Chapter XXII
already gave the reason (see p. 335, above). This world
_might_ be a world in which all things differed, and in which what properties there were were ultimate and had no farther predicates. In such a world there would be as many kinds as there were separate things. We could never subsume a new thing under an old kind; or if we could, no consequences would follow. Or, again, this might be a world in which innumerable things were of a kind, but in which no concrete thing remained of the same kind long, but all objects were in a flux. Here again, though we could subsume and infer, our logic would be of no practical use to us, for the subjects of our propositions would have changed whilst we were talking. In such worlds, logical relations would obtain, and be known (doubtless) as they are now, but they would form a merely theoretic scheme and be of no use for the conduct of life. But our world is no such world. It is a very peculiar world, and plays right into logic's hands. _Some_ of the things, at least, which it contains are of the same kind as other things; _some_ of them remain always of the kind of which they once were; and some of the properties of them cohere indissolubly and are always found together. _Which_ things these latter things are we learn by experience in the strict sense of the word, and the results of the experience are embodied in 'empirical propositions.' Whenever such a thing is met with by us now, our sagacity notes it to be of a certain kind; our learning immediately recalls that kind's kind, and then _that_ kind's kind, and so on; so that a moment's thinking may make us aware that the thing is of a kind so remote that we could never have directly perceived the connection. The flight to this last kind _over the heads of the intermediaries_ is the essential feature of the intellectual operation here. Evidently it is a pure outcome of our sense for apprehending serial increase; and, unlike the several propositions themselves which make up the series (and which may all be empirical), it has nothing to do with the time- and space-order in which the things have been experienced.
MATHEMATICAL RELATIONS.
So much for the _a priori_ necessities called systematic classification and logical inference. The other couplings of data which pass for _a priori_ necessities of thought are the _mathematical_ judgments, and certain metaphysical propositions. These latter we shall consider farther on. As regards the mathematical judgments, they are all 'rational propositions' in the sense defined on p. 644, for they express results of comparison and nothing more. The mathematical sciences deal with similarities and equalities exclusively, and not with coexistences and sequences. Hence they have, in the first instance, no connection with the order of experience. The comparisons of mathematics are between numbers and extensive magnitudes, giving rise to arithmetic and geometry respectively.
* * * * *
_Number_ seems to signify primarily the strokes of our attention in discriminating things. These strokes remain in the memory in groups, large or small, and the groups can be compared. The discrimination is, as we know, psychologically facilitated by the mobility of the thing as a total (p. 173). But within each thing we discriminate parts; so that the number of things which any one given phenomenon may be depends in the last instance on our way of taking it. A globe is one, if undivided; two, if composed of hemispheres. A sand-heap is one thing, or twenty thousand things, as we may choose to count it. We amuse ourselves by the counting of _mere_ strokes, to form rhythms, and these we compare and name. Little by little in our minds the number-series is formed. This, like all lists of terms in which there is a direction of serial increase, carries with it the sense of those mediate relations between its terms which we expressed by the axiom "the more than the more is more than the less." That axiom seems, in fact, only a way of stating that the terms do form an increasing series. But, in addition to this, we are aware of certain other relations among our strokes of counting. We may interrupt them where we like, and go on again. All the while we feel that the interruption does not alter the strokes themselves. We may count 12 straight through; or count 7 and pause, and then count 5, but still the strokes will be the same. We thus distinguish between our acts of counting and those of interrupting or grouping, as between an unchanged matter and an operation of mere shuffling performed on it. The matter is the original units or strokes; which all modes of grouping or combining simply give us back unchanged. In short, _combinations of numbers are combinations of their units_, which is the fundamental axiom of arithmetic,[542] leading to such consequences as that 7 + 5 = 8 + 4 because both = 12. The general axiom of mediate equality, that equals of equals are equal, comes in here.[543] The principle of constancy in our meanings, when applied to strokes of counting, also gives rise to the axiom that the same number, operated on (interrupted, grouped) in the same way will always give the same result or be the same. How shouldn't it? Nothing is supposed changed.
_Arithmetic and its fundamental principles are thus independent of our experiences or of the order of the world._ The matter of arithmetic is _mental matter_; its principles flow from the fact that the matter forms a series, which can be cut into by us wherever we like without the matter changing. The empiricist school has strangely tried to interpret the truths of number as results of coexistences among outward things. John Mill calls number a physical property of things. 'One,' according to Mill, means one sort of passive sensation which we receive, 'two' another, 'three' a third. The same things, however, can give us different number-sensations. Three things arranged thus, ---, for example, impress us differently from three things arranged thus, -_-. But experience tells us that every real object-group which can be arranged in one of these ways can always be arranged in the other also, and that 2 + 1 and 3 are thus modes of numbering things which 'coexist' invariably with each other. The indefeasibility of our belief in their 'coexistence' (which is Mill's word for their equivalence) is due solely to the enormous amount of experience we have of it. For all things, whatever other sensations they may give us, give us at any rate number-sensations. Those number-sensations which the same thing may be successively made to arouse are the numbers which we deem equal to each other; those which the same thing refuses to arouse are those which we deem unequal.
