Part 43
This is the scheme of “otherness” which, in the case of the Neo-Kantians, particularly the French section, represented by M. Pillon, M. Renouvier, and others, has proved such a snare. To these thinkers, (as indeed to the late Prof. T. H. Green, of Oxford, though in a less degree,) the so-called external world lies in “other” thinking subjects—in “foreign centres of representations.” The free-trade doctrine has verily penetrated to the philosophic region—the wholesale admission of foreign wares to the detriment of home products. Why should I place the content of that so-called external world, which, external or internal, is my very own inalienably, in a centre of representation other than my own, thus making my cognition of it rest entirely upon the “ejective” plane? It is only when I discover, as I must sooner or later, that there is nothing in the report of an “outsider” (or in any number of them) beyond what I credit him or her with in my own consciousness; and that the outsider is on the same plane as other objects, it is only then that the mystification is cleared up. I do _not_ cognise, or recognise, the external at second-hand. The “note” of otherness is simply another term more or less in the cosmical series.
It is, however, not only with the familiar “other selves” of ordinary life that we are confronted in Synechism. In the creed of animism
“Millions of spiritual creatures walk the earth,”
and Mr. Peirce speaks of “spiritual influences” (p. 559) as having at least no hindrance presented to them by his doctrine. But he has some other shadowy personalities at command, which, it must be confessed, are well calculated to give us pause. “There should be something like[78] personal consciousness in bodies of men who are in intimate and intensely sympathetic communion.... None of us can fully realise which the minds of corporations are.... But the law of mind clearly points to the existence of such personalities.” It is probably true that the “minds of corporations,” must ever present an insoluble riddle of perversity to the suburban dweller, vexed with the mockery of paving and lighting. But we need not linger over this speculation, for there are other shades behind.
“If such a fact is capable of being made out anywhere it should be in the Church.... Surely a personality ought to have developed in that Church, in that ‘bride of Christ,’ as they call it.” (“Man’s Glassy Essence,” pp. 21-22.)
A PERSONAL CREATOR.
Bearing our ecclesiastical divisions in mind, it is difficult to conceive the unity of a “corporate personality” of this kind, but, to let that pass, it may be remarked that, when any one begins to imagine that there are others in the universe besides himself, he is not, as a rule, content with two or three companions of his solitude. They come in battalions. Thus, behind the other selves, corporate personalities and spiritual influences of Synechism, there looms a transcendent personality. “A genuine evolutionary philosophy,” we are told, “... is so far from being antagonistic to the idea of a personal Creator, that it is really inseparable from that idea.” And a philosophy of pseudo-evolutionism is “hostile to all hopes of personal relations to God.” (“The Law of Mind,” p. 557.)
Mr. Peirce thus assigns to his first cause a place in the _continuum_ of ideas, and says that if there is a personal God we must have a direct perception of that person and “indeed be in personal communication with him.” The difficulty, he admits, is that if this be so, how is it possible that the existence of this being should ever have been doubted by anybody. And the only answer he can at present make is, that “facts that stand before our face and eyes, and stare us in the face, are far from being in all cases the ones most easily discerned. That,” he adds, “has been remarked from time immemorial.” (“The Law of Mind,” pp. 558-559.)
One of the ablest of living philosophical writers, Professor Veitch, of Glasgow University, puts it somewhat similarly, though with his own realistic coloring, when he says:
“God, if at all, must rise above the line of finite regress; He cannot be a cause in that; He cannot be a cause dependent on another cause; He must be somewhere, or at some point, in the line of an otherwise endless scientific regress, there, above it, yet related to it, and in it; otherwise He is nothing for us.” (“Knowing and Being,” p. 320.)
The parallelism is worth noting. Those views embody what has been the contention of the present writer throughout this paper, _with this most notable difference_: that no term of a series may thus transcend the series, or be other than on a level with the other terms, being itself only a term, a link, in the series itself. And with this falls forever the idea of a cause uncaused.
Yet am _I_ not _in_ the series? For all that is in the series is mine every percept, every concept; so that, “extravagant” as it may appear, it is _I who am the series_. In other words, the ego is the universe-synthesis, and the universe-synthesis the ego.
Is Mr. Peirce prepared to take the consequences of that which his Synechism leads to?
G. M. MCCRIE.
FOOTNOTES:
[67] From τύχη, chance.
[68] _Tychism_ again comes to the front in the succeeding number of _The Monist_, (Vol. III, No. 1,) in an article by Mr. Peirce, entitled “Man’s Glassy Essence.”
