The Summer Hill Journal Of New Physics

The Two Faces Of Zero

Introduction

Our culturally-based assumptions about numbers have a long and varied history. We have constructed "rules" of mathematics, based on knowledge passed from generation to generation. Among these is the way we think of zero.

We commonly think of zero as a single number. In fact, it has 2 distinct meanings:

• It is used as an ordinal value between -1 and 1 as in the set { -1, 0, 1 }
• It is also a measure of quantity which may be nothing at all, as in the null set { }

The ordinal use of zero is a one-dimensional number.
The null view of zero is zero-dimensional.

We typically treat zero in calculations as a hybrid of these 2 meanings. While these assumptions and rules work most of the time, there are cases where they fail. This is not to say that mathematics fails, simply our assumptions about the way it works.

It is our convention to treat zero as a real number half-way between -1 and 1. This is the case where it is used as an ordinal value, but is not necessarily correct when it is used as a quantity.

Our conventions can be maintained most of the time, but there are several cases where our assumptions falter, such as:

• Division by zero
• 0⁰ (0^0 or zero raised to the 0th power)
• Planck's Constant h=E/v, when E=0
• Infinities in physics

These give different results if we assume that, for the purposes of multiplying or dividing a quantity, zero is a null {  } or nothing rather than a real number between -1 and 1 { 0 }. That is, if it treated as the absence, rather than the presence of a number.

Cultural Considerations

There are 2 different versions of zero and there are different types of infinities.
In Western culture we treat both representations of zero as "0" and all infinities interchangeably as "∞". We often don't even have words in our vocabulary to differentiate between them.

We are culturally conditioned from an early age to demand a result from every equation and to "know" that 1/0 = "infinity" and that infinity+1=infinity.

The difference in views of zero didn't really matter too much until the mathematics of Planck or Einstein's times. It rarely seemed to matter much in Newton's day, when many of our current "rules" were made up or formalised from a variety of Greek, Indian, Arabic and other traditions.

{ 0 } as zero worked for centuries, although there was always some discomfort about zero and the whole "infinity" thing due to inconsistent results and odd exceptions to rules.

Below we look at the consequences of viewing zero as the absence of any number. That is, as "null" or as an empty set { }, rather than as the set of a single number { 0 }. We compare the results of modern physics with our conventional views.  These examples are often, almost by definition, at the very limits and at boundary conditions as this is where the differences arise.

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