On Sameness and Identity: A Psychological Study / Being a Contribution to the Foundations of a Theory of Knowledge

by George Stuart Fullerton

Now it is very evident that Dr. McCosh, in his anxiety to prove an extra-mental world, is actuated by a desire to retain real things. He is under the impression that, unless the extra-mental is known, our knowledge is confined to shadows and unrealities. He combats the Idealist, because he supposes him to deny the body of which we are conscious; whereas, all that the Idealist is denying (if he be consistent with his principles) is the hypothetical representative of the body, assumed to exist within the body, and to which consciousness does not testify. It is this that is the unreality. The body to which the Idealist holds is the very body to which Dr. McCosh thinks consciousness testifies; but this body is not beyond consciousness, nor in any proper sense of the words extra-mental. The above argument for the extra-mental is consequently due to a misconception—to the misconception that the body revealed by consciousness is the extra-mental body, and that the only body left to an Idealist is an unreal phantom of this body, and distinct from it. And it is the attempt to make this body revealed by consciousness both in mind and out of mind that has occasioned the difficulties and inconsequences of the reasoning I have quoted. This attempt is due to a confusion of sameness in sense seventh with sameness in sense first. My excuse for so minute a criticism of this plainly untenable position is that we have here a representative instance of an error quite common, and indeed characteristic of a certain stage of reflection.

Sec. 36. The last confusion of samenesses that I shall discuss lies at the bottom of the common opinion on the infinite divisibility of space, and causes the antinomies which arise from it. The position I shall criticize is well set forth in Professor W. K. Clifford's popular lecture entitled "Of Boundaries in General."[72] From this I take a few passages which will suffice to illustrate his doctrine.

"Now the idea expressed by that word continuous is one of extreme importance; it is the foundation of all exact science of things; and yet it is so very simple and elementary that it must have been almost the first clear idea that we got into our heads. It is only this: I cannot move this thing from one position to another, without making it go through an infinite number of intermediate positions. Infinite; it is a dreadful word, I know, until you find out that you are familiar with the thing which it expresses. In this place it means that between any two positions there is some intermediate position; between that and either of the others, again, there is some other intermediate; and so on without any end. Infinite means without any end. If you went on with that work of counting forever, you would never get any further than the beginning of it. At last you would only have two positions very close together, but not the same; and the whole process might be gone over again, beginning with those as many times as you like."

* * * * "When a point moves, it moves along some line; and you may say that it traces out or describes the line. To look at something definite, let us take the point where this boundary of red on paper is cut by the surface of water. I move all about together. Now you know that between any two positions of the point there is an infinite number of intermediate positions. Where are they all? Why, clearly, in the line along which the point moved. That line is the place where all such points are to be found."

* * * * "It seems a very natural thing to say that space is made up of points. I want you to examine very carefully what this means, and how far it is true. And let us first take the simplest case, and consider whether we may safely say that a line is made up of points. If you think of a very large number—say, a million—of points all in a row, the end ones being an inch apart; then this string of points is altogether a different thing from a line an inch long. For if you single out two points which are next one another, then there is no point of the series between them; but if you take two points on a line, however close together they may be, there is an infinite number of points between them. The two things are different in kind, not in degree."

* * * * "When a point moves along a line, we know that between any two positions of it there is an infinite number (in this new sense[73]) of intermediate positions. That is because the motion is continuous. Each of those positions is where the point was at some instant or other. Between the two end positions on the line, the point where the motion began and the point where it stopped, there is no point of the line which does not belong to that series. We have thus an infinite series of successive positions of a continuously moving point, and in that series are included all the points of a certain piece of line-room. May we say then that the line is made up of that infinite series of points?

"Yes; if we mean no more than that the series makes up the points of the line. But no, if we mean that the line is made up of those points in the same way that it is made up of a great many very small pieces of line. A point is not to be regarded as a part of a line, in any sense whatever. It is the boundary between two parts."

These extracts suffice, I think, to show what the common doctrine is, and to show also the unavoidable difficulties connected with it. These were clearly seen long ago. Motion, argues Zeno of Elea,[74] cannot begin, because a body in motion must pass through an infinite number of intermediate places before it can arrive at any other place. Achilles can never overtake the tortoise, for by the time that he has reached the place where it was, it has always moved a little beyond. If Professor Clifford could not move a thing from one position to another, without making it go though an infinite number of intermediate positions, if these positions must be gone through with successively, and if infinite really mean without any end, then the final member of the series could never have been reached, for the plain reason that there is no final member to an endless series. If the new position is reached without passing through every member of the series and leaving none farther to pass through, it is not reached by passing through an infinite number of intermediate positions. The difficulty here is a hopeless one; either the series has a final member, and then it is not infinite; or it has not, and then one cannot come to the end.

The attempt sometimes made to avoid this difficulty by calling upon a precisely similar one for aid is of not the least avail. The time of the motion, it is said, is divisible just as is the space over which the body moves; the spaces and the times then vary together, and as the spaces become very small the times become very small; infinitesimal spaces are passed over in infinitesimal times, and all these infinitesimals are included in the finite space and finite time of the motion. But if there be a difficulty in arriving at the end of an endless series of places or positions, there is surely no less a difficulty in reaching the end of an endless series of times. If the series of times to be successively exhausted be truly endless, then an end of the motion can never be reached. Quibbling over the size of the members of the series in the case of either space or time is useless. Whether things are big or little, if the supply of them is truly endless, one can never get to the end of the supply. The rapidity with which they are exhausted has nothing to do with the question, for an increase in rapidity has obviously no effect in facilitating an approach to what is assumed not to exist, a final term. It is, then, perfectly clear that, if, in order to move a body, I must come to the end of an endless series, I may reasonably conclude that I cannot move a body. Granting the assumption upon which it is based, Zeno's argument is unanswerable. It is not a question of an ordinary difficulty, a trifling evil; it is a question of an impossibility, a flat contradiction; to move an inch, to endure for a minute, one is to accomplish the feat of reaching the end of the endless. One thing is quite certain; no rival doctrine can present a greater difficulty.

It is possible that some one may wish to find a way out of this difficulty by distinguishing, as Clifford has done, between the points of the line and the parts of the line. But this distinction is of no service. All these points are declared to be on the line, and anything that passes over the whole line must exhaust them one by one until it arrives at the final point. By hypothesis, there is no final point to the series—the series is without any end. Unless, then, the line can be passed over without passing over the points, there would seem to be no help in turning to line pieces. Moreover, it appears reasonable to assume that there are as many parts to the line as there are points. For all these points are on the line, and no two of them are in precisely the same position on the line; they must consequently be on different parts of the line. If it be objected that, having no extension, they cannot properly be said to be on parts of the line, I answer that, even on this hypothesis, they must be at different parts of the line, in order to be distinguished from each other. The part of the line between any two of them is certainly not the same as the part between any other two. It follows that the number of parts of which the line is made up is at least as great as the number of points less one, if we refuse to say that the points are on the line; and is as great as the number of points, if we are willing to say that they are on the line. To move over the whole line, then, a point must come within one term of the end of an endless series, or it must pass over an endless number of small pieces of line until it comes to the very end. Does this seem a sensible doctrine?