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CHAPTER XI
ALGEBRAIZERS

"The Evidence of Things not seen."

Much of the tragedy of religious conflict is due to lack of comprehension of the process of Algebraization

I have pointed out how a question which seemed, on its own level, hopelessly unsolvable, is often solved at once by reference to an order of considerations higher than that involved in the question itself. This is especially the case when a Law of Thought is appealed to, to settle a discussion about things. Laws of things we cannot be said really to know, except in a fumbling and empirical manner; when we have true knowledge, it is because we have discovered the Law of Thought which presided at the Genesis of the Things. I wish to speak in all humility and reverence; but I cannot say less than I mean. We get clues, suggestions of Laws of Thought, by studying things; but whenever we truly know, what we know is a Law of Thought, which we have arrived at by discharging from our observation of particular finite things all that made them finite and particular. The elementary geometrician who first conceived, the idea of the circle caught his suggestion from looking at things whose forms were approximately round; but, as soon as he had discovered the law of roundness within his own mind, he was able to express roundness in a new material, to state it generally (by scratching it on the sand) in a manner which afforded no clue to the objects from which the suggestion had come to him. And the law of circularity, thus formulated, was henceforth master within him, and governed his appreciation of things. He did not test his Ideal circle by comparing it with the sun or with an apple; on the contrary, he tested the circularity of a fruit by comparing it with his abstract or Ideal circle. His circle then was an algebraization of the round outline of the sun or moon, or of a fruit.

In the same way we discover a law of number, first, by thinking of some particular numbers; but as soon as we know the Law, we can state it Algebraically, i.e., in a manner which conveys no information as to what were the particular numbers of which we happened to be thinking when we discovered it. The particular-numbers suggested the law to our consciousness; they do not prove it to our reason. When once it has been suggested, it carries its own evidence, independently of particular numbers. And as soon as we have formulated a Law thus algebraically, it is henceforth master within us. Particular statements about number are referred to it; and our opinion as to the truth of those statements is controlled by it. For instance, the law that one number multiplied by a second always comes to the same result as is obtained by multiplying the second by the first was of course suggested to the consciousness first by the observation of some particular pairs of numbers; but it is not proved by reference to any special numbers; it is general and algebraic. And no student thoroughly understands it as a law of number until he understands it in its algebraic statement: ab=ba. As soon as he understands the algebraic statement, it becomes master of his thought.

A similar process has been going on throughout the history of religion. For instance, the idea of God as Father suggested itself first in connection with the love of animals or men for their progeny. But the latter contains no proof of the former. The idea is suggested to the consciousness by particular facts; but those who understand it at all, accept it on its own evidence; and forthem it henceforth governs their ideas of particularfacts.

Superstition has always been trying to cumber our ideas about God with the pomposity, or vengefulness, or jealousy, or whatever vice happened to prevail in the parents of the particular era. Philosophy has always been trying to algebraize our conception of Fatherhood; to present the normal or Ideal Father; and to induce human fathers to conform their conduct to this Ideal.

The difference between our imperfect and fumbling knowledge of things, and our absolute and perfect knowledge about Laws of Thought, has been thus described:—

"The general Laws of Nature are not, for the most part, immediate objects of perception. They are either inductive inferences from a large body of facts,[1] the common truth in which they express, or, in their origin at least, physical hypotheses of a causal nature, serving to explain phenomena and to predict new combinations of them. They are in all cases, and in the strictest sense of the term, probable conclusions; approaching, indeed, ever and ever nearer to certainty, as they receive more and more of the confirmation of experience; but of the character of probability, in the strict and proper sense of that term, they are never wholly divested. On the other hand, the knowledge of the laws of mind does not require as its basis any extensive collection of observations. The general truth is seen in the particular instance, and it is not confirmed by the repetition of instances. That formula of reasoning which is called the dictum of Aristotle de omni et nullo expresses a general truth in Logic; now that truth is made manifest in all its generality by reflection upon a single instance of its application. And this is both an evidence that the particular principle or formula in question is founded upon some general law or laws of the mind, and an illustration of the doctrine that the perception of such general truths is not derived from an induction from many instances, but is involved in the clear apprehension of a single instance. In connection with this truth is seen the not less important one that our knowledge of the laws upon which the science of the intellectual powers rests, whatever may be its extent or its deficiency, is not probable knowledge. For we not only see in the particular example the general truth, but we see it as a certain truth, — a truth our confidence in which will not continue to increase with increasing experience of its verifications. . . . Shall we then err in regarding that as the true science of Logic which, laying down certain elementary laws, confirmed by the very testimony of the mind, permits us then to deduce, by uniform processes, the entire chain of its secondary consequences, and furnishes, for its practical applications, methods of perfect generality ? "[2]

