The Zero Delusion v0.99x

incomplete number sets with cardinality infinitely countable and "takes on many
of the aspects of a religion" [189]. We, physicists, profess this religion.
In the complex setting, the situation gets worse. One of the most severe
problems in C is that the polar angle for the origin is undefined; new kinds of
indeterminate and undefined forms appear [20], their values varying depending
on the approaching direction to zero. For instance, consider the function f (z) = ¯z
(z’s conjugate) at the origin; along the imaginary axis, f (z) behaves like the
function −z with derivative limit (f(z)−f(0)) /z = −z /z = −1, whereas along the
real axis, f (z) equals z with derivative limit 1. ¯z is what the theory of functions
of a complex variable denominates a non-holomorphic (non-analytic) function
around 0, where it does not behave regularly. More generally, complex analysis
studies points s where limf (z) or lim 1/f(z) is undefined. If neither of these limits
z→s z→s
exists, s is an essential singularity for f and 1 /f; for example, the origin is an
essential singularity for e 1 /z
, e 1 /¯z
, z z , and their reciprocal functions. To resolve a
singularity of any type, we must approach it from different directions through a
vanishing sequence of distances |z − s|.
Animosity towards zero increases with the complexity of the setting. In the
quaternions H and octonions O, we find new undefined expressions related to
zero; for example, the n-th root of quaternions q with negative R [q] and vanishing
I [q], such as √ −1, is undefined [128]. Furthermore, a quaternionic universe and
an octonionic universe would be non-commutative, i.e., ∃p, q ∈ H such that
pq − qp does not vanish. Likewise, an octonionic cosmos would not be even
associative, i.e., ∃p, q, r ∈ O such that (pq) r ≠ p (qr) [7].
Zero belongs to Q, R, C, H, and O despite not fitting in with these relevant
number sets. It constitutes a pervasive algebraic anomaly whose intractability
cannot be solved, an "oddball", and an "obnoxious bugger in a lot of ways"
[35]. Zero does not exist per se, but it is a linear scale’s hole, i.e., a universal
singularity we must handle as a limiting value. However, as a logarithmic scale’s
fixed mark, zero can be.
5.2 Beability
Our proposal denies zero as an actual number and takes it as a potential number.
Consider the following probability mass function for an integer random variable
{ (
)
Z ∈ Ž Pr (Z) = 1
Z =
(2Z) 2 (1)
else (Pr (0) = 1 − ½ζ (2))
where ζ (2) is the value of the Riemann zeta function at 2. It is well-defined
because the probabilities sum to 1
∑ −1
Z=−∞ (2Z)−2 = ∑ ∞
Z=1 (2Z)−2 = ζ(2) /4.
The expected value of Z is undefined
(
Ê (Z) = Ê (N) − Ê ∑∞ (Z− ) = Z /4
Z=1 Z−2 − ∑ )
−1
Z=−∞ Z−2 = ∞ − ∞,