Four-Dimensional Vistas.
by Claude Fayette Bragdon.
INTRODUCTION
There are two notable emanc.i.p.ations of the mind from the tyranny of mere appearances that have received scant attention save from mathematicians and theoretical physicists.
In 1823 Bolyai declared with regard to Euclid"s so-called axiom of parallels, "I will draw two lines through a given point, both of which will be parallel to a given line." The drawing of these lines led to the concept of the curvature of s.p.a.ce, and this to the idea of _higher_ s.p.a.ce.
The recently developed Theory of Relativity has compelled the revision of the time concept as used in cla.s.sical physics. One result of this has been to introduce the notion of _curved_ time.
These two ideas, of curved time and higher s.p.a.ce, by their very nature are bound to profoundly modify human thought. They loosen the bonds within which advancing knowledge has increasingly labored, they lighten the dark abysses of consciousness, they reconcile the discoveries of Western workers with the inspirations of Eastern dreamers; but best of all, they open vistas, they offer "glimpses that may make us less forlorn."
FOUR-DIMENSIONAL VISTAS
I THE QUEST OF FREEDOM
THE UNDISCOVERED COUNTRY
Expectancy of freedom is the dominant note of to-day. Amid the crash of armies and the clash of systems we await some liberating stroke which shall release us from the old dreary thralldoms. As Nietzsche says, "It would seem as though we had before us, as a reward for all our toils, a country still undiscovered, the horizons of which no one has yet seen, a beyond to every country and every refuge of the ideal that man has ever known, a world so overflowing with beauty, strangeness, doubt, terror and divinity, that both our curiosity and our l.u.s.t of possession are frantic with eagerness."
Should a name be demanded for this home of freedom, there are those who would unhesitatingly call it _The Fourth Dimension of s.p.a.ce_.
For such readers as may be ignorant of the amazing content of this seemingly meaningless phrase, any summary attempt at enlightenment will lead only to deeper mystification. To the question, where and what is the fourth dimension, the answer must be, it is here--in us, and all about us--in a direction toward which we can never point because at right angles to all the directions that we know. Our s.p.a.ce cannot contain it, because it contains our s.p.a.ce. No walls separate us from this demesne, not even the walls of our fleshly prison; yet we may not enter, even though we are already "there." It is the place of dreams, of living dead men: it is _At the Back of the North Wind_ and _Behind the Looking Gla.s.s_.
So might one go on, piling figure upon figure and paradox upon paradox, to little profit. The effective method is the ordered and deliberate one; therefore the author asks of his reader the endurance of his curiosity pending certain necessary preparations of the mind.
MIRACLES
Could one of our aviators have landed in ancient Athens, doubtless he would have been given a place in the Greek Pantheon, for the old idea of a demiG.o.d was a man with wings. Why, then, does a flying man so little amaze us? Because we know about engines, and the smell of gasoline has dulled our sense of the sublime. The living voice of a dead man leaves us unterrified if only we can be sure that it comes from a phonograph; but let that voice speak to us out of vacancy and we fall a prey to the same order of alarm that is felt by a savage at the report of a gun that he has never seen.
This ill.u.s.tration very well defines the nature of a miracle: it is a manifestation of power new to experience, and counter to the current thought of the time, Miracles are therefore always in order, they always happen. It is nothing that the sober facts of to-day are more marvellous than the fictions of Baron Munchausen, so long as we understand them: it is everything that phenomena are multiplying, that we are unable to understand. This increasing pressure upon consciousness _from a new direction_ has created a need to found belief on something firmer than a bottomless gullibility of mind.
This book is aimed to meet that need by giving the mind the freedom of new s.p.a.ces; but before it can even begin to do so, the reader must be brought to see the fallacy of attempting to measure the limits of the possible by that faculty known as common sense. And by common sense is meant, not the appeal to abstract reason, but to concrete experience.
THE FAILURE OF COMMON SENSE
Common sense had scarce had its laugh at Bell, and its shout of "I told you so!" at poor Langley, when lo! the telephone became the world"s nervous system, and aeroplanes began to multiply like summer flies. To common sense the alchemist"s dream of trans.m.u.ting lead into gold seems preposterous, yet in a hundred laboratories radium is breaking down into helium, and the new chemistry bids fair to turn the time-honored jeer at the alchemists completely upside down.
A wife whose mind was oriented in the new direction effectually silenced her husband"s ridicule of what he called her credulity by reminding him that when wireless telegraphy was first suggested he had exclaimed, "Ah, that, you know, is one of the things that is not possible!" He was betrayed by his common sense.
