The process of statistical measurement is used to make precise the evaluation of a claim relies upon our assumptions about the sampling measurement process and the empirical phenomena measured. The independence of the sampling measurements leads to the normal distribution, which allows the confidence of statistical estimations to be calculated. This is the metric used to gauge the validity of the tested hypothesis and therefore the empirical claim proposed. While usually the question of independence of variables arises in relation to the different quantities measured for each repeated sample, we ask now about the independence of the measurement operation from the measured quantity, and thus the effect of the observer, i.e. subjectivity, on the results found, i.e. objectivity. When there is an interaction between the observing subjectivity and the observed object, the normal distribution does not hold and thus the objective validity of the sampling test is under question. Yet, this is the reality of quantum (small) measurements and measurements in the social world. If we consider the cognitive bias of decreasing marginal utility, we find that samples of marginal utility will decrease with each consumption of a good, making the discovery of an underlying objective measurement of the subjective preference impossible. This assumption of independence of the Measurer from the Measurement is inherited from Descartes.

Descartes created the
modern *mathematical sciences* through the development of
a *universal mathematics* that would apply to all the other sciences to find certain
validity with exactitude and a rigor of proof, for which essays can be found in
his early writings developing these subject-oriented reflections. In his *Meditations*, one finds two
‘substances’ clearly and distinctly after his ‘doubting of everything, to know
what is true’ – *thinking* & *extension. *This separation of thinking
and extension places measurement as objective, without acknowledging the
perspective, or reference frame, of the subjective observer, leading to the
formulation of the person as ‘a thinking thing,’ through *cogito*, *ergo sum*, ‘I think, I am.’ Just as with the detachment of mathematics
from the other sciences – a pure universal science – and therefore the concrete
particularity of scientific truth, the mind becomes disconnected from the
continuum of reality (i.e. ’the reals,’ cc: Cantor) of the extended body as
subjectivity infinitely far from objectivity, yet able to measure it
perfectly. This would lead to the
Cartesian Plane of XY independence as a generalization of Euclidean Geometry
from the 2D Euclidean Plane where the parallel (5th) postulate was retained:

*Euclid’s 5th Postulate: For an infinitely extended straight line
and a point outside it, there exist only one parallel (non-intersecting) line
going through the point.*

This became the objective coordinate system of the extended world, apart from the subjective consciousness that observed each dimension in its infinite independence, since it was itself independent of all extended objects of the world. All phenomena, it was said could be embedded within this geometry to be measured using the Euclidean-Cartesian metrics of distance. For centuries, attempts were made to prove this postulate of Euclid, but none successful. The 19th century jurist, Schweikart, no doubt following up millennia of ancient law derived from cosmo-theology, wrote to Gauss a Memorandum (below) of the first complete hyperbolic geometry as “Astral Geometry” where the geometry of the solar system was worked out by internal relationships between celestial bodies rather than through imposing a Cartesian-Euclidean plane.

(p.76, *Non-Euclidean Geometry*, Bonola, 1912)

This short Memorandum convinced Gauss to take the existence of non-Euclidean geometries seriously, developing differential geometry into the notion of curvature of a surface, one over Schweikart’s Constant. This categorized the observed geometric trichotomy of hyperbolic, Euclidean, and elliptical geometries to be distinguished by negative, null, and positive curvatures. These geometries are perspectives of measurement – internal, universally embedding, and external – corresponding to the value-orientations of subjective, normative, and objective. From within the Solar System, there is no reason to assume the ‘infinite’ Constant of Euclidean Geometry, but can instead work out the geometry around the planets, leading to an “Astral” geometry of negative curvature. The question of the horizon of infinity in the universe, and therefore paralleling, is a fundamental question of cosmology and theology, hardly one to be assumed away. Yet, it may practically be conceived as the limit of knowledge in a particular domain space of investigation. In fact, arising at a similar time as the Ancient Greeks (i.e. Euclid), the Mayans worked out a cosmology similar to this astral geometry, the ‘4-cornered universe’ (identical to Fig. 42 above) using circular time through modular arithmetic, only assuming the universal spatial measurement when measuring over 5,000 years of time. The astral geometry of the solar system does not use ‘universal forms’ to ‘represent’ the solar system – rather, it describes the existing forms by the relation between the part and the whole of that which is investigated. The Sacred Geometries of astrology have significance not because they are ‘perfectly ideal shapes’ coincidently found in nature, but because they are the existing shapes and numbers found in the cosmos, whose gravitational patterns, i.e. internal geometry, determine the dynamics of climate and thus the conditions of life on Earth.

