Chapter XXXI (1)

REFLECTIONS ON THE RESULTS AND LIMITS OF SCIENTIFIC METHOD.

Before concluding a work on the Principles of Science, it will not be inappropriate to add some remarks upon the limits and ultimate bearings of the knowledge which we may acquire by the employment of scientific method. All science consists, it has several times been stated, in the detection of identities in the action of natural agents. The purpose of inductive inquiry is to ascertain the apparent existence of necessary connection between causes and effects, expressed in the form of natural laws. Now so far as we thus learn the invariable course of nature, the future becomes the necessary sequel of the present, and we are brought beneath the sway of powers with which nothing can interfere.

By degrees it is found, too, that the chemistry of organised substances is not entirely separated from, but is continuous with, that of earth and stones. Life seems to be nothing but a special form of energy which is manifested in heat and electricity and mechanical force. The time may come, it almost seems, when the tender mechanism of the brain will be traced out, and every thought reduced to the expenditure of a determinate weight of nitrogen and phosphorus. No apparent limit exists to the success of scientific method in weighing and measuring, and reducing beneath the sway of law, the phenomena both of matter and of mind. And if mental phenomena be thus capable of treatment by the balance and the micrometer, can we any longer hold that mind is distinct from matter? Must not the same inexorable reign of law which is apparent in the motions of brute matter be extended to the subtle feelings of the human heart? Are not plants and animals, and ultimately man himself, merely crystals, as it were, of a complicated form? If so, our boasted free will becomes a delusion, moral responsibility a fiction, spirit a mere name for the more curious manifestations of material energy. All that happens, whether right or wrong, pleasurable or painful, is but the outcome of the necessary relations of time and space and force.

Materialism seems, then, to be the coming religion, and resignation to the nonentity of human will the only duty. Such may not generally be the reflections of men of science, but I believe that we may thus describe the secret feelings of fear which the constant advance of scientific investigation excites in the minds of many. Is science, then, essentially atheistic and materialistic in its tendency? Does the uniform action of material causes, which we learn with an ever-increasing approximation to certainty, preclude the hypothesis of a benevolent Creator, who has not only designed the existing universe, but who still retains the power to alter its course from time to time?

To enter upon actual theological discussions would be evidently beyond the scope of this work. It is with the scientific method common to all the sciences, and not with any of the separate sciences, that we are concerned. Theology therefore would be at least as much beyond my scope as chemistry or geology. But I believe that grave misapprehensions exist as regards the very nature of scientific method. There are scientific men who assert that the interposition of Providence is impossible, and prayer an absurdity, because the laws of nature are inductively proved to be invariable. Inferences are drawn not so much from particular sciences as from the logical nature of science itself, to negative the impulses and hopes of men. Now I may state that my own studies in logic lead me to call in question such negative inferences. Laws of nature are uniformities observed to exist in the action of certain material agents, but it is logically impossible to show that all other agents must behave as these do. The too exclusive study of particular branches of physical science seems to generate an over-confident and dogmatic spirit. Rejoicing in the success with which a few groups of facts are brought beneath the apparent sway of laws, the investigator hastily assumes that he is close upon the ultimate springs of being. A particle of gelatinous matter is found to obey the ordinary laws of chemistry; yet it moves and lives. The world is therefore asked to believe that chemistry can resolve the mysteries of existence.

*The Meaning of Natural Law.*

Pindar speaks of Law as the Ruler of the Mortals and the Immortals, and it seems to be commonly supposed that the so-called Laws of Nature, in like manner, rule man and his Creator. The course of nature is regarded as being determined by invariable principles of mechanics which have acted since the world began, and will act for evermore. Even if the origin of all things is attributed to an intelligent creative mind, that Being is regarded as having yielded up arbitrary power, and as being subject like a human legislator to the laws which he has himself enacted. Such notions I should describe as superficial and erroneous, being derived, as I think, from false views of the nature of scientific inference, and the degree of certainty of the knowledge which we acquire by inductive investigation.

A law of nature, as I regard the meaning of the expression, is not a uniformity which must be obeyed by all objects, but merely a uniformity which is as a matter of fact obeyed by those objects which have come beneath our observation. There is nothing whatever incompatible with logic in the discovery of objects which should prove exceptions to any law of nature. Perhaps the best established law is that which asserts an invariable correlation to exist between gravity and inertia, so that all gravitating bodies are found to possess inertia, and all bodies possessing inertia are found to gravitate. But it would be no reproach to our scientific method, if something were ultimately discovered to possess gravity without inertia. Strictly defined and correctly interpreted, the law itself would acknowledge the possibility; for with the statement of every law we ought properly to join an estimate of the number of instances in which it has been observed to hold true, and the probability thence calculated, that it will hold true in the next case. Now, as we found (p. 259), no finite number of instances can warrant us in expecting with certainty that the next instance will be of like nature; in the formulas yielded by the inverse method of probabilities a unit always appears to represent the probability that our inference will be mistaken. I demur to the assumption that there is any necessary truth even in such fundamental laws of nature as the Indestructibility of Matter, the Conservation of Energy, or the Laws of Motion. Certain it is that men of science have recognised the conceivability of other laws, and even investigated their mathematical consequences. Airy investigated the mathematical conditions of a perpetual motion (p. 223), and Laplace and Newton discussed imaginary laws of forces inconsistent with those observed to operate in the universe (pp. 642, 706).

