Ideal Construction In Science
58. No priestly dogmas," says Hume, " invented on purpose to tame and subdue the rebellious reason of mankind, ever shocked common-sense more than the doctrine of the infinite divisibility of extension with its consequences, as they are pompously displayed by all geometricians and metaphysicians with a kind of triumph and exultation. A real quantity infinitely less than any finite quantity containing quantities infinitely less than itself, and so on in infinitum; this is an edifice so bold and prodigious that it is too weighty for any pretended demonstration to support because it shocks the clearest and most natural principle of human reason. But what renders the matter more extraordinary . is that these seemingly absurd opinions are supported by a chain of reasoning the clearest and most natural; nor is it possible to allow the premises without admitting the consequences."
59. This is an echo of the arguments put forward by Berkeley in his famous Analyst, wherein he endeavors to justify the incomprehensible dogmas of Theology by arraigning the not less incomprehensible dogmas accepted in Mathematics. Hume does not intend it simply as a retort, but as an argument to support the sceptical position that Reason is incompetent to solve her own doubts, and that these doubts, which cannot be answered by Philosophy, are nevertheless suppressed by Action. The fallacy of this argument. will appear when we see that the absurdities and incomprehensibilities with which Mathematics are arraigned do not exist. It is true that many fictions and conventions are introduced, but they' are never made to take the place of realities ; zero is made a quantity, and a curve a straight line, without any misgiving ; imaginary quantities and impossible quantities are freely employed; but the question is not whether conventions are made which deviate from common-sense, the question is, What are the uses to which these conventions are applied ? Every one admits that the language of Mathematicians is often contradictory and ambiguous ; we must also admit that their conceptions are sometimes wanting in the precision which would enable a logical justification to be given for operations which practice justifies. We need a Philosophy of Mathematics to show that an impossible quantity is, a possible operation on quantity ; and that infinity, which is indeed inconceivable as a magnitude, all magnitudes having fixed limits, is perfectly intelligible as a variable limit, dependent on pur will and pleasure. Literally interpreted, nothing can be less conceivable than infinitesimals which are sometimes treated as if they were real quantities, at other times as zeros, "and seem by their equivocal properties to be something between existence and nothing." Mathematically interpreted, however, they are operations on quantities, which may be made as small as we please without thereby altering the quantities of which we seek the relations.` They are instruments of construction, not elements of construction : hypotheses, not factors. In interpreting an algebraic operation it is the result, and not the operation, which fixes attention ; as in the construction of a palace it is the building itself, and not the scaffolding, which has to be judged.
From this point of view a logical justification may be reached for all the seemingly absurd artifices employed by mathematicians. If the theologian were to imitate the practice of the mathematician, and eliminate from his results all that was arbitrary and fictitious in his operations, not allowing his incomprehensible data to enter into the final equations, not allowing what was assumed in the premises to be more than an assumption in the conclusion, then indeed Berkeley's argument would be irrefragable.
60. But Hume is more directly concerned with the incompetence of Reason. Reason, no less than Perception and Intuition, is liable to error. The errors of each may be rectified through similar tests. When Perception errs, — when there is what men call an error of Sense,
how is it rectified ? Lafontaine charmingly says : —
" Quand l'eau courbe un baston ma raison le redresse, La raison décide en maîtresse, Mes yeux moyennant ce secours Ne me trompent jamais en men mentant toujours."
And Kant declares the vulgar objection against the veracity of the senses to be the foolishest that can be urged; "not because the senses always judge correctly, but because they never judge at all." f The stick which appears bent when seen in the water does not give a false impression of the stick in the water; but the inference that this stick will present this aspect when out of the water is precipitate and false. The falsity is shown by ' reducing Inference to Sensation, — by removing the stick from the water, and then looking at it. This once done, the memory of it enables Reason on any future occasion to redress, not the error of Sense but the error of Inference.
Intuition, when it errs, may be corrected in the same way. It is mental vision, and is as liable to error as optical vision. Reasoning is also Inference, mental vision, and is corrected by reducing judgments to their sensible elements.