[69] Dr. Carus, in his review of Mr. Peirce’s doctrines, (_The Monist_, Vol. II, No. 4, p. 575,) notes this positivistic-constructionism.
[70] Cf. T. H. Green, _Prolegomena to Ethics_, Ch. II, p. 63.
[71] Nos. 258, 59, 61, August, 1892. _Miss Naden’s World-Scheme._
[72] In a note to this passage was appended a quotation from a pamphlet by Dr. E. Cobham Brewer as a practical instance of the objective being, on the antiquated subject-object plane, actually superseded. Suppose a very remote star to become extinct, the “vibrations” would continue to “travel” towards a spectator situated on our planet for years, it may be for centuries. So that the spectator, ultimately, “sees” that which does not even exist. Dr. Brewer’s comment, which cannot be considered any contribution to a satisfactory _rationale_, is: “the objects, however, must have existed, or no messenger could have been sent from their courts.” Evidently, in this case, that which is sent is, at least, as good as the sender—is, in fact, the self-same thing. Only, in that case, what of the extinct object?
[73] Or to put it in another form, any one idea, and the timing of this idea are really two ideas, although, as we shall see later, they may be inseparable in practice.
[74] Cf., in this connection, the results of experiments by Cheselden, as far back as 1727 on congenitally blind persons, couched for double cataract.
[75] Much more inclusive, also, than the Relational Theory of the Neo-Kantians.
[76] _The Monist_, Vol. III, No. 1.
[77] Mr. Peirce uses the word “instant” to mean a point of time, and “moment” to mean an infinitesimal duration.
[78] The phrase, “something like,” is significant, when we remember, (see _ante_,) that with Mr. Peirce the excitants were “something like” the excited feelings.
THE FOURTH DIMENSION.
MATHEMATICAL AND SPIRITUALISTIC.
INTRODUCTORY.
The tendency to generalise long ago led mathematicians to extend the notion of three-dimensional space, which is the space of sensible representation, and to define aggregates of points, or spaces, of more than three dimensions, with the view of employing these definitions as useful means of investigation. They had no idea of requiring people to imagine four-dimensional things and worlds, and they were even still less remote from requiring of them to believe in the real existence of a four-dimensioned space. In the hands of mathematicians this extension of the notion of space was a mere means devised for the discovery and expression, by shorter and more convenient ways, of truths applicable to common geometry and to algebra operating with more than three unknown quantities. At this stage, however, the spiritualists came in, and coolly took possession of this private property of the mathematicians. They were in great perplexity as to where they should put the spirits of the dead. To give them a place in the world accessible to our senses was not exactly practicable. They were compelled, therefore, to look around after some _terra incognita_, which should oppose to the spirit of research inborn in humanity an insuperable barrier. The residence of the spirits had to be a place inaccessible to our senses and full of mystery to the mind. This property the four-dimensioned space of the mathematicians possessed. With an intellectual perversity which science has no idea of, these spiritualists boldly asserted, first, that the whole world was so situated in a four-dimensioned space as a plane might be situated in the space familiar to us, secondly, that the spirits of the dead lived in such a four-dimensioned space, thirdly, that these spirits could accordingly act upon the world and, consequently, upon the human beings resident in it, exactly as we three-dimensioned creatures can produce effects upon things that are two-dimensional; for example, such effects as that produced when we shatter a lamina of ice, and so influence some possibly existing two-dimensioned _ice_-world.
Since spiritualism, under the leadership of the Leipsic Professor Zöllner, thus proclaimed the existence of a four-dimensioned space, this notion, which the mathematicians are thoroughly master of,—for in all their operations with it, though they have forsaken the path of actual representability, they have never left that of the truth,—this notion has also passed into the heads of lay persons who have used it as a catchword, ordinarily without having any clear idea of what they or any one else mean by it. To clear up such ideas and to correct the wrong impressions of cultured people who have not a technical mathematical training, is the purpose of the following pages. A similar elucidation was aimed at in the tracts which Schlegel (Riemann, Berlin, 1888) and Cranz (Virchow-Holtzendorff’s Sammlung, Nos. 112 and 113) have published on the so-called fourth dimension. Both treatises possess indubitable merits, but their methods of presentation are in many respects too concise to give a lay mind any profound comprehension of the subject. The author, accordingly, has been able to add to the reflections which these excellent treatises offer, a great deal that appears to him necessary for a thorough explanation in the minds of non-mathematicians of the notion of the fourth dimension.
