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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Table of Contents
The Zero Delusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Julio Rives
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1 Scope and rationale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 The troubles with zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Zero in Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1 Elusiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Extent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Zero Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1 Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4 Zero Countability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.1 Emptiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2 Futility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5 Zero Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.1 Singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.2 Beability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
6 Zero Computability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.1 Rationality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.2 LFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
7 Zero Conformality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
7.1 Cross-ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
7.2 Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
8.1 Zero impression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
8.2 Zero future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
9 Postcript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
9.1 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
9.2 Statements and Declarations . . . . . . . . . . . . . . . . . . . . . . . . 52
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
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