Ever since a benevolent teacher initiated me into the mystery of the unknown x, and, in doing this, introduced me to mathematical abstraction, ever since he taught me its many and often incredible practical applications, I have believed in the existence of a virtual world—invisible, formal, and, what is more, multiply layered—since later I found it in various forms, not only in law, medicine, and the fine arts, but also in private and public life. I should not say merely that I believe in this virtual world: I see it, like everyone else; what is more, I have lived a part of my life in it, I have immersed myself in it.
I begin with mathematics because those who are strangers to its sublime beauties have a harder time perceiving this virtual world than those who, in grappling with mathematics, have run up against its stubborn resistance, its proud independence, and understood its real utility. The virtual world does not submit to our laws; quite to the contrary, we obey its laws, which are discovered rather than invented, and which give access to the laws of the physical world—the same laws, miraculously, as those of mathematics.
How could we survive without this absent world that shapes our innermost being, that inflames our imaginations, that shapes our relations with others, both individually and collectively, that enriches human perception—this world whose notoriously unreasonable effectiveness alleviates the sorrows of life and lightens the burdens of work? Is it too much to claim that our deliberate use of mathematics for so many purposes sets us apart from our animal cousins? To claim that our essence and virtue as human beings reside in our manipulation of this virtual world, whose subtle gradations call to mind the color spectrum, with all its infinitely fine shades?
Once our ancestors began to paint animals and signs on the walls of caves, which is to say once they invented representation (the very term indicates the difference between absence and presence: this is not an animal, and so on); once they began, in Malta, for example, some forty thousand years ago in the Upper Paleolithic, to sculpt an improbable man-lion, with two-bodied gods of good fortune later migrating more or less everywhere, from Egypt to Mexico, another world emerged, mythical, alien, formal, imaginary, aesthetic, symbolic—I do not know how to characterize it, except to say that it is a world different from the one our immediate sensations reveal, a world that no less surely constrains our freedom of maneuver. And with the advent of spoken language, the reference and meaning of words marked out the intersection between the world we live in and categories external to it.
Plastic, supple, fluent, labile, sometimes as dense and transparent as a diamond, this other world manifests itself in different ways, depending on the place, that unfold in the form of histories. The majority of living species display roughly analogous behaviors at all latitudes, whereas human cultures, languages, religions, conventions, contractual arrangements, and the like are liable to differ markedly over small distances and in similar climates. And yet each culture evolves in such a way that successive generations of a single group may differ from one another no less than neighboring peoples differ from one another. Thus culture takes over from nature by means of a sort of exo-Darwinism in which mutations and selection occur more quickly and with greater adaptive capacity than biological evolution itself. When spring arrives we take off our warm clothes in far less time than it takes for an animal to shed its coat. The formal world conditions how we perceive our own world and permits us to make it a more hospitable habitat, better suited to our needs.
Question: Is the spiritual world evoked by religion just one among many others, or does it show—does it form, I should say, by virtue of its distribution over all cultures and of its temporal antiquity—the trunk from which these other worlds branch off, the burning source of all the rest that subsequently cools? I am sometimes inclined to favor the latter alternative, for if these other worlds often shine like light on a translucent ice floe, the religious world not only shines, it burns with intense brilliance: light and energy both, of that there can be no doubt. Nevertheless some other virtual worlds are liable also to catch fire and then to start similar fires in their turn. The fertile heat of the religious flame brings forth much that is new, but sometimes only at the cost of appalling violence. A holy flame, to be sure, but a sacred one as well, a flame that encourages innumerable murderous sacrifices. If it did not burn so fiercely, religions would not have been able to recruit so many followers; from such inauspicious beginnings, they would not have been able to flourish for thousands of years. I do not know if this hypothesis is true; in fact I often suspect it is not. Even so, comparing various virtual worlds is well worth the effort, as I will now attempt to show.
Proofs that virtual worlds exist are to be found everywhere. Mathematical techniques—invisible and absent, but nonetheless inevitable—can decipher anything. How is that we can give meaning to signals that are propagated by waves? Where does the trust that we place in contracts come from? Why do we live so often in our imaginations, filled with dreams, memories of the past, and hopes for the future? Would our world exist, the one we know as human beings, without the mathematical world, since without this other world we would not be able to understand our own and make it livable? Equations and algorithms, triangles and polyhedrons, language and music, musical notation and alphabets, images—where are all these things to be found? Show me their place and their home.
But this other world, abstract, virtual, possible, whatever else it may be, sometimes manifests its existence in our world by suddenly emerging in places of exceptional heat that, once they have cooled, are so long-lasting that their traces lie outside the boundaries of historical time. Let us call them “hot spots.”
We now know how to detect the places where fire beneath the earth’s tectonic plates, here and there more intense than elsewhere, pierces them and causes volcanic eruptions similar to those that occur on the islands of Réunion and Hawaii, or leaves behind cooled remains, like the Maldives or the Deccan Traps in west central India.
