De Selby's Paradox

 In Flann O'Brien's posthumously-published surrealist satire, The Third Policeman, a side character of some importance is a certain crank scientist and mad philosopher by the name of de Selby (who also appears in this role in O'Brien's The Dalkey Archive). Through the use of various side-discourses on the fictional theorist's postulates (sometimes confined to footnotes), O'Brien (real name Brian O'Nolan) satirizes the scientific method. He does so specifically by having de Selby first point out a seeming paradox or inconsistency in our everyday working understanding of the world, and then using these difficulties to arrive at an utterly zany conclusion. 

As O'Brien's narrator describes the mad scientist's "customary line" in his own terms: he proceeds by "pointing out fallacies involved in existing conceptions and then quietly setting up his own design in place of the ones he claims to have demolished." (A real-life practitioner of the de Selby method might be found in the great Charles Fort, who used the inexplicable accounts of scientific anomalies he found in newspapers to argue, among other things, that intergalactic cross-dimensional vessels must sometimes breach our reality in order to lure us up for dinner. "Maybe we're fished for, by supercelestial beings" as a character in William Gaddis's The Recognitions summarizes the theory.)

De Selby, we might admit, has varying degrees of success in his ability to explode our work-a-day conceptions of reality. One might not be convinced, for instance, by his effort to prove by logical deduction that the Earth must not in fact be spherical, but rather, "sausage-shaped." (His reason being that, no matter where we start on the planet's surface, if we walk far enough we will eventually end up in the same place, and therefore humankind must in fact only ever be walking in one direction—namely, around the circumference of the cylinder-Earth. A break to the sides—a trip down the "barrel" of the sausage-Earth—would therefore, in de Selby's vision, be the ultimate act of human self-transcendence.)

Still less might one side with de Selby's conviction that day and night are not in fact caused by the sequential rotation of the planet, but rather by an industrial byproduct known as "black air," which accumulates in sufficient masses every twelve hours or so and acts to stifle us into a state of unconsciousness, which we mistake for "sleep." (How is it, then, that even during the nighttime, a sudden flash of illumination can banish so much of this "black air" so quickly? De Selby has an answer for that as well: the black air must be highly combustible, such that a struck match can quickly burn large pockets of it away, leaving the illuminated air of daytime, which—in his telling—is really just the ordinary air of a non-polluted Earth.)

We may not be persuaded so far, that is to say, that de Selby has wholly succeeded in his method. Part of the problem may be that he has not yet completed stage one of his two-part scientific process before moving on to the second. He rushed to supply his own zany hypothesis, that is to say, before adequately identifying a genuine fallacy in our existing conception of the world. After all, the fact that we always arrive back at one place, if we walk far enough on the Earth's surface, is explained equally well by the theory of a spherical planet as by the hypothesis that we are really only ever moving in one direction, and has the added advantage that it does not require the intervention of so many "hallucinations" of altered scenery.) Likewise for the alternation of day and night. 

But there is one area of his theorizing in which the fictional scientist seems to have stumbled upon a genuine paradox: the riddle of motion. As O'Brien summarizes his writings on this point: de Selby rejects as fallacious the ordinary human conceptions of movement through time and space. After all, "If one is resting at A [...] and desires to rest in a distant place B, one can only do so by resting for infinitely brief intervals in innumerable intermediate places." Thus, what appears to us to be either the passage of time or our motion through a tridimensional spatial grid is in reality merely "a succession of static experiences each infinitely brief" (De Selby pictures it as a kind of "cinematograph.")

Here, we may say that even if de Selby has not persuaded us to adopt his own theoretical conclusion, he has at least managed to identify a genuine paradox that has troubled minds throughout the ages. What he is talking about, after all, is really just a variant of Zeno's Paradox. The fundamental riddle—which de Selby's "cinematograph" and all the other many illustrations of the paradox are gesturing at in their myriad ways—boils down to this: in order to move from point A to point B (which we all agree things in our world appear to do), the intervening space they are moving through must either be discrete or continuous. This much is clear enough. But it instantly confronts us with a problem, either way we turn. 

Suppose the intervening space to be continuous, as most evidence indicates. Then, we are theoretically able to infinitely subdivide that space into ever-smaller units. No matter how intimately snuggled next to each other point A and point B might be, that is to say, so long as they are not actually touching, we can always mathematically postulate another point between them. In order to move from one to the other, therefore, one must cross through an infinite number of intervening points. And if the movement from any one point to another requires some time, and cannot happen instantaneously, then motion from any A to any B would seem to demand an infinite amount of time, because it is traversing an infinite number of intervening points. 

Okay, we concede, so that doesn't make sense. Therefore, space-time must be discrete, yes? But here we face new dilemmas. If space-time is not continuous after all, then is not movement through it no less paradoxical? Wouldn't motion then require leaps from one discrete instant or location to another, with no intermediate passage; thus, wouldn't our existence consist of a jerky succession of frozen states in different locations in precisely the way de Selby describes? How then is motion between these states possible, if nothing intervenes? Must it all just be an hallucination, as de Selby maintains? But even if we could admit this—and simply say that, for once, de Selby must be right—it would merely be to sidestep an even more fundamental problem with the theory of "discrete" space: the problem of incommensurability. 

At the very dawn of mathematics, after all, the ancients tried at first to describe all of space in terms of unit lengths. They soon confronted a problem. In even something as simple as a square, the sides of which are each posited to be one unit long, we realize that we cannot measure the length of the square's diagonal in terms of the same units. Rather, the measurement of the diagonal requires the intervention of the square root of two—an irrational number that repeats to infinity. Thus, we have to resort to the continuum, even to describe something as basic as a single unit square on a two-dimensional plane. Space cannot be discrete, without running afoul of the paradox that confounded the Pythagoreans millennia ago. 

