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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A function is "univariate rational" if and only if reduced to the lowest terms
is in form P (z) /Q(z), i.e., a quotient of two polynomials with coefficients taken in
a field F and values of the variable taken in a field G ⊃ F , where the greatest
common divisor of P (z) and Q (z) (after a reduction process) is a constant.
There cannot be indeterminate forms because Q (z) cannot be the null function,
and zero is not a univariate rational function because P (z) cannot be either the
null function.
The sum, difference, product, or quotient of two rational functions of the
same nonzero algebraic variable whose numerator and denominator are minimal
polynomials over a subfield of Ǎ is a univariate rational function. The zeroes and
poles of a rational function and its reciprocal are rationals or limiting values in
Ǎ, like 0 and ∞. The set of univariate rational functions taking values in Ǎ is
the closed field of fractions of the ring of minimal polynomials over the elements
of ˇQ or Ǎ, a basis for generating the field of meromorphic functions [175] over
Ǎ.
An LFT is a univariate rational function whose numerator and denominator
are linear (degree 1) polynomials. An LFT (with coefficients) over a subfield of
Ǎ taking values in Ǎ adjoining 0 and ∞ as projective opposite limiting values
to eliminate mapping discontinuities is a Möbius transformation. The Mobius
group is the automorphism group on the "algebraic projective line", of which
the Riemann sphere is a model [124]. Since the composition of two Möbius transformations
is a Möbius transformation, we can produce the effect of an ongoing
smooth process over this projective line by joining successive Möbius maps.
Given the coefficients {a, b, c, d} ∈ Ǎ, where ad − bc is nonzero, the rational
function of one variable z ∈ Ǎ
f ∠ (z) = a c
( ) z + b/a
z + d /c
= az + b
cz + d
represents a Möbius map or permutation. The requirement ad ≠ bc means
that the four points define a tetragon (quadrilateral or quadrangle) with a
nonzero area, ensuring that this LFT has inverse
f −1
∠
(z) = −d c
( ) z − b/d
z − a /c
= dz − b
−cz + a
If z approaches the origin ( lim ), we obtain f ∠ (0) → b /d and f −1
∠
(0) →
|z|→0
− b /a. If ad = bc (the parabolic transform), then b /a = d /c and b /d = a /c, and
hence their direct and inverse maps are (characteristic) constants of the Möbius
transformation taken as the limiting values of and from infinity, i.e., f ∠ (∞) → a /c
is the inverse pole and f −1
∠
(∞) → −d /c
{ is the direct pole. The requirement
z, f∠ (z) , f −1
∠
(z)} ∈ Ǎ makes these four (irreducible) rationals, namely b /d,
− b /a, a /c, and − d /c, limiting values where Ǎ is punctured.
Mind that 3 is the composition of a translation from the direct pole (5), an
inversion and reflection concerning the real axis (a conjugation 6), a nonzero
homothety (scaling) plus rotation (7), and a translation to the inverse pole (8),
namely
(3)
(4)
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