The lessons such things teach us are summed up in the reply of Arago, the great savant, to the wife of Daguerre. She asked him if he thought her husband was losing his mind because he was trying to make permanent the image in a mirror. Arago is said to have answered, "He who, outside of pure mathematics, says a thing is impossible, speaks without reason."
Common sense neither leads nor lags, but is ever limited to the pa.s.sing moment: the common knowledge of to-day was the mystery and enchantment of the day before yesterday, and will be the mere commonplace of the day after to-morrow. If common sense can so little antic.i.p.ate the ordinary and orderly advancement of human knowledge, it is still less able to take that leap into the dark which is demanded of it now. The course of wisdom is therefore to place reliance upon reason and intuition, leaving to common sense the task of guiding the routine affairs of life, and guiding these alone.
THE FUNCTION OF SCIENCE
In enlisting the aid of reason in our quest for freedom, we shall be following in the footsteps of mathematicians and theoretical physicists. In their arduous and unflinching search after truth they have attained to a conception of the background of phenomena of far greater breadth and grandeur than that of the average religionist of to-day. As a mathematician once remarked to a neo-theosophist, "Your idea of the ether is a more material one than the materialist"s own." Science has, however, imposed upon itself its own limitations, and in this connection these should be clearly understood.
Science is that knowledge which can be gained by exact observation and correct thinking. If science makes use of any methods but these it ceases to be itself. Science has therefore nothing to do with morals: it gives the suicide his pistol, the surgeon his life-saving lance, but neither admonishes nor judges them. It has nothing to do with emotion: it exposes the chemistry of a tear, the mechanism of laughter; but of sorrow and happiness it has naught to say. It has nothing to do with beauty: it traces the movements of the stars, and tells of their const.i.tution; but the fact of their singing together, and that "such harmony is in immortal souls," it leaves to poet and philosopher. The timbre, loudness, pitch, of musical tones, is a concern of science; but for this a Beethoven symphony is no better than the latest ragtime air from the music halls. In brief, science deals only with _phenomena_, and its gift to man is power over his material environment.
MATHEMATICS
The gift of pure mathematics, on the other hand, is primarily to the mind and spirit: the fact that man uses it to get himself out of his physical predicaments is more or less by the way. Consider for a moment this paradox. Mathematics, the very thing common sense swears by and dotes on, contradicts common sense at every turn. Common sense balks at the idea of _less than nothing_; yet the _minus_ quant.i.ty, which in one sense is less than nothing in that something must be added to it to make it equal to nothing, is a concept without which algebra would have to come to a full stop. Again, the science of quaternions, or more generally, a vector a.n.a.lysis in which the progress of electrical science is essentially involved, embraces (explicitly or implicitly) the extensive use of _imaginary_ or _impossible_ quant.i.ties of the earlier algebraists. The very words "imaginary" and "impossible" are eloquent of the defeat of common sense in dealing with concepts with which it cannot practically dispense, for even the negative or imaginary solutions of imaginary quant.i.ties almost invariably have some physical significance. A similar statement might also be made with regard to _transcendental_ functions.
Mathematics, then, opens up ever new horizons, and its achievements during the past one hundred years give to thought the very freedom it seeks. But if science is dispa.s.sionate, mathematics is even more austere and impersonal. It cares not for teeming worlds and hearts insurgent, so long as in the pure clarity of s.p.a.ce, relationships exist. Indeed, it requires neither time nor s.p.a.ce, number nor quant.i.ty. As the mathematician approaches the limits already achieved by study, the colder and thinner becomes the air and the fewer the contacts with the affairs of every day. The Promethean fire of pure mathematics is perhaps the greatest of all in man"s catalogue of gifts; but it is not most itself, but least so, when, immersed in the manifoldness of phenomenal life, it is made to serve purely utilitarian ends.
INTUITION
Common sense, immersed in the mere business of living, knows no more about life than a fish knows about water. The play of reason upon phenomena dissects life, and translates it in terms of inertia. The pure logic of mathematics ignores life and disdains its limitations, leading away into cold, free regions of its own. Now our desire for freedom is not to vibrate in a vacuum, but to live more abundantly.
_Intuition_ deals with life directly, and introduces us into life"s own domain: it is related to reason as flame is related to heat. All of the great discoveries in science, all of the great solutions in mathematics, have been the result of a _flash_ of intuition, after long brooding in the mind. _Intuition illumines_.
Intuition is therefore the light which must guide us into that undiscovered country conceded by mathematics, questioned by science, denied by common sense--_The Fourth Dimension of s.p.a.ce_.