In short, we are confronted with seeming incoherencies and inconceivabilities either way we turn. 

One may be thinking at this point that even if we ourselves cannot yet see a way out of the paradox, it concerns such fundamental aspects of reality that greater minds must long since have solved it. Surely it cannot be the case that de Selby has managed to confound science on such an elementary point as whether motion, space, or time exist! But in fact, the riddles propounded in Zeno's paradoxes have not been settled. The question of whether spacetime is discrete or continuous remains a subject of debate among philosophers of science. (To be sure, mathematical "solutions" to the paradox have been proposed. But as Jorge Luis Borges once wrote in an essay on the subject, they are on inspection closer to being restatements of the problem than solutions to it.) 

As if this weren't enough, though, the problem gets murkier still. We have already seen above that the concept of motion through a continuum (which we are compelled to adopt because of the obvious problems with the theory of discrete spacetime) nonetheless also involves us in the paradoxical notion of infinity (via the related notion of the infinitesimal). But even this is not the end of our troubles. For modern mathematics has layered a second paradox on top of the first. Not only is the continuum fraught with the paradox of infinity; it is a special kind of infinity. The continuum is shown to be so infinite, in fact, that it is more infinite than infinity. Even among infinities, that is to say, continuous spacetime would have to be especially infinite. The difference comes down to "countability."

David Foster Wallace spells out this progression of ideas in terms I could understand (and which will therefore no doubt be disputed by real mathematicians) in his book, Everything and More: A Compact History of Infinity. The crucial step in this development of thought came, as he tells it, from Cantor's researches into countable sets. 

If we take one of these sets—let us say, the set of positive integers—we realize that while it may be infinite, it is at least countable. We are not able to conceive of the largest possible positive integer, that is to say, but we at least know how to get there, and in what direction it lies. We simply have to move from one discrete element to the next in an ordered sequence to know that we are getting closer to it. There are an infinite number of other possible countable sets (let's take, say, the set of multiples of three), but they all have this same basic property. They can be arranged in a matrix such that they correspond to the set of positive integers. They are countable. So far so good.

Suppose now, however, that we try to set the continuum (i.e., the set of real numbers, which includes irrational numbers such as the square root of 2) in a similar matrix. We quickly realize that it cannot be made to correspond to the set of positive integers. Why? In essence, it's the old problem of incommensurability all over again. There is no way to break down the set of real numbers into discrete units. Between any two numbers, after all, there is an infinite set of other real numbers. So we cannot make each correspond to a positive integer in an ordered sequence. 

We are confronted once again with Zeno's Paradox in a new form. Yet, now we are in a position to see that the riddle of the continuum is even more imponderable than the riddle of the infinitely-great positive integer. There, with the positive integers, we had only a single infinity. Amongst the real numbers, we have as it were a double infinity: infinite magnitude, as well as infinite divisibility (by which we mean, the infinitesimal). 

De Selby is not wrong, therefore, in his contention that our ordinary working conception of the world is riddled with incoherencies and inconceivabilities. He has pointed out a genuine "fallacy" in our world-picture and used it to "demolish" our everyday understandings. 

For my part, I am content to say—with Hume—the problem cannot be solved. In the second part of his Enquiry Concerning Human Understanding, recall, Hume enfolds a brief discussion of the problem of the infinitesimal, which he approaches through a discussion of circles and tangent lines. On the one hand, he writes, nothing could appear simpler or more common in nature than curves and lines. (Just as nothing could be more elementary than distance and motion). Yet, upon closer inspection, these matters become "big with contradiction and absurdity," because they involve the infinitesimal. (As Hume points out, the angle between a curve and its tangent line is infinitely small in a way incommensurable with units of "rectilinear angles," and—despite already being infinitely small—becomes smaller still as the circle expands.)

Hume then turns to time and discovers there no smaller difficulties. Recall from above de Selby's "succession of static experiences each infinitely brief." Hume appears to be forced to something like the same conclusion by the same paradox. On the one hand: "An infinite number of real parts of time," he writes, "passing in succession, and exhausted one after another, appears so evident a contradiction, that no man, one should think, whose judgement is not corrupted, instead of being improved, by the sciences, would ever be able to admit of it." Yet, as we have seen, the infinite divisibility and continuousness of time is as necessary an axiom as the continuum of space (indeed, they are the same thing—as modern Einsteinian physics has taught us), and we are forced to adopt this absurdity as our own conception of the world, even while recognizing its absurdity (as Hume himself well knew). 

Hume was not trying by these means to say that our working conception of the world is false. Rather, he was pointing out that even the most basic concepts we hold about reality necessarily involve us in imponderable riddles. The solution, he thought, was not to reject as "hallucinations" the findings of our reason and senses (he was no follower of de Selby in this regard). Rather, it was to have a sense of humor about the whole dilemma. He recommended that we should be "the first to join in the laugh [...] to confess, that all [our] objections are mere amusement, and can have no other tendency than to show the whimsical condition of mankind, who must act and reason and believe; though they are not able, by their most diligent enquiry, to satisfy themselves concerning the foundation of these operations, or to remove the objections, which may be raised against them."

Which approach in turn, if not de Selby's method, seems to be O'Brien's own. There could be few better novels than The Third Policeman, after all, for teaching "the whimsical condition of mankind," and of prompting laughter at the insoluble riddles with which our own most diligent enquiries into the nature of reality and the universe must soon confront us. If de Selby stands as the exemplar of Hume's overweening skeptic, who pursues his own paradoxes to the point of ultimate absurdity, O'Brien stands as a modern paragon of the Humean spirit—full of laughter and delight at the circumstance in which we find ourselves: namely, that of being placed into a world that forces upon us at each turn the necessity of rational action, while persistently withholding the most basic certainties that would make such action possible.