The error of Descartes can
be found in his conception of mathematics as a purely universal subject, often
inherited in the bias of ‘pure mathematics’ vs. ‘applied mathematics.’ Mathematics may be defined as methods of
counting, which therefore find the universality of an object (‘does it exist in
itself as 1 or more?’), but always in a particular context. Thus, even as ‘generalized methods of
abstraction,’ mathematics is rooted in concrete scientific problems as the
perspectival position of an observer in a certain space. Absolute measurement can only be found in the
reduction of the space of investigation as all parallel lines are collapsed in
an elliptical geometry. Always, the
independence of dimensions in Cartesian Analysis is a presupposition given by
the norms of the activity in question.
Contemporary to Descartes, Vico criticized his mathematically universal
modern science as lacking in the common sense wisdom of the humanities in favor
of a science of rhetoric. While rhetoric
is often criticized as the art of saying something well over saying the truth,
it is fundamentally the art of argumentation and thus, like Mathematics, as the
art of measurement, neither are independent from the truth as the topic of what
is under question. The Greek into Roman
word for Senatorial debate was Topology, which comes from *topos* (topic) + *logos* (speech), thus using the
numeral system of mathematics to measure the relationships of validation
between claims made rhetorically concerning the public interest or greater
good. The science of topology itself
studies the underlying structures (‘of truth’) of different topics under
question.

*Together, Rhetoric and Mathematics enable Statistics, the art of
validation. Ultimately, Statistics
questions ask ‘What is the probability an empirical claim is true?’*

While it is often assumed
the empirical claim must be ‘objective,’ as independent of the observer,
quantum physics developing in Germany around WWI revealed otherwise. When we perform statistics on claims of a
subjective or normative nature, as commonly done in the human sciences, we must
adjust the geometry of our measurement spaces to correspond to internal and
consensual measurement processes. In
order to do justice to subjectivity in rhetorical claims, it may be that
hyperbolic geometry is the proper domain for most measurements of validity in
empirical statistics, although this is rarely used. Edmund Husserl, a colleague of Hilbert who
was formulating the axiomatic treatment of Euclid by removing the 5th
postulate, in his *Origins of Geometry*, described how Geometry is
a culture’s idealizations about the world and so their axioms can never be
self-grounded, but only assumed based upon the problems-at-hand as long as they
are internally consistent to be worked out from within an engaged activity of
interest – survival and emancipation.
Geometry is the basis of how things appear, so it encodes a way of
understanding time and moving within space, therefore conditioned on the
embedded anthropology of a people, rather than a human-independent universal
ideal – how we think is how we act.
Thus, the hypothesis of equidistance at infinity of parallel lines is an
assumption of independence of linear actions as the repeated trials of
sample-testing in an experiment (‘Normality’).
Against the universalistic concept of mathematics, rooted in Euclid’s
geometry, Husserl argued in *The Crisis of the European
Sciences*
for a concept of science, and therefore verification by mathematics, grounded
in the *lifeworld*, the way in which things appear through
intersubjective & historical processes – hardly universal, this geometry is
hyperbolic in its nature and particular to contextual actions. Post WWII German thinkers, including Gadamer
and Habermas, further developed this move in philosophy of science towards historical
intersubjectivity as the process of Normativity. The Geometry from which we measure the
validity of a statement (in relation to reality) encodes our biases as the
value-orientation of our investigation, making idealizations about the reality
we question. We cannot escape
presupposing a geometry as there must always be ground to walk on, yet through
the phenomenological method of questioning how things actually appear we can
find geometries that do not presuppose more than the problem requires and
through the hermeneutic method gain a historical interpretation of their
significance, why certain presuppositions are required for certain
problems. Ultimately, one must have a
critical approach to the geometry employed in order to question one’s own
assumptions about the thing under investigation.