61. Hume did not clearly understand that Science is essentially an ideal construction very far removed from a real transcript of facts. Its most absolute conclusions are formed from abstractions expressing modes of existence which never were, and never could be, real; and are very often at variance with sensible Experience. It not only deals with data that are extra-sensible, but with data avowedly fictitious: The point, the line, and the circle are abstractions ; they are elements of ideal, not of sensible space. Nevertheless out of these abstractions a science is constructed which is rigorously exact in itself, and is found to harmonize with that very Experience which it appears to contradict. In the presence of such a fact the question may well arise : How can such abstractions have a positive value, an objective validity, yet metempirical abstractions be rejected ? or, to put it in Berkeley's way, Why do you trust the mathematician, and distrust the theologian ? The answer to this question must be postponed till we have examined more closely the procedure of Science, and recognized its essentially ideal construction out of real experience.
62. I say ideal construction, and emphasize it, with the intention of meeting the vulgar objection, iterated from all sides, against the Experiential Method, whose followers are said "to believe only in what they can see and touch" ; whereas the truth is that Science mounts on the wings of Imagination into regions of the Invisible and Impalpable, peopling these regions with Fictions more remote from fact than the fantasies of the Arabian Nights are from the daily occurrences in Oxford Street. The fictions of the thinker differ from the fictions of the poet in not being wayward caprices ; they are constructed in obedience to rigorous canons, and moulded by the pressures of Reality ; two conditions absent in the fictions both of Fairyland and of Metempirics. It is worthy of remark that the two regions of indisputable certainty are the extremes of the mental world, — Sensation and Abstraction. There is no doubt possible in Sensation, whatever doubt may hover round Inference from it. There is no doubt possible in Abstraction, what-ever doubt may hover round its concrete reality. The intermediate region of Inference is the sphere of doubt.
63. Now inferences are hypotheses ; and these become less and less doubtful in the exact proportion of their reduction to Intuition or Sensation. We shall presently see the part played by Hypothesis ; here let us be content with the significant fact that Science is so truly ideal, without pretence of reflecting real existence; that it avowedly relies on data known not to be true, except within its own sphere of Abstraction.*
Note, however this essential point : the abstraction must not have been arbitrarily formed if it is to be subsequently applied to reality : it must have been formed from concretes (by the substitution of ideal limits for sensibles) ; and this condition having been fulfilled, the sensible concretes which are its elements not having been capriciously introduced, we may be certain that the abstraction is as true in its sphere, as the sensibles were in theirs. A ratio once abstracted from numbers is as true as the numbers from which it was abstracted. In this way an abstraction becomes a truth of Nature, though departing from the phenomena of Nature by its disregard of details.
64. The first law of Motion is an absolute truth. But the supposition that any real body will pursue an uniform movement in a straight line, is flagrantly at variance with all observation, and with what is even physically possible. No such phenomenon was ever seen. No such phenomenon could present itself in an universe like ours, where Motion is always accelerated or retarded, and always more or less divergent from a straight line. The ideal law is absolute within ideal space : it is the identical proposition that no change in velocity or direction can occur unless the factors of such a change are operant. But within real space the requisite conditions are unrealizable : the presence of other bodies in movement must always obstruct the realization of the conditions : the factors of a change are always present.
65. There is a real law of Motion, one to which all movements conform without variation ; it may be ex-pressed in the formula, Motion always pursues the line of least resistance. This formula has the utmost generality. It does not formulate the process as if Motion were necessarily rectilinear ; that is quite an arbitrary assumption, and is open to Comte's criticism ; * but it says that whatever the direction impressed, that, and that only, will be preserved, until another impress modify it. t
66. The ideal law sets aside all resistances. From the Pisgah of what is, the mind sees what will be, or what would be, if all conflicting movements were allowed to neutralize each other. Laws are indeed nothing but general formulae expressing general facts from which all disturbing particulars are eliminated. They do not describe the path which bodies actually pursue, but the path the bodies tend to take, and would take were the obstacles removed. Thus, to cite the law of the tides, there never is, in fact, sufficient time for the sea to assume the form towards which it tends, and which it would assume were the period longer ; the mathematician disregards this fact, and substitutes his ideal law in its place. If, on. this principle, a woman were to argue that her hunchback lover had a form of graceful symmetry, because his back would be straight were there no curvature of the spine, we should point out that she confounded the concrete with the abstract, the real with the ideal. In like manner were any one to declare the law of the tides to be false, because the observed facts did not conform to it, we should point out that laws being ideal constructions are not transcripts of real particulars.
