The truth, entirely to the contrary, is on the side of the imagination; as the immensity continues to escape us, though, we fix a few partial symbols by means of calculation.
Mathematics, like the fine arts, is merely an algebraic representation of reality, but, while the fine arts summarize qualitative relations borrowed from life itself, mathematics only conceives quantitative relations based in a game of numbers that is fixed in advance.
Calculation is anthropomorphic.
Now, what finer anthropomorphic monument, elevated beyond man in man’s own image, could we find than the mathematical edifice? Based on the number 1, the only one that we find in our consciousness, it is comprised of this number reproduced infinitely around us, as if by mirrors: a comfortable, even fecund, hypothesis—a reassuring hypothesis, like so many others, since it permits us to obtain a certain victory, by playing a game whose rules we have established—but a limited hypothesis, incapable of integrating the continuity of the universe by means of its limited numbers.
Calculation only glimpses a fraction of reality.
I know full well that the ancients excluded arithmetical signs from their calculations, and that it was only after the Renaissance—or indeed, in more practical terms, after the invention of the metric system—that numerical measures were introduced into formulas. It is no less true, however, that mathematics, today as yesterday, even as it is elevated to the heights of philosophical speculation, never expresses anything but numerical grandeur.
Thanks to new physical observations, mathematics has been enriched by new formulas, just as a library is enriched by new books, but no library ever created a thought or gave birth to a book.
The imagination is a closer approximation of life.
The imagination is, therefore, the reality of which mathematics is only the memory.
Now, a memory is limited in certain respects, while a reality is not, in a world where everything is continuous and connected, and what we call illusion is often a better approximation of a superior reality than what we call certainty.
Let us attempt to make this notion more precise by means of some simple examples.
Perspective is a better approximation of our consciousness than a plan.
An architect imagines a monument: the colonnade of the Louvre, for instance. He conceives all its elements equally and in equilibrium; geometry and calculation immediately furnish him with the formulas corresponding to that intuition, and the monument is constructed.
A painter then comes with his easel to make a view of the monument, and the necessity of designing it in perspective immediately imposes itself upon his mind. Geometry and arithmetic, which are good daughters, and live at everyone’s expense, immediately translate his desires into formulas and give him, with the same certainty, an elevation quite different from the one that they provide to the architect, with receding lines, unequal and deformed, but no less certain and exact. The monument is the same, but two opposed mathematical certainties render accounts of it according to the relative viewpoints of two observers.
But, you might say, reality is on the side of the architect, and optical illusion on the side of the painter. How do you know that, if not by the experience of your senses—as the phrase “optical illusion” indicates—and how do you know that perspective is not opening a domain to us even more real, more elevated and more universal in the material of art?
Laws of attraction of lines and masses, vaguely anticipated by ancient architects, led them to warp the two superior lines of a triangular fronton into a skyward bulge, to resist in their extremities the attraction of the base-line. These lines now appear to us to be rigorously straight, whereas, if they really were straight, they would appear to dip at their extremities, as in the modern fronton of the Madeleine. Analogous observations led them to incline their outermost columns in order that they should appear straight.
I ask you, where is the reality? In the conception that deforms, under the pretext of reality, or in the one that satisfies the secret laws of art, and renders a masterpiece perfect?
Other sensory deformations providing superior realities.
Another example: let us observe a moving automobile passing before us at top speed. The sound of the motor as it comes towards us is high-pitched, and immediately declines in pitch as it draws away. Nothing is simpler to explain, relative to us, but it requires a new notion of speed, independent of that of sound. Which is the more complete truth with respect to our consciousness? Evidently, the one that renders the more complete account of the sensations.
But that is nothing; this is what is important: the car’s chassis, traveling at speed, appears to us to be shorter than it really is. As for the wheels, which we know to be round, they present to us the appearance of elongated ovals, inclined at the top in the direction of movement. A mere illusion, obviously, which is modified according to the position of the observer—but an illusion that is shared by the mechanical eye of a photographic apparatus, and an illusion that calculation can reduce to a formula. It is evident, in fact, that the top of the wheel is moving more rapidly than the bottom, since its movement is added to that of the car instead of being restricted by its backward movement—I mean, of course, relative to the observer, since it is obvious that the wheel’s own movement is constant in all its parts, relative to its hub.
What can be deduced from such an apparently facile observation?
Simply this:
Firstly, that all observation in the physical domain is relative, and only has value in relation to the observer.
Secondly, that is it not appropriate to speak in terms of illusions or realities, reality being no more than an observation which one unthinkingly believes to hold all the necessary elements, and an illusion an observation from which one is aware that it is appropriate to detach new and unknown elements.
Thirdly, that the role of science is not to explain that unknown, but to attach a symbol to our observation that fixes the unknown in place, thus facilitating further research.
What might an optical deformation be in a two-dimensional world? A superior truth.