This is as clear a restatement as I can make of Mill's doctrine.[544] And its failure is written upon its front. Woe to arithmetic, were such the only grounds for its validity! The same real things are countable in numberless ways, and pass from one numerical form, not only to its equivalent (as Mill implies), but to its other, as the sport of physical accidents or of our mode of attending may decide. How could our notion that one and one are eternally and necessarily two ever maintain itself in a world where every time we add one drop of water to another we get not two but one again? in a world where every time we add a drop to a crumb of quicklime we get a dozen or more?--had it no better warrant than such experiences? At most we could then say that one and one are _usually_ two. Our arithmetical propositions would never have the confident tone which they now possess. That confident tone is due to the fact that they deal with abstract and ideal numbers exclusively. _What we mean_ by one plus one _is_ two; we _make_ two out of it; and it would mean two still even in a world where _physically_ (according to a conceit of Mill's) a third thing was engendered every time one thing came together with another. We are masters of our meanings, and discriminate between the things we mean and our ways of taking them, between our strokes of numeration themselves, and our bundlings and separatings thereof.
Mill ought not only to have said, "All things are numbered." He ought, in order to prove his point, to have shown that they are _unequivocally_ numbered, which they notoriously are not. Only the abstract numbers themselves are unequivocal, only those which we create mentally and hold fast to as ideal objects always the same. A concrete natural thing can always be numbered in a great variety of ways. "We need only conceive a thing divided into four equal parts (and all things may be conceived as so divided)," as Mill is himself compelled to say, to find the number four in it, and so on.
The relation of numbers to experience is just like that of 'kinds' in logic. So long as an experience will keep its kind we can handle it by logic. So long as it will keep its number we can deal with it by arithmetic. _Sensibly_, however, things are constantly changing their numbers, just as they are changing their kinds. They are forever breaking apart and fusing. Compounds and their elements are never numerically identical, for the elements are sensibly many and the compounds sensibly one. Unless our arithmetic is to remain without application to life, we must somehow _make_ more numerical continuity than we spontaneously find. Accordingly Lavoisier discovers his weight-units which remain the same in compounds and elements, though volume-units and quality-units all have changed. A great discovery! And modern science outdoes it by denying that compounds exist at all. There is no such thing as 'water' for 'science;' that is only a handy name for H_{2} and O when they have got into the position H-O-H, and then affect our senses in a novel way. The modern theories of atoms, of heat, and of gases are, in fact, only intensely artificial devices for gaining that constancy in the numbers of things which sensible experience will not show. "Sensible things are not the things for me," says Science, "because in their changes they will not keep their numbers the same. Sensible qualities are not the qualities for me, because they can with difficulty be numbered at all. These hypothetic atoms, however, are the things, these hypothetic masses and velocities are the qualities for me; they will stay numbered all the time."
By such elaborate inventions, and at such a cost to the imagination, do men succeed in making for themselves a world in which real things shall be coerced _per fas aut nefas_ under arithmetical law.
* * * * *
The other branch of mathematics is _geometry_. Its objects are also ideal creations. Whether nature contain circles or not, I can know what I mean by a circle and can stick to my meaning; and when I mean two circles I mean two things of an identical kind. The axiom of constant results (see above, p. 645) holds in geometry. The same forms, treated in the same way (added, subtracted, or compared), give the same results--how shouldn't they? The axioms of mediate comparison (p. 645), of logic (p. 648), and of number (p. 654) all apply to the forms which we imagine in space, inasmuch as these resemble or differ from each other, form kinds, and are numerable things. But in addition to these general principles, which are true of space-forms only as they are of other mental conceptions, there are certain axioms relative to space-forms exclusively, which we must briefly consider.
Three of them give marks of identity among straight lines, planes, and parallels. Straight lines which have two points, planes which have three points, parallels to a given line which have one point, in common, coalesce throughout. Some say that the certainty of our belief in these axioms is due to repeated experiences of their truth; others that it is due to an intuitive acquaintance with the properties of space. It is neither. We experience lines enough which pass through two points only to separate again, only we won't call them straight. Similarly of planes and parallels. We have a definite idea of what we mean by each of these words; and when something different is offered us, we see the difference. Straight lines, planes, and parallels, as they figure in geometry, are mere inventions of our faculty for apprehending serial increase. The farther continuations of these forms, we say, _shall_ bear the same relation to their last visible parts which these did to still earlier parts. It thus follows (from that axiom of skipped intermediaries which obtains in all regular series) that parts of these figures separated by other parts must agree in direction, just as contiguous parts do. This uniformity of direction throughout is, in fact, all that makes us care for these forms, gives them their beauty, and stamps them into fixed conceptions in our mind. But obviously if two lines, or two planes, with a common segment, were to part company beyond the segment, it could only be because the direction of at least one of them had changed. Parting company in lines and planes _means_ changing direction, means assuming a new relation to the parts that pre-exist; and assuming a new relation means ceasing to be straight or plane. If we mean by a parallel a line that will never meet a second line; and if we have one such line drawn through a point, any new line drawn through that point which does not coalesce with the first must be inclined to it, and if inclined to it must approach the second, i.e., cease to be parallel with it. No properties of outlying space need come in here: only a definite conception of uniform direction, and constancy in sticking to one's point.
