We must not forget, however, that, even in the mathematical sciences, before a beaten path was attained, disputes as to the significance of numbers and the cogency of proofs were sufficiently common. And we must bear in mind that even to-day, where the beaten path does not seem wholly satisfactory, men seem to be driven to reflect upon the significance of their assumptions and the nature of their method.
Thus, we find it not unnatural that a man should be led to ask; What is a minus quantity really? Can anything be less than nothing? or that he should raise the questions: Can one rightly speak of an infinite number? Can one infinite number be greater than another, and, if so, what can greater mean? What are infinitesimals? and what can be meant by different orders of infinitesimals?
He who has interested himself in such questions as these has betaken himself to philosophical reflection. They are not answered by employing mathematical methods.
Let us now turn to logic. And let us notice, to begin with, that it is broader in its application than the mathematical sciences. It is concerned to discover what constitutes evidence in every field of investigation.
There is, it is true, a part of logic that may be developed somewhat after the fashion of mathematics. Thus, we may examine the two statements: All men are mortal, and Caesar is a man; and we may see clearly that, given the truth of these, we must admit that Caesar is mortal. We may make a list of possible inferences of this kind, and point out under what circumstances the truth of two statements implies the truth of a third, and under what circumstances the inference cannot be made. Our results can be set forth in a system of symbols. As in mathematics, we may abstract from the particular things reasoned about, and concern ourselves only with the forms of reasoning. This gives us the theory of the syllogism; it is a part of logic in which the mathematician is apt to feel very much at home.
But this is by no means all of logic. Let us consider the following points:—
(1) We are not concerned to know only what statements may be made on the basis of certain other statements. We want to know what is true and what is false. We must ask: Has a man the right to set up these particular statements and to reason from them? That some men accept as true premises which are repudiated by others is an undoubted fact. Thus, it is maintained by certain philosophers that we may assume that any view of the universe which is repellant to our nature cannot be true. Shall we allow this to pass unchallenged? And in ethics, some have held that it is under all circumstances wrong to lie; others have denied this, and have held that in certain cases—for example, to save life or to prevent great and unmerited suffering—lying is permissible. Shall we interest ourselves only in the deductions that each man makes from his assumed premises, and pay no attention to the truth of the premises themselves?
(2) Again. The vast mass of the reasonings that interest men are expressed in the language that we all use and not in special symbols. But language is a very imperfect instrument, and all sorts of misunderstandings are possible to those who express their thoughts in it.
Few men know exactly how much is implied in what they are saying. If I say: All men are mortal, and an angel is not a man; therefore, an angel is not mortal; it is not at once apparent to every one in what respect my argument is defective. He who argues: Feathers are light; light is contrary to darkness; hence, feathers are contrary to darkness; is convicted of error without difficulty. But arguments of the same kind, and quite as bad, are to be found in learned works on matters less familiar to us, and we often fail to detect the fallacy.
Thus, Herbert Spencer argues, in effect, in the fourth and fifth chapters of his "First Principles," as follows:—