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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We can use the unit ball of E n centered at the origin as a coding space having
as a boundary the unit (n − 1)-sphere. Depending on the conformal transformation,
the open side (infinity) can be the origin or the boundary. Using the
Euclidean norm ‖‖, the Poincaré ball model defines the conformal distance d C
between two points z 2 and z 3 in the n-dimensional unit ball, connected by the
geodesic that intersects the unit (n − 1)-sphere at the ideals z 1 = and z 4 , as
‖z 3−z 1‖
d C (z 2 , z 3 ) = ½
∣ ln ‖z 3−z 2‖
‖z 4−z 1‖
∣
‖z 4−z 2‖
The problem is that the cross-ratio provides decreasing information and increasing
flexibility with the number of dimensions, mainly owing to the growth
of possible geometric configurations of distinct tuples of points and the complication
of specifying when they are in "general position".
Physics has a great interest in conformality [104]. Random fluctuations are
conformal in a plane, i.e., conformal invariance is a property of Brownian motion.
In all dimensions, conformality seems to reflect causality and scale invariance,
hence a means of natural renormalization. The set of all causality-preserving
transformations in a point vicinity is equivalent to the conformal group. This
set is a local Lie group that can be globally extended by "conformal compactification",
which allows studying the boundary and singularities of a topologically
open space confined into a conformally equivalent closed (finite) space
with adjoining limiting points. In two dimensions, the compactified spaces are
the three-dimensional models of the Möbius, Laguerre, and Minkowski planes
that we introduced at the end of the previous section. In all dimensions, the
conformal compactification of Euclidean space requires adding a point at infinity
in one more dimension and stereographic projection, which is a conformal
map itself. The flat Minkowski spacetime extension into a Lorentzian "infinitesheeted"
cylindric covering, i.e., the "Einstein cylinder", is another application
of conformal compactification.
More specifically, physicists use conformal transformations in theories of gravity
and cosmology [47]. A point-dependent local rescaling of the metric tensor
g ab → Ω 2 g ab affects the time and space lengths but respects the causal structure
of spacetime; Ω (x) = е −ω(x) in the case of the Weyl map, where ω (x) is a scalar
field w at the spacetime event x. Thus, we get a new source of Einstein gravity
and a power-law potential of cosmological inflation, which allows quantizing
gravity when the spacetime metric is conformally flat (e.g., η ab → Ω −2 g ab , where
η ab is the flat Minkowski metric). Ω cannot vanish, so a conformal model can
get around curvature singularities, for example, to describe the universe before
the Big Bang. Conformal invariance is also crucial for QFT in curved spaces and
superstring and supergravity theories, giving rise to the famous correspondence
between a QG theory in 5-dimensional Anti-de Sitter space with a 4-dimensional
supersymmetric conformal-invariant QFT via isometries of H 3 .
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