstract Dynamics

The demurrer of modern scientific thought, though illegitimately, yet practically, forecloses theistic inquiries. A discussion of its fundamental positions therefore called for in the interest of such inquiries.

Natural knowledge to be examined (i) formally as knowledge, (ii) as a body of real principles. Beginning with the latter, we have (a) the mechanical theory of Nature, (b) the theory of Evolution, and (c) the psychophysical theory of Body and Mind.

A. The Mechanical Theory:—The Laplacean calculator; different views of him; he excludes the teleological. Abstract dynamics, a strictly mathematical science, the basis of this theory, which thus divests itself of the real categories of Substance and Cause, and substitutes for them the quantitative terms ‘Mass’ and ‘Force’ (or Mass-acceleration). But if this be so, Laplace's calculator never attains to real knowledge.

ANY attempt in these days to discuss the problem of theism is, we have seen, liable to demurrers more or less emphatic from what we may fairly call the spirit of the age. Naturalism, speaking in the name of science, declares the problem superfluous, and agnosticism, professing to represent reason, declares it to be insoluble. This attitude we have traced to that positivist conception of knowledge which the rapid advances of science and the repeated failures of philosophy have jointly encouraged. Referring to this conception G. H. Lewes remarks: “A new era has dawned. For the first time in history an explanation of the world, society, and man is presented which is thoroughly homogeneous and at the same time thoroughly in accordance with accurate knowledge; having the reach of an all-embracing system, it condenses human knowledge into a Doctrine, and coördinates all the methods by which that knowledge has been reached, and will in future be extended.… Its basis is science—the positive knowledge we have attained, and may attain, of all phenomena whatever. Its superstructure is the hierarchy of the sciences, i.e., that distribution and coördination of general truths which transforms the scattered and independent sciences into an organic whole, wherein each part depends on all that precede and determines all that succeed.”1 In the last lecture we made a cursory examination of this soi-disant organic whole of phenomenal knowledge. Even that brief survey would justify us in saying: First, that it is not in itself a homogeneous and organic whole; for the dualism of matter and mind, at any rate, runs through it, and is only evaded by desperate means. Materialism itself is repudiated, but the materialistic terminology is retained as primary and fundamental. Secondly, that it is not a whole of accurate, positive, knowledge; for it confessedly involves postulates and methods, which it is the business of no one of ‘the scattered and independent sciences’ to scrutinise, and which they all alike, therefore, accept in a naïve and uncritical fashion. Finally, that it is not an all-embracing system. Hamilton has supplied it with a Virgilian motto: Rerumque ignarus, imagine gaudet. The ‘accurate and strict’ knowledge of appearances implicates an indefinite but still positive consciousness of an ultimate Reality; for, says Mr. Spencer, “it is rigorously impossible to conceive that our knowledge is a knowledge of Appearances only without at the same time conceiving a Reality of which they are appearances, for appearance without reality is unthinkable.”2

But since the theistic problem deals primarily with spirit, not with matter, since further it involves those fundamental principles of knowledge which science is not concerned to discuss, and since finally it belongs to that extra-scientific or supernatural region of ‘nescience’ which science allows to be, but to lie forever beyond its pale, we might, if so disposed, reasonably contend that the demurrer both of Naturalism and of Agnosticism is altogether ultra vires; we might politely request science to mind its own business and proceed at once to our own. In so doing, too, we could safely count on the approval and good-will of many eminent representatives of science in every department. For, after all, agnosticism and naturalism are not science, but, so to say, a philosophy of knowing and being which is specially plausible to, and hence is widely prevalent among, scientific men. But just for this reason it would ill become us to treat them with cavalier disdain. If Gifford Lectures were less numerous, I might not perhaps be justified in devoting a whole course to these initial objections; but as every university in Scotland has always its Gifford Lecturer, I venture to think such restriction is not only allowable but desirable.

Our knowledge of nature, as unified and systematised according to the naturalistic scheme, may be considered from two sides. We may examine it formally, as knowledge, in respect, that is to say, of its postulates, categories, and methods. Or, taking these for granted, as science itself does, we may examine those of its real principles to which its supposed unity and completeness are ascribed. Some odd instances of confusion could be cited due to a mingling of these two points of view—a favourite practice with those who, like Huxley and Tyndall, are at once fervent naturalists and pronounced agnostics. We may know where we are when matter is spoken of throughout as an objective fact, or throughout as a mental symbol, but it is bewildering to find it posing in both characters at once. To begin with, let us then, postpone any attempt to get behind the plain deliverances of science by epistemological reflexions; let us give our attention first to its real principles.

