Why Does the World Exist?: An Existential Detective Story Read Online Free Page A

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 + x 2 + x 3 + x 4 + x 5 + …
    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.