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Raindrops on a Car

“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.”

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From the chapter “The Arithmetic of Nothingness” in Jim Holt’s new book Why Does the World Exist?: An Existential Detective Story.

The photograph: Rain on the hood of a car. Houston, Texas.