The other two axioms peculiar to geometry are that figures can be moved in space without change, and that no variation in the way of subdividing a given amount of space alters its total quantity.[545] This last axiom is similar to what we found to obtain in numbers. 'The whole is equal to its parts' is an abridged way of expressing it. A man is not the same biological whole if we cut him in two at the neck as if we divide him at the ankles; but geometrically he is the same whole, no matter in which place we cut him. The axiom about figures being movable in space is rather a postulate than an axiom. _So far as they are_ so movable, then certain fixed equalities and differences obtain between forms, _no matter where placed_. But if translation through space warped or magnified forms, then the relations of equality, etc., would always have to be expressed with a position-qualification added. A geometry as absolutely certain as ours could be invented on the supposition of such a space, if the laws of its warping and deformation were fixed. It would, however, be much more complicated than our geometry, which makes the simplest possible supposition; and finds, luckily enough, that it is a supposition with which the space of our experience seems to agree.
By means of these principles, all playing into each other's hands, the mutual equivalences of an immense number of forms can be traced, even of such as at first sight bear hardly any resemblance to each other. We move and turn them mentally, and find that parts of them will superpose. We add imaginary lines which subdivide or enlarge them, and find that the new figures resemble each other in ways which show us that the old ones are equivalent too. We thus end by expressing all sorts of forms in terms of other forms, enlarging our knowledge of the kinds of things which certain other kinds of things are, or to which they are equivalent.
The result is a new system of mental objects which can be treated as identical for certain purposes, a new series of _is_'s almost indefinitely prolonged, just like the series of equivalencies among numbers, part of which the multiplication-table expresses. And all this is in the first instance regardless of the coexistences and sequences of nature, and regardless of whether the figures we speak of have ever been outwardly experienced or not.
CONSCIOUSNESS OF SERIES IS THE BASIS OF RATIONALITY.
Classification, logic, and mathematics all result, then, from the mere play of the mind comparing its conceptions, no matter whence the latter may have come. The essential condition for the formation of all these sciences is that we should have grown capable of apprehending series as such, and of distinguishing them as homogeneous or heterogeneous, and as possessing definite directions of what I have called 'increase.' This consciousness of series is a human perfection which has been gradually evolved, and which varies greatly from man to man. There is no accounting for it as a result of habitual associations among outward impressions, so we must simply ascribe it to the factors, whatever they be, of inward cerebral growth. Once this consciousness attained to, however, _mediate_ thought becomes possible; with our very awareness of a series may go an awareness that dropping terms out of it will leave identical relations between the terms that remain; and thus arises a perception of relations between things so naturally separate that we should otherwise never have compared them together at all.
The axiom of skipped intermediaries applies, however, only to certain
## particular series, and among them to those which we have considered,
in which the recurring relation is either of difference, of likeness, of kind, of numerical addition, or of prolongation in the same linear or plane direction. It is therefore not a purely formal law of thinking, but flows from the nature of the matters thought about. It will not do to say universally that in all series of homogeneously related terms the remote members are related to each other as the near ones are; for that will often be untrue. The series A is not B is not C is not D.... does not permit the relation to be traced between remote terms. From two negations no inference can be drawn. Nor, to become more concrete, does the lover of a woman generally love her beloved, or the contradictor of a contradictor contradict whomever he contradicts. The slayer of a slayer does not slay the latter's victim; the acquaintances or enemies of a man need not be each other's acquaintances or enemies; nor are two things which are on top of a third thing necessarily on top of each other.
All skipping of intermediaries and transfer of relations occurs within homogeneous series. But not all homogeneous series allow of intermediaries being skipped and relations transferred. It depends on which series they are, on what relations they contain.[546] Let it not be said that it is a mere matter of verbal association, due to the fact that language sometimes permits us to transfer the _name_ of a relation over skipped intermediaries, and sometimes does not; as where we call men 'progenitors' of their remote as well as of their immediate posterity, but refuse to call them 'fathers' thereof. There are relations which are _intrinsically_ transferable, whilst others are not. The relation of _condition_, e.g., is intrinsically transferable. What conditions a condition conditions what it conditions--"cause of cause is cause of effect." The relations of negation and _frustration,_ on the other hand, are not transferable: what frustrates a frustration does not frustrate what it frustrates. No changes of terminology would annul the intimate difference between these two cases.
