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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vertical parabolas on the cylinder to vertical parabolas in the plane [21], just as
a Möbius transformation maps circles of the Riemann sphere onto circles of the
plane. Likewise, an LFT over the split-algebraic numbers adjoining two lines of
points at infinity is a Minkowski transformation. This "split-algebraic projective
line" is isomorphic to the geometry of plane sections of a hyperboloid of one
sheet. Minkowski maps preserve the hyperbola, which approximates a line when
the curvature at its vertex vanishes. These projective lines are usually known
as the Möbius, Laguerre, and Minkowski classical "planes" [68], isomorphic to
the geometry of plane sections of their corresponding three-dimensional model,
to wit, the sphere, the cylinder, and the hyperboloid of one sheet, from which
accomplish "stereographic projection" onto the plane.
7 Zero Conformality
On the importance of conserving concyclicity and angles as much as possible,
especially locally.
7.1 Cross-ratio
Möbius, Laguerre, and Minkowski transformations are homographies, i.e., isomorphisms
of projective spaces. These mappings hold the structural or configurational
relations among the elements of the original space in the new space, although
some information can be lost; every homography presents a new perspective
or connects two perspectives of a given system. Specifically, homographies
are LFT when we identify the "projective line" over a ring with its adjoining
"infinite points", i.e., points at infinity.
Projective geometry deals with proportions and assumes that any two lines
intersect. We can take the incidence points as projective line landmarks. Since
the cross-ratio is a central rational construct that, in plain language, calculates
the positioning of a pair of points concerning another, a planar homography
preserves the cross-ratio of four distinct points, which we define in vectorial
form as
(z 1 , z 2 ; z 3 , z 4 ) =
z 3−z 1
z 3−z 2
z 4−z 1
z 4−z 2
(11)
If one of these vectors points to infinity, i.e., represents an infinite point, we
erase the two differences associated with it. If z 1 → z 3 or z 2 → z 4 , the cross-ratio
vanishes. If z 1 → z 2 or z 3 → z 4 , the cross-ratio tends to the unit. If z 2 → z 3 or
z 1 → z 4 , the cross-ratio diverges.
Cross-ratios are invariant under LFT over rings [197]. In particular, Ǎ∪{0, ∞}
rules this projective invariance via Möbius maps according to the expression
(z 1 , z 2 ; z 3 , z 4 ) = (f ∠ (z 1 ) , f ∠ (z 2 ) ; f ∠ (z 3 ) , f ∠ (z 4 )) (12)
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