Even as thus expanded, the definition yet excludes many a mathematical realm of vast, nay, infinite extent. Consider, for that immense class of things known as opera tions. These are limitless alike in number and in kind. Now it so happens that there are systems of operations such that any two opera tions of a given system, if thought as follow ing one another, together thus produce the same effect as some other single operation of the system. For an illustration, think of all possible straight motions in space. The oper ation of going from a point A to a point B, followed by the operation of going from B to a point C,' is equivalent to the single operation of going from A to C. Thus the system of such operations is a closed system: combina tion of any two operations yields a third not without hut within the system. Now the theory of such closed systems — called groups (see GROUPS) of operations—is a mathematical theory of colossal proportions. But it is obvi ous that an abstract operation, though a very real thing, is neither a position nor a magnitude.
This way of trying to come at an adequate conception of mathematics, viz., by attempt ing to characterize in succession its distinct domains, or varieties of content, or modes of activity, is not likely to prove successful. For it demands an exhaustive enumeration, not only of the fields now occupied by the science, but also of those destined to be conquered by it in the future, and such an achievement would require a prevision that none may claim.
Fortunately there are other paths of ap proach that seem more promising. Every one has observed that mathematics, whatever it may be, possesses a certain mark, namely, a de gree of certainty not found elsewhere. So it is, proverbially, the exact science par excel lence. Exact, no doubt, but in what sense? To this an excellent answer is found in a defi nition of the science given about one gener ation ago by a distinguished American mathe matician, Prof. Benjamin Pierce: mathematics is the science which draws necessary conclu sions — a formulation of like significance with the fine mot by Prof. William Benjamin Smith, to *wit: mathematics is the universal art apo dictic. These statements, though neither of them may be entirely adequate, are, both of them, telling approximations, at once foreshad owing and neatly summarizing for popular use the conclusion reached by the creators of mod ern logic (see SYMBOLIC LoGic), that mathe matics is included in, and, in a profound sense, may be said to be identical with, symbolic logic. Observe that the emphasis falls on the equality of being enecessary,s or logically correct. Naught is said about the conclusions being true. That is another matter for subsequent consideration.
But why are mathematical conclusions cor rect? Is it that the mathematician has a rea soning faculty essentially different in kind from that of other men? By no means. What,
then, is the secret? 'Reflect that conclusion im plies premises, that premises involve terms, that terms stand for ideas or concepts or no tions, and that these latter are the ultimate material with which the spiritual architect, called the reason, designs and builds. Here, then, one may expect to find light. The apo dictic quality of mathematical thought, the cor rectness of its conclusions, are due, not to any special mode of ratiocination, but to the char acter of the concepts with which it deals. What is that distinctive characteristic? The answer is: precision, sharpness, completeness of determination. But how comes the mathe matician by such completeness? There is no mysterious trick involved: some ideas admit of such precision and completeness of deter mination, others do not; and the mathemati cian is one who deals with those that do. Law, says Blackstone, is a rule of action pre scribed by the supreme power of a state com manding what is right and prohibiting what is wrong. But what are a state, and supreme power, and right and wrong? If all such terms admitted of complete determination as do, for example, such terms as triangle and circle, then the science of law would be a branch of pure mathematics. And such, too, to take another example, would be psychology, were conscious ness, mind, perception, imagination, and all kindred terms, as completely determinable as the notion sphere. It will be asked, does not the lawyer sometimes arrive at correct con clusions? It may be admitted that he does sometimes, and so, too, of the psychologist or historian or sociologist. When this happens, however, when these students arrive, it is not meant at truth, for that may be by happy chance or by intuition, but when, strictly speaking, they arrive at conclusions that are correct, at conclusions, i.e., that follow log ically from completely ascertained data or premises, then that is because they have been for the time acting in all literalness the part of mathematician. That is not to aggrandize the science of mathematics. Rather is it for credit to all serious thinking that, in any con siderable garment of thought, one may find here and there, rarely enough sometimes, a mathetic fiber, woven in some perhaps excep tional moment of precise conception and rigor ous reason. To think aright is no character istic striving of a class of men. It is a com mon aspiration. Only, as before said, the stuff of thought is mostly intractable, formless, nebu lous, like some milky way awaiting analysis into distinct star forms of completely determi nate ideas.