"Now the idea expressed by that wordcontinuousis 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 onwithout 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 carefullywhat 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 differentin 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 thepointsof the line. Butno, 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 apartof 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 connectedwith 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 meanwithout 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 theseries 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 thepointsof the line and thepartsof 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 beonparts of the line, I answer that, even on this hypothesis, they must beatdifferent parts of the line, in order to be distinguished from each other. Thepart 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?The rival doctrine, sometimes called the Berkeleyan, contains no such difficulties, and it makes evident that the difficulties discussed above arise simply out of a confusion of samenesses, and are gratuitous. Its discussion demands that I call to mind a few distinctions already made.One must bear in mind, in the first place, that a line immediately known, existing in consciousness, is the same with an "external" line corresponding to it, not in sense first, but in sense seventh. That is, they are two lines, not one, and in the interests of clearness they should be considered separately.One should remember, in the second place, that a line in consciousness at one moment is not, in the strictest sense, the same with a line in consciousness at another moment. One may stand for the other and thus be the same with it in sense fifth; or the two may be regarded as both belonging to the one series of experiences, which, taken together, represent to us "an object," in which case they are the same in sense third. A thing the same with another thing in either of these senses is not necessarily much like it. It must only be able to serve as its representative.Now I see a line about an inch long on the paper before me. It is a certain distance from my eyes. I shall concern myselffor the present only with the line immediately perceived, which means for me so much sensation. If I move this line (which remains the same in sense third), nearer to me or farther away, I do not perceive the identical thing that I did before. My quantity of sensation is increased or diminished. If I keep the line at the same distance and change none of the conditions, the quantity of sensation remains presumably the same. The question arises, Is this line as actually experienced at this moment infinitely divisible or not? I can certainly conceive of it as divisible to some extent, for I see part out of part, and I can think of these parts as separated. But if this line were divided, the division would soon result in parts which could be seen, but which could not be seen to consist of part out of part. Were these apparently non-extended parts (they would remain the same in sense third), approached to the eyes, they, too, would be seen to consist of part out of part, but then I should simply have substituted for the apparently non-extended a representative which was extended. This would not prove that what was before in consciousness was extended and could be divided. Consciousness certainly seems to testify that any particular line in consciousness is composed of a limited number of indivisible parts, and when one adds to this reflection the consideration that a point moving over a given line does not appear to have an endless task before it, but soon arrives at the final term, one is irresistibly impelled to the conclusion that the parts of the line are not infinite, but that the division results in the indivisible, the simple element of sensation, which, joined with other such elements, makes an extended object, but which taken alone is not extended at all. The whole difficulty lies in keeping to the line and the parts with which one started. It is so easy to pass from sameness in sense first to sameness in sense third or sense fifth; it is so natural to bring an object which is, as wesay, imperfectly seen, closer to the eye and thus substitute for what was seen before a new experience connected with it in the order of nature, confident that any system of relations derived from the latter may safely be carried over to all possible experiences connected with the former; one does this so instinctively that a man may very readily suppose that he is still busied about the apparently non-extended element with which he started, when he is in reality dividing and sub-dividing its representative, which is evidently extended. But the question is not whether, when one has divided a line until the parts cannot be seen to consist of parts, one may substitute for these parts what evidently does consist of parts, and go on dividing that. The question is, whether an apparently non-extended element of a line in consciousness is divisible or not. Any argument from the possibility of dividing its substitutes evidently has nothing to do with this point.It is plain that this doctrine, which makes any particular finite line in consciousness to consist of a limited number of simple parts, is not open to the objection that it necessitates the absurdity of exhausting an endless series. Moving along such a line, Achilles could overtake the tortoise, for the successively diminishing distances between them do not constitute an endless series. The descending series results after a limited number of terms in the simple, and the series is broken, for the simple does not consist of parts. In this there is at least no contradiction. It remains to see what other objections may lie against it.It may be argued, first, as it often is argued, that it is impossible to conceive of any part of a line as not itself extended and having parts. It may be admitted that the small parts arrived at do not seem to have part out of part; that these sub-parts are not observed in them, but still it is said that one who thinks about them cannot but think of them as really having suchparts. I ask one who puts forward this objection to look into his own mind and see whether he does not mean by "thinking about them," bringing them in imagination nearer to the eye, or by some means substituting for them what can be seen to have part out of part. That one can do this no one would think of denying, but, as I have said, this does not prove the original parts to be extended.It may be objected again that extension can never be built up out of the non-extended—that if one element of a given kind has, taken alone, no extension at all, two or more such elements together cannot have any extension either. I answer that a straight line has no angularity at all, and yet two straight lines may obviously make an angle; that one man is not in the least a crowd, but that one hundred men may be; that no single tree is a forest, but that many trees together do make a forest; that a uniform expanse of color is in no sense a variegated surface, but that several such together do make a variegated surface. It may be that extension is simply the name we give to several simple sense-elements of a particular kind taken together. One cannot say off-hand that it is not.Should one object, finally, that, if a given line in consciousness be composed of a limited number of indivisible elements of sensation, consciousness ought to distinguish these single elements and testify as to their number; I answer that what is in consciousness is not necessarily in a clear analytical consciousness, nor well distinguished from other elements. For example, I am at present conscious of a stream of sensations which I connect with the hand that holds my pen. The single elements in this complex I cannot distinguish from each other, nor can I give