Article: 18820101006


Popular Science
THE theory of a fourth dimension of space has lately been brought forward somewhat prominently, under the imposing title of “Transcendental Physics,” by Mr. John Charles Frederick Zöllner, Professor of Physical Astronomy in the University of Leipsic, although the learned professor, it should be said, imputes the suggestion of the theory primarily to Kant, and secondarily to Gauss, the celebrated mathematician of Göttingen, both of whom, he says, struck out the thought.



THE theory of a fourth dimension of space has lately been brought forward somewhat prominently, under the imposing title of “Transcendental Physics,” by Mr. John Charles Frederick Zöllner, Professor of Physical Astronomy in the University of Leipsic, although the learned professor, it should be said, imputes the suggestion of the theory primarily to Kant, and secondarily to Gauss, the celebrated mathematician of Göttingen, both of whom, he says, struck out the thought. In this, it is possible, the professor does himself less than justice.

Gauss had large expectations from the geometry of position, but its development, as contemplated by him, does not appear to have included the development of a fourth dimension, and we know he expressed a lofty scorn for table-tippings and other things of the sort, to which the new dimension, it would seem, has thus far lent itself exclusively. Kant, it is true, admitted the possibility of other dimensions, but derived it from the impossibility of knowing what exists outside of the limits of our faculties, and consequently of legitimately affirming or denying anything about it. He suggested the possibility of other dimensions of space, and, for that matter, the possibility of no space, but not the possibility of the discovery and comprehension of either by the human mind. So far from this, he distinctly suggested the reverse, actually citing, as an example of truths “ always united with the consciousness of necessity,” the proposition, “ Space has only three dimensions ” ; and, since a necessary truth is one of which the contrary is inconceivable, he could not have suggested, without self-contradiction, that space may have more than three dimensions. If other dimensions exist, they exist, according to Kant, not for man, but for other orders of beings, endowed with corresponding forms of perception—exist as parts of the original furniture of the intelligences of other spheres—hypothetical properties of hypothetical creatures in hypothetical worlds, one and all of which transcend our faculties. This, however, is by no means the theory of Professor Zöllner.

“ In my first treatise, ‘ On Action at a Distance,’ ” says that distinguished physicist, “ I have discussed in detail the truth, first discovered by Kant, later by Gauss and the representatives of the anti-Euclidian geometry, viz., that our present conception of space, familiar to us by habit, has been derived from experience, i. e., from empirical facts, by means of the causal principle existing a priori in our intellect. This in particular is to be said of the three dimensions of our present conception of space. If from our childhood phenomena had been of daily occurrence, requiring a space of four dimensions for an explanation which should be free from contradiction, i. e., conformable to reason, we should be able to form a conception of space of four or more dimensions. It follows that the real existence of a four-dimensional space can only be decided by experience, i. e., by observation of facts. A great step has been made by acknowledging that the possibility of a four-dimensional development of space can be understood by our intellect, although, on account of reasons previously given, no corresponding image of it can be conceived by the mind.” Probably the first thought which will occur to the intelligent reader of this passage is that Professor Zöllner mistakes Kant ; and perhaps the second thought will be that the professor mistakes himself. If these impressions prove just, he may find one day that his theory of a fourth dimension has suddenly passed away more completely, if less mysteriously, than the book from the slate at Vienna, or the round-table from the sittingroom at Leipsic. Let us inquire whether or not they are just.

