by Jim Holt
Let us, then, dip briefly into that abyss, with full assurance that we will not come up empty-handed. For, as the old saying goes: Nothing seek, nothing find.
Interlude
The Arithmetic of Nothingness
Mathematics has a name for nothing, and that is “zero.” It is notable that the root of zero is a Hindu word: sunya, meaning “void” or “emptiness.” For it was among Hindu mathematicians that our notion of zero arose.
To the Greeks and Romans, the very idea of zero was inconceivable—how could a nothing be a something? Lacking a symbol for it in their number systems, they could not take advantage of convenient “positional” notation (in which, for example, 307 stands for 3 hundreds, no tens, and 7 ones). That’s one reason why multiplying with roman numerals is hell.
The idea of emptiness was familiar to Indian mathematicians from Buddhist philosophy. They had no difficulty with an abstract symbol that signified nothing. Their notation was transmitted westward to Europe during the Middle Ages by Arab scholars—hence our “arabic numerals.” The Hindu sunya became the Arabic sifr, which shows up in English in both the words “zero” and “cipher.”
Although European mathematicians welcomed zero as a notational device, they were at first chary of the concept behind it. Zero was initially regarded more as a punctuation mark than as a number in its own right. But it soon began to take on greater reality. Oddly enough, the rise of commerce had something to do with this. When double-entry bookkeeping was invented in Italy around 1340, zero came to be viewed as a natural dividing point between credits and debits.
Whether discovered or invented, zero was clearly a number to be reckoned with. Philosophical doubts about its nature receded before the virtuoso calculations of mathematicians such as Fibonacci and Fermat. Zero was a gift to algebraists when it came to solving equations: if the equation could be put in the form ab = 0, then one could deduce that either a = 0 or b = 0.
As for the origin of the numeral “0,” that has eluded historians of antiquity. On one theory, now discredited by scholars, the numeral comes from the first letter of the Greek word for “nothing,” ouden. On another theory, admittedly fanciful, its form derives from the circular impression left by a counting chip in the sand—the presence of an absence.
Suppose we let 0 stand for Nothing and 1 stand for Something. Then we get a sort of toy version of the mystery of existence: How can you get from 0 to 1?
In higher mathematics, there is a simple sense in which the transition from 0 to 1 is impossible. Mathematicians say that a number is “regular” if it can’t be reached via the numerical resources lying below it. More precisely, the number n is regular if it cannot be reached by adding up fewer than n numbers that are themselves smaller than n.
It is easy to see that 1 is a regular number. It cannot be reached from below, where all there is to work with is 0. The sum of zero 0’s is 0, and that’s that. So you can’t get from Nothing to Something.
Curiously, 1 is not the only number that is unreachable in this way. The number 2 also turns out to be regular, since it can’t be reached by adding up fewer than two numbers that are less than 2. (Try it and see.) So you can’t get from Unity to Plurality.
The rest of the finite numbers lack this interesting property of regularity. They can be reached from below. (The number 3, for example, can be reached by adding up two numbers, 1 and 2, each of which is itself less than 3.) But the first infinite number, denoted by the Greek letter omega, does turn out to be regular. It can’t be reached by summing up any finite collection of finite numbers. So you can’t get from Finite to Infinite.
But back to 0 and 1. Is there some other way of bridging the gap between them—the arithmetical gap between Nothing and Something?
As it happens, no less a genius than Leibniz thought he had found a bridge. Besides being a towering figure in the history of philosophy, Leibniz was also a great mathematician. He invented the calculus, more or less simultaneously with Newton. (The two men feuded bitterly over who was the true originator, but one thing is certain: Leibniz’s notation was a hell of a lot better than Newton’s.)
Among much else, the calculus deals with infinite series. One such infinite series that Leibniz derived is:
1/(1–x) = 1 + x + x2 + x3 + x4 + x5 + . . .
Showing remarkable sangfroid, Leibniz plugged the number –1 into his series, which yielded:
1/2 = 1–1 + 1–1 + 1–1 + . . .
