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.

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