A System of Logic, Ratiocinative and Inductive · John Stuart Mill
Chapter XIV.
Of The Limits To The Explanation Of Laws Of Nature; And Of Hypotheses. § 1. The preceding considerations have led us to recognize a distinction between two kinds of laws, or observed uniformities in nature: ultimate laws, and what may be termed derivative laws. Derivative laws are such as are deducible from, and may, in any of the modes which we have pointed out, be resolved into, other and more general ones. Ultimate laws are those which can not. We are not sure that any of the uniformities with which we are yet acquainted are ultimate laws; but we know that there must be ultimate laws; and that every resolution of a derivative law into more general laws brings us nearer to them.
Since we are continually discovering that uniformities, not previously known to be other than ultimate, are derivative, and resolvable into more general laws; since (in other words) we are continually discovering the explanation of some sequence which was previously known only as a fact; it becomes an interesting question whether there are any necessary limits to this philosophical operation, or whether it may proceed until all the uniform sequences in nature are resolved into some one universal law. For this seems, at first sight, to be the ultimatum toward which the progress of induction by the Deductive Method, resting on a basis of observation and experiment, is tending. Projects of this kind were universal in the infancy of philosophy; any speculations which held out a less brilliant prospect being in these early times deemed not worth pursuing. And the idea receives so much apparent countenance from the nature of the most remarkable achievements of modern science, that speculators are even now frequently appearing, who profess either to have solved the problem, or to suggest modes in which it may one day be solved. Even where pretensions of this magnitude are not made, the character of the solutions which are given or sought of particular classes of phenomena, often involves such conceptions of what constitutes explanation, as would render the notion of explaining all phenomena whatever by means of some one cause or law, perfectly admissible.
The ideal limit, therefore, of the explanation of natural phenomena (toward which as toward other ideal limits we are constantly tending, without the prospect of ever completely attaining it) would be to show that each distinguishable variety of our sensations, or other states of consciousness, has only one sort of cause; that, for example, whenever we perceive a white color, there is some one condition or set of conditions which is always present, and the presence of which always produces in us that sensation. As long as there are several known modes of production of a phenomenon (several different substances, for instance, which have the property of whiteness, and between which we can not trace any other resemblance) so long it is not impossible that one of these modes of production may be resolved into another, or that all of them may be resolved into some more general mode of production not hitherto recognized. But when the modes of production are reduced to one, we can not, in point of simplification, go any further. This one may not, after all, be the ultimate mode; there may be other links to be discovered between the supposed cause and the effect; but we can only further resolve the known law, by introducing some other law hitherto unknown, which will not diminish the number of ultimate laws.
In what cases, accordingly, has science been most successful in explaining phenomena, by resolving their complex laws into laws of greater simplicity and generality? Hitherto chiefly in cases of the propagation of various phenomena through space; and, first and principally, the most extensive and important of all facts of that description, mechanical motion. Now this is exactly what might be expected from the principles here laid down. Not only is motion one of the most universal of all phenomena, it is also (as might be expected from that circumstance) one of those which, apparently at least, are produced in the greatest number of ways; but the phenomenon itself is always, to our sensations, the same in every respect but degree. Differences of duration or of velocity, are evidently differences in degree only; and differences of direction in space, which alone has any semblance of being a distinction in kind, entirely disappear (so far as our sensations are concerned) by a change in our own position; indeed, the very same motion appears to us, according to our position, to take place in every variety of direction, and motions in every different direction to take place in the same.
And again, motion in a straight line and in a curve are no otherwise distinct than that the one is motion continuing in the same direction, the other is motion which at each instant changes its direction. There is, therefore, according to the principles I have stated, no absurdity in supposing that all motion may be produced in one and the same way, by the same kind of cause. Accordingly, the greatest achievements in physical science have consisted in resolving one observed law of the production of motion into the laws of other known modes of production, or the laws of several such modes into one more general mode; as when the fall of bodies to the earth, and the motions of the planets, were brought under the one law of the mutual attraction of all particles of matter; when the motions said to be produced by magnetism were shown to be produced by electricity; when the motions of fluids in a lateral direction, or even contrary to the direction of gravity, were shown to be produced by gravity; and the like. There is an abundance of distinct causes of motion still unresolved into one another: gravitation, heat, electricity, chemical action, nervous action, and so forth; but whether the efforts of the present generation of savants to resolve all these different modes of production into one are ultimately successful or not, the attempt so to resolve them is perfectly legitimate. For, though these various causes produce, in other respects, sensations intrinsically different, and are not, therefore, capable of being resolved into one another, yet, in so far as they all produce motion, it is quite possible that the immediate antecedent of the motion may in all these different cases be the same; nor is it impossible that these various agencies themselves may, as the new doctrines assert, all of them have for their own immediate antecedent modes of molecular motion.