The laws of nature, as I venture to regard them, are simply general propositions concerning the correlation of properties which have been observed to hold true of bodies hitherto observed. On the assumption that our experience is of adequate extent, and that no arbitrary interference takes place, we are then able to assign the probability, always less than certainty, that the next object of the same apparent nature will conform to the same laws.

*Infiniteness of the Universe.*

We may safely accept 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 thenceforth and for ever 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. The astronomer can calculate the positions of the heavenly bodies when thousands of generations of men shall have passed away, and in this fact we have some illustration, as Laplace remarks, of the power which scientific prescience may attain. Doubtless, too, all efforts in the investigation of nature tend to bring us nearer to the possession of that ideally perfect power of intelligence. Nevertheless, as Laplace with profound wisdom adds,[603] we must ever remain at an infinite distance from the goal of our aspirations.

[603] *Théorie Analytique des Probabilités*, quoted by Babbage,
*Ninth Bridgewater Treatise*, p. 173.

Let us assume, for a time at least, as a highly probable hypothesis, that whatever is to happen must be the outcome of what is; there then arises the question, What is? Now our knowledge of what exists must ever remain imperfect and fallible in two respects. Firstly, we do not know all the matter that has been created, nor the exact manner in which it has been distributed through space. Secondly, assuming that we had that knowledge, we should still be wanting in a perfect knowledge of the way in which the particles of matter will act upon each other. The power of scientific prediction extends at the most to the limits of the data employed. Every conclusion is purely hypothetical and conditional upon the non-interference of agencies previously undetected. The law of gravity asserts that every body tends to approach towards every other body, with a certain determinate force; but, even supposing the law to hold true, it does not assert that the body *will* approach. No single law of nature can warrant us in making an absolute prediction. We must know all the laws of nature and all the existing agents acting according to those laws before we can say what will happen. To assume, then, that scientific method can take everything within its cold embrace of uniformity, is to imply that the Creator cannot outstrip the intelligence of his creatures, and that the existing Universe is not infinite in extent and complexity, an assumption for which I see no logical basis whatever.

*The Indeterminate Problem of Creation.*

A second and very serious misapprehension concerning the import of a law of nature may now be pointed out. It is not uncommonly supposed that a law determines the character of the results which shall take place, as, for instance, that the law of gravity determines what force of gravity shall act upon a given particle. Surely a little reflection must render it plain that a law by itself determines nothing. It is *law plus agents obeying law which has results*, and it is no function of law to govern or define the number and place of its own agents. Whether a particle of matter shall gravitate, depends not only upon the law of Newton, but also upon the distribution of surrounding particles. The theory of gravitation may perhaps be true throughout all time and in all parts of space, and the Creator may never find occasion to create those possible exceptions to it which I have asserted to be conceivable. Let this be as it may; our science cannot certainly determine the question. Certain it is, that the law of gravity does not alone determine the forces which may be brought to bear at any point of space. The force of gravitation acting upon any particle depends upon the mass, distance, and relative position of all the other particles of matter within the bounds of space at the instant in question. Even assuming that all matter when once distributed through space at the Creation was thenceforth to act in an invariable manner without subsequent interference, yet the actual configuration of matter at any moment, and the consequent results of the law of gravitation, must have been entirely a matter of free choice.

Chalmers has most distinctly pointed out that the existing *collocations* of the material world are as important as the laws which the objects obey. He remarks that a certain class of writers entirely overlook the distinction, and forget that mere laws without collocations would have afforded no security against a turbid and disorderly chaos.[604] Mill has recognised[605] the truth of Chalmers’ statement, without drawing the proper inferences from it. He says[606] of the distribution of matter through space, “We can discover nothing regular in the distribution itself; we can reduce it to no uniformity, to no law.” More lately the Duke of Argyll in his well-known work on the *Reign of Law* has drawn attention to the profound distinction between laws and collocations of causes.

[604] *First Bridgewater Treatise* (1834), pp. 16–24.

[605] *System of Logic*, 5th edit. bk. III. chap. V. § 7; chap. XVI.
§ 3.

[606] *System of Logic*, vol. i. p. 384.

The original conformation of the material universe, as far as we can tell, was free from all restriction. There was unlimited space in which to frame it, and an unlimited number of material particles, each of which could be placed in any one of an infinite number of different positions. It should be added, that each particle might be endowed with any one of an infinite number of quantities of *vis viva* acting in any one of an infinite number of different directions. The problem of Creation was, then, what a mathematician would call *an indeterminate problem*, and it was indeterminate in a great number of ways. Infinitely numerous and various universes might then have been fashioned by the various distribution of the original nebulous matter, although all the particles of matter should obey the law of gravity.

