The Zero Delusion

Zero signifies absence or an amount of no measure. This mathematical object purportedly exemplifies one of humanity's most splendid insights. Endorsement of the continuum consolidated zero as a cultural latecomer that, at present, everybody uses daily as an indispensable number. Zero and infinity represent symmetric and complementary concepts; why did algebra embrace the former as a number and dismiss the latter? Why is zero an unprecedented number in arithmetic? Is zero a cardinal number? Is it an ordinal number? Is zero a "real" point? Has it a geometrical meaning? To what extent is zero naturalistic?

A preliminary analysis indicates that zero is short of numerical competence, contrived, and unsolvable. We find it elusive when we dig into zero's role in physics, especially in thermodynamics, quantum field theory, cosmology, and metrology. A minimal fundamental extent is plausible but hard to accept due to zero's long shade. In information theory, the digit 0 is inefficient; we should replace standard positional notation with bijective notation. In communication theory, the transmission of no bits is impossible, and information propagation is never error-free. In statistical mechanics, the uniform distribution is inaccessible. In set theory, the empty set is ontologically paradoxical. Likewise, other mathematical zeroes are semantically vacuous (e.g., the empty sum, zero vector, zero function, unknot). Because division by zero is intractable, we advocate for the nonzero rational numbers, Q-{0}, to build a new physics that reflects nature's countable character. We provide a zero-free and unique rational-based representation of the algebraic numbers punctured at the origin, A-{0}, the computable version of the complex numbers.

In a linear scale, we must handle zero as the limit of an asymptotically vanishing sequence of rationals or substitute it for the smallest possible nonzero rational. Zero, as such, is the predetermined power indicating the beginning of logarithmically encoded data via log(1). The exponential function decodes the logarithmic scale's beables back to the linear scale. The exponential map is crucial to understand advanced algebraic concepts such as the Lie algebra-group correspondence, the Laplace transform, and univariate rational functions in cross-ratio form. Specifically, linear fractional transformations over a ring lead to the critical notion of conformality, the property of a projection or mapping between spaces that preserves angles between intersecting conics. Ultimately, we define "coding space" as a doubly conformal transformation domain that allows for zero-fleeing hyperbolic (logarithmic) geometry while keeping relationships of structure and scale.