Nothing but the clear sight of the ideas themselves shows whether the axiom of skipped intermediaries applies to them or not. Their connections, immediate and remote, flow from their inward natures. We try to consider them in certain ways, to bring them into certain relations, and we find that sometimes we can and sometimes we cannot _The question whether there are or are not inward and essential connections between conceived objects as such, really is the same thing as the question whether we can get any new perception from mentally coupling them together, or pass from one to another by a mental operation which gives a result._ In the case of some ideas and operations we get a result; but no result in the case of others. Where a result comes, it is due exclusively to the _nature_ of the ideas and of the operation. Take blueness and yellowness, for example. We can operate on them in some ways, but not in other ways. We can compare them; but we cannot add one to or subtract it from the other. We can refer them to a common kind, color; but we cannot make one a kind of the other, or infer one from the other. This has nothing to do with experience. For we _can_ add blue _pigment_ to yellow _pigment_, and subtract it again, and get a result both times. Only we know perfectly that this is no addition or subtraction of the blue and yellow qualities or natures themselves.[547]
* * * * *
There is thus no denying the fact that _the mind is filled with necessary and eternal relations which it finds between certain of its ideal conceptions, and which form a determinate system, independent of the order of frequency in which experience may have associated the conception's originals in time and space._
Shall we continue to call these sciences 'intuitive,' innate,' or '_a priori_' bodies of truth, or not?[548] Personally I should like to do so. But I hesitate to use the terms, on account of the odium which controversial history has made the whole of their connotation for many worthy persons. The most politic way not to alienate these readers is to flourish the name of the immortal Locke. For in truth I have done nothing more in the previous pages than to make a little more explicit the teachings of Locke's fourth book:
"The immutability of the same relations between the same immutable things is now the idea that shows him that if the three angles of a triangle were once equal to two right angles, they will always be equal to two right ones. And hence he comes to be certain that what was once true in the case is always true; what ideas once agreed will always agree.... Upon this ground it is that particular demonstrations in mathematics afford general knowledge. If, then, the perception that the same ideas will eternally have the same habitudes and relations be not a sufficient ground of knowledge, there could be no knowledge of general propositions in mathematics.... All general knowledge lies only in our own thoughts, and consists barely in the contemplation of our abstract ideas. Wherever we perceive any agreement or disagreement amongst them, there we have general knowledge; and by putting the names of those ideas together accordingly in propositions, can with certainty pronounce general truths.... What is once known of such ideas will be perpetually and forever true. So that, as to all general knowledge, we must search and find it only in our own minds and it is only the examining of our own ideas that furnisheth us with that. Truths belonging to essences of things (that is, to abstract ideas) are eternal, and are to be found out only by the contemplation of those essences.... Knowledge is the consequence of the ideas (be they what they will) that are in our minds, producing there certain general propositions.... Such propositions are therefore called 'eternal truths,'... because, being once made about abstract ideas so as to be true, they will, whenever they can be supposed to be made again, at any time past or to come, by a mind having those ideas, always actually be true. For names being supposed to stand perpetually for the same ideas, and the same ideas having immutably the same habitudes one to another, propositions concerning any abstract ideas that are once true must needs be eternal verities."
But what are these eternal verities, these 'agreements,' which the mind discovers by barely considering its own fixed meanings, except what I have said?--relations of likeness and difference, immediate or mediate, between the terms of certain series. Classification is serial comparison, logic mediate subsumption, arithmetic mediate equality of different bundles of attention-strokes, geometry mediate equality of different ways of carving space. None of these eternal verities has anything to say about facts, about what is or is not in the world. Logic does not say whether Socrates, men, mortals or immortals _exist_; arithmetic does not tell us where her 7's, 5's, and 12's are to be _found_; geometry affirms not that circles and rectangles are _real_. All that these sciences make us sure of is, that _if_ these things are anywhere to be found, the eternal verities will obtain of them. Locke accordingly never tires of telling us that the
"universal propositions of whose truth or falsehood we can have certain knowledge, concern not existence.... These universal and self-evident principles, being only our constant, clear, and distinct knowledge of our own ideas more general or comprehensive, can assure us of nothing that passes without the mind; their certainty is founded only upon the knowledge of each idea by itself, and of its distinction from others; about which we cannot be mistaken whilst they are in our minds.... The mathematician considers the truth and properties belonging to a rectangle or circle only as they are in idea in his own mind. For it is possible he never found either of them existing mathematically, i.e., precisely true, in his life. But yet the knowledge he has of any truths or properties belonging to a circle, or any other mathematical figure, are nevertheless true and certain even of real things existing; because real things are no farther concerned nor intended to be meant by any such propositions, than as things really agree to those archetypes in his mind. Is it true of the idea of a triangle, that its three angles are equal to two right ones? It is true also of a triangle wherever it really exists. Whatever other figure exists that is not exactly answerable to that idea in his mind is not at all concerned in that proposition. And therefore he is certain all his knowledge concerning such ideas is real knowledge: because, intending things no farther than they agree with those his ideas, he is sure what he knows concerning those figures when they have barely an ideal existence in his mind will hold true of them also when they have a real existence in matter." But "that any or what bodies do exist, that we are left to our senses to discover to us as far as they can."[549]
Locke accordingly distinguishes between 'mental truth' and 'real truth.'[550] The former is intuitively certain; the latter dependent on experience. Only _hypothetically_ can we affirm intuitive truths of real things--by _supposing_, namely, that real things exist which correspond exactly with the ideal subjects of the intuitive propositions.