There are three fundamental theories which—as we have already noted—are held to be primarily concerned in the unity of nature: the mechanical theory, this comes first and ‘determines all that succeed’; the theory of evolution, which essays in terms homogeneous with this to ‘formulate’ the development of the world, society, and man; last, the theory of psychophysical parallelism, dealing with the relation of body and mind. To the first of these we may now pass.

There is a well-known passage at the beginning of Laplace's essay on Probability, which may serve as a basis for the remarks I have to offer on the MECHANICAL THEORY OF NATURE. Having enounced as an axiom—known, he says, as the principle of sufficient reason, that “a thing cannot begin to be without a cause to produce it,” and having summarily disposed of the notion of freewill as an easily explained illusion, Laplace proceeds: “We ought then to regard the present state of the universe as the effect of its antecedent state and as the cause of the state that is to follow. An intelligence, who for a given instant should be acquainted with all the forces by which nature is animated and with the several positions of the beings composing it, if further his intellect were vast enough to submit these data to analysis, would include in one and the same formula the movements of the largest bodies in the universe and those of the lightest atom. Nothing would be uncertain for him; the future as well as the past would be present to his eyes.” “The human mind,” he continues, “in the perfection it has been able to give to astronomy, affords a feeble outline of such an intelligence. Its discoveries in mechanics and in geometry, joined to that of universal gravitation, have brought it within reach of comprehending in the same analytical expressions the past and future states of the system of the world.… All its efforts in the search for truth tend to approximate it without limit to the intelligence we have just imagined.” So wrote Laplace in 1812, and his words have been classic among men of science ever since. As one instance among many schewing in what sense they have been understood, I may mention the Leipzig Address to the Deutscher Naturforscher Versammlung by Émile du Bois-Reymond, an address that has made more stir in its way than Tyndall's Belfast Address of a year or two later, which it seems to have inspired. “As the astronomer,” said the Berlin professor, “has only to assign to the time in the lunar equations a certain negative value to determine whether as Pericles embarked for Epidaurus there was a solar eclipse visible at the Piræus, so the spirit imagined by Laplace could tell us by due discussion of his world-formula who the man with the iron mask was or how the President came to be wrecked. As the astronomer foretells the day on which—years after—a comet shall reëmerge in the vault of heaven from the depths of cosmic space, so that spirit would read in his equations the day when the Greek cross shall glance again from the mosque of St. Sophia or England have burnt her last bit of coal. Let him put t = -∞ and there would be unveiled before him the mysterious beginning of all things. Or if he took t positive and increasing without limit, he would learn after what interval Carnot's Law will menace the universe with icy stillness. To such a spirit even the hairs of our heads would all be numbered and without his knowledge not a sparrow would fall to the ground.”3

Spite of these scriptural allusions, it would be a mistake to imagine any connexion between the knowledge of this Laplacean intelligence and Divine omniscience. How God knows, or even what knowledge means when attributed to the Supreme Being, few of us will pretend to understand. But this imaginary intelligence of Laplace knows, as we know, by calculation and inference based on observation. To God the secret thoughts of man's heart are supposed to lie open; from this Laplacean spirit they would be forever hidden, were it not that he can calculate the workings of the brain. Human free will and divine foreknowledge have been held to be not incompatible: but free will and mechanical prediction are avowedly contradictory. Laplace therefore is careful to exclude free will. Before the future can be in this way deduced from the past, all motives must admit of mechanical statement and the motions of matter and its configurations be the sole and sufficient reasons of all change.

It would be a mistake again to confound this mechanical theory of the universe with doctrines such as those of Newton, Clarke, Butler, Chalmers, and other Christian apologists. They too refer to events in the material world as “brought about, not by insulated interpositions of divine power exerted in each particular case, but through the establishment of general laws.”4 But they none the less regard the laws and properties of matter as but “the instruments with which God works.”5 Such language may be open to serious criticism, but that just now is not the point. It is enough if we realise that whoever holds the notion of the Living God as paramount can never maintain that exact acquaintance with his instruments is enough to make plain all that God will do or suffer to be done. Thus Newton, at the close of his Opticks, declares that the various portions of the world, organic or inorganic, “can be the effect of nothing else than the wisdom and skill of a powerful ever-living Agent who, being in all places, is more able by his will to move the bodies within his boundless uniform sensorium, and thereby to form and re-form the parts of the universe than we are by our will to move the parts of our own bodies.” To men like Laplace and the French Encyclopædists, of course, this bold anthropomorphism would mean nothing; such strictly voluntary movement being for them a delusion. But coming from Newton, who did not regard man as a machine or conscious automaton, these words shew plainly that he would not have subscribed to the mechanical theory, although he laid what are taken to be its foundations.

