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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5 Zero Consistency
On the zero duality; why zero is not but can be.
5.1 Singularity
Within Q, the polarity between 0 and ∞ is even more evident. If infinity is
outside the rationals, so should zero as a reciprocal counterpart of infinity [78].
Since zero is as troublesome as infinity, the rationals should treat them on par
with each other as the embodiment of abstract extreme magnitudes. However,
note that "arbitrarily little" (the smallest computable rational) is not the same
as "inconceivably little", which is a metaphysical claim about the nothingness
opposed to "unboundedly large" and admitting no regular implementation.
The exceptionality of zero stands out in a rational context more than in Z
because it can be a fraction’s numerator, not a fraction’s denominator. A way to
sort out this algebraic disruption is the convention that a rational expression will
be assumed to be vacuous if a choice of variables involves a zero denominator
[188]. An alternative solution is seeking an interpretation of "division by zero",
but this approach also assumes that zero demands differentiated handling. Any
strategy to give zero a global meaning is in vain; "there is no uniformly satisfactory
solution" (sections 8.5 to 8.7 in [163]). The Solomonic decision of banning
the expression x /0 only if x ≠ 0 does not work because the undefined forms
0/0 in arithmetic and 0 0 in calculus remain. Trying to figure out a meaning for
these expressions leads again to absurdity. Q’s completions (e.g., R) and their
extensions inherit these "irregularities" to the point that they are handled by
computer programming languages and software libraries differently!
Inconsistencies disappear if we concede that zero is not a number of a linear
scale, hence neither the numerator nor the denominator of a fraction. The
nonzero rational numbers ˇQ ≡ Q − {0} form a field because an arithmetic expression
of rational variables that evaluates to zero vanishes. Moreover, we have
explained in subsections 1.1 and 3.2 that thinking of a rational number as a
pair of interchangeable components is an almighty perspective, inhibited owing
to our obsession with zero. Despite this potential, lightness stands a blot on the
reputation of the rationals because they are negligible compared in number with
the irrationals, not to mention compared with the real transcendental numbers.
Specifically, Q has a null Hausdorff dimension in the unit segment of the reals,
although it is unclear why this measure is a preference criterium to choose among
number sets; for instance, the rationals have Minkowski dimension one in the
unit segment of the reals.
How come is the heaviness of the R-line useful for calculability? The reals are
so unthinkably thick that divisibility, contraction, and expansion become pointless;
quite the opposite, we claim that countability is a condition of non-rigid
transformations, e.g., deformations. So, we find no solid empirical-argumentative
basis to sustain this enthusiasm for the reals and wonder who abhors a vacuum
[78], nature or human beings? However, modern mathematics trusts the impeccable
reals as though they were an elixir to cure almost all the flaws of "minor"
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