Lucretius tells us how in the original rain of atoms some of these little bodies diverged from the rectilinear direction, and coming into contact with other atoms gave rise to the various combinations of substances which exist. He omitted to tell us whence the atoms came, or by what force some of them were caused to diverge; but surely these omissions involve the whole question. I accept the Lucretian conception of creation when properly supplemented. Every atom which existed in any point of space must have existed there previously, or must have been created there by a previously existing Power. When placed there it must have had a definite mass and a definite energy. Now, as before remarked, an unlimited number of atoms can be placed in unlimited space in an unlimited number of modes of distribution. Out of infinitely infinite choices which were open to the Creator, that one choice must have been made which has yielded the Universe as it now exists.

It would be a mistake, indeed, to suppose that the law of gravity, when it holds true, is no restriction on the distribution of force. That law is a geometrical law, and it would in many cases be mathematically impossible, as far as we can see, that the force of gravity acting on one particle should be small while that on a neighbouring particle is great. We cannot conceive that even Omnipotent Power should make the angles of a triangle greater than two right angles. The primary laws of thought and the fundamental notions of the mathematical sciences do not seem to admit of error or alteration. Into the metaphysical origin and meaning of the apparent necessity attaching to such laws I have not attempted to inquire in this work, and it is not requisite for my present purpose. If the law of gravity were the only law of nature and the Creator had chosen to render all matter obedient to that law, there would doubtless be restrictions upon the effects derivable from any one distribution of matter.

*Hierarchy of Natural Laws.*

A further consideration presents itself. A natural law like that of gravity expresses a certain uniformity in the action of agents submitted to it, and this produces, as we have seen, certain geometrical restrictions upon the effects which those agents may produce. But there are other forces and laws besides gravity. One force may override another, and two laws may each be obeyed and may each disguise the action of the other. In the intimate constitution of matter there may be hidden springs which, while acting in accordance with their own fixed laws, may lead to sudden and unexpected changes. So at least it has been found from time to time in the past, and so there is every reason to believe it will be found in the future. To the ancients it seemed incredible that one lifeless stone could make another leap towards it. A piece of iron while it obeys the magnetic force of the loadstone does not the less obey the law of gravity. A plant gravitates downwards as regards every constituent cell or fibre, and yet it persists in growing upwards. Life is altogether an exception to the simpler phenomena of mineral substances, not in the sense of disproving those laws, but in superadding forces of new and inexplicable character. Doubtless no law of chemistry is broken by the action of the nervous cells, and no law of physics by the pulses of the nervous fibres, but something requires to be added to our sciences in order that we may explain these subtle phenomena.

Now there is absolutely nothing in science or in scientific method to warrant us in assigning a limit to this hierarchy of laws. When in many undoubted cases we find law overriding law, and at certain points in our experience producing unexpected results, we cannot venture to affirm that we have exhausted the strange phenomena which may have been provided for in the original constitution of matter. The Universe might have been so designed that it should go for long intervals through the same round of unvaried existence, and yet that events of exceptional character should be produced from time to time. Babbage showed in that most profound and eloquent work, *The Ninth Bridgewater Treatise*, that it was theoretically possible for human artists to design a machine, consisting of metallic wheels and levers, which should work invariably according to a simple law of action during any finite number of steps, and yet at a fixed moment, however distant, should manifest a single breach of law. Such an engine might go on counting, for instance, the natural numbers until they would reach a number requiring for its expression a hundred million digits. “If every letter in the volume now before the reader’s eyes,” says Babbage,[607] “were changed into a figure, and if all the figures contained in a thousand such volumes were arranged in order, the whole together would yet fall far short of the vast induction the observer would have had in favour of the truth of the law of natural numbers.... Yet shall the engine, true to the prediction of its director, after the lapse of myriads of ages, fulfil its task, and give that one, the first and only exception to that time-sanctioned law. What would have been the chances against the appearance of the excepted case, immediately prior to its occurrence?”

[607] *Ninth Bridgewater Treatise*, p. 140.

As Babbage further showed,[608] a calculating engine, after proceeding through any required number of motions according to a first law, may be made suddenly to suffer a change, so that it shall then commence to calculate according to a wholly new law. After giving the natural numbers for a finite time, it might suddenly begin to give triangular, or square, or cube numbers, and these changes might be conceived theoretically as occurring time after time. Now if such occurrences can be designed and foreseen by a human artist, it is surely within the capacity of the Divine Artist to provide for analogous changes of law in the mechanism of the atom, or the construction of the heavens.

[608] *Ibid.* pp. 34–43.