Professor Zöllner thinks “ our present conception of space ” has been derived from experience, “ by means of tbe causal principle existing a priori in our intellect.” Kant beíd, on the contrary, that our idea of space is not a mere conception, but a pure intuition ; that it belongs to the faculty of sense ; that it is in fact one of the necessary conditions of sensation, without which experience itself is impossible ; and, finally, that the “ causal principle ” originates in the higher faculty of the understanding, to which sense delivers up its presentations ready cast in the mold of space. In the philosophy of Kant, therefore, the “ causal principle ” has nothing to do with the origin of the idea of space, which emerges before sensation rises to the sphere of that principle. Kant, among his endless subtilties, distinguished indeed between the form of the intuition and the formal intuition, in which space is represented as an object, and with which unity of representation is given through the understanding ; but the determination of this unity, he taught, is contained a priori in the intuition, not developed a posteriori, gradually or otherwise. As a formal intuition, space, like every other sensuous intuition, Kant maintained, is subject to the categories of the understanding, causality included, but the subjection, besides extending to all sensuous intuitions alike, is a priori, and as such incapable of expansion or contraction by experience, the possibility of which presupposes it. In the Kantian theory causality has nothing special to do in any mode with the idea of space in any aspect. In common with the other categories, it is simply presupposed in every intuition of sense, that of space with the rest. It lies at the bottom of the possibility of experience in general. Moreover, an a priori principle, as already intimated, is not a germ susceptible of growth, but rather a die, which for ever impresses the same form. A new kind of impression necessitates a new kind of die, and, if you would have a new kind of determination, you must get a new a priori principle ; a given principle can not be altered to suit emergencies. Experience may sharpen but not remodel it : as well expect the metal to remodel the die that cuts it. An a priori principle reconstructed a posteriori is an article which the author of the “ Critique of Pure Reason ” happened never to turn out from his workshop, although, if it had found entrance, he infallibly would have turned it out in doublequick time.

Yet Professor Zöllner thinks not only that the “ causal principle” is a special agent in producing our conception of space, but that “ this, in particular, is to be said of the three dimensions of our present conception of space,” implying thereby, in the teeth of the Kantian dogma, that we have arrived by degrees at a conception of three dimensions, halting for a time at twTo, if not halting before at one. Indeed, Professor Crookes, expounding in “The Quarterly Journal of Science” Professor Zöllner’s theory, affirms as much explicitly. “ The totality of all empirical experience,” he states, doubtless repeating the language of his theorist, “is communicated to the intellect by the senses, i. e., by organs which communicate to the mind all the sensuous impressions which are received at the surface of our bodies. These impressions are a reality to us, and their sphere is two-dimensional, acting not in our body, but only on its surface. We have only attained the conception of a world of objects with three dimensions by an intellectual process. What circumstances, we may ask, have compelled our intellect to come to this result ? If a child contemplate its hand, it is conscious of its existence in a double manner—in the first place, by its tangibility; in the second, by its image on the retina of the eye. By repeated groping about and touching, the child knows by experience that his hand retains the same form and extension through all the variations of distance and positions under which it is observed, notwithstanding that the form and extension of the image on the retina constantly change with the diiferent position and distance of the hand in respect to the eye. The problem is thus set to the child’s understanding : IIow to reconcile to its comprehension the apparently contradictory facts of the invariableness of the object, together w*ith the variableness of its appearance. This is only possible within space of three dimensions, in which, owing to perspective distortions and changes, these variations of projection can be reconciled with the constancy of the form of a body.” The italics in this extract, as in the preceding one, are the author’s.

Professor Zöllner describes the problem which he conceives to be set before a child. The problem which the professor seems to have set before himself may be described thus : Given a tatter of Kantism, a scrap from the received doctrine of our acquired perceptions, and the rickety figment of a fourth dimension, to evolve a theory which shall save the fourth dimension at all hazards. And we have here the outcome. His theory is, that our idea of space is not, what Kant declared it to be, a pure intuition, constituting one of the necessary conditions of experience, but a conception gradually arising from experience, modified by the “ causal principle,” or, more precisely, modifying the “ causal principle,” which last the professor apparently regards as a kind of store-house of potential dimensions, projecting into space a new dimension whenever hard pressed by the contradictions of things, very much, one may suppose, as M. Faure’s secondary battery gives out its store of energy at the touch of the operator. In short, Professor Zöllner holds that our idea of space is not intuitive, but ratiocinative, limited only by the diversity of contradictions that present themselves to thought, each of which, if otherwise irreconcilable, is pregnant with a new dimension. A child, for instance, because sensuous impressions are received at the surface of its body (and partly, we may presume, because it takes logical contradictions easy, and is not particularly strong in ratiocination anyway), perceives only two dimensions, we are told, until such time, indeed, as, grown impatient of contradictions, and withal a mighty cause-hunter, it seeks to reconeile the invariableness of an object with the variableness of its appearance at different distances, when, presto, forth from the battery of the “ causal principle ” streams a third dimension, to smooth away the contradiction, and solve the problem ; although how, meanwhile, the two-dimensional little one contrives to crawl over the house-dog, or to keep its own body in its own sight when lifted out of the cradle, or to “ contemplate its hand,” and go through with “ repeated groping about and touching,” without stumbling on the third dimension, is a puzzle that must tax the resources even of transcendental physics. To them I leave it.