I must confess to some surprise on finding Jevons, who must certainly be counted on the theistic side as a strenuous opponent of naturalism, nevertheless seeming to approve of Laplace's “mechanical mythology,” as it has been called. “We may safely accept,” says Jevons, “as a satisfactory scientific hypothesis the doctrine so grandly put forth by Laplace, who asserted that a perfect knowledge of the universe, as it existed at any given moment, would give a perfect knowledge of what was to happen henceforth and forever after. Scientific inference is impossible, unless we may regard the present as the outcome of what is past, and the cause of what is to come. To the view of perfect intelligence nothing is uncertain.”6 I must again repeat, that neither the intelligence conceived by Laplace, nor the knowledge attributed to it, is in any sense entitled to be called perfect. Laplace himself, though accounted hardly second to Newton as a mathematician, was hopelessly incompetent in the region of moral evidence. After a few weeks in office as Minister of the Interior, his master Napoleon sent him about his business,7 declaring him fit for nothing but solving problems in the infinitely little. His imaginary intelligence was only an indefinite magnification of himself, commanding an appalling amount of differential detail and possessed of the means of integrating it; but there is nothing to shew that the incapacity of this Colossus may not in other respects have been as sublime as his capacity for calculation. Jevons's inconsequence in accepting this Laplacean conceit is possibly due to a misunderstanding. A reference to Newton's first law of motion will make my meaning clear. When it is there said that a body left to itself perseveres in its state of rest or of uniform motion in a straight line, what is affirmed is a tendency, not a fact, for no body ever is left to itself. Similarly it might be said of the material universe, if left to itself, that its state thenceforth and ever after would be the outcome of its state at the given moment. So understood, Laplace's ‘doctrine’ would formulate a tendency, but would not assert a fact. That it is in the former sense that Jevons interprets it is plain, for he says expressly: “The same Power, which created material nature, might, so far as I can see, create additions to it, or annihilate portions which do exist.… The indestructibility of matter, and the conservation of energy, are very probable scientific hypotheses, which accord satisfactorily with experiments of scientific men during a few years past; but it would be a gross misconception of scientific inference to suppose that they are certain in the sense that a proposition in Geometry is certain.”8 But this was assuredly not Laplace's meaning; and from the illustrations used it was clearly not what Du Bois-Reymond understood him to mean. And lastly, it is certainly not in any such tentative and provisional sense that the mechanical theory now holds sway among scientific men and ‘weighs,’ as Huxley put it, ‘like a night-mare’ on the minds of many.

