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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f 1 (z) = z + d /c (5)
f 2 (z) = 1 /z (6)
f 3 (z) = (bc−ad)z /c 2 (7)
f 4 (z) = z + a /c (8)
f 4 (z) ◦ f 3 (z) ◦ f 2 (z) ◦ f 1 (z) = a c + (bc−ad) /c 2
= f
z + d ∠ (z) (9)
/c
If ad → bc, the rotation effect dissipates, f ∠ (z) → a /c, and f −1
∠
(z) → −d /c.
Likewise, note that a Möbius map is the Laplace transform of an impelling
Dirac force (proportional to the inverse pole) and a steady exponential (decaying
as the direct pole), i.e., impulse plus friction
{ }
a (bc − ad)
L δ (t) +
c c 2 е − d c t (z) = f ∠ (z) (10)
For a /c and bc − ad are nonzero, no system can unfold a Möbius map without
exerting a punctual shove or prod followed by growth or decline. This insight
has profound implications in physics; a stimulus with reinforcement or cessation
transforms the space.
The Möbius transformations of Ǎ constitute the "physical" Möbius Group.
It is noticeably a supergroup of the Modular Group, which we redefine as the set
of LFT over Ž that acts transitively on the points of Ž2 visible from the origin,
i.e., the irreducible fractions, preserving the form of polygonal shapes.
A Möbius map preserves generalized circles (lines or circles) in the complex
plane. Unlike the complex plane, the Ǎ plane, or Möbius plane, does not allow
flat subspaces so that straight lines exist just in the limit; for instance, reflection
at a line is inversion at the asymptote of a circle with diverging radius. Ǎ handles
a line like the real-algebraic projective line. Hence, Ǎ focuses on the circle only, a
set of points z at radius r from a center point o, i.e., |z − o| = r. After squaring,
the circle becomes z¯z − ōz − o¯z + oō = r 2 , or in general, the vanishing expression
Az¯z + Bz + ¯B¯z + C, with A, C ∈ E, B ∈ Ǎ, and B ¯B > AC. A straight line
corresponds to case A → 0. A circle is invariant under translations, homotheties,
and rotations, while inversion at it implies the change of variable 1 /z = s, yielding
the vanishing expression Cs¯s + ¯Bs + B¯s + A, another circle, only that AC > B ¯B
in this case.
We can generalize an LFT to projective lines of rings. The LFT analog of
Möbius transformations over the dual-algebraic numbers is a Laguerre transformation,
which acts on the dual-algebraic plane adjoining a line of points at
infinity, so topologically becoming an infinite cylinder. Much as Möbius transformations
preserve the circle, parabolas are invariant under Laguerre transformations;
lines are arbitrarily flattened parabolas. A Laguerre transformation maps
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