Professor Zöllner’s theory, it goes without saying, is important, if true ; for, if one contradiction may determine a new dimension of space, another contradiction, by the same token, may determine another new property, and so on, till everything shall be made plain, and we “ be as gods.” The theory, as will be observed, is propounded in its seminal form ; but its capabilities of explanation are restricted only by the number of possible contradictions inexplicable without it, so that, if capable of solving a single problem insoluble in any other way, it is capable of solving every insoluble problem—is a philosophical menstruum, a logical catholicon, a key to all the mysteries that mind is heir to. It is this, or nothing ; but this, alas ! is quackery. Yet Professor Zöllner is no quack. He is a trained scientist, enthusiastic without doubt, but equally without doubt sincere. What, then, is at fault ? To be sure, he is a man with a theory, and that means a good deal ; but it does not cover the whole ground. He, above all, is a man with an impossible fact. By this he has fallen. Theorizing to explain a fact is hazardous enough, and theorizing to prove a fact more hazardous still, but the man who theorizes to prove a fact beyond the bounds of human knowledge is lost; and this is the predicament of our worthy professor. Through his zealous efforts to prove a fourth dimension he has got psychology at sixes and sevens. In framing his theory of space, he unconsciously has made hash of Kant’s theory, as in expounding it he has unconsciously perverted Berkeley’s theory of vision ; his theory, accordingly, is itself a hash. In other words, Professor Zöllner, having mutilated Kant, and hacked the fragments, has mixed up with the remains a sprinkling of Berkeley (after vitiating it), and taken the resulting hodge-podge as his theory of space, which, like the honey in Samson’s riddle, comes out of the carcass of the lion he has rent, though in his case, unhappily, the bees have swarmed from his own bonnet. It would be superfluous to characterize seriously his account of the origin of our idea of space. It is enough to say that the account has no warrant in the teachings of Kant, or of any other metaphysician, from Zeno to Spencer. It is not Kantian in the least, but purely Zöllnerian ; and, as John Wilkes acknowledged that he was never much of a Wilkesite, so Professor Zöllner, if he some day becomes as well versed in psychology as he is in physics, will be tempted to deny that he was ever a Zöllnerite at all ; and, should he yield to the temptation, the recording angel might fitly blot with a tear the record of the lapse. At all events, Professor Zöllner does not understand Kant.