OUR SENSE OF s.p.a.cE
s.p.a.ce has been defined as "room to move about." Let us accord to this definition the utmost liberty of interpretation. Let us conceive of s.p.a.ce not alone as room to move ponderable bodies in, but as room to think, to feel, to strike out in unimaginable directions, to overtake felicities and knowledges unguessed by experience and preposterous to common sense. s.p.a.ce is not measurable: we attribute dimensionality to s.p.a.ce because such is the method of the mind; and that dimensionality we attribute to s.p.a.ce is progressive because progression is a law of the mind. The so-called dimensions of s.p.a.ce are to s.p.a.ce itself as the steps that a climber cuts in the face of a cliff are to the cliff itself. They are not necessary to the cliff: they are necessary only to the climber.
Dimensionality is the mind"s method of mounting to the idea of the infinity of s.p.a.ce. When we speak of the fourth dimension, what we mean is the fourth stage in the apprehension of that infinity. We might as legitimately speak of a fifth dimension, but the profitlessness of any discussion of a fifth and higher stages lies in the fact that they can be intelligently approached only through the fourth, which is still largely unintelligible. The case is like that of a man promised an increase of wages after he had worked a month, who asks for his second month"s pay before he is ent.i.tled to the first.
THE SUBJECTIVITY OF s.p.a.cE
Without going deep into the doctrine of the ideality--that is, the purely subjective reality--of s.p.a.ce, it is easy to show that we have arrived at our conception of a s.p.a.ce of three dimensions by an intellectual process. The sphere of the senses is two-dimensional: except for the slight aid afforded by binocular vision, sight gives us moving pictures _on a plane_, and touch contacts _surfaces_ only.
What circ.u.mstances, we may ask, have compelled our intellect to conceive of _solid_ s.p.a.ce? This question has been answered as follows:
"If a child contemplates his hand, he is conscious of its existence in a double manner--in the first place by its tangibility, the second by its image on the retina of his 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 position under which it is observed, notwithstanding that the form and extension of the image on the retina constantly change with the different position and distance of his hand in respect to his eye. The problem is thus set to the child"s understanding: how to reconcile to his comprehension the apparently contradictory facts of the _invariableness_ of the object together with the _variableness_ of its appearance. This is only possible within a s.p.a.ce 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."
Thus we have come to the idea of a three-dimensional s.p.a.ce in order to overcome the apparent contradictoriness of facts of sensible experience. Should we observe in three-dimensional s.p.a.ce contradictory facts our reason would be forced to reconcile these contradictions, also, and if they could be reconciled by the idea of a four-dimensional s.p.a.ce our reason would accept this idea without cavil. Furthermore, if from our childhood, phenomena had been of daily occurrence requiring a s.p.a.ce of four or more dimensions for an explanation conformable to reason, we should feel ourselves native to a s.p.a.ce of four or more dimensions.
Poincare, the great French mathematician and physicist, arrived at these same conclusions by another route. By a process of mathematical reasoning of a sort too technical to be appropriately given here, he discovers an order in which our categories range themselves naturally, and which corresponds with the points of s.p.a.ce; and that this order presents itself in the form of what he calls a "three circuit distribution board." "Thus the characteristic property of s.p.a.ce," he says, "that of having three dimensions, is only a property of our distribution board, _a property residing, so to speak, in human intelligence_." He concludes that a different a.s.sociation of ideas would result in a different distribution board, and that might be sufficient to endow s.p.a.ce with a fourth dimension. He concedes that there may be thinking beings, living in our world, whose distribution board has four dimensions, and who do consequently think in hypers.p.a.ce.
THE NEED OF AN ENLARGED s.p.a.cE-CONCEPT
It is the contrariety in phenomena already referred to, that is forcing advanced minds to entertain the idea of higher s.p.a.ce.
Mathematical physicists have found that experimental contradictions disappear if, instead of referring phenomena to a set of three s.p.a.ce axes and one time axis of reference, they be referred to a set of four interchangeable axes involving four h.o.m.ogeneous co-ordinates.
In other words, _time_ is made the fourth dimension. Psychic phenomena indicate that occasionally, in some individuals, the will is capable of producing physical movements for whose geometrico-mathematical definition a four-dimensional system of co-ordinates is necessary. This is only another step along the road which the human mind has always travelled: our conception of the cosmos grows more complete and more just at the same time that it recedes more and more beneath the surface of appearances.
Far from the Higher s.p.a.ce Hypothesis complicating thought, it simplifies by synthesis and co-ordination in a manner a.n.a.logous to that by which plane geometry is simplified when solid geometry becomes a subject of study. By immersing the mind in the idea of many dimensions, we emanc.i.p.ate it from the idea of dimensionality.
But the mind moves most readily, as has been said, in ordered sequence. Frankly submitting ourselves to this limitation, even while recognizing it as such, let us learn such lessons from it as we can, serving the illusions that master us until we have made them our slaves.