Something from nothing

Recently reading the book Why Does The World Exist by Jim Holt I came across the idea of an arithmetic nothingness. Holt explores the mystery of existence including taking a philosophical tour and quantum based spontaneous creation of something from nothing and so on…

So back to the arithmetic – and in particular Zero and One (0 and 1). To the Greeks and Romans the very idea of zero was inconceivable how could a nothing be a something.

If we let 0 stand for nothing and 1 for something then we get a sort of toy version of the mystery of existence. How can you get from 0 to 1. In advanced mathematics there is a simple sense in which the transition from 0 to 1 is impossible.

A number is regular if it can 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.

We can see 1 is regular because it can’t be reached from below. The sum of 0s is 0.So the big question is whether there is a way of bridging the gap between 0 and 1 – between nothing and something.

Leibniz thought he had found an answer and this is fascinating. One infinite series is 1/(1.x) = 1 + x + x(2) + x(3) + x(4) + ……

By plugging in -1 into this series

½ = 1-1 + 1-1 + 1-1 + 1-1  + …..

½ = (1-1) + (1-1) + (1-1) + (1-1) + ……

and so

½ = 0 + 0 + 0 + 0 ….

The equation seems to suggest that something can appear from nothing. Had Leibniz stumbled on something?

This is of course invalid. The infinite series isn’t convergent and so therefore a series jumping between 0 and 1 makes no sense.