67. Again : the path of a planet is said to be an ellipse. Every one knows that the real orbit is nothing of the kind. The ellipse is not to be found in the heavens, but in the calculations of astronomers. The path would be elliptical if there were only one planet moving round the sun ; but as, in fact, there are many planets, all acting on each other by forces varying with their varying positions, the planets cannot move in exact ellipses. the radius vector of each does not pass over equal areas in equal times. The orbit is not only not an ellipse, it is not any regularly formed curve ; nor is the same curve described in successive revolutions.
Are then Kepler's laws illusions ? By no means : they are abstractions ; they are Types erected by scientific Imagination, which moulds the elements of concrete observation into abstractions by getting rid of all perturbing particulars. The planet is supposed to move in an ellipse, by assuming the elements of the ellipse to have been perpetually altering. The supposition is a fiction, and is justified by its results. The reader sees at once that by similar fictions Ptolemy and his successors represented the movements of the planets, adding epicycle on cycle to make theory approximate to observation.
Sir John Herschel reminds us that Kepler's laws are to be regarded as only first approximations to the much more complicated ones which actually prevail, and that " to bring the remote observations into rigorous and mathematical accordance with each other, and at the same time retain the extremely convenient nomenclature and relations of the elliptic system, it becomes necessary to modify to a certain extent our verbal expression of the laws, and to regard the numerical data or elliptic elements of the orbits as not absolutely permanent, but subject to a series of slow and almost imperceptible changes. These changes may be neglected when we consider only a few revolutions but going on from century to century and continually accumulating, they at length produce material departures in the orbits from their original state." f
68. Another fiction is that by which solids are distinguished from fluids ; it assigns to the molecules of fluids an independence as respects cohesion, enabling them to move freely among each other. This is needful for calculation, though obviously untrue in reality, many of the observed phenomena of fluids being due to the cohesion of their molecules, a cohesion less energetic but similar in kind to that in solids ; and this fact occasions many discrepancies between theory and practice, notably in the flow of liquids from an orifice. Indeed it is perfectly well understood that all the applications of theory to practice are only approximations. " Take for instance the very simple case of a crow-bar employed to move a heavy mass. The ac-curate mathematical investigation of the action would involve the simultaneous treatment of the motions of every part of bar, fulcrum, and mass raised ; and from our almost complete ignorance of the nature of mat-ter and molecular forces, it is clear that such a treatment of the problem is impossible. It is a result of observation that the particles of the bar, fulcrum, and mass separately retain throughout the process nearly the same relative positions. Hence the idea of solving, instead of the above impossible problem, another in reality quite different, but while infinitely simpler obviously leading to nearly the same results as the former. The new form is given at once by the experimental result of the trial. Imagine the masses involved to be perfectly rigid (i. e. incapable of changing their form or dimensions) and the infinite series of forces really acting may be left out of consideration; so that the mathematical investigation deals with a finite (and generally small) number of forces instead of a practically infinite number."
69. Were the whole circle of the sciences to pass be-fore us, each would in turn display the essentially ideal nature of its construction, and wide departure from reality, either in its abstractions or in its hypotheses. The abstractions necessarily disregard particulars. The laws, usually accepted as absolutely exact (and justly so in the region of Abstraction), are "only general truths always more or less falsified in every particular case."
The hypotheses are fictions, provisional guesses. So far from Facts, Perceptions, constituting the material of Science, as is often said or implied, they are simply the elements out of which its material is constructed. Perception gives the naked fact of Sense, isolated, unconnected, merely juxtaposed with other facts, and without far-reaching significance. To the brute simplicity of Sensation must be added the artifice of Construction. Science looks through the brute fact, to contemplate the Abstraction which gives it connection, significance. Renee the paradox that we understand the fall of bodies only through the movements of the planets ; the growth of a plant only through biological laws. It is through the manifold ideal constructions of the Possible that we learn to appreciate the Actual. Facts are mere letters which have their meaning only in the words they form ; and these words again have their meaning, not in them-selves alone, but in their positions in the sentence.