With appropriate bracketing, this yielded the interesting equation:
1/2 = (1–1) + (1–1) + (1–1) + . . .
or:
1/2 = 0 + 0 + 0 + . . .
Leibniz was transfixed. Here was a mathematical analogue of the mystery of creation! The equation seemed to prove that Something could indeed issue from Nothing.
Alas, he was deceived. As mathematicians soon came to appreciate, such series made no sense unless they were convergent series—unless, that is, the infinite sum in question eventually homed in on a single value. Leibniz’s oscillating series failed to meet this criterion, since its partial sums kept jumping from 0 to 1 and back again. Thus his “proof” was invalid. The mathematician in him must surely have suspected this, even as the metaphysician in him rejoiced.
But perhaps something can be salvaged from this conceptual wreckage. Consider a simpler equation:
0 = 1–1
What might it represent? That 1 and –1 add up to zero, of course.
But that is interesting. Picture the reverse of the process: not 1 and –1 coming together to make 0, but 0 peeling apart, as it were, into 1 and –1. Where once you had Nothing, now you have two Somethings! Opposites of some kind, evidently. Positive and negative energy. Matter and antimatter. Yin and yang.
Even more suggestively, –1 might be thought of as the same entity as 1, only moving backward in time. This is the interpretation seized on by the Oxford chemist (and outspoken atheist) Peter Atkins. “Opposites,” he writes, “are distinguished by their direction of travel in time.” In the absence of time, –1 and 1 cancel; they coalesce into zero. Time allows the two opposites to peel apart—and it is this peeling apart that, in turn, marks the emergence of time. It was thus, Atkins proposes, that the spontaneous creation of the universe got under way. (John Updike was so struck by this scenario that he used it in the conclusion of his novel Roger’s Version as an alternative to theism as an explanation for existence.)
All that from 0 = 1–1. The equation is more ontologically fraught than one might have guessed.
Simple arithmetic is not the only way that mathematics can build a bridge between Nothingness and Being. Set theory also furnishes the materials. Quite early in their mathematical education, indeed often in grade school, children are introduced to a curious thing called the “empty set.” This is a set that has no members at all—like the set of female U.S. presidents preceding Barack Obama. It is conventionally denoted by {}, the set brackets with nothing inside of them, or by the symbol Ø.
Children sometimes bridle at the idea of the empty set. How, they ask, can a collection that contains nothing really be a collection? They are not alone in their skepticism. One of the greatest mathematicians of the nineteenth century, Richard Dedekind, refused to regard the empty set as anything more than a convenient fiction. Ernst Zermelo, a creator of set theory, called it “improper.” More recently, the great American philosopher David K. Lewis mocked the empty set as “a little speck of sheer nothingness, a sort of black hole in the fabric of Reality itself . . . a special individual with a whiff of nothingness about it.”
Does the empty set exist? Can there be a Something whose essence—indeed, whose only feature—is that it encompasses Nothing? Neither believers nor skeptics have produced any strong arguments for or against the empty set. In mathematics it is simply taken for granted. (Its existence can be proved from the axioms of set theory, on the assumption that there is at least one other set in the universe.)
Let’s be metaphysically liberal and say that the
empty set does exist. Even if there’s nothing, there must be a set that contains it.
Admit that, and a regular ontological orgy gets under way. For, if the empty set Ø exists, so does a set that contains it: {Ø}. And so does a set that contains both Ø and {Ø}: {Ø, {Ø}}. And so does a set that contains that new set, plus Ø and {Ø}: {Ø, {Ø}, {Ø, {Ø}}}. And on and on.
Out of sheer nothingness, a remarkable profusion of entities has come into being. These entities are not made out of any “stuff.” They are pure, abstract structure. They can mimic the structure of the numbers. (In the preceding paragraph, we “constructed” the numbers 1, 2, and 3 out of the empty set.) And numbers, with their rich web of interrelations, can mimic complicated worlds. Indeed, they can mimic the entire universe. At least they can if, as thinkers like the physicist John Archibald Wheeler have speculated, the universe consists of mathematically structured information. (This view is captured by the slogan “it from bit.”) The whole show of reality can be generated out of the empty set—out of Nothing.