We are bound, I think, carefully to distinguish these two views: the one regarding the universe, so far at least as we can know it, as a vast automatic mechanism, and the other regarding the ‘laws of nature’ as but the instrument of Nature's God. But it is important to observe, too, that they have a certain common ground in the recognition of laws as ‘secondary causes.’ In this the naturalism of modern science and the supernaturalism of popular theology are so far at one; although the naturalist stops at the laws, and the theologian advances to a Supreme Cause beyond them and distinct from them. Now, it is, I think, inevitable, so far as the question of theism is argued out from such premisses, that theism will get the worst of it. Unquestionably it has had the worst of it on these lines so far; of this we noted many instances in the last lecture. Not a few temples to the Deity founded on some impressive fact supposed to be safely beyond the reach of scientific explanation have been overtaken and secularised by the unexpected extension of natural knowledge. Chalmers's now classic distinction between the laws and the collocation of matter, familiar at least to every reader of Mill's Logic, may serve to illustrate this point. “The tendency of atheistical writers,” says Chalmers, “is to reason exclusively on the laws of matter, and to overlook its dispositions. Could all the beauties and benefits of the astronomical system be referred to the single law of gravitation, it would greatly reduce the strength of the argument for a designing cause.”9 “When Professor Robison felt alarmed by the attempted demonstration of Laplace, that the law of gravitation was an essential property of matter, lest the cause of natural theology should be endangered by it, he might have recollected that the main evidence for a Divinity lies, not in the laws of matter, but in the collocations.”10 “Though we conceded to the atheist the eternity of matter and the essentially inherent character of all its laws, we would still point out to him, in the manifold adjustments of matter, its adjustments of place and figure, and magnitude, the most impressive signatures of a Deity.”11 But what would become of this ‘main evidence for a Divinity’ if the laws of matter themselves explained its collocations? They can never explain them completely, of course. Till a definite configuration is given him the physicist has no problem; but with such data he professes to deduce the motions and redistributions that according to the laws of matter must ensue. So, if science by the help of these laws should trace the course of the universe backwards, it must halt at some configuration or other; and of the configuration at which it halts it can give no account. “The laws of nature,” says Chalmers, “may keep up the working of the machinery—but they did not and could not set up the machine.”12 This final configuration reached by the scientific regress, then—let it be noted—is “the machine.” That—provisionally at all events—science cannot explain; so much is true. But meanwhile two things are noteworthy. First, in innumerable cases, as I have said, what was formerly taken to be part of the machine turns out to be due to the workings of its machinery. Secondly, as a consequence of this, those constructive interventions, which are held “to demonstrate so powerfully the fiat and finger of a God,” become rapidly fewer in number, and recede farther and farther into the deep darkness of the infinite past. It was surely a short-sighted procedure to rest the theistic argument on a view of nature that must inevitably reduce the strength and diminish the impressiveness of that argument at every advance of natural science. When, too, those who adopt such a line of reasoning themselves allow this fatal weakness, as we have seen that Chalmers did, the proceeding becomes almost fatuous. Indeed, it would hardly be an exaggeration to say that the naturalism of to-day is the logical outcome of the natural theology of a century ago. I do not forget a rejoinder on the old lines that one frequently hears now that the theories of Lyall and Darwin and Spencer are supposed to have become established truths—a sort of dernier ressort where direct attacks have failed. After all, it is said, the more a machine can direct itself and repair itself the more wonderful its first construction must have been. To have so created and disposed the primal elements of the world as to insure by the steadfast working of unvarying laws the emergence in due time of all the life and glory of the round ocean and the teeming earth, is not this after all “the most impressive signature of a Deity”? This seems to me very like asking whether, after all, infinity times nothing is not greater than n times m? In other words such an argument points logically either to the machine being nothing and God all, or to God being nothing and the machine everything. But which? That depends where we start: if from God, the machine is throughout dependent; but if from the machinery, we may never reach God at all. For the avowed pantheist, who knows neither secondary laws nor machinery, it is, of course, God that is all.

“That God, which ever lives and loves—

One God, one law, one element,

And one far-off divine event.”

For those, on the other hand, anxious, perhaps, like Chalmers, to keep clear of what he calls ‘the metaphysical obscurity’ concerning the origination of matter and its essential properties, and content to “discern in the mere arrangements of matter the most obvious and decisive signatures of the artist hand which has been employed in it,”13 for such, it is God that vanishes. Logically and actually on their premisses we find in the words of Huxley already quoted “that matter and law have banished spirit and spontaneity.”14

This then is the Laplacean conception that we have first to examine, and if it lead us in the end into ‘metaphysical obscurity,’ let us be warned not to shrink from the task. In the beginning, however, it will rather be certain physical commonplaces that must claim our attention. As to these it behoves me to say at once and emphatically that I make no pretence to special knowledge. But I shall take care to refer to nothing—unless it be generally known—without expressly mentioning my authority.

First of all, it will be remembered that Laplace regarded the universe as composed of a number of beings having assignable positions and movements, and ranging in size from the largest celestial bodies down to the lightest atoms. He assumed that all these, whether masses or molecules, whether of finite or of infinitesimal dimensions, are amenable to the same mechanical laws; and this assumption is still regarded as “the axiom on which all modern physics is founded.”15 None the less there are some striking differences in the methods followed in the two cases, i.e. according as the masses to be dealt with are of sensible or of insensible dimensions. With sensible masses the physicist's procedure is in the main abstract, and any exactness he may attain is attained in this manner. But he at least knows the bodies he is investigating, say the sun or the moon, the bob of a pendulum or the screw of a steamship. In dealing with molecules or atoms, on the other hand, such identification and individualisation is impossible. His procedure here, if I may so say, is predominantly idealistic. Actual perception is replaced by ideal conception. Moreover, the ideal atoms or molecules are often wholly hypothetical, and when not this—as in chemistry, perhaps—are still rather statistical means or averages than actual existences. Further, the exactness which it is known cannot be affirmed of mechanisms of sensible mass, except after manifold abstractions, is assumed, not unfrequently, to apply literally to the hypothetical mechanisms of which atoms and molecules and other ideal conceptions form the working parts. We shall thus have to consider the abstract theory first in itself, next in its application to sensible masses, and lastly in its application to insensible masses.