Physical science, so far as its highest speculations can be trusted, gives some indication of a change of law in the past history of the Universe. According to Sir W. Thomson’s deductions from Fourier’s *Theory of Heat*, we can trace down the dissipation of heat by conduction and radiation to an infinitely distant time when all things will be uniformly cold. But we cannot similarly trace the heat-history of the Universe to an infinite distance in the past. For a certain negative value of the time the formulæ give impossible values, indicating that there was some initial distribution of heat which could not have resulted, according to known laws of nature,[609] from any previous distribution.[610] There are other cases in which a consideration of the dissipation of energy leads to the conception of a limit to the antiquity of the present order of things.[611] Human science, of course, is fallible, and some oversight or erroneous simplification in these theoretical calculations may afterwards be discovered; but as the present state of scientific knowledge is the only ground on which erroneous inferences from the uniformity of nature and the supposed reign of law are founded, I am right in appealing to the present state of science in opposition to these inferences. Now the theory of heat places us in the dilemma either of believing in Creation at an assignable date in the past, or else of supposing that some inexplicable change in the working of natural laws then took place. Physical science gives no countenance to the notion of infinite duration of matter in one continuous course of existence. And if in time past there has been a discontinuity of law, why may there not be a similar event awaiting the world in the future? Infinite ingenuity could have implanted some agency in matter so that it might never yet have made its tremendous powers manifest. We have a very good theory of the conservation of energy, but the foremost physicists do not deny that there may possibly be forms of energy, neither kinetic nor potential, and therefore of unknown nature.[612]

[609] Professor Clifford, in his most interesting lecture on “The
First and Last Catastrophe” (*Fortnightly Review*, April 1875,
p. 480, reprint by the Sunday Lecture Society, p. 24), objects that I
have erroneously substituted “known laws of nature” for “known laws
of conduction of heat.” I quite admit the error, without admitting
all the conclusions which Professor Clifford proceeds to draw; but I
maintain the paragraph unchanged, in order that it may be discussed
in the Preface.

[610] Tait’s *Thermodynamics*, p. 38. *Cambridge Mathematical
Journal*, vol. iii. p. 174.

[611] Clerk Maxwell’s *Theory of Heat*, p. 245.

[612] Maxwell’s *Theory of Heat*, p. 92.

We can imagine reasoning creatures dwelling in a world where the atmosphere was a mixture of oxygen and inflammable gas like the fire-damp of coal-mines. If devoid of fire, they might have lived through long ages unconscious of the tremendous forces which a single spark would call into play. In the twinkling of an eye new laws might come into action, and the poor reasoning creatures, so confident about their knowledge of the reign of law in their world, would have no time to speculate upon the overthrow of all their theories. Can we with our finite knowledge be sure that such an overthrow of our theories is impossible?

*The Ambiguous Expression, “Uniformity of Nature.”*

I have asserted that serious misconception arises from an erroneous interpretation of the expression Uniformity of Nature. Every law of nature is the statement of a certain uniformity observed to exist among phenomena, and since the laws of nature are invariably obeyed, it seems to follow that the course of nature itself is uniform, so that we can safely judge of the future by the present. This inference is supported by some of the results of physical astronomy. Laplace proved that the planetary system is stable, so that no perturbation which planet produces upon planet can become so great as to cause disruption and permanent alteration of the planetary orbits. A full comprehension of the law of gravity shows that all such disturbances are essentially periodic, so that after the lapse of millions of years the planets will return to the same relative positions, and a new cycle of disturbances will then commence.

As other branches of science progress, we seem to gain assurance that no great alteration of the world’s condition is to be expected. Conflict with a comet has long been the cause of fear, but now it is credibly asserted that we have passed through a comet’s tail without the fact being known at the time, or manifested by any more serious a phenomenon than a slight luminosity of the sky. More recently still the earth is said to have touched the comet Biela, and the only result was a beautiful and perfectly harmless display of meteors. A decrease in the heating power of the sun seems to be the next most probable circumstance from which we might fear the extinction of life on the earth. But calculations founded on reasonable physical data show that no appreciable change can be going on, and experimental data to indicate a change are wholly wanting. Geological investigations show indeed that there have been extensive variations of climate in past times; vast glaciers and icebergs have swept over the temperate regions at one time, and tropical vegetation has flourished near the poles at another time. But here again the vicissitudes of climate assume a periodic character, so that the stability of the earth’s condition does not seem to be threatened.

All these statements may be reasonable, but they do not establish the Uniformity of Nature in the sense that extensive alterations or sudden catastrophes are impossible. In the first place, Laplace’s theory of the stability of the planetary system is of an abstract character, as paying regard to nothing but the mutual gravitation of the planetary bodies and the sun. It overlooks several physical causes of change and decay in the system which were not so well known in his day as at present, and it also presupposes the absence of any interruption of the course of things by conflict with foreign astronomical bodies.