We need not extend our illustration to other cases, as, for instance, to the propagation of light, sound, heat, electricity, etc., through space, or any of the other phenomena which have been found susceptible of explanation by the resolution of their observed laws into more general laws. Enough has been said to display the difference between the kind of explanation and resolution of laws which is chimerical, and that of which the accomplishment is the great aim of science; and to show into what sort of elements the resolution must be effected, if at all.(159) § 3. As, however, there is scarcely any one of the principles of a true method of philosophizing which does not require to be guarded against errors on both sides, I must enter a caveat against another misapprehension, of a kind directly contrary to the preceding. M. Comte, among other occasions on which he has condemned, with some asperity, any attempt to explain phenomena which are “evidently primordial” (meaning, apparently, no more than that every peculiar phenomenon must have at least one peculiar and therefore inexplicable law), has spoken of the attempt to furnish any explanation of the color belonging to each substance, “la couleur élémentaire propre à chaque substance,” as essentially illusory. “No one,” says he, “in our time attempts to explain the particular specific gravity of each substance or of each structure. Why should it be otherwise as to the specific color, the notion of which is undoubtedly no less primordial?”(160)
Now although, as he elsewhere observes, a color must always remain a different thing from a weight or a sound, varieties of color might nevertheless follow, or correspond to, given varieties of weight, or sound, or some other phenomenon as different as these are from color itself. It is one question what a thing is, and another what it depends on; and though to ascertain the conditions of an elementary phenomenon is not to obtain any new insight into the nature of the phenomenon itself, that is no reason against attempting to discover the conditions. The interdict against endeavoring to reduce distinctions of color to any common principle, would have held equally good against a like attempt on the subject of distinctions of sound; which nevertheless have been found to be immediately preceded and caused by distinguishable varieties in the vibrations of elastic bodies; though a sound, no doubt, is quite as different as a color is from any motion of particles, vibratory or otherwise. We might add, that, in the case of colors, there are strong positive indications that they are not ultimate properties of the different kinds of substances, but depend on conditions capable of being superinduced upon all substances; since there is no substance which can not, according to the kind of light thrown upon it, be made to assume almost any color; and since almost every change in the mode of aggregation of the particles of the same substance is attended with alterations in its color, and in its optical properties generally.
The really weak point in the attempts which have been made to account for colors by the vibrations of a fluid, is not that the attempt itself is unphilosophical, but that the existence of the fluid, and the fact of its vibratory motion, are not proved, but are assumed, on no other ground than the facility they are supposed to afford of explaining the phenomena. And this consideration leads to the important question of the proper use of scientific hypotheses, the connection of which with the subject of the explanation of the phenomena of nature, and of the necessary limits to that explanation, need not be pointed out. § 4. An hypothesis is any supposition which we make (either without actual evidence, or on evidence avowedly insufficient) in order to endeavor to deduce from it conclusions in accordance with facts which are known to be real; under the idea that if the conclusions to which the hypothesis leads are known truths, the hypothesis itself either must be, or at least is likely to be, true. If the hypothesis relates to the cause or mode of production of a phenomenon, it will serve, if admitted, to explain such facts as are found capable of being deduced from it. And this explanation is the purpose of many, if not most hypotheses. Since explaining, in the scientific sense, means resolving a uniformity which is not a law of causation, into the laws of causation from which it results, or a complex law of causation into simpler and more general ones from which it is capable of being deductively inferred, if there do not exist any known laws which fulfill this requirement, we may feign or imagine some which would fulfill it; and this is making an hypothesis.
An hypothesis being a mere supposition, there are no other limits to hypotheses than those of the human imagination; we may, if we please, imagine, by way of accounting for an effect, some cause of a kind utterly unknown, and acting according to a law altogether fictitious. But as hypotheses of this sort would not have any of the plausibility belonging to those which ally themselves by analogy with known laws of nature, and besides would not supply the want which arbitrary hypotheses are generally invented to satisfy, by enabling the imagination to represent to itself an obscure phenomenon in a familiar light, there is probably no hypothesis in the history of science in which both the agent itself and the law of its operation were fictitious. Either the phenomenon assigned as the cause is real, but the law according to which it acts merely supposed; or the cause is fictitious, but is supposed to produce its effects according to laws similar to those of some known class of phenomena.