First, as to the abstract method. A few sentences from a standard text-book will make clear what is meant by this. In Thomson and Tait's Natural Philosophy the division entitled Abstract Dynamics begins as follows:—

“Until we know thoroughly the nature of matter and the forces which produce its motion, it will be utterly impossible to submit to mathematical reasoning the exact conditions of any physical questions.… Take, for instance, the very simple case of a crowbar employed to move a heavy mass. The accurate mathematical investigation of the action would involve the simultaneous treatment of the motions of every part of bar, fulcrum, and mass raised; but our ignorance of the nature of matter and molecular forces, precludes any such complete treatment of the problem.… Hence, the idea of solving, instead of the complete but infinitely transcendent problem, another in reality quite different, but which, while amply simple, obviously leads to practically the same results as the former, so far as concerns… the bodies as a whole.… Imagine the masses involved to be perfectly rigid, that is, incapable of changing form or dimensions. Then the infinite multiplicity of the forces really acting may be left out of consideration.” After some further elucidation the writers conclude: “Enough, however, has been said to show, first, our utter ignorance as to the true and complete solution of any physical question by the only perfect method, that of the consideration of the circumstances which affect the motion of every portion, separately, of each body concerned; and, second, the practically sufficient manner in which practical questions may be attacked by limiting their generality, the limitations introduced being themselves deduced from experience.”

The method above referred to as ‘the only perfect method,’—in which the motions of every particle concerned are taken into account—is obviously the very method that Laplace's imaginary spirit is supposed to apply to the universe. We seem meant to assume that this method is not abstract—a very questionable assumption to which we shall be brought back later. Meanwhile, turning to the confessedly abstract method with which the actual physicist has to content himself, let us note in what respects the simple question he actually solves differs from the concrete and really quite different question that is propounded. This refers to a particular crowbar, a particular fulcrum, and a particular material body to be raised at a particular place and date. Assuming that raising the load at a given place means moving it against the gravitational forces at that place,—though in fact these will not be the only forces concerned,—we shall be told that the answer to the question on this score alone will in general vary for every different place, and even, in general, at every different date. But abstract dynamics knows nothing of places and dates; these are the affair of topography and chronology: it knows only of abstract space, time, and motion, as dealt with by geometry and kinematics. Accurately to ascertain the actual forces existing at any place or time would require precise measurements of a complex kind, and precise measurement in the simplest case is, strictly speaking, an impossibility. Abstract dynamics is a mathematical science and therefore does not measure; there would be an end of all exactness if it did. We should be requested accordingly to state what the weight of the load is, or at any rate what it may be taken to be. For the same reason the lengths of the two arms of the lever must be given, then the power to be applied can be found. Let us next suppose that the lever is made of lead or of lancewood, and that the load consists of dynamite, sheet-glass, or putty. The exponent of abstract mechanics will object again: You are proposing here millions, nay billions, of problems, instead of one. The properties of the lever as a simple machine being in question, we are entitled to replace the material crowbar by a line of equal length fixed at the point answering to the fulcrum, and to regard it as unalterable in form and dimensions. And as to the load, dynamics can deal only with the mass of that; it does not recognise the qualitative differences of material bodies. “In abstract dynamics”—to quote Maxwell—“matter is considered under no other aspect than as that which can have its motion changed by the application of force. Hence any two bodies are of equal mass if equal forces applied to these bodies produce, in equal times, equal changes of velocity. This is the only definition of equal masses which can be admitted in dynamics, and it is applicable to all material bodies, whatever they may be made of.”16

The gulf between this final abstraction of ‘mass’ and the material bodies which it replaces is so great that even the physicists to whom it is due often fail to realise how much they have stripped off. The notion of mass leaves far behind it not merely all the diversities of chemical classification, where iron and carbon, oxygen and chlorine are placed wide apart; not merely the variety of secondary qualities, colour, taste, smell, and the like, whereby sensible objects are commonly described; not merely the physical distinction of solid, liquid, and gaseous states, in one or other, of which all material bodies are found. A mass has no chemical nature, no physical properties, not even a weight. Even its relation to space differs from that of sensible bodies. Matter has often been defined as that which can, or that which must, occupy space. Whatever these definitions may be worth, they cannot at all events be applied directly to mass as just defined. A mass must have position or it could not be moved, but it may be of finite amount and yet have no size, or it may be of any size whatever. It is true that all bodies of sensible dimensions are found to resist compression, or deformation, or both. But these characteristics are due not to mass, but to forces. Moreover, when such changes in the configuration of a body are under investigation, the body is regarded as a system of mass-elements or mass-points, and these either as continuous or discontinuous, as circumstances may determine. Inasmuch, however, as changes of configuration are conceivable in every material body of finite dimensions, the logical implication is that all such bodies consist of mass-points. Thus the question whether matter is discrete, consisting ultimately of atoms, or is continuous and so indefinitely divisible, is not a question that concerns mass. Indeed, not only may a mass of finite volume be divisible as long as that volume itself is divisible; but even if we suppose ourselves to have reached the geometrical point or limit of spatial divisibility, which has neither parts nor magnitude, this puts no limit to the divisibility of mass. As already said, such a geometrical point may be regarded as the seat of a mass that still has both parts and magnitude. “In certain astronomical investigations,” as Maxwell points out, “the planets, and even the sun, may be regarded each as a material particle,”17 or mass-point. Yet these masses require a very great number to express them when our customary units of mass are used. On the other hand, “even an atom, when we consider it as capable of rotation, must be regarded as consisting of many material particles” or mass-points—although its total mass in gravitation measure be less than the billionth part of a gramme.