It is now acknowledged by astronomers that there are at least two ways in which the *vis viva* of the planets and satellites may suffer loss. The friction of the tides upon the earth produces a small quantity of heat which is radiated into space, and this loss of energy must result in a decrease of the rotational velocity, so that ultimately the terrestrial day will become identical with the year, just as the periods of revolution of the moon upon its axis and around the earth have already become equal. Secondly, there can be little doubt that certain manifestations of electricity upon the earth’s surface depend upon the relative motions of the planets and the sun, which give rise to periods of increased intensity. Such electrical phenomena must result in the production and dissipation of heat, the energy of which must be drawn, partially at least, from the moving bodies. This effect is probably identical (p. 570) with the loss of energy of comets attributed to the so-called resisting medium. But whatever be the theoretical explanation of these phenomena, it is almost certain that there exists a tendency to the dissipation of the energy of the planetary system, which will, in the indefinite course of time, result in the fall of the planets into the sun.

It is hardly probable, however, that the planetary system will be left undisturbed throughout the enormous interval of time required for the dissipation of its energy in this way. Conflict with other bodies is so far from being improbable, that it becomes approximately certain when we take very long intervals of time into account. As regards cometary conflicts, I am by no means satisfied with the negative conclusions drawn from the remarkable display on the evening of the 27th of November, 1872. We may often have passed through the tail of a comet, the light of which is probably an electrical manifestation no more substantial than the aurora borealis. Every remarkable shower of shooting stars may also be considered as proceeding from a cometary body, so that we may be said to have passed through the thinner parts of innumerable comets. But the earth has probably never passed, in times of which we have any record, through the nucleus of a comet, which consists perhaps of a dense swarm of small meteorites. We can only speculate upon the effects which might be produced by such a conflict, but it would probably be a much more serious event than any yet registered in history. The probability of its occurrence, too, cannot be assigned; for though the probability of conflict with any one cometary nucleus is almost infinitesimal, yet the number of comets is immensely great (p. 408).

It is far from impossible, again, that the planetary system may be invaded by bodies of greater mass than comets. The sun seems to be placed in so extensive a portion of empty space that its own proper motion would not bring it to the nearest known star (α Centauri) in less than 139,200 years. But in order to be sure that this interval of undisturbed life is granted to our globe, we must prove that there are no stars moving so as to meet us, and no dark bodies of considerable size flying through intervening space unknown to us. The intrusion of comets into our system, and the fact that many of them have hyperbolic paths, is sufficient to show that the surrounding parts of space are occupied by multitudes of dark bodies of some size. It is quite probable that small suns may have cooled sufficiently to become non-luminous; for even if we discredit the theory that the variation of brightness of periodic stars is due to the revolution of dark companion stars, yet there is in our own globe an unquestionable example of a smaller body which has cooled below the luminous point.

Altogether, then, it is a mere assumption that the uniformity of nature involves the unaltered existence of our own globe. There is no kind of catastrophe which is too great or too sudden to be theoretically consistent with the reign of law. For all that our science can tell, human history may be closed in the next instant of time. The world may be dashed to pieces against a wandering star; it may be involved in a nebulous atmosphere of hydrogen to be exploded a second afterwards; it may be scorched up or dissipated into vapour by some great explosion in the sun; there might even be within the globe itself some secret cause of disruption, which only needs time for its manifestation.

There are some indications, as already noticed (p. 660), that violent disturbances have actually occurred in the history of the solar system. Olbers sought for the minor planets on the supposition that they were fragments of an exploded planet, and he was rewarded with the discovery of some of them. The retrograde motion of the satellites of the more distant planets, the abnormal position of the poles of Uranus and the excessive distance of Neptune, are other indications of some violent event, of which we have no other evidence. I adduce all these facts and arguments, not to show that there is any considerable probability, as far as we can judge, of interruption within the scope of human history, but to prove that the Uniformity of Nature is theoretically consistent with the most unexpected events of which we can form a conception.

*Possible States of the Universe.*

When we give the rein to scientific imagination, it becomes apparent that conflict of body with body must not be regarded as the rare exception, but as the general rule and the inevitable fate of each star system. So far as we can trace out the results of the law of gravitation, and of the dissipation of energy, the universe must be regarded as undergoing gradual condensation into a single cold solid body of gigantic dimensions. Those who so frequently use the expression Uniformity of Nature seem to forget that the Universe might exist consistently with the laws of nature in the most diverse conditions. It might consist, on the one hand, of a glowing nebulous mass of gaseous substances. The heat might be so intense that all elements, even carbon and silicon, would be in the state of gas, and all atoms, of whatever nature, would be flying about in chemical independence, diffusing themselves almost uniformly in the neighbouring parts of space. There would then be no life, unless we can apply that name to the passage through each part of space of similar average trains of atoms, the particular succession of atoms being governed only by the theory of probability, and the law of divergence from a mean exhibited in the Arithmetical Triangle. Such a universe would correspond partially to the Lucretian rain of atoms, and to that nebular hypothesis out of which Laplace proposed philosophically to explain the evolution of the planetary system.

According to another extreme supposition, the intense heat-energy of this nebulous mass might be radiated away into the unknown regions of outer space. The attraction of gravity would exert itself between each two particles, and the energy of motion thence arising would, by incessant conflicts, be resolved into heat and dissipated. Inconceivable ages might be required for the completion of this process, but the dissipation of energy thus proceeding could end only in the production of a cold and motionless universe. The relation of cause and effect, as we see it manifested in life and growth, would degenerate into the constant existence of every particle in a fixed position relative to every other particle. Logical and geometrical resemblances would still exist between atoms, and between groups of atoms crystallised in their appropriate forms for evermore. But time, the great variable, would bring no variation, and as to human hopes and troubles, they would have gone to eternal rest.

