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.