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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If A, B, C, and D are four distinct points on a circle in the Möbius plane,
11 reduces to (A, B; C, D) = (|CA| |DB|) /(|CB| |DA|), where a pair of vertical bars
indicates the Euclidean length of the line segment connecting the pair of points.
In this form, we regard the cross-ratio as a measure of the extent to which the
ratio with which point B divides AC is proportional to the ratio with which B
divides AD.
Keep in mind that the cross-ratio’s imaginary part I [(z 1 , z 2 ; z 3 , z 4 )] vanishes
if, and only if, the four points lie on the same circle. Since the action of the Möbius
group is "simply transitive" on a triple of Ǎ, the unique Möbius transformation
m ∠ to arrive in {−1, 1, ∞} from any triple of distinct points {z 3 , z 2 , z 4 } is
m ∠ (z) ≡ (m ∠ (z) , 1; −1, ∞) = ( z, m −1
∠
(1) ; m−1
∠
(−1) , m−1
∠ (∞)) = (z, z 2 ; z 3 , z 4 )
and the only Möbius transformation m −1
∠ to come in {z 3, z 2 , z 4 } from {−1, 1, ∞}
m −1
∠
(z) ≡ ( m −1
∠ (z) , 1; −1, ∞) = (z, m ∠ (1) ; m ∠ (−1) , m ∠ (∞)) = (z, z 2 ; z 3 , z 4 )
The cross ratio of a "concyclic" quadruple of algebraic points is a realalgebraic
number because −1, 1, and ∞ are points of the real-algebraic projective
line, so (z 1 , z 2 ; z 3 , z 4 ) is real-algebraic if m ∠ (z 1 ) or m −1
∠
(z 1) are real-algebraic.
Since a point and a line are dual and interchangeable by the "principle of
plane duality" in a planar projective space, if a, b, c, and d are four distinct lines
emanating from a point, their cross-ratio is (a, b; c, d) = (sin ∠ c a sin ∠d b)/(sin ∠ c b sin ∠d a).
In this disguise, a sine plays the role of a distance, and the interpretation of a
cross-ratio is the same, i.e., a measure of how much the quadruple deviates from
the ideal proportion 1. Therefore, the cross-ratio applies to points and lines
equally, in agreement with the precepts of projective geometry.
If the four vectors involved in 11 point to infinity, only characterized by their
slopes l, m, p, and q, their cross-ratio is
(z 1 , z 2 ; z 3 , z 4 ) =
p−l
p−m
q−l
q−m
For these infinite points are not on a circle with radius r ∈ Ǎ, we cannot
expect the cross-ratio to be a real-algebraic number. This remark also applies
to points of the Laguerre and Minkowski planes because our approach excludes
the vanishing modulus numbers, i.e., null vectors, from the non-extended rings.
How can this slope-based formula result in a specific algebraic, split-algebraic,
or dual-algebraic number with a non-vanishing imaginary part? In the case of
split-algebraic numbers, we must make ±j represent the diverging slopes of two
lines meeting infinity. Thus
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