Science is not really adequate to proving that such is the inevitable fate of the universe, for we can seldom trust our best-established theories far from their data. Nevertheless, the most probable speculations which we can form as to the history, especially of our own planetary system, is that it originated in a heated revolving nebulous mass of gas, and is in a state of excessively slow progress towards the cold and stony condition. Other speculative hypotheses might doubtless be entertained. Every hypothesis is pressed by difficulties. If the whole universe be cooling, whither does the heat go? If we are to get rid of it entirely, outer space must be infinite in extent, so that it shall never be stopped and reflected back. But not to speak of metaphysical difficulties, if the medium of heat undulations be infinite in extent, why should not the material bodies placed in it be infinite also in number and aggregate mass? It is apparent that we are venturing into speculations which surpass our powers of scientific inference. But then I am arguing negatively; I wish to show that those who speak of the uniformity of nature, and the reign of law, misinterpret the meaning involved in those expressions. Law is not inconsistent with extreme diversity, and, so far as we can read the history of this planetary system, it did probably originate in heated nebulous matter, and man’s history forms but a brief span in its progress towards the cold and stony condition. It is by doubtful and speculative hypotheses alone that we can avoid such a conclusion, and I depart least from undoubted facts and well-established laws when I assert that, whatever uniformities may underlie the phenomena of nature, constant variety and ever-progressing change is the real outcome.

*Speculations on the Reconcentration of Energy.*

There are unequivocal indications, as I have said, that the material universe, as we at present see it, is progressing from some act of creation, or some discontinuity of existence of which the date may be approximately fixed by scientific inference. It is progressing towards a state in which the available energy of matter will be dissipated through infinite surrounding space, and all matter will become cold and lifeless. This constitutes, as it were, the historical period of physical science, that over which our scientific foresight may more or less extend. But in this, as in other cases, we have no right to interpret our experience negatively, so as to infer that because the present state of things began at a particular time, there was no previous existence. It may be that the present period of material existence is but one of an indefinite series of like periods. All that we can see, and feel, and infer, and reason about may be, as it were, but a part of one single pulsation in the existence of the universe.

After Sir W. Thomson had pointed out the preponderating tendency which now seems to exist towards the conversion of all energy into heat-energy, and its equal diffusion by radiation throughout space, the late Professor Rankine put forth a remarkable speculation.[613] He suggested that the ethereal, or, as I have called it, the *adamantine* medium in which all the stars exist, and all radiation takes place, may have bounds, beyond which only empty space exists. All heat undulations reaching this boundary will be totally reflected, according to the theory of undulations, and will be reconcentrated into foci situated in various parts of the medium. Whenever a cold and extinct star happens to pass through one of these foci, it will be instantly ignited and resolved by intense heat into its constituent elements. Discontinuity will occur in the history of that portion of matter, and the star will begin its history afresh with a renewed store of energy.

[613] *Report of the British Association* (1852), Report of Sections,
p. 12.

This is doubtless a mere speculation, practically incapable of verification by observation, and almost free from restrictions afforded by present knowledge. We might attribute various shapes to the adamantine medium, and the consequences would be various. But there is this value in such speculations, that they draw attention to the finiteness of our knowledge. We cannot deny the possible truth of such an hypothesis, nor can we place a limit to the scientific imagination in the framing of other like hypotheses. It is impossible, indeed, to follow out our scientific inferences without falling into speculation. If heat be radiated into outward space, it must either proceed *ad infinitum*, or it must be stopped somewhere. In the latter case we fall upon Rankine’s hypothesis. But if the material universe consist of a finite collection of heated matter situated in a finite portion of an infinite adamantine medium, then either this universe must have existed for a finite time, or else it must have cooled down during the infinity of past time indefinitely near to the absolute zero of temperature. I objected to Lucretius’ argument against the destructibility of matter, that we have no knowledge whatever of the laws according to which it would undergo destruction. But we do know the laws according to which the dissipation of heat appears to proceed, and the conclusion inevitably is that a finite heated material body placed in a perfectly cold infinitely extended medium would in an infinite time sink to zero of temperature. Now our own world is not yet cooled down near to zero, so that physical science seems to place us in the dilemma of admitting either the finiteness of past duration of the world, or else the finiteness of the portion of medium in which we exist. In either case we become involved in metaphysical and mechanical difficulties surpassing our mental powers.

