Now whenever we become thus conscious, we may become aware of an additional relation which is of the highest intellectual importance, inasmuch as upon it the whole structure of logic is reared. _The principle of mediate predication or subsumption_ is only the axiom of skipped intermediaries applied to a series of successive predications. It expresses the fact that any earlier term in the series stands to any later term in the same relation in which it stands to any intermediate term; in other words, that _whatever has an attribute has all the attributes of that attribute;_ or more briefly still, that _whatever is of a kind is of that kind's kind._ A little explanation of this statement will bring out all that it involves.
We learned in the chapter on Reasoning what our great motive is for abstracting attributes and predicating them. It is that our varying practical purposes require us to lay hold of different angles of the reality at different times. But for these we should be satisfied to 'see it whole,' and always alike. The purpose, however, makes one aspect essential; so, to avoid dispersion of the attention, we treat the reality as if for the time being it were nothing but that aspect, and we let its supernumerary determinations go. In short, we substitute the aspect for the whole real thing. _For our purpose_ the aspect _can_ be substituted for the whole, and the two treated as the same; and the word _is_ (which couples the whole with its aspect or attribute in the categoric judgment) expresses (among other things) the identifying operation performed. The predication-series _a_ is _b, b_ is _c, c_ is _d_,...closely resembles for certain practical purposes the equation-series _a = b, b = c, c = d_, etc.
But what is our purpose in predicating? Ultimately, it may be anything we please; but proximately and immediately, it is always the gratification of a certain curiosity as to whether the object in hand is or is not _of a kind_ connected with that ultimate purpose. Usually the connection is not obvious, and we only find that the object S is of a kind connected with P, after first finding that it is of a kind M, which itself is connected with P. Thus, to fix our ideas by an example, we have a curiosity (our ultimate purpose being conquest over nature) as to how Sirius may move. It is not obvious whether Sirius is a kind of thing which moves in the line of sight or not. When, however, we find it to be a kind of thing in whose spectrum the hydrogen-line is shifted, and when we reflect that _that_ kind of thing is a kind of thing which moves in the line of sight; we conclude that Sirius does so move. Whatever Sirius's attribute is, Sirius is; its adjective's adjective can supersede its own adjective in our thinking, and this with no loss to our knowledge, _so long as we stick to the definite purpose in view._ Now please note that this elimination of intermediary kinds and transfer of _is_'s along the line, results from our insight into the very meaning of the word _is_, and into the constitution of any series of terms connected by that relation. It has naught to do with what any particular thing is or is not; but, _whatever_ any given thing may be, we see that it also is whatever _that_ is, indefinitely. To grasp in one view a succession of _is_'s is to apprehend this relation between the terms which they connect; just as to grasp a list of successive equals is to apprehend their _mutual_ equality throughout. The principle of mediate subsumption thus expresses relations of ideal objects as such. It can be discovered by a mind left at leisure with any set of meanings (however originally obtained), of which some are predicable of others. The moment we string them in a serial line, that moment we see that we can drop intermediaries, treat remote terms just like near ones, and put a genus in the place of a species. This shows that _the principle of mediate subsumption has nothing to do with the particular order of our experiences, or with the outer coexistences and sequences of terms._ Were it a mere outgrowth of habit and association, we should be forced to regard it as having no universal validity; for every hour of the day we meet things which we consider to be of this kind or of that, but later learn that they have none of the kind's properties, that they _do not_ belong to the kind's kind. Instead, however, of correcting the principle by these cases, we correct the cases by the principle. We say that if the thing we named an M has not M's properties, then we were either mistaken in calling it an M, or mistaken about M's properties; or else that it is no longer M, but has changed. But we never say that it is an M without M's properties; for by conceiving a thing as of the kind M I mean that it _shall_ have M's properties, be of M's kind, even though I should never be able to find in the real world anything which is an M. The principle emanates from my perception of what a lot of successive is's _mean_. This perception can no more be confirmed by one set, or weakened by another set, of outer facts, than the perception that black is not white can be confirmed by the fact that snow never blackens, or weakened by the fact that photographer's paper blackens as soon as you lay it in the sun.
The abstract scheme of successive predications, extended indefinitely, with all the possibilities of substitution which it involves, is thus an immutable system of truth which flows from the very structure and form of our thinking. _If_ any real terms ever do fit into such a scheme, they will obey its laws; _whether_ they do is a question as to nature's facts, the answer to which can only be empirically ascertained. _Formal logic_ is the name of the Science which traces in skeleton form all the remote relations of terms connected by successive _is_'s with each other, and enumerates their possibilities of mutual substitution. To our principle of mediate subsumption she has given various formulations, of which the best is perhaps this broad expression, that _the same can be substituted for the same in any mental operation._[540] The ordinary logical series contains but three terms--"Socrates, man, mortal." But we also have 'Sorites'--Socrates, man, animal, machine, run down, mortal, etc.--and it violates psychology to represent these as syllogisms with terms suppressed. The ground of there being any logic at all is our power to grasp any series as a whole, and the more terms it holds the better. This synthetic consciousness of an uniform direction of advance through a multiplicity of terms is, apparently, what the brutes and lower men cannot accomplish, and what gives to us our extraordinary power of ratiocinative thought. The mind which can grasp a string of _is_'s as a whole--the objects linked by them may be ideal or real, physical, mental, or symbolic, indifferently--can also apply to it the principle of skipped intermediaries. _The logic-list is thus in its origin and essential nature just like those graded classificatory lists which we erewhile described._ The 'rational proposition' which lies at the basis of all reasoning, the _dictum de omni et nullo_ in all the various forms in which it may be expressed, the fundamental law of thought, is thus _only the result of the function of comparison_ in a mind which has come by some lucky variation to apprehend a series of more than two terms at once.[541] So far, then, _both Systematic Classification and Logic are seen to be incidental results of the mere capacity for discerning difference and likeness,_ which capacity is a thing with which the _order of experience_, properly so styled, has absolutely nothing to do.
* * * * * But how comes it (it may next be asked) when systematic classifications have so little ultimate theoretic importance--for the conceiving of things according to their mere degrees of resemblance always yields to other modes of conceiving when these can be obtained--that the logical relations among things should form such a mighty engine for dealing with the facts of life?
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