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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and rationals to themselves), while the action of the Möbius map swaps the
"algebraic projective line" [185].
Like a Möbius map, a conformal map (e.g., the exponential function) leaves
angles unchanged around every domain point or singularity and stretches equally
in all directions ([101]’s section 2.2 and chapter 6), so the nearer a figure is, the
better its shape is preserved. We also say that a manifold is conformally flat
around a point if a diffeomorphism maps its neighborhood onto flat space, meaning
that the angle between intersecting geodesics rather than distances is what
locally matters. The proportionality (conformal) factor between the manifold
and the flat metrics is physically an exponential function of a scalar potential
that never dissipates. Therefore, conformality generally makes proximity a
relative condition, allowing us naturally to deal with the small and the large
interchangeably while fleeing flat (and infinitely curved) spaces.
Although we use zero as an ordinary number of a linear scale, the Dirac
delta distribution, the Laplace transform, the Möbius group, the cross-ratio,
and the conformal transformations point to handling zero and infinity together
as unreachable binary limits of an algebraic sequence. This belief is evident to
Descartes, who argues that [19] "There is no imaginable extension which is so
great that we cannot understand the possibility of an even greater one, and so
we shall describe it as Indefinite. Again, however many parts a body is divided
into, each of the parts can still be understood to be divisible, and so we shall
hold that quantity is indefinitely divisible." Moreover, we must take as many
steps forward from zero to get infinity as steps back from infinity to get zero.
This symmetry reflects that these dual antagonists or complementary mates of
a number set are two sides of the same coin.
In summary, the troubles with zero are its irrationality, unreality, and inseparability
from infinity.
2 Zero in Physics
On why physics must distinguish the arbitrarily small from the incommensurably
small and believe in a minimal natural extent.
2.1 Elusiveness
No clue so far demonstrates that zero is physical. The Third Law of Thermodynamics
stays current; we cannot reduce the entropy of a closed system to
its final zero. A particle’s lifetime can be very near but never precisely naught.
A spectral line has nonzero linewidth and extends over a range of frequencies,
not a single frequency. According to Electromagnetic Theory, if a photon had
zero wavelength, it would have boundless energy. Although a photon has no rest
mass, nobody has ever gauged it strictly zero [195]. Indeed, some theoretical
and exploratory studies indicate that the photon’s rest mass can be nonzero
[60]. Experimentally, a strong force’s gluon rest mass is < 1.3 MeV. Is it a negligible
value? The answer is troublesome because [196] "it is unclear whether the
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