their number. It does not follow that I am to assume the number to be infinite. Much less should I be impelled to make this assumption, if it necessitated my accepting as true what Isee to be flatly contradictory, as in the case under discussion. It was because of this vagueness and lack of discrimination in the testimony of consciousness that I said, some distance back, that consciousnessseemsto testify that any finite line in it is composed of simple parts. If the testimony were quite clear, the matter would be settled at once. As it is not quite clear, the matter has to be settled on a deductive basis. The most reasonable solution appears to be the Berkeleyan.So much for the line immediately perceived, the line in consciousness. What shall we say to one who is willing to admit that this line is not infinitely divisible, but is composed of simple sense-elements; and yet who maintains that there exists an "external" line corresponding to it, which is not immediately perceived, and is infinitely divisible? We may begin by suggesting to him that an "external" point moving over this "external" line must perform the wholly impossible feat to which Clifford condemns a point moving over a line; and we may farther suggest that, if the "external" world be an intelligible world at all, a contradiction may be as much out of place in it as anywhere else. And if the existence of this world be problematic, a thing not self-evident, it seems quite reasonable to demand very good proof indeed of the existence, of that which contains in its very conception such excellent reasons for believing in its non-existence. This proof, the student of the history of speculation will testify, has not as yet been forthcoming.Sec. 37.With this I close my analysis of samenesses, and of confusions which have resulted in needless embarrassments and gratuitous difficulties. More instances of the latter could be given, of course. The reader will be able to furnish, I presume, many like them. Those which I have given seem to me quite sufficient to prove the need of much greater care and exactitudethan one commonly finds in metaphysical reasonings. Loose reasoning is bad reasoning, and leads to bad results. Its one virtue is that it does not require much mental application on the part of either author or reader. On the other hand, the attempt to be cautious and exact, to distinguish between things easily confounded, and to keep strictly to the thing in dispute through a long discussion, these things are wearisome to all concerned. Although I am quite conscious of this fact, I have tried to do these things: with what result, my fellow-analysts must judge. I feel reasonably sure that I have succeeded in being wearisome, and for this I make due apology.
"Now the idea expressed by that wordcontinuousis 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 onwithout 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 carefullywhat 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 differentin 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 thepointsof the line. Butno, 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 apartof 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 connectedwith 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 meanwithout 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 theseries 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 thepointsof the line and thepartsof 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 beonparts of the line, I answer that, even on this hypothesis, they must beatdifferent parts of the line, in order to be distinguished from each other. Thepart 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?
The rival doctrine, sometimes called the Berkeleyan, contains no such difficulties, and it makes evident that the difficulties discussed above arise simply out of a confusion of samenesses, and are gratuitous. Its discussion demands that I call to mind a few distinctions already made.
One must bear in mind, in the first place, that a line immediately known, existing in consciousness, is the same with an "external" line corresponding to it, not in sense first, but in sense seventh. That is, they are two lines, not one, and in the interests of clearness they should be considered separately.
One should remember, in the second place, that a line in consciousness at one moment is not, in the strictest sense, the same with a line in consciousness at another moment. One may stand for the other and thus be the same with it in sense fifth; or the two may be regarded as both belonging to the one series of experiences, which, taken together, represent to us "an object," in which case they are the same in sense third. A thing the same with another thing in either of these senses is not necessarily much like it. It must only be able to serve as its representative.
Now I see a line about an inch long on the paper before me. It is a certain distance from my eyes. I shall concern myselffor the present only with the line immediately perceived, which means for me so much sensation. If I move this line (which remains the same in sense third), nearer to me or farther away, I do not perceive the identical thing that I did before. My quantity of sensation is increased or diminished. If I keep the line at the same distance and change none of the conditions, the quantity of sensation remains presumably the same. The question arises, Is this line as actually experienced at this moment infinitely divisible or not? I can certainly conceive of it as divisible to some extent, for I see part out of part, and I can think of these parts as separated. But if this line were divided, the division would soon result in parts which could be seen, but which could not be seen to consist of part out of part. Were these apparently non-extended parts (they would remain the same in sense third), approached to the eyes, they, too, would be seen to consist of part out of part, but then I should simply have substituted for the apparently non-extended a representative which was extended. This would not prove that what was before in consciousness was extended and could be divided. Consciousness certainly seems to testify that any particular line in consciousness is composed of a limited number of indivisible parts, and when one adds to this reflection the consideration that a point moving over a given line does not appear to have an endless task before it, but soon arrives at the final term, one is irresistibly impelled to the conclusion that the parts of the line are not infinite, but that the division results in the indivisible, the simple element of sensation, which, joined with other such elements, makes an extended object, but which taken alone is not extended at all. The whole difficulty lies in keeping to the line and the parts with which one started. It is so easy to pass from sameness in sense first to sameness in sense third or sense fifth; it is so natural to bring an object which is, as wesay, imperfectly seen, closer to the eye and thus substitute for what was seen before a new experience connected with it in the order of nature, confident that any system of relations derived from the latter may safely be carried over to all possible experiences connected with the former; one does this so instinctively that a man may very readily suppose that he is still busied about the apparently non-extended element with which he started, when he is in reality dividing and sub-dividing its representative, which is evidently extended. But the question is not whether, when one has divided a line until the parts cannot be seen to consist of parts, one may substitute for these parts what evidently does consist of parts, and go on dividing that. The question is, whether an apparently non-extended element of a line in consciousness is divisible or not. Any argument from the possibility of dividing its substitutes evidently has nothing to do with this point.