The point here insisted on has always been familiar to philosophers in each particular case, but I am not aware of any philosopher having boldly generalized the observation, and proclaimed the introduction of Fiction to be a necessary procedure of Research. The dread lest the admission of Fiction should throw doubt over the certainty of the conclusions reached by its aid, may probably have prevented the generalization ; especially when no sharp distinction had been drawn between the fictions of Science and the fictions of Poetry. But perhaps the most deterrent influence has been due to the erroneous conception, almost universal, of the phenomena of Nature being determined by Law. This must be replaced by the more accurate conception of the Law being determined by the phenomena. What we call Laws of Nature are not objective existences, but, subjective abstractions, —formulae in which the multifarious phenomena are stripped of their variety and reduced to unity.
70. Before proceeding to give precision to this perhaps paradoxical view of Law, we may pursue the illustration of the scientific employment of Fiction, especially in the creation of Abstract Types. It has already been shown that the first law of Motion is no expression of the actual movements, but simply the ideal standard by which all movements may be measured.
The mathematician knows that when a point moves along a curve there is inaccuracy in saying that it moves in any one direction through any arc however small. But a straight line may be found at every point which more nearly than any other straight line represents the direction of the motion. In the same way no motion can be uniform, no velocity can be uniform, but we can at every instant assign an uniform velocity which shall more nearly than any other represent the rate at which the body is moving. This is obviously an ideal construction, not a real transcription. Assuming that if there were no force traversing the direction of a body, then the body would proceed in a straight line, we are enabled to estimate the forces in any deviation from that straight line.. Rectilinear Motion, though never possible, is thus the ideal Type to which all actual motions are ideally made to conform. We must regard it purely in abstraction, — the ideal limit replacing the real limit, — other-wise the` law is false.
71. Schelling well says * that " the necessary tendency of all Science is to pass from Nature to Intelligence. The highest perfection of research would be the thorough spiritualization of natural laws, reducing them to laws of Intuition and Thought. The material phenomena must give place to their laws. Optical phenomena are nothing but a geometry whose lines are drawn by light ; and this light itself is of dubious materiality."
It may not at first be apparent why, since we have al-ways to deal with concretes, we must always transform them into abstractions ; why, having to understand the phenomena presented to Sense, we effect this through Laws that are intelligible but not sensible. An examination, however, of the conditions of Knowledge discloses that Science differs from Sensation in being indirect and constructive, — that Abstraction is a primary condition of Perception, — that Inference, or hypothesis, is largely mingled in what seems simple sensible experience. Thus it appears that among the preliminaries of exact knowledge there are two which have the paradoxical aspect of looking away from the data directly presented, and of guessing the presence of other data, — looking away from the particular object, to find some general object which includes this particular one without including its individual characters, seeking Man in Socrates and the Rodent in this rabbit. Instead, therefore, of vaguely warning Philosophy against the dangers of Abstraction and Hypothesis, — real as these dangers are, - we should openly avow them to be indispensable aids, and, by a clear recognition of the aid they furnish, learn wherein their dangers lurk.
72. Let us glance at one or two transcendental conceptions, and note the value of a purely imaginary Type. And first of the conception by which the great poet Goethe illuminated the whole of Vegetal Morphology, one of those germinal conceptions which change the state of a science. Amid all the diversities of sensible experience he saw the typical form of the Leaf present in every organ of the Plant, and conceived the Plant itself to be only a variously transformed Leaf, a Type which, developed in spirals in the stem, was also developed or aborted in calyx, stamen, and pistil, but always under every variety presenting constant relations, and preserving one typical order. Schiller's objection, which irritated him so much, that the Leaf was an idea, though true enough in fact, was irrelevant as an objection. The Leaf was an idea, but an idea which had sensibles for its concrete elements. A similar conception was applied by Goethe to the skeleton of vertebrates ; and, in the hands of his successors, the Vertebral Type has been a potent instrument of morphological research. Of the same order is the conception of the Animal Series, first suggested by Aristotle, but brought into effective clearness by Lamarck, Geoffroy St. Hilaire, and still more luminously by Mr. Darwin.
