The Zero Delusion v0.99x

inarticulate and still inconsistent (e.g., what is the complement to ˆR?). Likewise,
we must redefine critical concepts like "neighborhood" and "limit" before doing
calculus in ˆR. All in all, 0 and ∞ cause difficulties as real or complex numbers
without a benefit.
Therefore, extended algebraic objects like ˆR and Ĉ are inspiring but do not
satisfy the field axioms; 0 and ∞ are antipodal points that lead to complicated
arithmetic and rambling calculus. If dealing with a line at infinity is a
requirement, then the proper scenario is projective geometry [4], where everything
works with lines of sight, planes of reality, and planes of representation
instead of distances. This geometry is an actual fraction-oriented framework
where coordinates have the consideration of ratios. If "observational perspective",
e.g., incidence, is not an issue or we need to preserve sizes or angles, an
algebraic setting like ˇQ or Ǎ is the fitting choice.
Renouncing the points 0 and ∞ as numbers from the beginning, we assume
that zero and infinity are unreal curvature values. This solution achieves a consistent,
rational geometry of Euclidean spaces, comparable with their dual nonmetrical
projective spaces, that does not impede using 0 and ∞ as the limit
of every sequence of rational numbers whose absolute values are unboundedly
decreasing and increasing, respectively. Moreover, this approach successfully extends
a rational function, i.e., the ratio of two minimal polynomials, to a "continuous
function" from Ǎ to itself. In the following section, we analyze "conformality",
a unique property of this smooth algebraic action in any dimension.
A last clarification before going on. How is Euclidean space if it is not a
"real" space? A real-algebraic number is a minimal polynomial’s root with an
identically vanishing imaginary component (e.g., the real root of 2). The set of
real-algebraic numbers forms a field (Chapter 7’s first note in [129]), and so do
the "nonzero real-algebraic numbers". We claim that precisely this set is the onedimensional
Euclidean space E. Note that Ǎ is different from E2 ; while the former
has a complex structure and only exiles the origin, the latter has a less elaborated
structure, namely E × E, where "×" is the "direct product", banishing all the
points on the plane’s axes. This result agrees with experiments designed to falsify
quantum mechanics based on splitting the concerned wavefunction into their
real and imaginary parts [142]; a formulation of quantum mechanics in terms
of complex numbers is necessary [34], and complex translates into algebraic.
In general, the coordinates of a point in n-dimensional Euclidean space E n are
all nonzero, and the field operations of this compartmentalized space are the
coordinate-wise field operations of the nonzero real-algebraic numbers.
6.2 LFT
We have expounded why zero hinders progress in algebra and calculus. Instead,
everything works fine if we concede mathematical uncertainty, i.e., managing zero
as an implicit limit of a sequence of evaporating values instead of a number. For
instance, we have explained in the previous section that an algebraic number
inserted in its minimal polynomial provides us with a mere approximation to
zero. Can we move this way to treat zero to other algebraic (computable) objects?