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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recursive depth could be a predetermined value or limited to several possible
values given by an inwards quantum number, much as the location of a particle
confines itself to a countable number of spatial positions. IT leads to a similar
judgment; "Whether the inevitable limit on precision is simply a limit on the
number of bits that can be invoked in physics or is more complex and statistical
is unclear" [107]. Whether absolute or relative, the deepest zoom-in would ultimately
reach a discontinuity, an indivisible dull spot, an uncertainty bubble, or
a demarcated lattice cell.
Mind that discreteness and finite resolution of spacetime are often mistakenly
coupled. Discreteness is insufficient to justify an absolute minimum length because
we could always find a lower and lower scale as a lattice spacing approaches
zero (e.g., as a fractal), and vice versa, we can have a structural resolution limit
without discreteness (e.g., in String Theory). So, to guarantee that natural extents
are nonzero, a quantum theory of spacetime should state an additional
assumption or deduce that a fuzzy or crisp granularity exists. This approach
is the most common when research unites GR and QFT. String theory, Loop
Quantum Gravity, Asymptotically Safe Gravity, and non-commutative geometries,
nowadays the most renowned roads to QG, feature a minimal length scale.
Ultimately, the physicality of a minimum observable extent weakens the notions
of locality and coherence; in [55,56], a model for spacetime foam associates the
minimal length scale with decoherence in terms of nonlocal interactions to explain
quantum gravitational effects.
Another issue is whether the minimum natural distance is the Planck length.
We cannot anticipate if the Planck scale will play an essential role in a robust
theory of QG or whether it will involve another minimal size, surely invariant.
The Planck length stands for the diameter of the minor possible black hole,
i.e., where a mass’ Compton wavelength and Schwarzschild radius coincide [6],
or the minimum ball of spacetime accuracy [29]. Consequently, GR and QFT
interact intensely at the Planck scale to produce gravitation. Some physicists
argue that putting the Planck length and the minimal length on the same level
implies modifying SR (e.g., as in [3]); otherwise, "Planck-pixels" would violate
Lorentz symmetry [98], and hence we could observe lengths contracted below the
Planck supposed boundary from some inertial frame, and even electromagnetic
waves squished boundlessly. Alternatively, we can turn to de Sitter Invariant
Special Relativity to preserve observer-invariance. This theory exploits the invariant
length parameter that the de Sitter group naturally incorporates, which
is the order of the Planck length [1,2]. Regardless, rest-frame equivalence and a
minimum size do not clash. We do not negate physics beyond the Planck scale,
only that the indetermination would be so significant that thinking of the distance
between two points [76] or a spacetime topology [73] no longer would make
sense.
What is the role of the Planck length, assuming that it is a universal minimum,
in our (at least) four-dimensional universe?Admittedly, for something to
exist, it must have a nonzero quantum value of surface area (discrete multiples
of the Planck length squared) and volume (discrete multiples of the Planck
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