73. It is, however, a profound mistake in regard to Nature, and no less in regard to Method, when such Types are wrested from their position among Ideals, and offered as Reals. There is not in Nature, there never was, a typical Leaf, a primitive Vertebra, or an existent Series, from which all plants, vertebrae, and animals have successively varied. There never was a Plan laid down, according to which the organic world was constructed, after the manner of a plan prearranged by an architect for the builder's guidance. On the contrary, this Plan and these Types are our after-thoughts, abstractions formed out of the sensible data presented by various plants, vertebrae, and animals ; they are ideal constructions from reals, obtained by the mind's grouping together the dominant resemblances, and setting aside all the many diversities. The theologian and metaphysician, by a procedure familiar to them, seize hold of these Types and present them as indices of a Plan in Creation. But this is the fallacy of supposing a resultant to have been the determinant. All that Experience war-rants is the assertion that the original protoplasm, which was wholly destitute of plant-form, leaf or other, and the germinal membrane equally destitute of vertebral form, did, in the successive stages of evolution, pass through many forms, each new form being determined by that which preceded it, and by the external pressures of the medium in which it was evolved ; consequently, in so far as these external influences had a general resemblance, the resultant forms were necessarily similar. The Type is an abstract expression of this general similarity.
Such is the positive doctrine of Morphology. The speculative doctrine which finds favor with theologians and metaphysicians teaches that the Type pre-existed in the Divine Mind, or at any rate in Nature ; or teaches that somewhere there was a -vertebra formed, and from this vertebra all the other bones were constructed by modification of its special parts, and so the skull was a modification of the spinal column, etc.
74. The scientific value of Types is that of being ideal guides, not real facts. They are standards by which deviations may be appreciated. It is because so few writers, even of those who adopt the Evolution Hypothesis, remember that it is only an hypothesis, and being an Ideal cannot be accepted as a Real, that opponents demand - and advocates endeavor to supply — evidence of its reality. The Animal Series is an ideal construction. Writers who forget this, not content with the inductive data for a speculative insight, demand—and evolutionists endeavor to supply— evidence which, could it be furnished, would at once transform the hypothesis into a demonstration, the problem into a theorem. Thus one of the commonest objections urged is, that were the hypothesis true we ought to find a gradual and continuous line of organic development, from one group to another, and one species to another; whereas, in point of fact, what we do find is group sharply demarcated from group, species separated by an unbridgeable gulf from species; and these gaps the imagination is baffled in attempting to fill up, so as to render the transition from one form to the other apparent to Sense. The objection is wholly irrelevant; and although one cannot but be grateful for the interesting researches of those zoologists and paleontologists who endeavor to supply the evidence of "missing links,"—grateful because all extensions of our knowledge of organic forms is valuable, — one cannot applaud them for thus attempting to answer the objection, or for evading it by refuge in our geological ignorance. The objection is based on a twofold misconception. Continuity of form —.in the sense demanded - is incompatible with that variety of form which Evolution postulates and Observation discloses. Such continuity would make the whole organic world one form. The Type would cease to be an abstraction, and degenerate into a concrete sensible. Between any two forms, however similar, short of identity, there must, ex vi termini, be a solution of continuity ; if the incident forces which determine the form be unequal, the resultant must necessarily be a variation. Thus, sup-pose we have two samples of protoplasm identical in all respects, but subject to forces which vary in some respects, the resultant forms must be separated by a gap which is indefinite. It is an elementary deduction from mechanical principles, that when a body susceptible of various positions of stable equilibrium is moved in various directions on a plane, the changes from one position to another must be abrupt, without any stable intermediates. If two forms differ, and they must if they are two, then their difference is a solution of continuity, though it may be accompanied by resemblances. There can be no stable transition between an exogen and an endogen, between an animal with a shell and an animal without a shell, more than between a crystal and its solution, or between sugar and oil, both hydrocarbons. We may regard all sensations as modifications of Sensibility, but there is no transition between one sensation and another; and just as Sensibility is the abstract expression for all concrete sensations, and comes into existence with them, so Animal is the abstract expression for all concrete animals.