But that, of course, presumes there is Nothing to start with.
3
A BRIEF HISTORY OF NOTHING
Hartley told his Mother, that he was thinking all day—all the morning, all the day, all the evening—“what it would be, if there were Nothing! if all the men, & women, & Trees, & grass, and birds & beasts, & the Sky, & the Ground, were all gone: Darkness & Coldness—& nothing to be dark & cold.”
—SAMUEL TAYLOR COLERIDGE, letter to Sara (“Asra”) Hutchinson, June 1802 (Hartley was Coleridge’s son.)
NOTHING! thou elder brother even to shade
That hadst a being ere the world was made,
And (well fixed) are alone of ending not afraid.
—JOHN WILMOT, EARL OF ROCHESTER, “Upon Nothing”
Nothing,
said Heidegger,
the modernist
eminence,
noths.
—ARCHILOCHUS JONES, “Metaphysics Explained for You”
What is nothing? Macbeth answered this question with admirable concinnity: “Nothing is, but what is not.” My dictionary puts it somewhat more paradoxically—“nothing (n.): a thing that does not exist.” Although Parmenides, the ancient Eleatic sage, declared that it was impossible to speak of what is not—thereby violating his own precept—the plain man knows better. Nothing is popularly held to be better than a dry martini, but worse than sand in the bedsheets. A poor man has it, a rich man needs it, and if you eat it for a long time, it’ll kill you. On occasion, nothing could be further from the truth, but it is not clear how much further. It can be both black and white all over at the same time. Nothing is impossible for God, yet it is a cinch for the rankest incompetent. No matter what pair of contradictory properties you choose, nothing seems capable of embodying them. From this it might be concluded that nothing is mysterious. But that would only mean that everything is obvious—including, presumably, nothing.
That, perhaps, is why the world abounds with people who know, understand, and believe in nothing. But beware of speaking blasphemously of nothing, for there are also many bumptious types about—call them “nullophiles”—who are fond of declaring that, to them, nothing is sacred.
Ex nihilo nihil fit, averred the ancient philosophers, and King Lear agreed: nothing comes of nothing. This maxim would appear to attribute to nothing a remarkable power: that of generating itself—of being, like God, causa sui. The philosopher Leibniz paid nothing another compliment when he observed that it was “simpler and easier than something.” (Hard experience teaches the same lesson: nothing is simple, nothing is easy.) Indeed, it was the alleged simplicity of nothing that moved Leibniz to ask why there is something rather than nothing. If there were nothing, after all, there would be nothing to be explained—and no one to demand an explanation.
If nothing is so simple, so natural, then why, one wonders, does it seem so deeply mysterious? In the 1620s, John Donne, speaking from the pulpit, furnished a plausible answer: “The less anything is, the less we know it: how invisible, how unintelligible a thing, then, is this Nothing!”
And why should such a simple (albeit unintelligible) thing strike others as so sinister? Take the Swiss theologian Karl Barth, one of the most profound and brave thinkers of the twentieth century. What, asked Barth, is Nothing? It is “that which God does not will.” In his massive and unfinished life’s work, Church Dogmatics, Barth wrote, “The character of nothingness derives from its ontic peculiarity. It is evil.” Nothing rose up simultaneously with Something when God created the world, according to Barth. The two are rather like a pair of ontological twins, though contrary in moral character. It is nothingness that accounts for man’s perverse tendency to do evil, to rebel against divine goodness. For Barth, nothingness was downright Satanic.
The existentialists, though godless themselves, regarded nothingness with similar dread. “Nothingness haunts being,” declared Jean-Paul Sartre, in his ponderous treatise Being and Nothingness. For Sartre, the world was like a little sealed container of being floating on a vast sea of nothingness. Not even a Parisian café—on a good day a “fullness of being,” with its booths and mirrors, its smoky atmosphere, animated voices, clinking wine glasses, and rattling saucers—could afford sure refuge from nullity. Sartre drops into the Café de Flore to keep a rendezvous with his friend Pierre. But Pierre is not there! Et voilà: a little pool of nothingness has seeped into the realm of being from the great néant that surrounds it. Since it is through dashed hopes and thwarted expectations that nothingness intrudes into the world, our very consciousness must be to blame. Consciousness, says Sartre, is nothing less (or more?) than a “hole at the heart of being.”