Does he understand himself ? “ If from our childhood,” he says,

“phenomena had been of daily occurrence, requiring a space of four dimensions for an explanation which should be free from contradictions, i. e., conformable to reason, we should be able to form a conception of space of four or more dimensions. It follows that the real existence of a four-dimensional space can only be decided by eocj^erience, i. e., by observation of facts.” Yes, it all depends on experience, which, as every good Kantist will tell you, furnishes the matter of our cognitions, while the mind furnishes the form, the two elements uniting in the act of cognition ; insomuch that the form without the matter is void, as the matter without the form is undetermined. Wherefore, we can have no conception the matter of which has not been furnished by experience, or, as a strict Kantist would say, by sensibility. But what experience, known or conceivable, can furnish the matter of the conception of a fourth dimension ? Contradictions, which nothing but a fourth dimension will reconcile, answers Professor Zöllner—such as the tying of knots in endless cords, disappearance and reappearance of solid objects, passage of a shell through a table, impressions of feet on the inner sides of a closed book-slate, and other “ contradictions ” emerging or appearing to emerge in the presence of Dr. Slade. But contradictions are objects not of the sensitive faculty but of the understanding, and, howrever amazing, do not furnish new matter of conception ; they merely call for new applications or new combinations of the conceptions we have. A knot tied in an endless cord is, so far as presented to us, a phenomenon within three dimensions, and, though we may not know how the knot was tied, our ignorance of the way of tying it can give us no new matter of thought. It is an appeal to our sense, not to our senses ; and the office of our sense is to receive and organize the matter of thought, not give it. The mind has no portal through which a fourth dimension can enter, no chamber in which it may lodge. To assume that, because the contradictions in question are irreconcilable, they ipso facto present to our mind a new dimension of space, in the perception of which only they become reconcilable, is to make the simple inexplicability of a problem not only the verification of a given solution of it, but the occasion of a new experience in sense, and a new form in thought—a change, that is, in the constitution of our faculties. If a theory explains a phenomenon, the theory is so far confirmed ; but, if the phenomenon is comprehensible only by a theory which is incomprehensible, the phenomenon itself is incomprehensible, and, a sensible person would say, there is an end of it. With Professor Zöllner, however, there is only the beginning of it, the incomprehensibility of the phenomenon, he contends, verifying the incomprehensible theory, which in return explains the phenomenon “ conformably to reason.” This is arguing in a circle as vicious as ever a bedlamite whirled in. From the point of view of any system of philosophy under heaven, it is vile metaphysics, and viler physics !

Could Kant have been a Zöllnerist, instead of a Kantist, we should never have heard of his wrestle with the famous antinomies, which in that event would have disappeared before a new world and a new mind, evoked by the contradictions themselves, from which he would have freed his reason by the cheap and easy method of a new dimem sion of space, and, if necessary, a new dimension of time, with as many more transcendent discharges from the “causal principle” as the pinch required ; but the philosopher of Königsberg knew not this master-key of mysteries, and, had it been put before him, we can faintly imagine with what pulverizing force he would have brought down on it his Thor-hammer, unless, forsooth, it vanished before the “ whiff and wind ” of the ponderous weapon. Germany is responsible for some wild philosophers, but Professor Zöllner, it must be confessed, has “ bettered the instruction ” of the wildest of them. What Kant styled “ the error of inactive reason,” Zöllner runs into “ the very error of the moon.” Assuredly he leads the canons of scientific inquiry a pretty dance. It is not reckoned sound philosophy to multiply causes unnecessarily, or to assume unknown causes before exhausting known ones ; but what should be thought of a method which, not content with brushing aside the law of parcimony, goes on to account for an effect by assuming a cause beyond the grasp of the intellect—to explain what is not understood by what can not be conceived—to make known the unknown by the unknowable ; and which, not stopping at this, turns round, and from the mystery of the effect infers not merely the reality of the inconceivable cause, but the evolution of a new faculty to match it? If ever there was pressing need for the application of Occam’s razor, it surely is afforded by the theory of this German scientist, whose general method in his professional domain, we may be sure, it does not fairly exemplify. If it did, his occupancy of a chair in the University of Leipsic would be almost as great a marvel as the fourth dimension itself.

“A great step has been made,” Professor Zöllner congratulates himself, “ by acknowledging that the possibility of a four-dimensional development of space can be understood by our intellect.” This possibility our professor, of course, fancies that he himself understands ; and no doubt he is as capable of understanding it as anybody else is. But does he understand it? Nay, can he think of it? To think of the possibility of a thing is to put together in thought the terms of the proposition asserting the possibility. In the case under notice, can Professor Zöllner do this ? He owns that he can not. “ No corresponding image of it can be conceived by the mind,” he avows. If he can not imagine the thing, he can not imagine the possibility of the thing, the imagination of which involves the imagination of the thing itself ; possibility, as every Kantist should know, being simply a mode of conceiving the thing. And wdiat he can not imagine he will scarcely insist that he understands. As a rule, we do not understand what we can not conceive. They may order things differently in Germany ; but, as yet, Professor Zöllner is the only evidence of it that wre have, and it is now tolerably clear that he does not think what he thinks he thinks. He deceives himself. He will not venture to say, for example, that he understands the possibility of a round triangle, or of a whole equal to its part, or of three and two making seven. Yet it is certain that he understands any one of these propositions as truly as he understands the possibility of a fourth dimension : which is as good as saying that the latter proposition is without meaning to him, as to every other human being, and can have no meaning to any mortal so long as the constitution of our minds remains wThat it is. Professor Zöllner, able and accomplished though he be, is the dupe of words. It is not possible to understand the possibility of a fourth dimension. Fourth we know, and dimension we know ; but what is fourth dimension f The realities denoted by the words can not be united in thought. The phrase is perfectly empty—a sign that signifies nothing. To use a Wall-Street figure, it is a metaphysical kite, not worth the breath that flies it.