But all this will become plainer, and the extreme abstractions involved in the notion of mass more apparent, if we recur again to its definition, regarding it this time synthetically rather than analytically. It is possible to describe the motions of points or figures and the composition or resolution of such motions in a purely formal manner, just as in geometry their situations and constructions are formally described. In this way kinematics, as the science of abstract motion, covers all the ground implied in change of position or change of speed in any body or system of bodies, so far as such motion involves only pure or abstract space and time. By abstract space and time, it need hardly be said, is meant, as I have already incidentally remarked, the space and time of mathematics, not the variously filled space and time of our concrete perceptual experience. Kinematics is then in the strictest sense a branch of pure mathematics, and not an empirical science. But we pass, it may be supposed, from the mathematical to the real, when, in place of merely describing motion, we ask what is moved and what are the causes of such actual motion. The categories of substance and cause here seem to come upon the scene, and they surely transcend the range of the purely mathematical. But is mass conceived by abstract mechanics as a thing or substance; or is force conceived as a cause? The answer, I think, must be negative to both questions. But deferring the question as to force, it must be noted that mass is by no means synonymous with matter, though sometimes used as if it were. “We must be careful to remember,” Maxwell tells us, “that what we sometimes, even in abstract dynamics, call matter, is not that unknown substratum of real bodies against which Berkeley directed his arguments, but something as perfectly intelligible as a straight line or a sphere.” “Why, then,” he asks, “should we have any change of method when passing from kinematics to abstract dynamics? Why should we find it more difficult to endow moving figures with mass than to endow stationary figures with motion? The bodies we deal with in abstract dynamics are just as completely known to us as the figures in Euclid. They have no properties whatever except those which we explicitly assign to them.”18 In entire accord with this we have the statement of Professor Tait,—all the more impressive because of his well-known hankering after the metaphysical,—that “we do not know and are probably incapable of discovering what matter is.”19 Matter as substance is, in short, as rigorously excluded from modern physics as mind, as substance, is banished from modern psychology; indeed, matter is not merely excluded but abused as a ‘metaphysical quagmire,’ ‘fetish,’20 and the like.

In dealing with mass, then, we are only dealing with a property; and, since it is a property that varies continuously in quantity, it is one that admits of mathematical treatment. Mass, in short, is but another name for quantity of inertia. By inertia the physicist denotes the fact, or to be strictly accurate I should say the well-grounded inference, that a body, so long as it is left to itself, preserves strictly in respect of motion its status quo. We can perfectly well imagine any number of such bodies of the most various sizes and shapes moving severally in all possible directions, and all at different speeds, that of zero speed or rest being one. Referred to some defined origin and axes, their apparent changes of size, shape, relative position after a given interval, as well as their apparent changes of speed, could all be dealt with by kinematics. Such motions, in accordance with Newton's First Law, might be called, perhaps have been called, free, or independent, or unconstrained motions. But this is not all that kinematics could do. We might arbitrarily assign to any or all the bodies under contemplation any deviations from uniform rectilinear motion or from rest; and the resulting positions after a given interval might still be found as before. Such deviation from uniform rectilinear motion or from rest is, of course, in actual fact the rule; and the kinematical problems of abstract dynamics—if I might so call them—differ from such arbitrary problems only in not being arbitrary. “The new idea appropriate to dynamics (then) is ”—I quote Maxwell—“that the motions of bodies are not independent of each other, but that, under certain conditions, dynamical transactions take place between two bodies whereby the motions of both bodies are affected.”21