I.
THE CONCEPT OF DIMENSION.
Many text-books of stereometry begin with the words: “Every body has three dimensions, length, breadth, and thickness.” If we should ask the author of a book of this description to tell us the length, breadth, and thickness of an apple, of a sponge, or of a cloud of tobacco smoke, he would be somewhat perplexed and would probably say, that the definition in question referred to something different. A cubical box, or some similar structure, whose angles are all right angles and whose bounding surfaces are consequently all rectangles is the only body of which it can at all be unmistakably asserted that there are three principal directions distinguishable in it, of which any one can be called the length, any other the breadth, and any third the thickness. We thus see that the notions of length, breadth, and thickness are not sufficiently clear and universal to enable us to derive from them any idea of what is meant when it is said that every body possesses three dimensions, or that the space of the world is three-dimensional.
This distinction may be made sharper and more evident by the following considerations: We have, let us suppose, a straight line on which a point is situated, and the problem is proposed to determine the position of the point on the line in an unequivocal manner. The simplest way to solve this is, to state how far the point is removed in the one or the other direction from some given fixed point; just as in a thermometer the position of the surface of the mercury is given by a statement of its distance in the direction of cold or heat from a predetermined fixed point—the point of freezing water. To state, therefore, the position of a point on a straight line, the sole datum necessary is a single number, for beforehand we have fixed upon some standard line, like the centimetre, and some definite point to which we give the value zero, and have also previously decided in what direction from the zero-point, points must be situated whose position is expressed by positive numbers, and also in what direction those must lie whose position is expressed by negative numbers. This last mentioned fact, that a _single_ number is sufficient to determine the place of a point in a straight line, is the real reason why we attribute to the straight line or to any part of it a single dimension.
More generally, we call every totality or system, of infinitely numerous things, _one_-dimensional, in which _one_ number is all that is requisite to determine and distinguish any particular one of these things amidst the entire totality. Thus, time is one-dimensional. We, as inhabitants of the earth, have naturally chosen as our unit of time, the period of the rotation of the earth about its axis, namely, the day, or a definite portion of a day. The zero-point of time is regarded in Christian countries as the year of the birth of Christ, and the positive direction of time is the time _subsequent_ to the birth of Christ. These data fixed, all that is necessary to establish and distinguish any definite point of time amid the infinite totality of all the points of time, _is a single number_. Of course this number need not be a whole number, but may be made up of the sum of a whole number and a fraction in whose numerator and denominator we may have numbers as great as we please. We may, therefore, also say that the totality of all conceivable numerical magnitudes, or of only such as are greater than one definite number and smaller than some other definite number, is one-dimensional.
We shall add here a few additional examples of one-dimensioned magnitudes presented by geometry. First, the circumference of a circle is a one-dimensional magnitude, as is every curved line, whether it returns into itself or not. Further, the totality of all equilateral triangles which stand on the same base is one-dimensional, or the totality of all circles that can be described through two fixed points. Also, the totality of all conceivable cubes will be seen to be one-dimensional, provided they are distinguished, not with respect to position, but with respect to magnitude.
In conformity with the fundamental ideas by which we define the notion of a one-dimensional manifoldness, it will be seen that the attribute _two_-dimensional must be applied to all totalities of things in which _two_ numbers are necessary (and sufficient) to distinguish any determinate individual thing amid the totality. The simplest two-dimensioned complex which we know of is the plane. To determine accurately the position of a point in a plane, the simplest way is to take two axes at right angles to each other, that is, fixed straight lines, and then to specify the distances by which the point in question is removed from each of these axes.
