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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7.2 Coding
Once described the power of the cross-ratio and its transformational approach,
let us tackle the notion of conformality. Roughly, a conformal transformation
preserves the angles between intersecting conics and between the cords of four
points on a conic. Besides, a conformal transformation preserves conics and the
cross-ratio of concyclic points, understanding concyclicity as the condition of a
set of points on the same conic. For instance, the circle inversion map 1 /z = s in
the Möbius plane is "anticonformal" according to our description in subsection
6.2, meaning that angles keep their value but reverse direction. In contrast, the
genuinely conformal function 1 /¯z = s preserves angles and orientation.
In two dimensions, conformal transformations in the Möbius plane are holomorphic
(analytic, regular) maps such as polynomial, exponential, trigonometric,
logarithmic, and power functions that preserve local angles. Any conformal
mapping of a variable taking values in Ǎ with continuous partial derivatives is
holomorphic, and conversely, a holomorphic function is conformal at any point
where its derivative does not vanish. A Möbius map has continuous derivatives
except at z = − d /c, so from this point, the Möbius map 3 is conformal throughout
Ǎ. Any three points in the Möbius plane define a conformal map; in particular,
by fixing 0 (nothing), 1 (something), and ∞ (everything), or −1 (minus something),
1 (plus something), and ∞, we can define metric-independent conformal
maps that relativize the definition of a point "neighborhood" by adapting the
size and curvature of the figures around a point to conserve their shape.
A generalized conformal map is a concatenation of LFT over rings. In addition
to the elliptic rotations ascribed to Ǎ, the extension contemplates the
parabolic rotations in the Laguerre (right) plane and the hyperbolic cycles in
the Minkowski (right) plane. Null vectors are uncomputable and hence considered
limiting values; Möbius, Laguerre, and Minkowski planes ban the origin, the
imaginary axis, and the diagonals ∝ (1 ± j), respectively. Conformality follows
again from the fact that generalized rotations by a (f → af), translations by
b (f → f + b), and inversions (f → 1 /f) preserve angles. The null map is not
conformal because a and b are nonzero.
Cross-ratios are the basis of conformal geometry because we can write conformal
maps in cross-ratio form, e.g., (z − 1, 0; (z + 1, 0; z, ∞) , ∞) ≡ z 2 − z + 1
and ((−1, z; 1, ∞) , z, 0, ∞) ≡ 2 /(z−z 2 ), where z belongs to a ring extended with
(null vectors and) infinite points. The two open (for differentiability) subsets in
the plane that a conformal mapping connects are essentially indistinguishable
from the cross-ratio invariant projective geometry’s viewpoint, so we can study
functions with given properties on a somewhat complicated region by first mapping
it to a simpler one, conserving the properties of the function in the two-way
transference between both subsets.
This scenario suggests that we can use a subset of the domain as a region to
encode the information received from the "external" world. A concyclic quadruple’s
cross-ratio is a real-algebraic number ranging from 1 to ∞ if the elements
are consecutive, except when all the four points are infinite. Since hyperbolic
functions are rational functions whose numerator and denominator are linear
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