*The Divergent Scope for New Discovery.*

In the writings of some recent philosophers, especially of Auguste Comte, and in some degree John Stuart Mill, there is an erroneous and hurtful tendency to represent our knowledge as assuming an approximately complete character. At least these and many other writers fail to impress upon their readers a truth which cannot be too constantly borne in mind, namely, that the utmost successes which our scientific method can accomplish will not enable us to comprehend more than an infinitesimal fraction of what there doubtless is to comprehend.[614] Professor Tyndall seems to me open to the same charge in a less degree. He remarks[615] that we can probably never bring natural phenomena completely under mathematical laws, because the approach of our sciences towards completeness may be asymptotic, so that however far we may go, there may still remain some facts not subject to scientific explanation. He thus likens the supply of novel phenomena to a convergent series, the earlier and larger terms of which have been successfully disposed of, so that comparatively minor groups of phenomena alone remain for future investigators to occupy themselves upon.

[614] Mr. C. J. Monroe objects that in this statement I do injustice
to Comte, who, he thinks, did impress upon his readers the inadequacy
of our mental powers compared with the vastness of the subject matter
of science. The error of Comte, he holds, was in maintaining that
science had been carried about as far as it is worth while to carry
it, which is a different matter. In either case, Comte’s position is
so untenable that I am content to leave the question undecided.

[615] *Fragments of Science*, p. 362.

On the contrary, as it appears to me, the supply of new and unexplained facts is divergent in extent, so that the more we have explained, the more there is to explain. The further we advance in any generalisation, the more numerous and intricate are the exceptional cases still demanding further treatment. The experiments of Boyle, Mariotte, Dalton, Gay-Lussac, and others, upon the physical properties of gases, might seem to have exhausted that subject by showing that all gases obey the same laws as regards temperature, pressure, and volume. But in reality these laws are only approximately true, and the divergences afford a wide and quite unexhausted field for further generalisation. The recent discoveries of Professor Andrews have summed up some of these exceptional facts under a wider generalisation, but in reality they have opened to us vast new regions of interesting inquiry, and they leave wholly untouched the question why one gas behaves differently from another.

The science of crystallography is that perhaps in which the most precise and general laws have been detected, but it would be untrue to assert that it has lessened the area of future discovery. We can show that each one of the seven or eight hundred forms of calcite is derivable by geometrical modifications from an hexagonal prism; but who has attempted to explain the molecular forces producing these modifications, or the chemical conditions in which they arise? The law of isomorphism is an important generalisation, for it establishes a general resemblance between the forms of crystallisation of natural classes of elements. But if we examine a little more closely we find that these forms are only approximately alike, and the divergence peculiar to each substance is an unexplained exception.

By many similar illustrations it might readily be shown that in whatever direction we extend our investigations and successfully harmonise a few facts, the result is only to raise up a host of other unexplained facts. Can any scientific man venture to state that there is less opening now for new discoveries than there was three centuries ago? Is it not rather true that we have but to open a scientific book and read a page or two, and we shall come to some recorded phenomenon of which no explanation can yet be given? In every such fact there is a possible opening for new discoveries, and it can only be the fault of the investigator’s mind if he can look around him and find no scope for the exercise of his faculties.

*Infinite Incompleteness of the Mathematical Sciences.*

There is one privilege which a certain amount of knowledge should confer; it is that of becoming aware of the weakness of our powers compared with the tasks which they might undertake if stronger. To the poor savage who cannot count twenty the arithmetical accomplishments of the schoolboy are miraculously great. The schoolboy cannot comprehend the vastly greater powers of the student, who has acquired facility in algebraic processes. The student can but look with feelings of surprise and reverence at the powers of a Newton or a Laplace. But the question at once suggests itself, Do the powers of the highest human intellect bear a finite ratio to the things which are to be understood and calculated? How many further steps must we take in the rise of mental ability and the extension of mathematical methods before we begin to exhaust the knowable?

I am inclined to find fault with mathematical writers because they often exult in what they can accomplish, and omit to point out that what they do is but an infinitely small part of what might be done. They exhibit a general inclination, with few exceptions, not to do so much as mention the existence of problems of an impracticable character. This may be excusable as far as the immediate practical result of their researches is in question, but the custom has the effect of misleading the general public into the fallacious notion that mathematics is a *perfect* science, which accomplishes what it undertakes in a complete manner. On the contrary, it may be said that if a mathematical problem were selected by chance out of the whole number which might be proposed, the probability is infinitely slight that a human mathematician could solve it. Just as the numbers we can count are nothing compared with the numbers which might exist, so the accomplishments of a Laplace or a Lagrange are, as it were, the little corner of the multiplication-table, which has really an infinite extent.

I have pointed out that the rude character of our observations prevents us from being aware of the greater number of effects and actions in nature. It must be added that, if we perceive them, we should usually be incapable of including them in our theories from want of mathematical power. Some persons may be surprised that though nearly two centuries have elapsed since the time of Newton’s discoveries, we have yet no general theory of molecular action. Some approximations have been made towards such a theory. Joule and Clausius have measured the velocity of gaseous atoms, or even determined the average distance between the collisions of atom and atom. Thomson has approximated to the number of atoms in a given bulk of substance. Rankine has formed some reasonable hypotheses as to the actual constitution of atoms. It would be a mistake to suppose that these ingenious results of theory and experiment form any appreciable approach to a complete solution of molecular motions. There is every reason to believe, judging from the spectra of the elements, their atomic weights and other data, that chemical atoms are very complicated structures. An atom of pure iron is probably a far more complicated system than that of the planets and their satellites. A compound atom may perhaps be compared with a stellar system, each star a minor system in itself. The smallest particle of solid substance will consist of a great number of such stellar systems united in regular order, each bounded by the other, communicating with it in some manner yet wholly incomprehensible. What are our mathematical powers in comparison with this problem?

