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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1 Introduction
We present this essay and explain why zero is problematic.
1.1 Scope and rationale
This work is a generalistic investigation of the number zero’s role in diverse
momentous fields of mathematics and physics. Because our society champions
zero, one can hardly find academic articles about its usage or facets, except for
some research on zero’s genesis and a few analyses of the nothingness. We feel
we are entering an untrodden area upon questioning zero in earnest.
Our motivation derives from a series of inquiries. How did zero’s historical
and sociological context affect its endorsement in science? Is it real? How zero
differs from other numbers? Are we conceptualizing, applying, and computing
zero appropriately? Is this invention necessary or advantageous? We aim to answer
these questions and explain why zero is delusive to a degree. Our long-term
goal is to raise awareness of the unfavorable consequences of overlooking foundational
issues in mathematics critical to make headway in physics at the most
fundamental level.
Zero has a history "full of intrigue, disguise, and mistaken identity" [89], allegedly
culminating in one of humanity’s most splendid findings. "The most useful
symbol in the world, the naught" [65], is only a recent cultural asset because
humans since forever ignored or deemed it nonintuitive and unsettling before its
adoption by the Europeans in the 16th century. The late Sumer, Babylonian,
Chinese, Maya, and Inca civilizations employed a place-value numeral system
where a distinctive mark played the role of the current digit 0 [87]. However,
it was not used alone as the current number zero. Likewise, with minor exceptions,
Ancient Egypt, Ancient Greece, and Rome did not recognize zero. Today,
we utilize zero daily as an intellectual resource, but the extent to which it is
universal and serves scientific purposes is unclear.
As a philosophical conception, zero symbolizes absence or void, whereas as
a number or value, it "is an integer representing a quantity amounting to nothing"
[59]. In this paper, however, we will not stress the dissimilarity between the
abstract meaning and its representation for clarity and eloquence of our exposition.
Moreso, some theorists [111] introduce a further disparity between the
numerical digit lone and 0 as a vacancy between other symbols of a numeral
codeword (e.g., a string of digits) in positional notation (PN). Since the role of 0
has the same computational essence alone or accompanied, we will also dismiss
such a difference. Consequently, we will adopt the widespread assumption that
zero designates "a count of null balance for the variable or class of objects in
hand" for analysis purposes.
Is zero authentic or fictitious? If the universe (or multiverse) is infinite, zero
finds no "space" to exist; else, nothingness "is" out of our finite universe. In the
latter case, can zero also live within our finite universe (or multiverse)? This
prospect depends strongly on whether our universe is natural, which is unclear
[30]. If it is not, zero might be a product of chance. Assuming that our finite
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