Sartre’s fellow existentialist Martin Heidegger was filled with Angst at the very thought of nothing, although this did not keep him from writing copiously about it. “Anxiety reveals the Nothing,” he observed—his italics. Heidegger distinguished between fear, which has a definite object, and anxiety, a vague sense of not being at home in the world. What, in our anxious states, are we afraid of? Nothing! Our existence issues from the abyss of nothingness and ends in the nothingness of death. Thus the intellectual encounter each of us has with nothingness is suffused with the dread of our own impending nonbeing.
As to the nature of nothingness, Heidegger was wildly vague. “Nothing is neither an object nor anything that is at all,” he sensibly declared at one point. Yet in order to avoid saying, Das Nichts ist—“Nothing is”—he was driven to an even more peculiar locution, Das Nichts nichtet: “Nothing noths.” Instead of being an inert object, nothingness would appear to be a dynamic thing, a sort of annihilating force.
The American philosopher Robert Nozick took Heidegger’s idea a step further. If nothing is an annihilating force, Nozick conjectured, it might just “noth” itself, thereby giving rise to a world of being. He imagined nothing as “a vacuum force, sucking things into non-existence or keeping them there. If this force acts upon itself, it sucks nothingness into nothingness, producing something or, perhaps, everything.” Nozick recalled the vacuum cleaner–like beast in the movie Yellow Submarine that goes around sucking up all it encounters. After hoovering away everything else on the movie screen, it ultimately turns on itself and sucks itself into nonexistence. With a pop, the world reappears, along with the Beatles.
Nozick’s speculations about nothing, though playful in spirit, left some of his fellow philosophers exasperated. They felt he was willfully sliding into nonsense. One of them, the Oxford philosopher Myles Burnyeat, commented, “By the time one has struggled through this wild and woolly attempt to find a category beyond existence and non-existence, and marvelled at such things as the graph showing ‘the amount of Nothingness Force it takes to nothing some more of the Nothingness Force being exerted,’ one is ready to turn logical positivist on the spot.”
The logical positivists, indeed, dismissed all such speculation as much ado about nothing. One of the most distinguished of them, Rudolf Carnap, obser
ved that the existentialists had been fooled by the grammar of “nothing”: since it behaves like a noun, they assumed, it must refer to an entity—a something. This is the same blunder that the Red King makes in Lewis Carroll’s Through the Looking-Glass: if Nobody had passed the messenger on the road, the Red King reasoned, then Nobody must have arrived first. Treating “nothing” as the name of a thing allows one to generate endless paradoxical twaddle, as the opening paragraphs of this very chapter attest.
THE IDEA THAT it is nonsensical to talk about nothing goes back to the dawn of Western philosophy. It was Parmenides, the greatest of the pre-Socratics, who was most emphatic on this point. Parmenides is a somewhat mysterious figure. A native of Elea, in southern Italy, he flourished in the mid-fifth century BCE. As an elderly man, he reputedly met the young Socrates. Plato described him as “venerable and awesome.” Parmenides was the first Greek philosopher to set out a sustained logical argument about the nature of reality, and thus he might be regarded as the original metaphysician. Curiously, he chose to present his argument in the form of a long allegorical poem, of which some 150 lines survive. In the poem, an unnamed goddess offers the narrator a choice between two paths: the path of being, and the path of nonbeing. But the latter path proves to be illusory, since nonbeing can be neither thought about nor spoken of. Just as “seeing nothing” is not seeing, speaking or thinking of nothing is not speaking or thinking at all, and approaching nothing is failing to make progress.
The Parmenidean line certainly seems to deflate the mystery of existence. If we cannot speak meaningfully of “nothing,” then we cannot meaningfully ask why there is something rather than nothing. The words would have no more sense than the bubbles issuing from the mouth of a fish.