If our senses corroborate the supposition all goes well. But note the strange descent in Locke's hands of the dignity of _a priori_ propositions. By the ancients they were considered, without farther question, to reveal the constitution of Reality. Archetypal things existed, it was assumed, in the relations in which we had to think them. The mind's necessities were a warrant for those of Being; and it was not till Descartes' time that scepticism had so advanced (in 'dogmatic' circles) that the warrant must itself be warranted, and the veracity of the Deity invoked as a reason for holding fast to our natural beliefs.
But the intuitive propositions of Locke leave us as regards outer reality none the better for their possession. We still have to "go to our senses" to find what the reality is. The vindication of the intuitionist position is thus a barren victory. The eternal verities which the very structure of our mind lays hold of do not necessarily themselves lay hold on extra-mental being, nor have they, as Kant pretended later,[551] a legislating character even for all possible experience. They are primarily interesting only as subjective facts. They stand waiting in the mind, forming a beautiful ideal network; and the most we can say is that we _hope_ to discover outer realities over which the network may be flung so that ideal and real may coincide.
* * * * *
And this brings us back to 'science' from which we diverted our attention so long ago (see p. 640). Science thinks that she has discovered the outer realities in question. Atoms and ether, with no properties but masses and velocities expressible by numbers, and paths expressible by analytic formulas, these at last are things over which the mathematico-logical network may be flung, and by supposing which instead of sensible phenomena science becomes yearly more able to manufacture for herself a world about which rational propositions may be framed. Sensible phenomena are pure delusions for the mechanical philosophy. The 'things' and qualities we instinctively believe in do not exist. The only realities are swarming solids in everlasting motion, undulatory or continued, whose expressionless and meaningless changes of position form the history of the world, and are deducible from initial collocations and habits of movement hypothetically assumed. Thousands of years ago men started to cast the chaos of nature's sequences and juxtapositions into a form that might seem intelligible. Many were their ideal prototypes of rational order: teleological and æsthetic ties between things, causal and substantial bonds, as well as logical and mathematical relations. The most promising of these ideal systems at first were of course the richer ones, the sentimental ones. The baldest and least promising were the mathematical ones; but the history of the latter's application is a history of steadily advancing successes, whilst that of the sentimentally richer systems is one of relative sterility and failure.[552] Take those aspects of phenomena which interest you as a human being most, and class the phenomena as perfect and imperfect, as ends and means to ends, as high and low, beautiful and ugly, positive and negative, harmonious and discordant, fit and unfit, natural and unnatural, etc., and barren are all your results. In the ideal world the kind 'precious' has characteristic properties. What is precious should be preserved; unworthy things should be sacrificed for its sake; exceptions made on its account; its preciousness is a reason for other things' actions, and the like. But none of these things need happen to your 'precious' object in the real world. Call the things of nature as much as you like by sentimental, moral, and æsthetic names, no natural consequences follow from the naming. They may be of the kinds you allege, but they are not of '_the kind's kind_': and the last great system-maker of this sort, Hegel, was obliged explicitly to repudiate logic in order to make any inferences at all from the names he called things by.
* * * * *
But when you give things mathematical and mechanical names and call them just so many solids in just such positions, describing just such paths with just such velocities, all is changed. Your sagacity finds its reward in the verification by nature of all the deductions which you may next proceed to make. Your 'things' realize all the _consequences_ of the names by which you classed them. The modern mechanico-physical philosophy of which we are all so proud, because it includes the nebular cosmogony, the conservation of energy, the kinetic theory of heat and gases, etc., etc., begins by saying that the _only_ facts are collocations and motions of primordial solids, and the only laws the changes of motion which changes in collocation bring. The ideal which this philosophy strives after is a mathematical world-formula, by which, if all the collocations and motions at a given moment were known, it would be possible to reckon those of any wished-for future moment, by simply considering the necessary geometrical, arithmetical, and logical implications. Once we have the world in this bare shape, we can fling our net of _a priori_ relations over all its terms, and pass from one of its phases to another by inward thought-necessity. Of course it is a world with a very minimum of rational _stuff_. The sentimental facts and relations are butchered at a blow. But the rationality yielded is so superbly complete in _form_ that to many minds this atones for the loss, and reconciles the thinker to the notion of a purposeless universe, in which all the things and qualities men love, _dulcissima mundi nomina_, are but illusions of our fancy attached to accidental clouds of dust which will be dissipated by the eternal cosmic weather as carelessly as they were formed.
The popular notion that 'Science' is forced on the mind _ab extra_, and that our interests have nothing to do with its constructions, is utterly absurd. The craving to believe that the things of the world belong to kinds which are related by inward rationality together, is the parent of Science as well as of sentimental philosophy; and the original investigator always preserves a healthy sense of how plastic the materials are in his hands.
