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“bound information,” J, belong to the interior of the N-dimensional hypercube.
If these identifications are correct, the following generalizations would follow:
All crisp answers are uncertain.
(5.33)
All crisp answers have non-zero Kosko entropies.
(5.34)
No crisp answers can be complete.
(5.35)
Reality cannot be completely represented.
(5.36)
The ultimate reality is ineffable.
(5.37)
12. Einstein stated (cited, e.g., in Kosko 1993, p. 29) that
As far as the laws of mathematics refer to reality, they are not certain; and as far as they are
certain, they do not refer to reality. (5.38)
Since Statement 5.38 is very often cited by physicists and seems to embody
truth, it deserves to be given a name. I here take the liberty of referring to
Statement 5.38 as the
Einstein's Uncertainty Thesis
(EUT).
EUT can be accommodated by the Knowledge Uncertainty Principle (KUP)
as expressed in Statements 5.33-5.38, if
we identify the volume or the interior
of the N-dimensional hypercube with “reality”
as already alluded to in (11)
and
its surface (
i.e.
, some of its vertices) as the “laws of mathematics”.
Again, we
may locate crisp articulations of all sorts (including mathematical laws and
logical deductions) on the vertices of the N-dimensional hypercube and the
“ineffable reality” in the interior or on the edges of the hypercube. If this
interpretation is correct, at least for some universes of discourse, we may have
here a possible
algebraic-geometric (or geometro-algebraic) rationale
for
referring to the
N-dimensional hypercube
defined in Table
5.4
as the “reality
hypercube (RH)” or as “a N-dimensional geometric representation of reality,”
and Inequality
5.27
and Statement 5.38 as the keystones of a new theory that
may be called the “Algebraic Geometric Theory of Reality (AGTR).” It is
hoped that RH and AGTR will find useful applications in all fields of inquiries
where uncertainties play an important role, including not only physics (see (13)
below) but also biology, cognitive neuroscience, risk assessment, pharmacol-
an objective and visual theoretical framework for reasoning.
13. The wave-particle duality of light (see Sect.
2.3.1
) served as a model of the
complementarity pair in the construction of the philosophy of complementarity
by N. Bohr in the mid-1920s (Plotnitsky 2006; Bacciagaluppi and Valenti
2009), although it was later replaced with the more general “kinematics-
dynamics complementarity pair” (Murdoch 1987). Assuming that the wave-
particle duality of light embodies an uncertainty principle (in addition to a
complementarity principle to a certain degree), it will be analyzed based on the
KUP, Eq.
5.29
. The analysis involves the following steps:
1.
Classical concepts
: The concepts of
waves
and
particles
have been well
established in human language, having developed over thousands of years as
a means to facilitate communication among humans about physical processes.