Professor Zöllner’s diagrams—intended to show how a twist in a cord, which we three-dimensional beings can do or undo by turning over a part of the cord, could not be done or undone by two-dimensional beings without making one end describe a circle, and, by means of this showing, to illustrate the possibility of a four-dimensional creature tying and untying knots in an endless cord as easily as we do and undo twists in it—are sheer delusions. A cord, wdiether laid on itself or extended in only one direction, and though conceived of the utmost conceivable thinness, can not be conceived with less than three dimensions. Nor can a line or a point. When we think of a mathematical plane or line or point, we do nothing more than fix our attention on length and breadth, regardless of thickness ; or on length, regardless of both breadth and thickness ; or on position, regardless of all three : we think away from wbat remains, but we do not think it away. It is thrust off, but not out—minimized, not annihilated. No effort of thought can annihilate it. Professor Zöllner either mistakes the hyperboles of geometry for literal expressions, or supposes that they are as valid for what he calls transcendental physics as for physics, forgetting that in the former, if we vex them at all, we must pass behind symbols to the things symbolized, which, if inconceivable, are of no use in aiding us to conceive anything else. And that is the trouble with his diagrams. They symbolize inconceivable things, whereas, to answer his purpose, they should symbolize conceivable ones ; seeing that the ordinary acceptation of lineal diagrams, although sufficiently true for the conclusions of mathematics, does not cut deep enough for a fourth dimension, which, if diagrams are to facilitate our comprehension of it, needs the exact truth of what they postulate. Less than this will leave us where it finds us—“ cabined, cribbed, confined ” in three dimensions. Professor Zöllner will pardon a little rigor in the interpretation of figures employed to throw light on a theory so occult. For the purpose he has in hand, which lies considerably deeper than the lowest foundation of mathematical reasoning, we are bound to demand that his diagrams shall represent in reality exactly what they purport to represent ; that, in plain words, they shall stand for real surfaces without thickness, and real lines without breadth or thickness. But these things do not exist for the human mind. They are incapable of representation in thought. If they were conceivable, they might then assist us to comprehend a fourth dimension, because, if we once cross the absolute limits of conceivability, there is no telling what we might or might not comprehend : presumably new heavens and a new earth would open on us, and certainly a new mind would open in us. But they are not conceivable, and the speculation is idle. For this reason, if no other, Professor Zöllner’s digrammatic illustrations do not illustrate. They but offer one inconceivable thing to help us conceive another. They begin and end in nothing.

It turns out as was foreseen. Professor Zöllner understands himself even less than he understands Kant. It is very sad ; yet the sadness has a silver lining. His merit is greater than he has dreamed. Although evidently a modest man, his modesty can not be spared. He is himself the author of his theory. He owes nothing to Kant. Let justice be done, though the blushes rise. But justice requires another word. Professor Zöllner can not imagine the possibility of the truth of his theory. He can not represent to his own mind the theory itself. It is unthinkable. Not to put too fine a point on it, his theory, in philosophical strictness, is pure nonsense. Let justice be done, though the blushes spread. We are now in a position to answer the question at the head of this paper, and the answer can hardly be put more aptly than in the terms of a definition of metaphysics familiar to every civilized country, and claimed as its own, I believe, by each in turn, if not simultaneously : When Zöllner tells the world what the world can’t understand, and what Zöllner doesn’t understand himself, that is transcendental physics.