"Once for all," says Helmholtz in beginning that little work of his which laid the foundations of the 'conservation of energy,' "it is the task of the physical sciences to seek for laws by which
## particular processes in nature may be referred to general rules,
and deduced from such again. Such rules (for example the laws of reflection or refraction of light, or that of Mariotte and Gay-Lussac for gas-volumes) are evidently nothing but generic-concepts for embracing whole classes of phenomena. The search for them is the business of the experimental division of our Science. Its theoretic division, on the other hand, tries to discover the unknown causes of processes from their visible effects; tries to understand them by the law of causality.... The ultimate goal of theoretic physics is to find the last _unchanging_ causes of the processes in Nature. Whether all processes be really ascribable to such causes, whether, in other words, _nature be completely intelligible,_ or whether there be changes which would elude the law of a necessary causality, and fall into a realm of spontaneity or freedom, is not here the place to determine; but at any rate it is clear that the Science whose aim it is to make nature appear intelligible [_die Natur zu begreifen_] must start with the _assumption_ of her intelligibility, and draw consequences in conformity with this assumption, until irrefutable facts show the limitations of this method.... The postulate that natural phenomena must be reduced to changeless ultimate causes next shapes itself so that _forces unchanged by time_ must be found to be these causes. Now in Science we have already found portions of matter with changeless forces (indestructible qualities), and called them (chemical) elements. If, then, we imagine the world composed of elements with inalterable qualities, the only changes that can remain possible in such a world are spatial changes, i.e. movements, and the only outer relations which can modify the action of the forces are spatial too, or, in other words, the forces are motor forces dependent for their effect only on spatial relations. More exactly still: The phenomena of nature must be reduced to [_zurückgeführt_, conceived as, classed as] motions of material points with inalterable motor forces acting according to space-relations alone.... But points have no mutual space-relations except their distance,... and a motor force which they exert upon each other can cause nothing but a change of distance--i.e. be an attractive or a repulsive force.... And its intensity can only depend on distance. So that at last the task of Physics resolves itself into this, to refer phenomena to inalterable attractive and repulsive forces whose intensity varies with distance. The solution of this task would at the same time be the condition of Nature's complete intelligibility."[553]
The subjective interest leading to the assumption could not be more candidly expressed. What makes the assumption 'scientific' and not merely poetic, what makes a Helmholtz and his kin _discoverers_, is that the things of Nature turn out to act as if they _were_ of the kind assumed. They behave as such mere drawing and driving atoms would behave; and so far as they have been distinctly enough translated into molecular terms to test the point, so far a certain fantastically ideal object, namely, the mathematical sum containing their mutual distances and velocities, is found to be constant throughout all their movements. This sum is called the total energy of the molecules considered. Its constancy or 'conservation' gives the name to the hypothesis of molecules and central forces from which it was logically deduced.
Take any other mathematico-mechanical theory and it is the same. They are all translations of sensible experiences into other forms, substitutions of items between which ideal relations of kind, number, form, equality, etc., obtain, for items between which no such relations obtain; coupled with declarations that the experienced form is false and the ideal form true, declarations which are justified by the appearance of new sensible experiences at just those times and places at which we logically infer that their ideal correlates ought to be. Wave-hypotheses thus make us predict rings of darkness and color, distortions, dispersions, changes of pitch in sonorous bodies moving from us, etc.; molecule-hypotheses lead to predictions of vapor-density, freezing point, etc.,--all which predictions fall true.
Thus the world grows more orderly and rational to the mind, which passes from one feature of it to another by deductive necessity, as soon as it conceives it as made up of so few and so simple phenomena as bodies with no properties but number and movement to and fro.
METAPHYSICAL AXIOMS.
But alongside of these ideal relations between terms which the world verifies, there are other ideal relations not as yet so verified. I refer to those propositions (no longer expressing mere results of comparison) which are formulated in such metaphysical and æsthetic axioms as "The Principle of things is one;" "The quantity of existence is unchanged;" "Nature is simple and invariable;" "Nature acts by the shortest ways;" "_Ex nihilo nihil fit;_" "Nothing can be evolved which was not involved;" "Whatever is in the effect must be in the cause;" "A thing can only work where it is;" "A thing can only affect another of its own kind;" "_Cessante causa, cessat et effectus;_" "Nature makes no leaps;" "Things belong to discrete and permanent kinds;" "Nothing is or happens without a reason;" "The world is throughout rationally intelligible;" etc., etc., etc. Such principles as these, which might be multiplied to satiety,[554] are properly to be called _postulates of rationality_, not propositions of fact. If nature _did_ obey them, she _would_ be _pro tanto_ more intelligible; and we seek meanwhile so to conceive her phenomena as to show that she does obey them. To a certain extent we succeed. For example, instead of the 'quantity of existence' so vaguely postulated as unchanged, Nature allows us to suppose that curious sum of distances and velocities which for want of a better term we call 'energy.' For the effect being 'contained in the cause,' nature lets us substitute 'the effect _is_ the cause,' so soon as she lets us conceive both effect and cause as the same molecules, in two successive positions.--But all around these incipient successes (as all around the molecular world, so soon as we add to it as its 'effects' those illusory 'things' of common-sense which we had to butcher for its sake), there still spreads a vast field of irrationalized fact whose items simply _are_ together, and from one to another of which we can pass by no ideally 'rational' way.