Now one of these conditions is that the said transactions between two bodies—as Maxwell picturesquely calls them—are in no ways affected by, and in no ways affect, other dynamical transactions which either or both the bodies may have with other bodies. In a word, the results of all such transactions are additive. All the principles involved may therefore be learnt by considering such a transaction in a single case. Another condition is that such transaction between two bodies takes place along the line joining them; also, that the changes of motion or the accelerations of each body along this line, in which the said transaction or mutual stress consists, are in opposite directions. But how far is each to shunt from its original direction, how much is each to alter its original speed, that is to say, what share in the whole transaction is each to take? The answer to this question gives the meaning of mass. To each body a number is to be assigned, such that the changes of their motion are inversely proportional to these numbers. Such number answers to the mass of the body to which it belongs. Its determination, of course, in any real case involves measurement, and is the business, not of abstract dynamics, but of experimental physics. The actual number again depends on the standard employed, but, once so determined, by dynamical transaction with the standard, it is determined once for all for every other dynamical transaction with other masses numbered according to the same unit. The appropriateness of defining mass as quantity of inertia, i.e. as the measure of that tendency to persistence of the motor status quo which preceded the particular dynamical transaction under investigation, is thus evident. For the greater the mass, the less in any given case the change of motion that ensues; the less the mass the greater the change of motion—kinematically estimated, of course. Thus, if the mass of one of the two bodies is infinite, its kinematic circumstances are unaltered; if the mass of one be zero, that of the other, however small, undergoes no acceleration; where both are equal, the accelerations of both are equal; and so for every other case. So far then from falling under the category of substance, a mass as it occurs in abstract dynamics is but a coefficient affecting the value of the acceleration to which it is affixed. True the phrase “mass of a body” is constantly recurring; but then the body, apart from the mass, is but a moving point or figure.

There still remains the correlative term Force. How, it may be asked, can the bodies of abstract dynamics be conceived as merely geometrical figures moving according to rule, if they are collectively endowed with all the forces of nature: gravitation, light, heat, electricity, chemical attraction, etc.? What are these if they are not the active properties of material bodies? The investigation of the nature of matter or of the properties of real bodies, we shall be told, is entirely the business of experimental physics; abstract dynamics takes account of no properties but those expressed by its definitions. But by definition a body is endowed with no essential properties but mass and mobility. Force, as understood by dynamics, cannot then be an inherent and permanent property of any given body, dynamically considered. On the contrary, no mass, though infinite, has any force by itself. A force in the dynamical sense cannot appear till there are two masses in dynamical relation, and then there will be two equal and opposite forces, let the masses differ as much as they may. A force is but the name for a mass-acceleration, i.e. for either side of the dynamical transaction between two bodies, which we have already considered; and a moment's recurrence to that transaction will make the purely mathematical character of such forces plain. Instead of the moving geometrical point of kinematics, we have in dynamics a mass-point in motion. This mass-motion for a given direction is called momentum; momentum being the product of the number of units of mass into the number of units of speed. It remains, so long as the body is left to itself, a constant quantity. When two masses are said to interact, the momentum of each changes, and the rate of this change for one of the bodies is called the moving force on that body; this again is a quantity, the product, as said, of mass into acceleration. In short, the old qualitative definition of force as “whatever changes or tends to change the motion of a body” is discarded by modern dynamics, which professes to leave the question of the causes of such change entirely aside. Force for it means simply the direction in which, and the rate at which, this change takes place. It answers, says Kirchhoff, in mathematical language to the second differential coefficient of the distance as a function of the time; is, as Tait puts it, no more an objective entity than say five per cent per annum is a sum of money.22

How completely the theory of mechanics has divested itself of the conceptions of substance and cause, in assuming its present strictly mathematical form, is brought home to us by one striking fact; the fact, I mean, that mass and force, in which these categories are supposed to be implied, are but dependent variables in certain general equations. In 7+5=12 or tan 45°=1, we cannot say that one side of these equations is more than the other effect or consequent, that other being the cause or essence whence it proceeds. It would be equally arbitrary to attempt any such distinction when we have the equation mv=Ft, or ms=Ft2/2 or Fs=mv2/2. In these, the fundamental equations of dynamics, we have four quantities so connected, that if any three are known the fourth can be found. In this respect one term is no more real than another, and the dependence is not temporal or causal or teleological, but mathematical simply. The sole use of such equations, it is contended, is “to describe in the exactest and simplest manner such motions as occur in nature.” So Kirchhoff defined the object of mathematical physics in his universally lauded textbook, and his definition has recently been made the motto of a manifesto on the part of Professor Mach. “It is said,” Mach remarks, “description leaves the sense of causality unsatisfied. In fact, many imagine they understand motions better when they picture to, themselves pulling forces, and yet the accelerations, the facts, accomplish more, without superfluous additions. I hope that the science of the future will discard the idea of cause and effect, as being formally obscure; and in my feeling that these ideas contain a strong tincture of fetishism, I am certainly not alone.”23

