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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We have argued that removing zero from the natural numbers is acceptable
and necessary. An additive constant is not a great deal, especially if we can define
it as the sum of two multiplicative constants, one for the negative and one for
the positive integers. We can take −1 as a precursor of the negative integers,
much as 1 is an inductor of the positive ones, both fixed under reciprocation, and
0 ≡ 1 − 1 vanishes. Alas, our mathematics accepts 0 to make 5 + 0 and 2 − 2 = 0
well-defined arithmetic expressions at the expense of dealing with undefined
forms such as x /0. What good is it? We have not found any compelling reason
to involve zero in the arithmetics of the sets N, Z, and Q. Similarly, fields and
rings based on Q-completions such as R, C, H, and O are troublesome because
zero has no multiplicative inverse, and evaluating to zero is uncomputable.
The critical problem with zero is rooted in the ideas of infinity and ultradensity,
which are very far from the (quantum) reality. Physics increasingly discovers
new evidence of the cosmos’ compartmentalized character, although the
impossibility of endless divisibility at the lower scales will remain untestable for
decades. Empirically, 0 is as exceptional as ∞ is, only acquiring significance as
the limiting value of an asymptotically vanishing sequence. As Liangkang puts
it, "zero represents the horizon of metaphysics: we can forever approach it, but
we cannot ultimately arrive at it." Moreover, the irresistible instinct to insert
a fulcrum that fills the hole between a line’s negative and positive sides has a
simple explanation as a sheer manifestation of the "horror vacui".
With all respect to Brahmagupta, antiquity was correct in denying zero as a
number of the linear scale. Zero initially seemed to provide us with new rudiments
of ken but has been afterward disappointing. Although a significant part of the
effort of the last two centuries in physics pivots around the number zero, the
outcome has been less fruitful than disturbing. Today, zero covers mathematics
and physics in mud, making them less easy to teach and learn. In general, because
zero is exceptional, vacuous, and inoperable, science must reconsider its meaning
and how to utilize it.
8.2 Zero future
Nonetheless, zero is possible and has a reasonable probability of existing as a
reference value in prospect rather than a number. From this outlook, we can give
zero a geometrical meaning; zero symbolizes flatness, specifically the straight line
constant, much like P, е, and π are the constants of the perfect parabolic, elliptic,
and hyperbolic unit cycles. Nothing can interact with 0 due to the same reason
why an ideal parabola, circle, or hyperbola is unattainable. The continuum, 0,
P, π, е, and ∞, only exist in the offing. Algebraically, zero is the transcendental
number 0.0 · · · , whose expansion in standard PN is zero between the decimal
point and "the end of the sequence". Consequently, although mathematics has
adopted zero as the abstraction of "inconceivably minute", physics must take it
as a computable "arbitrarily small" number or the smallest nonzero number in
hand. No special apparatus is needed to carry out this actualist idea, for computers
already make us keep our feet on the ground, giving us enough power and
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