CHAPTER X
SINGULAR SOLUTIONS
"I have made out what puts the whole subject of Singular Solutions into a state of Unity."
All metaphysicians have known of the existence of a law of antithesis or contradiction in Thought. "We form the conception of man by contrast with that of not-man; we know good by seeing it against a background of evil; the child learns to realize its own personality by coming in contact with things that are not itself." All this is the commonplace of metaphysics. But modern mathematics is leading to the perception that we need something more definite than to see x against a vague background of not-x; we need the special and definite antithetic conception to x within the particular "Universe of Thought" to which x properly belongs. And often we have to investigate a class of things as forming a part of more than one Universe of Thought, and contrast it with not-x in each. As, for instance, we may clear up our conception of "black sheep" by contrasting it, first, with "sheep that are not black"; secondly, with "black objects that are not sheep"; and so on. The Differential Calculus, however, shows that the mind is feeling after even a more definite mode of contrast than that indicated above. We cannot truly know the Law which governs an organic group of phenomena, unless we allow ourselves to conceive the idea of a Singular Solution of that Law, i.e., an individual which is constant in some relations as to which the other individuals are variable, and variable in some relations as to which the others are constant.
Now the so-called Messianic Seers are persons in whom is developed to a high degree the tendency to study ordinary human beings by the method of contrast with an Ideal Man, a sort of Singular Solution of Humanity; of whom they speak as constant in those relations to the Unseen in which we are variable, and fluent in those relations to the visible as to which each of us has his fixed prejudices. This Ideal Man, they assert, is the indispensable Mediator between them and any true knowledge of Humanity. When they have jumped to the conclusion that their need of conceiving an Ideal Man is any proof of the actual existence of such an Ideal, visible or Invisible, they err, of course. But, nevertheless, these Messianic Seers are neither dreamers nor maniacs; they are revealing, in the domain of emotion, a Law which the mathematical doctrine of Singular Solutions reveals to the intellect, viz. that the process of conceiving an antithetical exception is a necessary medium between the mind of man and any very thorough comprehension of the Laws which govern Humanity.
The study of the phenomenon of the Messianic Seerhood became the absorbing interest of George Boole's later years. The chapter on Singular Solutions in his text-book of Differential Equations contains much genuine metaphysical truth expressed in mathematical terminology. He made a special study of the sermons of the strange preacher, F. D. Maurice; and in the last year of his life wrote: "I have made out what puts the whole subject of Singular Solutions into a state of Unity." What strange pulsation of thought must have thrilled from the mind of the great Seer of Messianic Singular Solution, into that of the logician of the Unity, before those words were written, who can guess? The MS. on which George Boole was engaged at the time lies undeciphered in the archives of the Royal Society; no mere mathematician can understand it: and no theologian cares to try. Perhaps it may be deciphered yet; when further progress has been made in knowledge of the correspondence between the normal action of the human mind and its automatic expression by means of notations.