The author has omitted to notice one fact which, from his point of view, probably seemed too self-evident to be worth mentioning, but the ignoring of which causes much confusion in Science; viz. that whereas he who is teaching a Law of things must illustrate his remarks by reference to actual facts, or his whole argument falls to the ground, a principle of Algebraization can just as well be illustrated by reference to imaginary as to real occurrences. When the law of circularity has once been suggested to the mind, it matters nothing whether the object which suggested it was a real or a painted fruit; nor does it matter how nearly it approximated to the circular form. In the same way, the spiral law of thought-progress is equally well illustrated by the narrative of Abraham, whether it be a narration of actual facts or an imaginary tale.

The kind of confusion which is caused in literature by this principle not being understood, will be described in a future Chapter.

There was a time when even mathematical algebraization was considered impious and dangerous. We have grown accustomed to it now. But the algebraizers of the moral world are still the objects of the hatred of other classes of men (and that is by no means the worst part of their destiny, as we shall presently see). Ordinarily a tolerably strong antagonism is kept up between three classes : the supporters of convention, who desire to make some sort of practical life possible, and who object to whatever disturbs popular ideas; the so-called Free-thinkers, who refuse to acknowledge the validity of convention, and who do not believe in the possibility of knowing the true Law of any human relation; the Idealists, who create an ideal moral code according to some standard agreeable to their feelings. These three classes, in ordinary times, hate each other with very sufficient cordiality. There is a fourth class—honest cautious men who have picked up their notions of Logic from the study of what is called "Natural Science," and who imagine that no Law can be truly known unless it be generalized from a large number of instances. The Algebraizer is a mark for the sneers of all four classes. To the conventional he seems to be defying Law, because he ignores rules made in ignorance of the the true Law. To the self-styled Free-thinkers he seems to be attacking freedom, because he asserts that there is a Law which cannot be broken, and which will avenge attempts to ignore it. To Idealists he seems to be extinguishing Light, because their fancy-lamps fade when he dazzles them by opening the shutters and revealing the Sun. And the so-called "scientific" despise him as rash, because he illustrates his meaning by reference to only one or two instances, which (as matters of historical fact) may be of doubtful authenticity. He can never do right. If he speaks, he is arrogant, in that he professes to know more than other men; if he keeps silence, he is contemptuous and proud. He longs to give freely that wonder of joy which has been freely given to him. He longs to give freely, but Humanity will not have his gift; yet it reproaches him for hardness, in that he takes no trouble to slake its thirst for the living Truth. If he be of philosophic temperament, he retires into his study, and confides to his wife, or perhaps to some favourite dog or cat, refined satires about the absurd inconsistency of mankind. But if he be tender-hearted, his heart breaks, and he dies in despair. O! Jerusalem, that stonest the Prophets!

It may be, however, that he loves the Unseen Revealer more than either abstract Science or concrete Humanity. If so, he is preserved from both cynicism and despair. He possesses his soul in patience until it pleases God to make him understood.

In any case, the worst of his doom is still to come. After his death some follower distorts his meaning, so as to make it palatable to the unthinking masses; and uses his name to trample out truth revealed to his successors. An algebraizer of our own day used often to say: "One would accept being crucified; it does not last long; but to have one's words made into an excuse for trampling down the truth revealed through other Seers, that is a doom to make the stoutest heart quail." Of this awful torture Jesus has had eighteen centuries already; and Moses thirty-three. O! Jerusalem, that buildest the tombs of the Prophets, saying: "If we had lived in our fathers' days, we would not have partaken of their crimes."

  1. The author means "our mode of stating them are."
  2. Boole, Laws of Thought, ch. i. §§4, 5.