For this reason it is always a false step in mathematical science, a step over its brink into the abyss beyond, when we try to reduce its elements to anything not essentially sensible. Intuition must continue to furnish the subject of discourse, the axioms, and the ultimate criteria and sanctions. Calculation and transmutation can never make their own counters or the medium in which they move. So that space, number, continuity, and every other elementary intuition remains at bottom opaque—opaque, that is, to mathematical science; for it is no paradox, but an obvious necessity, that the data of a logical operation should not be producible by its workings. Reason would have nothing to do if it had no irrational materials. Saint Augustine’s rhetoric accordingly covered—as so often with him—a profound truth when he said of time that he knew what it was when no one asked him, but if any one asked him he did not know; which may be restated by saying that time is an intuition, an aspect of crude experience, which science may work with but which it can never arrive at.
Constancy and progress in intent.
When a concretion is formed in discourse and an intent is attained in consciousness, predicates accrue to the subject in a way which is perfectly empirical. Dialectic is not retrospective; it does not consist in recovering ground previously surveyed. The accretion of new predicates comes in answer to chance questions, questions raised, to be sure, about a given theme. The subject is fixed by the mind’s intent and it suffices to compare any tentative assertion made about it with that intent itself to see whether the expression suggested for it is truly dialectical and thoroughly honest. Dialectic verifies by reconsideration, by equation of tentative results with fixed intentions. It does not verify, like the sciences of existence, by comparing a hypothesis with a new perception. In dialectic no new perception is wanted; the goal is to understand the old fact, to give it an aureole and not a progeny. It is a transubstantiation of matter, a passage from existence to eternity. In this sense dialectic is “synthetic a priori”; it analyses an intent which demanded further elucidation and had fixed the direction and principle of its expansion. If this intent is abandoned and a new subject is introduced surreptitiously, a fallacy is committed; yet the correct elucidation of ideas is a true progress, nor could there be any progress unless the original idea were better expressed and elicited as we proceeded; so that constancy in intent and advance in explication are the two requisites of a cogent deduction.
The question in dialectic is always what is true, what can be said, about this; and the demonstrative pronoun, indicating an act of selective attention, raises the object it selects to a concretion in discourse, the relations of which in the universe of discourse it then proceeds to formulate. At the same time this dialectical investigation may be full of surprises. Knowledge may be so truly enriched by it that knowledge, in an ideal sense, only begins when dialectic has given some articulation to being. Without dialectic an animal might follow instinct, he might have vivid emotions, expectations, and dreams, but he could hardly be said to know anything or to guide his life with conscious intent. The accretions that might come empirically into any field of vision would not be new predicates to be added to a known thing, unless the logical and functional mantle of that thing fell upon them and covered them. While the right of particulars to existence is their own, granted them by the free grace of heaven, their ability to enlarge our knowledge on any particular subject—their relevance or incidence in discourse—hangs on their fulfilling the requirements which that subject’s dialectical nature imposes on all its expressions.
Intent determines the functional essence of objects.
It is on this ground, for instance, that the image of a loaf of bread is so far from being the loaf of bread itself. External resemblance is nothing; even psychological derivation or superposition is nothing; the intent, rather, which picks out what that object’s function and meaning shall be, alone defines its idea; and this function involves a locus and a status which the image does not possess. Such admirable iridescence as the image might occasionally put on—in the fine arts, for instance—would not constitute any iridescence or transformation in the thing; nor would identity of aspect preserve the thing if its soul, if its utility, had disappeared. Herein lies the ground for the essential or functional distinction between primary and secondary qualities in things, a distinction which a psychological scepticism has so hastily declared to be untenable. If it was discovered, said these logicians, that space was perceived through reading muscular sensations, space, and the muscles too, were thereby proved to be unreal. This remarkable sophism passed muster in the philosophical world for want of attention to dialectic, which might so easily have shown that what a thing means is spatial distinction and mechanical efficacy, and that the origin of our perceptions, which are all equally bodily and dependent on material stimulation, has nothing to do with their respective claims to hypostasis. It is intent that makes objects objects; and the same intent, defining the function of things, defines the scope of those qualities which are essential to them. In the flux substances and shadows drift down together; it is reason that discerns the difference.
Purposes need dialectical articulation as much as essences do, and without an articulate and fixed purpose, without an ideal, action would collapse into mere motion or conscious change. It is notably in this region that elucidation constitutes progress; for to understand the properties of number may be less important than empirically to count; but to see and feel the values of things in all their distinction and fulness is the ultimate fruit of efficiency; it is mastery in that art of life for which all the rest is apprenticeship. Dialectic of this sort is practised intuitively by spiritual minds; and even when it has to be carried on argumentatively it may prove very enlightening. That the excellence of courage is identical with that of wisdom still needs to be driven home; and that the excellence of poetry is identical with that of all other things probably sounds like a blind paradox. Yet did not all excellences conspire to one end and meet in one Life of Reason, how could their relative value be estimated, or any reflective sanction be found for them at all? The miscellaneous, captious fancies of the will, the menagerie of moral prejudices, still call for many a Socrates to tame them. So long as courage means a grimace of mind or body, the love of it is another grimace. But if it meant the value, recognisable by reason and diffused through all life, which that casual attitude or feeling might have, then we should be launched upon the quest for wisdom.
The want of integration in moral views is like what want of integration would be in arithmetic if we declared that it was the part of a man and a Christian to maintain that my two equals four or that a green fifteen is a hundred. These propositions might have incidental lights and shades in people’s lives to make them plausible and precious; but they could not be maintained by one who had clarified his intent in naming and adding. For then the arithmetical relations would be abstracted, and their incidental associates would drop out of the account. So a man who is in pursuit of things for the good that is in them must recognise and (if reason avails) must pursue what is good in them all. Strange customs and unheard-of thoughts may then find their appropriate warrant; just as in higher mathematical calculations very wonderful and unforeseen results may be arrived at, which a man will not accept without careful reconsideration of the terms and problem before him; but if he finds the unexpected conclusion flowing from those premises, he will have enlarged his knowledge of his art and discovered a congenial good. He will have made progress in the Socratic science of knowing his own intent.
Double status of mathematics.