The Zero Delusion v0.99x
Zero signifies absence or an amount of no dimension and allegedly exemplifies one of humanity's most splendid insights. Nonetheless, it is a questionable number. Why did algebra embrace zero and dismiss infinity despite representing symmetric and complementary concepts? Why is zero exceptional in arithmetic? Is zero a "real" point? Has it a geometrical meaning? Is zero naturalistic? Is it universal? Digit 0 is unnecessary in positional notation (e.g., bijective numeration). The uniform distribution is unreachable, transmitting nill bits of information is impossible, and communication is never error-free. Zero is elusive in thermodynamics, quantum field theory, and cosmology. A minimal fundamental extent is plausible but hard to accept because of our acquaintance with zero. Mathematical zeroes are semantically void (e.g., empty set, empty sum, zero vector, zero function, unknot). Because "division by zero" and "identically zero" are uncomputable, we advocate for the nonzero algebraic numbers to build new physics that reflects nature's countable character. In a linear scale, we must handle zero as the smallest possible nonzero rational or the limit of an asymptotically vanishing sequence of rationals. Instead, zero is a logarithmic scale's pointer to a being's property via log(1)). The exponential function, which decodes the encoded data back to the linear scale, is crucial to understanding the Lie algebra-group correspondence, the Laplace transform, linear fractional transformations, and the notion of conformality. Ultimately, we define a "coding space" as a doubly conformal transformation realm of zero-fleeing hyperbolic geometry that keeps the structural and scaling relationships of the world.
Zero signifies absence or an amount of no dimension and allegedly exemplifies one of humanity's most splendid insights. Nonetheless, it is a questionable number. Why did algebra embrace zero and dismiss infinity despite representing symmetric and complementary concepts? Why is zero exceptional in arithmetic? Is zero a "real" point? Has it a geometrical meaning? Is zero naturalistic? Is it universal? Digit 0 is unnecessary in positional notation (e.g., bijective numeration). The uniform distribution is unreachable, transmitting nill bits of information is impossible, and communication is never error-free. Zero is elusive in thermodynamics, quantum field theory, and cosmology. A minimal fundamental extent is plausible but hard to accept because of our acquaintance with zero. Mathematical zeroes are semantically void (e.g., empty set, empty sum, zero vector, zero function, unknot). Because "division by zero" and "identically zero" are uncomputable, we advocate for the nonzero algebraic numbers to build new physics that reflects nature's countable character. In a linear scale, we must handle zero as the smallest possible nonzero rational or the limit of an asymptotically vanishing sequence of rationals. Instead, zero is a logarithmic scale's pointer to a being's property via log(1)). The exponential function, which decodes the encoded data back to the linear scale, is crucial to understanding the Lie algebra-group correspondence, the Laplace transform, linear fractional transformations, and the notion of conformality. Ultimately, we define a "coding space" as a doubly conformal transformation realm of zero-fleeing hyperbolic geometry that keeps the structural and scaling relationships of the world.
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massless theory is really the limit of the massive one." Nobody has ever measured
a body occupying no space or an infinite mass or charge density. The cosmological
curvature parameter of the universe is very close to but not "identically
zero" [137].
Nil reckoning is intricately related to the classical "infinitesimal", an indivisible
quantity "with arbitrarily small but nonzero width" [170]. Mathematical
research on the modern concept of infinitesimal dates back to Leibniz and
Nieuwentijdt. Nearly two centuries later, Cantor abominated the theory of infinitesimals,
which he referred to as the "cholera bacillus" of mathematics [91],
and the idea of getting them mixed with his theory of transfinite sets; from his
angle, the one approach can be understood no way inverse of the other, despite
traditionally if a is infinitesimal for b, b is infinite for a. However, inspired by
Fechner’s work on how a psychological sensation relates to the physical intensity
of a stimulus, Poincaré was [61] "who set out most clearly where debates about
the real numbers were to divide mathematicians and scientists." Poincaré said
(Mathematical Magnitude and Experiment in [138]) that "the rough results of
the experiments may be [...] regarded as the formula of the physical continuum"
and "has compelled us to invent the mathematical continuum." Regardless of
how Cantor or Poincaré came to their conclusions, the fact is that infinitesimal
became a quantity stemming from the R-continuum examined in the field of the
minimal [13], potentially perceptible but indistinguishable from zero. Thus, observables
became a prime mover of this ultra-dense medium, although, ironically,
a measurement is only accurate and precise to a degree.
The unobservability of zero remains an unsolvable pragmatical problem. For
instance, are rounding errors of the initial conditions or results in any reallife
numerical problem evitable? How can we ensure that an observed body’s
property, e.g., rotational speed, is zero? We must measure infinitely rigorously
to prove a value is nil [83]. Zero is unreachable except for the incidental zero
level of artificial scales, such as the Julian calendar, Celsius temperatures, or the
sound intensity in decibels. The azimuthal and magnetic quantum numbers in
atomic physics can be zero, but the axes used for the spherical coordinates are
haphazard. In general, "An infinite precise statement that there is zero change
[...] is completely untestable" because "We can never measure an infinity or a
zero" [38]. Additionally, the origin is invisible when we behave as observers.
From our outlook, we cannot see ourselves or beyond the last visible number;
we only capture what is subjectively at distances 1, 2, 3, · · · , N, conditioned by
our physical limitations.
Nonetheless, the reader can object that we usually run into zero as a solution
or singularity of a scientific or engineering problem. As a solution, zero is either
a trivial result or an outcome with negligible magnitude. Thus, it should come
with no physical units and would not strictly require a number representation.
As a singularity, zero deserves further analysis.
R-based laws of physics always break down at their conjoined singularities.
Informally, a singularity is a domain value that involves some blank magnitude
causing the law to behave strangely. A zero at the wrong place yields an unman-
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