Let us now suppose, if you will, that we belonged to a two-dimensional world, and that our eyes, unaware of the accommodation, could only perceive plane surfaces.
Imagine a square-shaped sign turning towards us about a diagonal axis.53 It appears to us progressively as an irregular diamond, and then vanishes completely, leaving only a straight line. Should we perhaps tell ourselves gravely that the plane has been reduced to infinity, or declare that it now only belongs to a world of one dimension? What would transpire then if a thinker, imagining the world in three dimensions, were to say that, not only did the original square still exist but that its revolution has encompassed a vaster and more comprehensive world? The defenders of reality would undoubtedly take him for a madman.
The Lorentz Transformation poses an analogous problem, solely accessible to mathematical vision.
An analogous adventure is in the process of turning contemporary science upside down.
We know that, in order to reconcile the phenomenon of Bradley’s aberration (the image of a distant star being displaced in the telescope by reason of the movement of the Earth) with the no-less-certain Michelson and Morley experiment (establishing that the movement of the Earth has no influence on the velocity of light) Lorentz—anticipated by Fitzgerald—supposes that bodies contract in the direction of their movement. Every body in motion contracts, but that contraction is quite difficult to observe, since the Earth, for example, traveling at 30 kilometers a second, only suffers a contraction of 6 centimeters in a diameter of 12,740 kilometers—but that was sufficient to contract by one 50 millionth the bar bearing the mirrors in Michelson’s interferometer, and thus to affect the experiment.54
Einstein draws us closer to physical reality.
Thanks to Einstein, these convenient hypotheses have recently acquired a singular amplitude. Based on the observations of Michelson, Einstein takes the speed-li
mit of our universe to be 300,000 kilometers a second, which is that of light. If we compare light to a swimmer, we observe, in effect, with Michelson, that its speed is the same whether it is swimming with or against the current.
Bodies in motion contract.
One cannot travel faster than light and no assistance can permit light to travel any faster than it does. Bodies in motion contract in the direction of their movement. They become progressively flatter as their movement increases; at 260,000 kilometers a second, they diminish by half; when they attain the limiting velocity of 300,000 kilometers a second they become infinitely flat.
Einstein informs us, moreover, that light has mass, since celebrated astronomical verifications prove that it is subject to gravitation. The density of the luminous particle must, therefore, be infinite by definition, as we already know that the mass of a particle whose speed attains that of light becomes infinite, according to the theory of Kauffman and Max Abraham.55
Our Earth, being in motion, is contracted. The measuring instruments that are located there are contracted in the same proportion—including, like all the rest, those which measure time. Our measurements are, therefore, correct relative to a system in motion; they would not be the same in a system animated by a different motion. Time is therefore modified by reason of motion, and there is no absolute time.
Time is a function of velocity.
A year passed on Earth corresponds to a shorter time on a body moving more rapidly, and a terrestrial eternity corresponds to an absence of time in a world moving at the velocity of light. As for the providential and mysterious ether, a sort of graph-paper that has served until now as the fixed basis on which all hypotheses are inscribed, it does not exist. Mass or energy is nothing other than movement; it increases or diminishes with it in the opposite sense to time.
The time-motion relation varies between two infinities.
It therefore does not appear to us to be absurd to conclude that absolute immobility corresponds to eternity and an absence of mass occupying an infinitely large space, while absolute velocity corresponds to an absence of infinitely-dilated time and an immensely dense mass occupying an infinitely small space. Thus, our field of study extends beyond two infinities—which conforms quite well, it must be admitted, to a theory of relativity.
The interval ensures the equilibrium of the balance that weighs the universe.
Not the least praiseworthy aspect of Einstein’s powerful theories is the coherency of their assembly. Thanks to them, our universe attains a marvelous order and equilibrium; it folds back upon itself in the form of an egg, the luminous rays completing a curve, since they are subject to inertia, returning to their point of departure. No unknown remains, since time and space are interrelated, as the two pans of a balance are by the beam, by means of the supreme theory of the Interval, the masterpiece and keystone of the system that advantageously replaces the obsolete notions of fixed space and absolute time.
Time becomes the fourth dimension.
From there to envisage time as being the fourth dimension of the universe is only a single step, and that step has been lightly taken. The dimensions of objects are modified by velocity, so time, the final co-ordinate, must be added to the other three and we conceive henceforth of a four-dimensional universe. Moreover, as Minkowski has remarked, such has always been a matter of current observation, for no point in space has ever been seen save from a certain moment, nor time perceived other than from a point in space.56
The world is no longer Euclidean.
Let us finally add that our world is no longer Euclidean; the geodesic or alignment of the universe is no longer a straight line but a curve, which becomes the shortest distance between two points—and this leads us from the primitive principle of special relativity to the general theory of relativity that no longer recognizes any difference between a field of motion and a gravitational field.
Do Einstein’s theories correspond to physical reality?