This method of determining the position of a point in a plane suggested to the celebrated philosopher and mathematician Descartes the fundamental idea of analytical geometry, a branch of mathematics in which by the simple artifice of ascribing to every point in a plane two numerical values, determined by its distances from the two axes above referred to, planimetrical considerations are transformed into algebraical. So, too, all kinds of curves that graphically represent the dependence of things on time, make use of the fact that the totality of the points in a plane is two-dimensional. For example, to represent in a graphical form the increase of the population of a city, we take a horizontal axis to represent the time, and a perpendicular one to represent the numbers which are the measures of the population. Any two lines, then, whose lengths practical considerations determine, are taken as the unit of time, which we may say is a year, and as the unit of population, which we will say is one thousand. Some definite year, say 1850, is fixed upon as the zero point. Then, from all the equally distant points on the horizontal axis, which points stand for the years, we proceed in directions parallel to the other axis, that is, in the perpendicular direction, just so much upwards as the numbers which stand for the population of that year require. The terminal points so reached, or the curve which runs through these terminal points, will then present a graphic picture of the rates of increase of the population of the town in the different years. The rectangular axes of Descartes are employed in a similar way for the construction of barometer curves, which specify for the different localities of a country the amount of variation of the atmospheric pressure during any period of time. Immediately next to the plane the surface of the earth will be recognised as a two-dimensional aggregate of points. In this case geographical latitude and longitude supply the two numbers that are requisite accurately to determine the position of a point. Also, the totality of all the possible straight lines that can be drawn through any point in space is two-dimensional, as we shall best understand if we picture to ourselves a plane which is cut in a point by each of these straight lines and then remember that by such a construction every point on the plane will belong to some one line and, _vice versa_, a line to every point, whence it follows that the totality of all the straight lines which pass through the point assigned are of the same dimensions as the totality of the points of the imagined plane.
The question might be asked, In what way and to what extent in this case is the specification of _two_ numbers requisite and sufficient to determine amid all the rays which pass through the specified point a definite individual ray? To get a clear idea of the problem here involved, let us imagine the ray produced far into the heavens, where some quite definite point will correspond to it. Now, the position of a point in the heavens depends, as does the position of a point on all spherical surfaces, on two numbers. In the heavens these two numbers are ordinarily supplied by the two angles called altitude, or the distance above the plane of the horizon, and azimuth, or the angular distance between the circle on which the altitude is measured and the meridian of the observer. It will be seen thus that the totality of all the luminous rays that an eye, conceived as a point, can receive from the outer world is two-dimensional, and also that a luminous point emits a two-dimensional group of luminous rays. It will also be observed, in connection with this example, that the two-dimensional totality of all the rays that can be drawn through a point in space is something different from the totality of the rays that pass through a point but are required to lie in a given plane. Such a group of objects as the last-named one, is a one-dimensional totality.
Now that we have sufficiently discussed the attributes that are characteristic of one and two-dimensional aggregates, we may, without any further investigation of the subject, propose the following definition, that, generally, _an n-dimensional totality of infinitely numerous things is such, with respect to which the specification of n numbers is necessary and sufficient to indicate a definite individual amid the totality of all the infinitely numerous individuals of the group_.
Accordingly, the point-aggregate made up of the world-space which we inhabit, is a three-dimensional totality. To get true bearings in this space and to define any determinate point in it, we have therefore to lay through any point which we take as our zero-point three axes at right angles to each other, one running from right to left, one backwards and forwards, and one upwards and downwards. We then join each two of these axes by a plane and are enabled thus to specify the position of every point in space by the three perpendicular distances by which the point in question is removed in a positive or negative sense from these three planes. It is customary to denote the numbers which are the measures of these three distances by _x_, _y_, and _z_, the positive _x_, positive _y_, and positive _z_ ordinarily being reckoned in the right hand, the forward, and the upward directions from the origin. If now, with direct reference to this fundamental axial system, any particular specification of _x_, _y_, and _z_ be made, there will, by such an operation, be cut out and isolated from the three-dimensional manifoldness of all the points of space a totality of less dimensions. If, for example, _z_ is equal to seven units or measures, this is equivalent to a statement that only the two-dimensional totality of the points is meant, which constitute the plane that can be laid at right angles to the upward-passing _z_-axis at a distance of seven measures from the zero-point. Consequently, every imaginable equation between _x_, _y_, and _z_ isolates and defines a two-dimensional aggregate of points. If two different equations obtain between _x_, _y_, and _z_, two such two-dimensional totalities will be isolated from among all the points of space. But as these last must have some one-dimensional totality in common, we may say that the co-existence of two equations between _x_, _y_, and _z_ defines a one-dimensional totality of points, that is to say a straight line, a line curved in a plane, or even, perhaps, one curved in space. It is evident from this that the introduction of the three axes of reference forms a bridge between the theory of space and the theory of equations involving three variable quantities, _x_, _y_, _z_. The reason that the theory of space cannot thus be brought into connection with algebra in general, that is, with the theory of indefinitely numerous equations, but only with the algebra of three quantities, _x_, _y_, _z_, is simply to be sought in the fact that space, as we picture it, can only have three dimensions.