Why a mathematician thought this infinite series explained how God created the universe
Here’s a math problem that everybody can solve: what is one minus one? Zero. So far so good. If we then add a 1, the sum grows, but if we subtract yet another one, we’re back at zero. Let’s say, we keep doing this forever:. Paradoxes surround the problem because multiple seemingly sound arguments about the sum reach radically different conclusions. The first person to deeply investigate it thought it explained how God created the universe.
However, a slight shift of the brackets yields a different result. Instead of pairing off the first two terms and the next two terms, and so forth, if we set aside the first 1, then the second and third terms also cancel, and the fourth and fifth cancel:Again, all the parentheticals add up to 0, but we have this extra positive 1 at the beginning, which suggests that the whole expression sums to 1. in 1703.
Infinite series like this one have flummoxed thinkers dating back at least to the ancient Greeks with Zeno of Elea’s. In one famous example, Zeno observed that to walk a path, one must first traverse half of it, then must traverse half of the remaining distance , and then half of the remaining distance , and so on. One can keep subdividing forever, which suggests that every time we walk a path we complete an infinite number of actions in a finite amount of time—a paradox.
The resolution of Grandi’s series raises a sociological question. Why does the mathematical community accept the partial sums approach but not Leibniz’s probabilistic argument or some other definition of summing an infinite series? Though they may look alike and smell alike, summing anis not the same as addition. For a clear example, addition does not change when you shift parentheses around, e.g. 1 + = + 3, but many series including Grandi’s do.
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