In Chapter XII we saw that the mind can at successive moments mean the same, and that it gradually comes into possession of a stock of permanent and fixed meanings, ideal objects, or conceptions, some of which are universal qualities, like the black and white of our example, and some, individual things. We now see that not only are the objects permanent mental possessions, but the results of their comparison are permanent too. The objects and their differences together form an immutable system. The same objects, compared in the same way, always give the same results; if the result be not the same, then the objects are not those originally meant.
This last principle, which we may call the axiom of constant result, holds good throughout all our mental operations, not only when we compare, but when we add, divide, class, or infer a given matter in any conceivable way. Its most general expression would be "the Same operated on in the same way gives the Same." In mathematics it takes the form of "equals added to, or subtracted from, equals give equals," and the like. We shall meet with it again.
The next thing which we observe is that the operation of comparing may be repeated on its own results; in other words, that we can think of the various resemblances and differences which we find and compare them with each other, making differences and resemblances of a higher order. The mind thus becomes aware of sets of similar differences, and forms series of terms with the same kind and amount of difference between them, terms which, as they succeed each other, maintain a constant direction of serial increase. This sense of constant direction in a series of operations we saw in Chapter XIII (p. 490) to be a cardinal mental fact. "A differs from B differs from C differs from D, etc.," makes a series only when the differences are in the same direction. In any such difference-series all terms differ in just the same way from their predecessors. The numbers 1, 2, 3, 4, 5,... the notes of the chromatic scale in music, are familiar examples. As soon as the mind grasps such a series as a whole, it perceives that two terms taken far apart differ more than two terms taken near together, and that any one term differs more from a remote than from a near successor, and this no matter what the terms may be, or what the sort of difference may be, provided it is always the same sort.
This principle of mediate comparison might be briefly (though obscurely) expressed by the formula "more than the more is more than the less"—the words more and less standing simply for degrees of increase along a constant direction of differences. Such a formula would cover all possible cases, as, earlier than early is earlier than late, worse than bad is worse than good, east of east is east of west; etc., etc., ad libitum.[538] Symbolically, we might write it as a < b < c < d.... and say that any number of intermediaries may be expunged without obliging us to alter anything in what remains written.
The principle of mediate comparison is only one form of a law which holds in many series of homogeneously related terms, the law that skipping intermediary terms leaves relations the same. This axiom of skipped intermediaries or of transferred relations occurs, as we soon shall see, in logic as the fundamental principle of inference, in arithmetic as the fundamental property of the number-series, in geometry as that of the straight line, the plane and the parallel. It seems to be on the whole the broadest and deepest law of man's thought.
In certain lists of terms the result of comparison may be to find no-difference, or equality in place of difference. Here also intermediaries may be skipped, and mediate comparison be carried on with the general result expressed by the axiom of mediate equality, "equals of equals are equal," which is the great principle of the mathematical sciences. This too as a result of the mind's mere acuteness, and in utter independence of the order in which experiences come associated together. Symbolically, again: a = b = c = d..., with the same consequence as regards expunging terms which we saw before.