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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lustrates that nature is twofold, handling a linear and a logarithmic scale simultaneously.
Specifically, the poles of a Laplace Transform synthesize the critical
parameters of the generating signal, net information coding. The universality of
the exponential function is even more apparent when we consider maps of linear
fractions over the nonzero dual and split-algebraic numbers via concatenation of
translations, conjugations, and rotations. In these rings, the LFT’s singularities
are limiting values that adjoin the plane, like 0 and ∞. A generalized LFT preserves
a conic and the angle (area) between two intersecting conics, hence the
ring’s conformal structure (e.g., a figure’s shape).
When n ≥ 3, an n-dimensional Möbius transformation is a domain’s most
general conformal isometry, ensuring smooth translation moves, dilations, contractions,
rotations, and inversions. The main invariant of a conformal transformation
over a ring is the cross-ratio, the relative distance that separates a pair
of points from another, precisely the extent to which two ratios of differences
deviate from a proportion. The cross-ratio is a universal construct that ensures
angle invariance and allows contemplating an exponential map as a generalized
rotation, recursively defining all types of conformal maps, and qualifying coding
spaces ruled by hyperbolic geometries. Conformality enhances the thesis that our
universe handles (at least) two scales, and zero only fits in the logarithmic scale
of coding spaces. Thus, we have arrived at the suggestive insight of conformality
departing from the redefinition of rationality deprived of zero.
Our finitistic arguments have meaningful implications in physics. For one
thing, every lower bound must have its counterpart on the large as a maximum
value. For instance, we can state the Dual Uncertainty Principle, which limits
to what extent conjugate variables can lose (instead of retaining) their approximate
meaning by renouncing simultaneously ill-defined (instead of well-defined)
complementary properties expressed by a single value; for instance, σxσp / ≤ ,
where is a dimensionless absolute scaling limit. This "certainty principle" implies
that, in general, it is impossible to foretell a magnitude with arbitrary
uncertainty, resulting indiscernible on the microscopic scales humans experience
compared to the size of the universe. Our theory also predicts that a minimum
speed exists; none can observe that another frame of reference is static,
i.e., experimentally measuring zero relative speed would require infinite energy.
Additionally, if a discrete and finite universe is rational-oriented as we sustain,
nature would implement irrational numbers as computable algebraic expressions
whose main constants, e.g., √ 2, √ 3, and √ 5, might be built-in.
In the long-term future work, taking into consideration the content of the
essay, we point to a couple of additional issues that would deserve further investigation;
first, the prospect of a universal double logarithmic scale (i.e., primality
+ linear + logarithmic), second, whether the logarithmic scale connects with a
"natural" probability mass function for the rationals (and algebraic number),
and third, a discrete model of spacetime based on a regular space-filling tessellation
(or honeycomb) of four-dimensional Euclidean or hyperbolic space. A
premise for these research topics is to put zero aside.
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