I am quite aware that the elimination from natural science, of this so-called fetishism, which the categories of substance and cause are supposed to involve, has been gradual.24 But the history of mechanics shews conclusively that there at any rate this process of elimination has been steady, and now at length seems to be complete. The full significance of this deanthropomorphic tendency of science it will be best to defer, along with other epistemological reflections, till we have reached the end of this survey of the cardinal doctrines of modern science, which we have but just commenced. At this stage I will only venture the remark that those who seek to oppose this tendency—as Wundt and still more Sigwart, for example, seem to do—appear rather to mistake the issue. It is not a question of divesting the human mind of its most fundamental conceptions; it is simply a question of method and expediency, the propriety, in a word, of dividing natural science from natural philosophy. No doubt many of those who insist on this separation are privately of opinion, as we have seen, that natural science will make a whole of knowledge by itself. But in so thinking they are only playing the amateur philosopher. Such a declaration is no part of their business as scientific experts. As Mr. Bradley roundly puts it: “When Phenomenalism loses its head and, becoming blatant, steps forward as a theory of first principles, then it is really not respectable. The best that can be said of its pretensions is that they are ridiculous.”25 The sharper the division of labour, the more fragmentary becomes the contribution of each separate worker; but the more perfect also the finished production of their joint organisation. The ‘ragged edges’ of scientific knowledge ought to become more apparent the more strictly scientific they are; and the more defined these ragged edges are, the more effectively can philosophy enter upon the work it aspires to do, of articulating or connecting those sutures, of rounding off and unifying the whole.

No wonder Laplace could dispense with the hypothesis of a Deity, if his celestial mechanics turn out to be so abstract as to exclude the categories of substance and cause. A mathematical formula does not change its essential character by increasing in length and complexity. If the validity of an equation is by its very definition confined to what is mathematical, if it is only tentatively and approximately applicable to what is real, Laplace's world formula must be like the rest. On this question of the relation of abstract dynamics to actual phenomena, I propose to enter in the next lecture.

  • 1.

    History of Philosophy, 3d edition, vol. ii, p. 590.

  • 2.

    First Principles, stereo. ed., § 26, p. 88.

  • 3.

    Ueber die Grenzen des Naturerkennens, 4te Aufl., p. 6.

  • 4.

    Whewell, Bridgewater Treatise, 1847 edition, p. 356.

  • 5.

    Whewell, Bridgewater Treatise, 1847 edition, p. 357.

  • 6.

    Principles of Science, 2nd edition, p. 738.

  • 7.

    Whewell, Bridgewater Treatise, p. 338.

  • 8.

    Jevons, Principles of Science, p. 766.

  • 9.

    Bridgewater Treatise, vol. i, p. 17.

  • 10.

    Bridgewater Treatise, vol. i, p. 20, note.

  • 11.

    Bridgewater Treatise, vol. i, p. 21.

  • 12.

    Bridgewater Treatise, vol. i, p. 27.

  • 13.

    Bridgewater Treatise, vol. i, p. 25.

  • 14.

    Cf. above, Lecture 1.

  • 15.

    J. J. Thomson, Applications of Dynamics, p. 1.

  • 16.

    Matter and Motion, p. 40.

  • 17.

    Matter and Motion, art. vi, p. 11.

  • 18.

    Review of Thomson and Tait's Natural Philosophy, in Nature, vol. xx, p. 214; also Scientific Papers, vol. ii, p. 779.

  • 19.

    Properties of Matter, art. xx.

  • 20.

    Cf. Karl Pearson, Grammar of Science, passim.

  • 21.

    Nature, l.c.

  • 22.

    Cf. Tait on Force, Nature, vol. xvii, p. 459.

  • 23.

    Popular Scientific Lectures, Eng. trans., p. 253.

  • 24.

    Even in the time of Newton forces were regarded as powers inherent in substances. Their effects could be measured, but not the forces themselves. Still earlier the remora or echineis, though but a “little fish,” was credited with the power of stopping a ship by merely adhering to it. Cf. Whewell, History of Inductive Sciences, 3rd edition, vol. i, p. 189.

  • 25.

    Appearance and Reality, p. 126.