75. The distinction between Types and Beals was entirely overlooked in the famous controversy between Cuvier and Geoffroy St. Hilaire, and is equally so in the controversy now raging between the opponents and adherents of Darwinism. It will one day be likened to the controversy raised by the first promulgation of the Differential Calculus, the logical basis of which even Leibnitz himself very imperfectly conceived, and which even in our own day is generally acknowledged to be incomprehensible, because, as I hinted before, quantities are not discriminated from operations on quantity. Leibnitz when pressed by objections declared that he regarded infini tesimals as incomparables * which might be disregarded in reference to finite quantities as grains of sand in reference to the sea ; a defence which Comte remarks completely vitiates the analysis reducing it to a mere calculus of approximation, " Qui sous ce rapport serait radicalement vicieux puisqu'il serait impossible de prévoir à quel point les opérations successives peuvent grossir ces erreurs premières, dont l'accroissement pourrait même évidemment devenir ainsi quelconque." t
Without pausing here to exhibit the logical justification, let us ask, how did mathematicians practically justify the new Calculus ? By showing that it enabled them to solve problems hitherto insoluble. In like manner the adherents of the Evolution Hypothesis may answer the objections urged against it by showing — and they do show --the vast reach of organic phenomena, hitherto inexplicable, which are rendered intelligible by its aid. This will not prove the hypothesis to be true; but it proves it to be effective, which is all an hypothesis can pretend to be. Infinitesimals may not exist in Nature ; the Animal Series may have no real correspondent; but the calculus and the evolution hypothesis are ideal constructions of vast power in scientific research.
76. As a final example let us not omit to mention the creation of Moral Types, the standards for our conduct in life. It is often made an objection against moral and religious conceptions of Duty that they demand for their realization a perfection which is not human. Certainly no man ever did, or could, realize in conduct the exalted ideal of life which he may have formed, or accepted from others. " O, would that for one single day we had lived in this world as we ought !" is the passionate exclamation of A'Kempis (or whoever wrote the Imitation). It has been most keenly felt by those whose lives have been most free from reproach. The objection to ideals, on the ground of their surpassing human nature, is a misconception of their function. They are not the laws by- which we live, or can live, but the types by which we measure all deviations from a perfect life. The mind which has once placed before it an ideal of life has a pole-star by which to steer, although his actual course will be deter-mined by the winds and waves. The pole-star is not the helm, nor is the helm more than one of the active agents. Our passions and our ignorance constantly make us swerve from the path to which the pole-star points ; and thus the ideal of a Christian life, or the ideal of Marriage, are never wholly to be realized, yet who denies that such ideals are very potent influences in every soul that has clearly conceived them ? It is a truth, and not an idle phrase, that man does not live by bread alone ; that it is his privilege to live by aspiration, hope, and love, to be moved by ideal impulses which cause him to check the impulses of a lower self, to forego the transient pleasure of Sense, and passionately strive after the nobler pleasures of heart and intellect. We all place before ourselves the ideal of a noble life, the type of a grander character than our infirmities enable us to realize; and we do not look on that ideal as a fiction, on that type of character as a falsehood, because we fail to realize it. Like the typical laws of physical processes, these conceptions are solid truths although they exist only as ideals ; and he who imagines their validity impugned because human nature can but imperfectly realize them, is as ignorant of Life as he would be who should deny the validity of natural Laws, because of the perturbations observable in natural events.
The contrast between a real law and an ideal law, such as we find in the second law of Motion (formulated by Galileo in the parallelogram of forces) and the first law, which is only the formula of what would be the motion were all disturbing conditions absent, is equally exhibited in the moral law that " the habit of right action is the securest preparation for acting rightly under emergencies," — and the ideal law that " we should love our neighbors as ourselves." No moving body does move uniformly in a straight line ; no man does love his neighbor as himself. All bodies do move in the diagonal of the parallelogram of two incident forces ; and all men are trained to act rightly on emergencies by what is a kind of moral instinct, organized in previous habits of acting rightly.
77. It would be of eminent service if a classification of the Laws — real and ideal — were drawn up, so that in every case there might be distinct understanding whether we were dealing with a Type which pretended to no objective reality, or with a Notation of the real process observed, and only varying from observation as the general varies from the particular. To effect this it would be necessary first to settle the question mooted in the succeeding chapter.
( Originally Published 1874 )
Problems of Life and Mind:
The Reality Of Abstractions
Ideal Construction In Science
What Are Laws Of Nature?
The Use And Abuse Of Hypothesis
The Passage From The Abstract To The Concrete
Ideal Construction In Metaphysics
The Search After Causes
Intuition And Demonstration
Axioms And Their Validity
Read More Articles About: Problems of Life and Mind