It is not that these more metaphysical postulates of rationality are absolutely barren--though barren enough they were when used, as the scholastics used them, as immediate propositions of fact.[555] They have a fertility as ideals, and keep us uneasy and striving always to recast the world of sense until its lines become more congruent with theirs. Take for example the principle that 'nothing can happen without a cause.' We have no definite idea of what we mean by cause, or of what causality consists in. But the principle expresses a demand for _some_ deeper sort of inward connection between phenomena than their merely habitual time-sequence seems to us to be. The word 'cause' is, in short, an altar to an unknown god; an empty pedestal still marking the place of a hoped-for statue. _Any_ really inward belonging-together of the sequent terms, if discovered, would be accepted as what the word cause was meant to stand for. So we seek, and seek; and in the molecular systems we find a sort of inward belonging in the notion of identity of matter with change of collocation. Perhaps by still seeking we may find other sorts of inward belonging, even between the molecules and those 'secondary qualities,' etc., which they produce upon our minds.
It cannot be too often repeated that the triumphant application of any one of our ideal systems of rational relations to the real world justifies our hope that other systems may be found also applicable. Metaphysics should take heart from the example of physics, simply confessing that hers is the longer task. Nature _may_ be remodelled, nay, certainly will be remodelled, far beyond the point at present reached. Just how far?--is a question which only the whole future history of Science and Philosophy can answer.[556] Our task being Psychology, we cannot even cross the threshold of that larger problem.
* * * * *
Besides the mental structure which results in such metaphysical principles as those just considered, there is a mental structure which expresses itself in
ÆSTHETIC AND MORAL PRINCIPLES.
The æsthetic principles are at bottom such axioms as that a note sounds good with its third and fifth, or that potatoes need salt. We are once for all so made that when certain impressions come before our mind, one of them will seem to call for or repel the others as its companions. To a certain extent the principle of habit will explain these æsthetic connections. When a conjunction is repeatedly experienced, the cohesion of its terms grows grateful, or at least their disruption grows unpleasant. But to explain _all_ æsthetic judgments in this way would be absurd; for it is notorious how seldom natural experiences come up to our æsthetic demands. Many of the so-called metaphysical principles are at bottom only expressions of æsthetic feeling. Nature is simple and invariable; makes no leaps, or makes nothing but leaps; is rationally intelligible; neither increases nor diminishes in quantity; flows from one principle, etc., etc.,--what do all such principles express save our sense of how pleasantly our intellect would feel if it had a Nature of that sort to deal with? The subjectivity of which feeling is of course quite compatible with Nature also turning out objectively to be of that sort, later on.
The _moral_ principles which our mental structure engenders are quite as little explicable _in toto_ by habitual experiences having bred inner cohesions. Rightness is not _mere_ usualness, wrongness not _mere_ oddity, however numerous the facts which might be invoked to prove such identity. Nor are the moral judgments those most invariably and emphatically impressed on us by public opinion. The most characteristically and peculiarly moral judgments that a man is ever called on to make are in unprecedented cases and lonely emergencies, where no popular rhetorical maxims can avail, and the hidden oracle alone can speak; and it speaks often in favor of conduct quite unusual, and suicidal as far as gaining popular approbation goes. The forces which conspire to this resultant are subtle harmonies and discords between the elementary ideas which form the data of the case. Some of these harmonies, no doubt, have to do with habit; but in respect to most of them our sensibility must assuredly be a phenomenon of supernumerary order, correlated with a brain-function quite as secondary as that which takes cognizance of the diverse excellence of elaborate musical compositions. No more than the higher musical sensibility can the higher moral sensibility be accounted for by the frequency with which outer relations have cohered.[557] Take judgments of justice or equity, for example. Instinctively, one judges everything differently, according as it pertains to one's self or to some one else. Empirically one notices that everybody else does the same. But little by little there dawns in one the judgment "nothing can be right for me which would not be right for another similarly placed;" or "the fulfilment of my desires is intrinsically no more imperative than that of anyone else's;" or "what it is reasonable that another should do for me, it is also reasonable that I should do for him;"[558] and forthwith the whole mass of the habitual gets overturned. It gets _seriously_ overturned only in a few fanatical heads. But its overturning is due to a back-door and not to a front-door process. Some minds are preternaturally sensitive to logical consistency and inconsistency. When they have ranked a thing under a kind, they _must_ treat it as of that kind's kind, or feel all out of tune. In many respects we do class ourselves with other men, and call them and ourselves by a common name. They agree with us in having the same Heavenly Father, in not being consulted about their birth, in not being themselves to thank or blame for their natural gifts, in having the same desires and pains and pleasures, in short in a host of fundamental relations. Hence, _if these things be our essence,_ we should be substitutable for other men, and they for us, in any proposition in which either of us is involved. The more fundamental and common the essence chosen, and the more simple the reasoning,[559] the more wildly radical and unconditional will the justice be which is aspired to. Life is one long struggle between conclusions based on abstract ways of conceiving cases, and opposite conclusions prompted by our instinctive perception of them as individual facts. The logical stickler for justice always seems pedantic and mechanical to the man who goes by tact and the particular instance, and who usually makes a poor show at argument. Sometimes the abstract conceiver's way is better, sometimes that of the man of instinct. But just as in our study of reasoning we found it impossible to lay down any mark whereby to distinguish _right_ conception of a concrete case from _confusion_ (see pp. 336, 350), so here we can give no general rule for deciding when it is morally useful to treat a concrete case as _sui generis_, and when to lump it with others in an abstract class.[560]
