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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systematic error, particularly as nonzero decoherence if the observed system
is manifestly quantum, which is closely related to the measurement problem
[15,135]. On the other hand, Heisenberg’s Uncertainty Principle, a law independent
of the Schrödinger equation, claims a fundamental boundary to how well we
can predict the values for a pair of complementary variables of a (wave-like) system.
Note that the initial conditions of a transformation cannot be thoroughly
specified. Even if they were, it would be impossible to anticipate the exact value
of either of the conjugate properties (Fourier transforms of one another), ensuring
a minimum threshold for the product of their dispersion [146]. Then, Ozawa’s
inequality aggregates the observer effect’s systematic error to the Uncertainty
Principle’s statistical error [131]. Fujikawa’s relation combines these errors [53]
to state that the product of the inaccuracy of one variable and the subsequent
fluctuation in the other is nonzero, surpassing the modulus of the commutator’s
expectation value of the corresponding observable operators. The idea to bear in
mind is that not only are dual properties dependent on each other, but neither
can disappear, which supports the thesis that zero is unreal.
Like infinity, zero might be unnecessary to construct most physics as currently
utilized. Science should deal with zero as a beable projecting a property
of quantitative character instead of an actual concrete value. From this standpoint,
we can judge zero as a hole perfect in the abstraction of nothingness, the
immaterial, the unfinished, the imminent, "the unknown" (Hindus’ "sunya"),
an "inaccessible number" [23], "the unthinkable" [89], or rather an "undetermined
possibility" [125]. However, this sheer non-measurable potential can have
a nonzero probability of occurrence with implications in our view of the cosmos
(see the probability mass distribution of the integers below in 5.2).
2.2 Extent
The old discussion about whether nature is continuous or discrete [64] takes us
to the Quest for Fundamental Length in Modern Physics. On the one hand, the
structure for a continuous fabric of spacetime and matter is liable to be inexhaustible,
paving the way to zero and infinity but demanding limitless resources.
On the other hand, neither philosophy nor test data seem to impede fundamental
discreteness or a minimum physical extent.
Descartes [39] thought that "the nature of a body consists just in extension"
(2.4) and "nothingness cannot have any extension" (2.18), albeit "a body can
be divided indefinetely" (1.26). In Hume’s view, "no finite extension is infinitely
divisible", a statement that embraces space, time, and abstractions (Of the Ideas
of Space and Time, Book 1 in [84]). In the context of string theory, one cannot
compress a circle below a minimal stretch given a fundamental string tension,
which suggests that "smaller distances are not there" [192]. Smolin argues that
nature cannot contract distances ad infinitum [159]. Hossenfelder warns that
there might not be a minimal length, just a minimal length scale [80], as a lower
bound on the product of spatial and temporal extensions, for instance.
Furthermore, a minimal length scale would not necessarily appear as a spatial
resolution limit but could be noticeable at any layer. If this is the case, the
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