An instance of the first kind is afforded by the different suppositions made respecting the law of the planetary central force, anterior to the discovery of the true law, that the force varies as the inverse square of the distance; which also suggested itself to Newton, in the first instance, as an hypothesis, and was verified by proving that it led deductively to Kepler’s laws. Hypotheses of the second kind are such as the vortices of Descartes, which were fictitious, but were supposed to obey the known laws of rotatory motion; or the two rival hypotheses respecting the nature of light, the one ascribing the phenomena to a fluid emitted from all luminous bodies, the other (now generally received) attributing them to vibratory motions among the particles of an ether pervading all space. Of the existence of either fluid there is no evidence, save the explanation they are calculated to afford of some of the phenomena; but they are supposed to produce their effects according to known laws: the ordinary laws of continued locomotion in the one case, and in the other those of the propagation of undulatory movements among the particles of an elastic fluid.
According to the foregoing remarks, hypotheses are invented to enable the Deductive Method to be earlier applied to phenomena. But(161) in order to discover the cause of any phenomenon by the Deductive Method, the process must consist of three parts: induction, ratiocination, and verification. Induction (the place of which, however, may be supplied by a prior deduction), to ascertain the laws of the causes; ratiocination, to compute from those laws how the causes will operate in the particular combination known to exist in the case in hand; verification, by comparing this calculated effect with the actual phenomenon. No one of these three parts of the process can be dispensed with. In the deduction which proves the identity of gravity with the central force of the solar system, all the three are found. First, it is proved from the moon’s motions, that the earth attracts her with a force varying as the inverse square of the distance. This (though partly dependent on prior deductions) corresponds to the first, or purely inductive, step: the ascertainment of the law of the cause.
Secondly, from this law, and from the knowledge previously obtained of the moon’s mean distance from the earth, and of the actual amount of her deflection from the tangent, it is ascertained with what rapidity the earth’s attraction would cause the moon to fall, if she were no further off, and no more acted upon by extraneous forces, than terrestrial bodies are: that is the second step, the ratiocination. Finally, this calculated velocity being compared with the observed velocity with which all heavy bodies fall, by mere gravity, toward the surface of the earth (sixteen feet in the first second, forty-eight in the second, and so forth, in the ratio of the odd numbers, 1, 3, 5, etc.), the two quantities are found to agree. The order in which the steps are here presented was not that of their discovery; but it is their correct logical order, as portions of the proof that the same attraction of the earth which causes the moon’s motion causes also the fall of heavy bodies to the earth: a proof which is thus complete in all its parts. Now, the Hypothetical Method suppresses the first of the three steps, the induction to ascertain the law; and contents itself with the other two operations, ratiocination and verification; the law which is reasoned from being assumed instead of proved.
This process may evidently be legitimate on one supposition, namely, if the nature of the case be such that the final step, the verification, shall amount to, and fulfill the conditions of, a complete induction. We want to be assured that the law we have hypothetically assumed is a true one; and its leading deductively to true results will afford this assurance, provided the case be such that a false law can not lead to a true result; provided no law, except the very one which we have assumed, can lead deductively to the same conclusions which that leads to. And this proviso is often realized. For example, in the very complete specimen of deduction which we just cited, the original major premise of the ratiocination, the law of the attractive force, was ascertained in this mode; by this legitimate employment of the Hypothetical Method. Newton began by an assumption that the force which at each instant deflects a planet from its rectilineal course, and makes it describe a curve round the sun, is a force tending directly toward the sun. He then proved that if this be so, the planet will describe, as we know by Kepler’s first law that it does describe, equal areas in equal times; and, lastly, he proved that if the force acted in any other direction whatever, the planet would not describe equal areas in equal times. It being thus shown that no other hypothesis would accord with the facts, the assumption was proved; the hypothesis became an inductive truth. Not only did Newton ascertain by this hypothetical process the direction of the deflecting force; he proceeded in exactly the same manner to ascertain the law of variation of the quantity of that force. He assumed that the force varied inversely as the square of the distance; showed that from this assumption the remaining two of Kepler’s laws might be deduced; and, finally, that any other law of variation would give results inconsistent with those laws, and inconsistent, therefore, with the real motions of the planets, of which Kepler’s laws were known to be a correct expression.
