and the methods of
determining the numerical relations of things are gradually developed
before his eyes, beginning with operations of great simplicity.
Moreover, verification is possible, and within certain limits
verification by direct inspection.

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 th