To this we may add, that there has gradually been built up a fine system of unambiguous symbols, and it is possible for a man to know just what he is dealing with.

Thus, a certain beaten path has been attained, and a man may travel this very well without having forced on his attention the problems of reflective thought.  The knowledge of numbers with which he starts is sufficient equipment with which to undertake the journey.  That one is on the right road is proved by the results one obtains.  As a rule, disputes can be settled by well-tried mathematical methods.

There is, then, a common agreement as to initial assumptions and methods of work, and useful results are attained which seem to justify both.  Here we have the normal characteristics of a special science.

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:—­