Similarly, we have mapped the various places where meteorites have struck the earth, from Siberia to the Yucatán peninsula of Mexico. There the cosmos and the netherworld abruptly came into contact with the surface of our planet, causing it to burst into flames and spew lava, masses of molten rock, and clouds. In the extreme case the result may be equivalent to a nuclear winter. Light and darkness, creative and destructive energy—in this case violent energy.
By analogy, then, I call “hot spots” those places where, at a given moment, another world manifests itself in ours, those concrete images of contact with another reality, be it virtual, intelligent, spiritual, inspirational—perhaps dangerous as well.
Our Latin ancestors imagined that Jupiter, seated in his high Olympus, hurled down thunderbolts when he was angered. The Gauls, we are told, also dreaded this unforeseeable barrage. The Romans surrounded the place where lightning struck with a low wall of stone and bronze known as a puteal (from the Latin puteus) for its resemblance to a wellhead. Now, if a well puts a subterranean source into communication with the earth’s surface, a puteal marked the spot where heaven and earth came together in a blazing short circuit. Did those who constructed this barrier around the spot where lightning had struck fear that this spot diffused a lethal energy, like a star?
The word puteus meant both well and chimney, two vertical relations in one, fiery and aqueous, up and down, between another world and this one—earth beneath our feet, fire above in the heavens, water risen up from the depths. As for air, the word that designates the soul, invisible and incorporeal, expresses its animate character through the act of breathing. Are other worlds composed of the same elements as ours?
One evening a girl named Rebecca was drawing water from a well for her family’s meal and for their animals, as she did every evening, when Isaac appeared in the form of an intermediary, his servant, a traveler parched with thirst. Another girl, named Rachel, was likewise drawing water at a wellhead when Jacob appeared, thirsty as well. Each one drank from the pitcher that the girl offered him; each one took the girl who offered him water as his wife. From the love radiated by these wells there came a posterity as dense as the crown of a beech tree.
Many generations later, in the same fashion, a Samaritan woman met the Son of Man at another such wellhead. Jesus said to her: our ancestors drank this water and died; I will give you to drink the water of eternal life. From this well, in which water was transubstantiated into ambrosia, there radiated the resurrection of the dead.
The projected shadow of the sundial was useful less as a way of telling time, something to which our ancestors attached little importance, than of observing the relation between heavenly phenomena and a particular state of affairs on earth. To this, for example, we owe the invention of the latitude scale. The sundial therefore functioned less as a clock than as an astronomical observatory. The vertical axis intercepting sunlight—which is to say the part of the sundial that casts a shadow, the shaft (or pointer)—the Greeks called gnomon, a word that in their language meant knowing, understanding. Was this the first lightning rod?
Here we have not one but two blazing short circuits: between the sun and the ground by light and shadow, as our eyes see it, but above all between a vertical, material shaft and a decodable knowledge, as I like to call computer software; between the concrete, on the one hand, and the abstract, on the other; between the energy of light and the subtlety of information. A hot spot, in other words.
From this relation between light and shadow it was possible to acquire information about space and time on earth. Here we have one of the first realizations of an artificial intelligence: a metal shaft is said to be gnomon, that is, knowing. The sun came down to earth and, with its shadow, wrote on it marks, a kind of script, that had to be decoded. Thus the energy of the solar fire brought forth information.
Thales is said to have invented his theorem by comparing the shadow cast by one of the three great pyramids of Egypt with the one cast at the same hour by a man. This gnomonic story, as it might well be called, omits to note that anyone standing at the foot of Khufu, not merely Thales, could have seen that Khafre and Menkaure are similar figures. The theory of homothetic forms, which is to say figures having the same form but on different scales, is illustrated here in visible, tangible stones. Sundials, the story says. Objection: no matter that the height of a man and the length of the shadows cast by his upright body and the shaft (or gnomon) of a sundial were easily measured, the line that went to the exact center of the pyramid remained hidden beneath a thick wall of stone.
To make the line visible, it was necessary to conceive of an abstract volume, the tetrahedron—empty, luminous, and transparent; that is, the sun had to be brought into the very darkness of this edifice. Geometry therefore did not come into being solely by means of a gnomonic representation of light and shadow; it was also revealed in a wholly different manner, at the moment when the sun descended into a blind mass and penetrated it. Can we begin to imagine how much solar energy was needed to chase opacity from the stone?
Here, in a way that is readily visualized, we apprehend the dazzling discovery of the abstract. At the foot of Khufu, Thales contemplated a blinding short circuit between this world and another one: the sudden intuition of homotheticity and the black box of stone pierced by the light of the sun, as though it were lightning. It was for this reason that the Greeks named the structure “pyramid,” which is to say fire; and when Plato, in the Timaeus, describes the parallel between earthly elements and abstract polyhedrons, he associates fire with the tetrahedron, which is to say with the pyramid: fire comes down to earth and enters it. In order to invent geometry, by means of a theorem that retains its validity still today, after thousands of years, was it therefore necessary to surround a hot spot with a puteal?