I have said that in this case the verification fulfills the conditions of an induction; but an induction of what sort? On examination we find that it conforms to the canon of the Method of Difference. It affords the two instances, A B C, _a b c_, and B C, _b c_. A represents central force; A B C, the planets _plus_ a central force; B C, the planets apart from a central force. The planets with a central force give _a_, areas proportional to the times; the planets without a central force give _b c_ (a set of motions) without _a_, or with something else instead of _a_. This is the Method of Difference in all its strictness. It is true, the two instances which the method requires are obtained in this case, not by experiment, but by a prior deduction. But that is of no consequence. It is immaterial what is the nature of the evidence from which we derive the assurance that A B C will produce _a b c_, and B C only _b c_; it is enough that we have that assurance. In the present case, a process of reasoning furnished Newton with the very instances which, if the nature of the case had admitted of it, he would have sought by experiment.
It is thus perfectly possible, and indeed is a very common occurrence, that what was an hypothesis at the beginning of the inquiry becomes a proved law of nature before its close. But in order that this should happen, we must be able, either by deduction or experiment, to obtain _both_ the instances which the Method of Difference requires. That we are able from the hypothesis to deduce the known facts, gives only the affirmative instance, A B C, _a b c_. It is equally necessary that we should be able to obtain, as Newton did, the negative instance B C, _b c_; by showing that no antecedent, except the one assumed in the hypothesis, would in conjunction with B C produce _a_.
Now it appears to me that this assurance can not be obtained, when the cause assumed in the hypothesis is an unknown cause imagined solely to account for _a_. When we are only seeking to determine the precise law of a cause already ascertained, or to distinguish the particular agent which is in fact the cause, among several agents of the same kind, one or other of which it is already known to be, we may then obtain the negative instance. An inquiry which of the bodies of the solar system causes by its attraction some particular irregularity in the orbit or periodic time of some satellite or comet, would be a case of the second description. Newton’s was a case of the first. If it had not been previously known that the planets were hindered from moving in straight lines by some force tending toward the interior of their orbit, though the exact direction was doubtful; or if it had not been known that the force increased in some proportion or other as the distance diminished, and diminished as it increased, Newton’s argument would not have proved his conclusion. These facts, however, being already certain, the range of admissible suppositions was limited to the various possible directions of a line, and the various possible numerical relations between the variations of the distance, and the variations of the attractive force. Now among these it was easily shown that different suppositions could not lead to identical consequences.
Accordingly, Newton could not have performed his second great scientific operation: that of identifying terrestrial gravity with the central force of the solar system by the same hypothetical method. When the law of the moon’s attraction had been proved from the data of the moon itself, then, on finding the same law to accord with the phenomena of terrestrial gravity, he was warranted in adopting it as the law of those phenomena likewise; but it would not have been allowable for him, without any lunar data, to assume that the moon was attracted toward the earth with a force as the inverse square of the distance, merely because that ratio would enable him to account for terrestrial gravity; for it would have been impossible for him to prove that the observed law of the fall of heavy bodies to the earth could not result from any force, save one extending to the moon, and proportional to the inverse square.
It appears, then, to be a condition of the most genuinely scientific hypothesis, that it be not destined always to remain an hypothesis, but be of such a nature as to be either proved or disproved by comparison with observed facts. This condition is fulfilled when the effect is already known to depend on the very cause supposed, and the hypothesis relates only to the precise mode of dependence; the law of the variation of the effect according to the variations in the quantity or in the relations of the cause. With these may be classed the hypotheses which do not make any supposition with regard to causation, but only with regard to the law of correspondence between facts which accompany each other in their variations, though there may be no relation of cause and effect between them. Such were the different false hypotheses which Kepler made respecting the law of the refraction of light. It was known that the direction of the line of refraction varied with every variation in the direction of the line of incidence, but it was not known how; that is, what changes of the one corresponded to the different changes of the other. In this case any law different from the true one must have led to false results. And, lastly, we must add to these all hypothetical modes of merely representing or _describing_ phenomena; such as the hypothesis of the ancient astronomers that the heavenly bodies moved in circles; the various hypotheses of eccentrics, deferents, and epicycles, which were added to that original hypothesis; the nineteen false hypotheses which Kepler made and abandoned respecting the form of the planetary orbits; and even the doctrine in which he finally rested, that those orbits are ellipses, which was but an hypothesis like the rest until verified by facts.
Ideas discussed in this chapter
Text from Project Gutenberg, public domain in the United States.