An adequate treatment of the way in which we come by our æsthetic and moral judgments would require a separate chapter, which I cannot conveniently include in this book. Suffice it that these judgments express inner harmonies and discords between objects of thought; and that whilst outer cohesions frequently repeated will often seem harmonious, all harmonies are not thus engendered, but our feeling of many of them is a secondary and incidental function of the mind. Where harmonies are asserted of the real world, they are obviously mere postulates of rationality, so far as they transcend experience. Such postulates are exemplified by the ethical propositions that the individual and universal good are one, and that happiness and goodness are bound to coalesce in the same subject.
SUMMARY OF WHAT PRECEDES.
I will now sum up our progress so far by a short summary of the most important conclusions which we have reached.
The mind has a native structure in this sense, that certain of its objects, if considered together in certain ways, give definite results; and that no other ways of considering, and no other results, are possible if the same objects be taken.
The results are 'relations' which are all expressed by judgments of subsumption and of comparison.
The judgments of subsumption are themselves subsumed under the _laws of logic_.
Those of comparison are expressed in _classifications_, and in the _sciences of arithmetic and geometry_.
Mr. Spencer's opinion that our consciousness of classificatory, logical, and mathematical relations between ideas is due to the frequency with which the corresponding 'outer relations' have impressed our minds, is unintelligible.
Our consciousness of these relations, no doubt, has a natural genesis. But it is to be sought rather in the inner forces which have made the brain grow, than in any mere paths of 'frequent' association which outer stimuli may have ploughed in that organ.
But let our sense for these relations have arisen as it may, the relations themselves form a fixed system of lines of cleavage, so to speak, in the mind, by which we naturally pass from one object to another; and the objects connected by these lines of cleavage are often not connected by any regular time- and space-associations. We distinguish, therefore, between the empirical order of things, and this their rational order of comparison; and, so far as possible, we seek to translate the former into the latter, as being the more congenial of the two to our intellect.
Any classification of things into kinds (especially if the kinds form series, or if they successively involve each other) is a more rational way of conceiving the things than is that mere juxtaposition or separation of them as individuals in time and space which is the order of their crude perception. Any assimilation of things to terms between which such classificatory relations, with their remote and mediate transactions, obtain, is a way of bringing the things into a more rational scheme.
Solids in motion are such terms; and the mechanical philosophy is only a way of conceiving nature so as to arrange its items along some of the more natural lines of cleavage of our mental structure.
Other natural lines are the moral and æsthetic relations. Philosophy is still seeking to conceive things so that these relations also may seem to obtain between them.
As long as things have not successfully been so conceived, the moral and æsthetic relations obtain only between _entia rationis_, terms in the mind; and the moral and æsthetic principles remain but postulates, not propositions, with regard to the real world outside.
There is thus a large body of _a priori_ or intuitively necessary truths. As a rule, these are truths of _comparison_ only, and in the first instance they express relations between merely mental terms. Nature, however, acts as if some of her realities were identical with these mental terms. So far as she does this, we can make _a priori_ propositions concerning natural fact. The aim of both science and philosophy is to make the identifiable terms more numerous. So far it has proved easier to identify nature's things with mental terms of the mechanical than with mental terms of the sentimental order.
The widest postulate of rationality is that the world _is_ rationally intelligible throughout, after the pattern of _some_ ideal system. The whole war of the philosophies is over that point of faith. Some say they can see their way already to the rationality; others that it is hopeless in any other but the mechanical way. To some the very fact that there is a world at all seems irrational. Nonentity would be a more natural thing than existence, for these minds. One philosopher at least says that the relatedness of things to each other is irrational anyhow, and that a world of relations can never be made intelligible.[561]
With this I may be assumed to have completed the programme which I announced at the beginning of the chapter, so far as the _theoretic_ part of our organic mental structure goes. It can be due neither to our own nor to our ancestors' experience. I now pass to those practical parts of our organic mental structure. Things are a little different here; and our conclusion, though it lies in the same direction, can be by no means as confidently expressed.
To be as short and simple as possible, I will take the case of instincts, and, supposing the reader to be familiar with