After two centuries of continuous labour, the most gifted men have succeeded in calculating the mutual effects of three bodies each upon the other, under the simple hypothesis of the law of gravity. Concerning these calculations we must further remember that they are purely approximate, and that the methods would not apply where four or more bodies are acting, and all produce considerable effects upon each other. There is reason to believe that each constituent of a chemical atom goes through an orbit in the millionth part of the twinkling of an eye. In each revolution it is successively or simultaneously under the influence of many other constituents, or possibly comes into collision with them. It is no exaggeration to say that mathematicians have the least notion of the way in which they could successfully attack so difficult a problem of forces and motions. As Herschel has remarked,[616] each of these particles is for ever solving differential equations, which, if written out in full, might belt the earth.

[616] *Familiar Lectures on Scientific Subjects*, p. 458.

Some of the most extensive calculations ever made were those required for the reduction of the measurements executed in the course of the Trigonometrical Survey of Great Britain. The calculations arising out of the principal triangulation occupied twenty calculators during three or four years, in the course of which the computers had to solve simultaneous equations involving seventy-seven unknown quantities. The reduction of the levellings required the solution of a system of ninety-one equations. But these vast calculations present no approach whatever to what would be requisite for the complete treatment of any one physical problem. The motion of glaciers is supposed to be moderately well understood in the present day. A glacier is a viscid, slowly yielding mass, neither absolutely solid nor absolutely rigid, but it is expressly remarked by Forbes,[617] that not even an approximate solution of the mathematical conditions of such a moving mass can yet be possible. “Every one knows,” he says, “that such problems are beyond the compass of exact mathematics;” but though mathematicians may know this, they do not often enough impress that knowledge on other people.

[617] *Philosophical Magazine*, 3rd Series, vol. xxvi. p. 406.

The problems which are solved in our mathematical books consist of a small selection of those which happen from peculiar conditions to be solvable. But the very simplest problem in appearance will often give rise to impracticable calculations. Mr. Todhunter[618] seems to blame Condorcet, because in one of his memoirs he mentions a problem to solve which would require a great and impracticable number of successive integrations. Now, if our mathematical sciences are to cope with the problems which await solution, we must be prepared to effect an unlimited number of successive integrations; yet at present, and almost beyond doubt for ever, the probability that an integration taken haphazard will come within our powers is exceedingly small.

[618] *History of the Theory of Probability*, p. 398.

In some passages of that remarkable work, the *Ninth Bridgewater Treatise* (pp. 113–115), Babbage has pointed out that if we had power to follow and detect the minutest effects of any disturbance, each particle of existing matter would furnish a register of all that has happened. “The track of every canoe--of every vessel that has yet disturbed the surface of the ocean, whether impelled by manual force or elemental power, remains for ever registered in the future movement of all succeeding particles which may occupy its place. The furrow which it left is, indeed, instantly filled up by the closing waters; but they draw after them other and larger portions of the surrounding element, and these again, once moved, communicate motion to others in endless succession.” We may even say that “The air itself is one vast library, on whose pages are for ever written all that man has ever said or even whispered. There, in their mutable but unerring characters, mixed with the earliest as well as the latest sighs of mortality, stand for ever recorded, vows unredeemed, promises unfulfilled, perpetuating in the united movements of each particle the testimony of man’s changeful will.”

When we read reflections such as these, we may congratulate ourselves that we have been endowed with minds which, rightly employed, can form some estimate of their incapacity to trace out and account for all that proceeds in the simpler actions of material nature. It ought to be added that, wonderful as is the extent of physical phenomena open to our investigation, intellectual phenomena are yet vastly more extensive. Of this I might present one satisfactory proof were space available by pointing out that the mathematical functions employed in the calculations of physical science form an infinitely small fraction of the functions which might be invented. Common trigonometry consists of a great series of useful formulæ, all of which arise out of the relation of the sine and cosine expressed in one equation, sin ^{2}*x* + cos ^{2}*x* = 1. But this is not the only trigonometry which may exist; mathematicians also recognise hyperbolic trigonometry, of which the fundamental equation is cos ^{2}*x* - sin ^{2}*x* = 1. De Morgan has pointed out that the symbols of ordinary algebra form but three of an interminable series of conceivable systems.[619] As the logarithmic operation is to addition or addition to multiplication, so is the latter to a higher operation, and so on without limit.

[619] *Trigonometry and Double Algebra*, chap. ix.