An Introduction to Philosophy

by George Stuart Fullerton

No, the imperceptible world of atoms and molecules is not by any means totally different from the world of things in which the plain man lives. These little objects and groups of objects are discussed very much as we discuss the larger objects and groups of objects to which we are accustomed. We are still concerned with things which exist in space and move about in space; and even if these things are small and are not very familiarly known, no intellectual revolution is demanded to enable a man to understand the words of the scientist who is talking about them, and to understand as well the sort of reasonings upon which the doctrine is based.

9. MATHEMATICS.—Let us now turn to take a glance at the mathematical sciences. Of course, these have to do with things sooner or later, for our mathematical reasonings would be absolutely useless to us if they could not be applied to the world of things; but in mathematical reasonings we abstract from things for the time being, confident that we can come back to them when we want to do so, and can make use of the results obtained in our operations.

Now, every civilized man who is not mentally deficient can perform the fundamental operations of arithmetic. He can add and subtract, multiply and divide. In other words, he can use numbers. The man who has become an accomplished mathematician can use numbers much better; but if we are capable of following intelligently the intricate series of operations that he carries out on the paper before us, and can see the significance of the system of signs which he uses as an aid, we shall realize that he is only doing in more complicated ways what we have been accustomed to do almost from our childhood.

If we are interested, not so much in performing the operations, as in inquiring into what really takes place in a mind when several units are grasped together and made into a new unit,—for example, when twelve units are thought as one dozen,—the mathematician has a right to say: I leave all that to the psychologist or to the metaphysician; every one knows in a general way what is meant by a unit, and knows that units can be added and subtracted, grouped and separated; I only undertake to show how one may avoid error in doing these things.

It is with geometry as it is with arithmetic. No man is wholly ignorant of points, lines, surfaces, and solids. We are all aware that a short line is not a point, a narrow surface is not a line, and a thin solid is not a mere surface. A door so thin as to have only one side would be repudiated by every man of sense as a monstrosity. When the geometrician defines for us the point, the line, the surface, and the solid, and when he sets before us an array of axioms, or self-evident truths, we follow him with confidence because he seems to be telling us things that we can directly see to be reasonable; indeed, to be telling us things that we have always known.

The truth is that the geometrician does not introduce us to a new world at all. He merely gives us a fuller and a more exact account than was before within our reach of the space relations which obtain in the world of external objects, a world we already know pretty well.

Suppose that we say to him: You have spent many years in dividing up space and in scrutinizing the relations that are to be discovered in that realm; now tell us, what is space? Is it real? Is it a thing, or a quality of a thing, or merely a relation between things? And how can any man think space, when the ideas through which he must think it are supposed to be themselves non-extended? The space itself is not supposed to be in the mind; how can a collection of non-extended ideas give any inkling of what is meant by extension?

Would any teacher of mathematics dream of discussing these questions with his class before proceeding to the proof of his propositions? It is generally admitted that, if such questions are to be answered at all, it is not with the aid of geometrical reasonings that they will be answered.

10. THE SCIENCE OF PSYCHOLOGY.—Now let us come back to a science which has to do directly with things. We have seen that the plain man has some knowledge of minds as well as of material things. Every one admits that the psychologist knows minds better. May we say that his knowledge of minds differs from that of the plain man about as the knowledge of plants possessed by the botanist differs from that of all intelligent persons who have cared to notice them? Or is it a knowledge of a quite different kind?

Those who are familiar with the development of the sciences within recent years have had occasion to remark the fact that psychology has been coming more and more to take its place as an independent science. Formerly it was regarded as part of the duty of the philosopher to treat of the mind and its knowledge; but the psychologist who pretends to be no more than a psychologist is a product of recent times. This tendency toward specialization is a natural thing, and is quite in line with what has taken place in other fields of investigation.