Plato goes on later in the Timaeus to say that physical measurement, because it is always approximate, can never attain the ideal precision of geometry, unlike stereometry, the measurement of regular polyhedrons, which he was the first to reduce to five basic types. A sun must have penetrated and emptied these black volumes!
Another Greek legend relates the drowning of Hippasus of Metapontum, accused by the Pythagoreans of having divulged the secret, nervously guarded by the members of the brotherhood, concerning the discovery, shameful but inevitable, of irrational lengths and numbers whose infinite extension destroyed the serenity of logos, reason, and proportion. Are we to interpret Hippasus’s drowning at sea—precisely the bottomless well that we met with earlier—as a divine punishment or as a sort of collective lynching?
Yet another legend tells of Archimedes’s use of mirrors to set fire to the Roman fleet besieging Syracuse in the third century BCE by reflecting the rays of the sun. Here, with literally lightning suddenness, we pass from geometrical optics to combats in which soldiers, now transformed into torches, are killed; from gnomon to massacre. Information is returned to energy, energy assumes the form of violence.
Traveling around the island of Sicily we encounter next Empedocles, one of the earliest physicists, who, again according to legend, committed suicide by throwing himself into the crater of Etna, and then Ettore Majorana, who more than twenty centuries later mysteriously disappeared, no one knows why, though no doubt he was frightened at having come into possession, at least from a distance, of the quasi-solar fires of the atomic bomb. In this case we pass from physics to massive destruction, from an inventive puteal to a destructive one, from the abstract to the concrete, from information to energy, and finally to violence: volcano, incendiary mirrors, the lucid and blind announcement of Hiroshima . . .
If the triangular island of Sicily, a scale model of our world and its history, displays three local hot spots, it may well be wondered whether today we are transforming our world into a global hot spot. We have long believed that the fires of science produce less violence than those of religion. We were mistaken.
Contrary to a persistent legend, neither Copernicus nor Galileo managed to establish the reality of heliocentrism, limited as they were by the relativity of motions. Pascal, Descartes, and Leibniz rightly reaffirmed the equivalence of their theories, asserted long before by ancient Greek astronomers. It was not until 1725, however, when James Bradley discovered the aberration of light, that the sun was actually shown to be at the center of the planetary system. Then and only then was Kant able to speak of a “Copernican revolution,” thus setting truth on the retrograde course from which the legend arose.
Galileo’s true originality, which was to prove decisive, has to do with the connection he established between mathematics and experiment. The Greeks had missed this point of intersection, with the result that they were unable to develop an exact science of the world. Galileo, by associating a particular equation with a particular experiment, succeeded in creating a blindingly fruitful short circuit between a formal, virtual world and the real world that we perceive. In so doing, he heralded the advent of modern science. His method uncovered a hot spot.
The philosopher of science Alexandre Kojève grasped what was really at stake in Galileo’s famous trial. Galileo invented mathematical physics, which is to say he brought a concrete experiment carried out in this world into contact with an equation that had been known for centuries, suspended in a virtual world, pure and abstract. His invention, which Einstein called a miracle and which I call a hot spot, showed that only mathematics, however formal, virtual, and absent it may be, is capable of decoding reality, of writing the great book of nature, as Galileo put it, in its own language.
Now, the Church of Rome taught the Incarnation of Jesus Christ, which is to say a short circuit, blindingly luminous and charged with a unique truth, between the real, incarnate world, on the one hand, and, on the other, a kingdom utterly separate from it—a hot spot, if ever there was one. Did Galileo discredit the doctrine that the earth is at the center of the universe by an analogous gesture? Did the importance of his trial derive from a similarity between this dogmatic puteal and the short circuit brought about by mathematical physics? Were these two hot spots, these two points of contact between the immanent and the transcendent, unforeseeable and unimaginable, themselves somehow related?
The universe of physical laws involves all of mathematics—geometry, topology, algebra, number theory and algorithms, probability calculus, and so on. The plurality of these laws traces a sort of silhouette of mathematics as a whole. Scientific discovery therefore creates a direct link between a precise locality in the realm of mathematics and a definite phenomenon of the real world, a thread of the fabric that, virtually at least, unites equations and experiments. Thus a short circuit produces another hot spot.
Kojève suspected that mathematical physics emerged—perhaps could only have emerged—in the context of the doctrine of the Incarnation. Pascal had the same intuition in connection with his search for a fixed point: no learning, not even the most rigorous, could furnish such a point; only Jesus Christ himself, the central body to which everything tends, could supply it. The totality of the sciences turns around this sun.
That there exists another world than ours should therefore be plain; that one should turn out to be older than all the others in no way proves that it is the cause of them, their stock or trunk—post hoc sed non propter hoc.