It is plain that this doctrine, which makes any particular finite line in consciousness to consist of a limited number of simple parts, is not open to the objection that it necessitates the absurdity of exhausting an endless series. Moving along such a line, Achilles could overtake the tortoise, for the successively diminishing distances between them do not constitute an endless series. The descending series results after a limited number of terms in the simple, and the series is broken, for the simple does not consist of parts. In this there is at least no contradiction. It remains to see what other objections may lie against it.
It may be argued, first, as it often is argued, that it is impossible to conceive of any part of a line as not itself extended and having parts. It may be admitted that the small parts arrived at do not seem to have part out of part; that these sub-parts are not observed in them, but still it is said that one who thinks about them cannot but think of them as really having suchparts. I ask one who puts forward this objection to look into his own mind and see whether he does not mean by "thinking about them," bringing them in imagination nearer to the eye, or by some means substituting for them what can be seen to have part out of part. That one can do this no one would think of denying, but, as I have said, this does not prove the original parts to be extended.
It may be objected again that extension can never be built up out of the non-extended—that if one element of a given kind has, taken alone, no extension at all, two or more such elements together cannot have any extension either. I answer that a straight line has no angularity at all, and yet two straight lines may obviously make an angle; that one man is not in the least a crowd, but that one hundred men may be; that no single tree is a forest, but that many trees together do make a forest; that a uniform expanse of color is in no sense a variegated surface, but that several such together do make a variegated surface. It may be that extension is simply the name we give to several simple sense-elements of a particular kind taken together. One cannot say off-hand that it is not.
Should one object, finally, that, if a given line in consciousness be composed of a limited number of indivisible elements of sensation, consciousness ought to distinguish these single elements and testify as to their number; I answer that what is in consciousness is not necessarily in a clear analytical consciousness, nor well distinguished from other elements. For example, I am at present conscious of a stream of sensations which I connect with the hand that holds my pen. The single elements in this complex I cannot distinguish from each other, nor can I give their number. It does not follow that I am to assume the number to be infinite. Much less should I be impelled to make this assumption, if it necessitated my accepting as true what Isee to be flatly contradictory, as in the case under discussion. It was because of this vagueness and lack of discrimination in the testimony of consciousness that I said, some distance back, that consciousnessseemsto testify that any finite line in it is composed of simple parts. If the testimony were quite clear, the matter would be settled at once. As it is not quite clear, the matter has to be settled on a deductive basis. The most reasonable solution appears to be the Berkeleyan.
So much for the line immediately perceived, the line in consciousness. What shall we say to one who is willing to admit that this line is not infinitely divisible, but is composed of simple sense-elements; and yet who maintains that there exists an "external" line corresponding to it, which is not immediately perceived, and is infinitely divisible? We may begin by suggesting to him that an "external" point moving over this "external" line must perform the wholly impossible feat to which Clifford condemns a point moving over a line; and we may farther suggest that, if the "external" world be an intelligible world at all, a contradiction may be as much out of place in it as anywhere else. And if the existence of this world be problematic, a thing not self-evident, it seems quite reasonable to demand very good proof indeed of the existence, of that which contains in its very conception such excellent reasons for believing in its non-existence. This proof, the student of the history of speculation will testify, has not as yet been forthcoming.
Sec. 37.With this I close my analysis of samenesses, and of confusions which have resulted in needless embarrassments and gratuitous difficulties. More instances of the latter could be given, of course. The reader will be able to furnish, I presume, many like them. Those which I have given seem to me quite sufficient to prove the need of much greater care and exactitudethan one commonly finds in metaphysical reasonings. Loose reasoning is bad reasoning, and leads to bad results. Its one virtue is that it does not require much mental application on the part of either author or reader. On the other hand, the attempt to be cautious and exact, to distinguish between things easily confounded, and to keep strictly to the thing in dispute through a long discussion, these things are wearisome to all concerned. Although I am quite conscious of this fact, I have tried to do these things: with what result, my fellow-analysts must judge. I feel reasonably sure that I have succeeded in being wearisome, and for this I make due apology.