+ 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,