The Senders had to know this. It was expected that the content of the signal from the stars would be mathematics. Great stock, as you know, was placed in the almighty Pythagorean triangles; we were going to greet, across space, other civilizations — with Euclid’s geometry. The Senders chose another way, and I believed that they were right. With ethnic language they could not break free of their planet, because every language is pinned to a local foundation. Mathematics, on the other hand, is a severance too complete. It cuts bonds not only locally; it parts with the limitations that have become parameters for villainies and virtues; it is the result of a search for a freedom that dispenses with every tangible verification. It is the act of builders whose wish is that the world should never be able, not in any way, to disturb their work. Consequently, with mathematics one can say nothing about the world — it is called “pure” for the very reason that it has been purified of all material dross, and its absolute purity is its immortality. But precisely therein lies its arbitrariness, for it can beget any sort of world, as long as that world is consistent. Out of the infinite number of possible mathematics we have chosen one; our history decided this for us with its various unique and irreversible vicissitudes.
With mathematics one may signal only that one Is, that one Exists. If one wishes to act more effectively at a distance, the sending of a “production recipe” becomes inevitable. But such a recipe presupposes a technology, and technology is a transient, mutable condition, a passing from one set of materials and methods to another. And what of a description of an “object”? But an object, too, may be described in an infinite number of ways. It was an impasse.
There was one thing that bothered me. The stellar code had been transmitted in a continuous fashion, in uninterrupted repetitions, and this made no sense, because it hindered recognition of the signal as a signal. Poor Laserowitz had not been altogether mad: zones of periodic silence indeed seemed necessary — more, imperative — as an indication of the artificial nature of the signal. Periods of quiet would have drawn the attention of any observer. Why, then, was this not done? The question haunted me. I tried turning it around: the lack of interruptions seemed a lack of information, information indicating the intelligent source of the emission. But what if actually that was additional information? What could such a thing mean? That the “beginning” and the “end” of the message were nonessential. That one could read it starting at any point.
The idea fascinated me. I understood now why my friends had been so careful not to tell me anything about the ways in which the “letter” had been attacked. I was, as they wanted me to be, entirely without preconceptions. At the same time I had to wage the battle, so to speak, on two fronts at once: the main “opponent,” of course, whose motives I tried to guess, was the unknown Sender, however, at the same time I could not help also thinking, at every step of my reasoning, about whether or not the mathematicians of the Project had taken the same path as I. All I knew about their work was that it had yielded no definitive result, not merely in the sense that they had failed to decipher the “letter,” but in the sense, too, that they remained uncertain — in other words, they had not proved — that the “letter” belonged to the category of information that had been hypothesized: the “thing-process.”
Quite like my predecessors, I felt that the code was overly laconic. It could have been supplied, after all, with an introductory part, showing, in simple statements, how one ought to read it. Or so it seemed. But the laconicism of the code was not an objective property of the code; it depended, rather, on the degree of knowledge of the receiver — or, more precisely, on the difference in knowledge possessed by sender and receiver. The same information could be found sufficient by one receiver and “too laconic” by another. Any object, the simplest object, contains, potentially, an infinite amount of information. Therefore, however much we detail a transmitted description, it will always be unnecessarily precise for some and fragmentary for others. The difficulty we were encountering only showed that the Sender was addressing parties more advanced than mankind at the given historical moment.
Information that is divorced from objects is not only incomplete; it invariably represents some kind of generalization. Its referent is never fully designated. On an everyday basis we are of another opinion, since this fuzziness in the designation of objects is, in ordinary life, barely perceptible. It is the same in science. Although we now know that speeds cannot be added arithmetically, we do not make a relativistic correction when we add the velocity of a ship to the car driving on its deck, because the correction, for speeds not near that of light, is so minuscule as to be meaningless. Now, there exists an informational equivalent to this relativistic effect: the notion of “life” is practically identical for two biologists, one of whom lives in Hawaii and the other in Norway. Yet the tremendous gulf between two alien civilizations has caused the seeming identity of many notions to fall apart. Certainly, had the Senders used, for designated objects, the set of heavenly bodies, there would not have been this problem. And if they designated atoms? Atoms as “things” to a considerable degree depend on one’s knowledge of them. Eighty years ago an atom was “very similar” to a miniature solar system. Today it no longer is.
Let us suppose that they send us a hexagon. In it one can see the plan for a chemical molecule, or for a bee’s honeycomb, or for a building. An infinite number of objects correspond to that geometrical information. One can determine what the Senders have in mind only by specifying the building material. If, say, the material is to be brick, the class of solutions will indeed be narrowed down, and yet we will still have a set of infinite magnitude, because it is possible, after all, to construct an endless number of hexagonal buildings. The transmitted blueprint ought to be provided with precise measurements. But there exists a material of which the bricks themselves determine the exact measurements. Atoms. In their bonding it is impossible to bring them closer at will, or to move them farther apart. Therefore, having before me only a hexagon, I would think that the Senders meant a molecule of a chemical compound, one constructed of six atoms or of six groups of atoms. Such a statement very significantly limits the field of further searching.
Let us assume — I said to myself — that the “letter” is a description of a thing, a description, moreover, on the molecular level. The kernel of this preliminary thinking was the consideration of the letter’s “content” as a thing having no beginning or end, and therefore circular. It could be either a “circular object” or a circular process. The distinction between the one and the other, as was pointed out, depends in part on the scale of observation. If we lived a billion times more slowly, and correspondingly longer, if a second — in this fancy — equaled an entire century, we would certainly conclude that the continents of the globe were processes, seeing with our own eyes how changeable they were, for they would be moving before us no less than waterfalls do, or ocean currents. And if, on the other hand, we lived a billion times faster, we would conclude that the waterfall was an object — because it would present itself to us as something highly immobile and immutable. The difference between “object” and “process,” therefore, gave no need for concern. It was now only necessary to prove, and not merely to speculate, that the “letter” was a “ring,” just as the molecular model of benzene is a ring. If I did not wish to send a two-dimensional image of that molecule, but chose, rather, to code it into some linear form, a series of successive signals, the place in the benzene ring from which I would begin my description would be unimportant. Every place would serve equally well.
It was from this position that I proceeded to the translation of the problem into the language of mathematics. What I did I cannot present plainly, since our everyday language lacks the required concepts and words. I can only say, in general, that I studied the purely formal properties of the “letter” — treating it as an object mathematically interpreted — for features that are of central interest in topological algebra and the algebra of groups. In doing this, I employed the transformation of transformational sets, which gives the so-called infragroups or Hogarth groups (named after me, since I was the one who discovered them). If I obtained, as a result, an “open” structure, that would still prove nothing, because it could be that I had simply introduced an error into my work, going on some false assumption (such an assumption might be, e.g., the assertion of the number of code signs in a single “unit” of the message). But it happened otherwise. The “letter” closed beautifully for me, like an object separated from the rest of the world, or like a circular process (to be more precise, like the DESCRIPTION, the MODEL of such a thing).