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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structure, while holistic information gets imprecise. Thus, a being handles only
imperfect information and struggles between the specific and the general. In
particular, a living being is never wholly ignorant or aware of its environment.
This argument is also why we can strictly discard neither the null nor the
alternative hypothesis in statistics. The most we can do is to minimize presumptions
about a system, such as configurational preconditions and restrictions. The
maximum entropy principle, or minimal information principle, claims that [191]
"the probability distribution appropriate to a given state of information is one
that maximizes entropy subject to the constraints imposed by the information
given", a bet for the less informative a priori distribution possible and hence the
most sensible of all "fair" random a priori distributions. This principle "affords
a rule of inductive inference of the widest generality" [160], a giant step toward
formalizing Occam’s razor as a rule, for instance, to choose a prior probability
distribution in Bayesian thinking. Note that we always believe something before
considering evidence; presuming nada is impossible. Likewise, the posterior
distribution is never thoroughly informative; we cannot cancel its information
entropy. These are new signs that we live in a countable and orderly world.
Information interchange, discussed above, preserves heat. However, what can
we learn from a dissipative process? Information loss is a physical process intricately
linked with irreversibility or unpredictability [86], so a being must constantly
acquire new information to postpone death. Landauer’s erasure principle
tells that any phase-space compressing irreversible computation, such as erasing
information from memory, "would result in a minimal nonzero amount of work
converted into heat dumped into the environment and a corresponding rise in
entropy" [105]. Then, a computational system continually creates entropy, and
constant entropy means no computation takes place.
Can a system have no entropy? Defining "entropy" as the logarithm of the
number of possible microstates N of a system and assuming that all have the
same occurrence probability, its entropy vanishes when the system has only one
microstate, i.e., one component (N = 1) with no freedom degrees, whence closed;
it would not be a system per se, but just a beable, if any. Moreover, a system
evaporates if N = 0, whence speaking of entropy is nonsense. Can a system be
simplistic in full? Suppose we measure the complexity of a being by the length
of the computer program in a predetermined formal programming language that
produces such a system as output. In that case, the only possibility of possessing
no (Kolmogorov [100]) complexity is no entropy, i.e., no information content.
Thus, the representation of the void, i.e., no freedom and no complexity, is log 1.
We finish this subsection by commenting on a few paramount investigations
that have consolidated the thermodynamics-information connection. Shannon’s
Source Coding Theorem limits the reliability of data coding [155]; an ordinary
(noisy) channel cannot transfer error-free data if the codewords are finite in
length, meaning that error is natural. Bell’s theorem unveils that locality is a
condition vaguer than classically assumed [14]; we had better regard phenomena
as neither absolute local nor global. IT Evans’ fluctuation theorem is crucial
to generalizing the 2nd law of thermodynamics to microscopic systems, with
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