A difficulty of interpretation nevertheless crops up, which has not failed to divide our scientists. It has been repeatedly said that the Einsteinian doctrine is not translatable into everyday language and that its comprehension is only possible in the realm of pure mathematics.
Now, we have made the observation that mathematics, transcendent as it is, does not create any new verity and only furnishes useful symbols to glimpsed physical realities. If so, however brilliant Einstein’s theories might be in their mathematical form, it is no less true that it is in physical reality that they must be verified.
The Lorentz contraction has a physical meaning.
This is so true that, for Einstein, the Lorentz deformation—or, better, transformation—is not merely a mathematical artifice, as its author thought, but that it has a physical meaning. He is less affirmative, however, where general relativity is concerned.
It is at this point that we arrive at the edge of the abyss whose existence I have not ceased to advertise since the beginning of this essay—that abyss or, more simply, that mysterious turning that we rediscover every time we take a tentative step towards the unknown.
Where, then, is the more important reality, as I asked above: in the plan of an architect or in the moving perspective of an architect? In the round form of an automobile wheel at rest or in its complex deformation at speed? In the cold representation of absolute time and space or in that of their variation and inter-connection? What shall I tell you now? What should we call the reality of preference: life in motion or its negation, immobility and death?
Perhaps, though, this physical meaning can only be translated in symbols.
But how can this moving and continuous life, which resides in the utmost depths of our subconscious, be transposed into broad daylight? How can its aspects be fixed, made graspable and intelligible? How can we make it communicable by one man to another, if not by means of sufficiently general symbols? These symbols, in the first place, are the words of the language, which have a different nuance for each of us, but which represent approximately the same things and the same ideas to everyone. Then there are the mathematical formulas which are to physical phenomena what the algebra of language is to ideas.
The reality is obviously not in the word or the formula, but it corresponds to that word or formula and remains inexpressible without it.
To immobilize ideas or things by means of verbal or mathematical formulas, as one fixes time in the divisions of a clock-face, is an act of primitive knowledge. To imagine, by contrast, in literature or in science, moving symbols representing in a more proximal fashion the moving mystery of nature, testifies, by contrast, to a superior culture—and the formula, in this case, becomes more powerful and more comprehensive than the word.
Words and formulas are only valuable in relation to reality, but they see further than our senses.
A mathematical formula is, therefore, a verb of superior power, but it is never any more than a word, whose power only comes from what it expresses and which, in defining the reality, entirely removes, by virtue of that very fact, its character of life and continuity.
If Einstein’s Time-Space relationship were nothing but a pure mathematical artifice not corresponding to any physical reality, it would be of no more value than a pun.
It is therefore necessary that it rests upon a physical reality, but it is not necessary—quite the contrary—that the physical reality in question be directly graspable other than by reasoning, since it surpasses the range of our senses.
It responds, moreover, to a mental need that, by purely literary means and well before the popularization of Einstein’s work, has suggested the necessity of such a relationship to us. One can take account of this by reading the chapters entitled “The Silent Soul,” “The Innumerable Diligence,” “Spatial Abstractions,” “The Transmutation of the Atoms of Time,” etc., absolute speed nevertheless being, for us, that of thought.
Time cannot be the fourth dimension completing the universe.
Is Time, therefor
e, the fourth dimension, as Einstein wishes? On this point, we cannot agree with him, and we think that literature, by the very reason of its imprecision, is more prudent and closer to the truth than mathematical formulation.
From the fourth dimension we expect, in fact, the explanation of all phenomena and their contraries, all qualities and their contraries—in a word, the total explanation of our world and its contrary. Now, Einstein’s Time cannot bring us such integral notions. It varies so meekly as a function of space, incorporating itself by that token to the three dimensions, that one is finally led to ask whether it has not disappeared beneath the scarcely-glorious name of the Third Dimension when a body, hurtling at a velocity of 300,000 kilometers per second, no longer sustains anything more than a plane two-dimensional surface. That it is a fourth dimension—relative, symbolic—we are not ready to admit.
Einstein’s Time has the marvelous virtue of being simultaneously time, energy, mass, density and velocity. In sum, what does it represent? All that is moving and mysterious in nature, all that elevates consciousness rather than the senses—with the exception, nevertheless, of consciousness itself. The union, by means of relative motion, of this moving domain with the three exhausted dimensions moving in space, thus reconstitutes the true model of a moving world of four dimensions.
Is this model complete? Will it permit us, for instance, to formularize qualities as well as quantities and to construct a work of art mathematically? It is permissible to doubt it. And even if we admitted it, would not that model of the world remain specifically relative to our system, since it does not conceive its contrary?
We rediscover here the eternal limit of mathematical hypotheses.
In truth, magnificent as the theories of Einstein are—permitting, for the first time, the conception in mathematical symbolism of all the physical forces—they appear to us, by the very reason of their mathematical character, to be subject to the traditional limits that the subtle Zeno assigned to the savants in pursuit of the tortoise.
Journey to the Land of the Fourth Dimension Page 26
