**MANIPULATIVE MATHEMATICS**.

The problem with modern science and mathematics is
the introduction of imprecision in the name of precision. The precise
definitions of terms have been relegated to the background by introducing
operational definitions that can be manipulated as one wants to suit his/her
requirement even for exactly opposite effects. History of mathematical science
is replete with many such examples. An axiom is a statement or proposition
which is regarded as being established, accepted, or self-evidently true to
serve as a premise or starting point for further reasoning and arguments. The
word "axiom" is a slightly archaic synonym for postulate compared to
conjecture or hypothesis, both of which connote apparently true but not
self-evident statements. The self-evident part means invariant in effect in
everyday experience, but has been manipulated to imply other meanings. The
fifth Axiom of Euclid and its manipulation by nineteenth century mathematicians
is one example. Before I discuss that, a word about mathematics itself.

The validity of a mathematical statement is judged
by its logical consistency. The validity of a physical statement is judged by
its correspondence to reality. Wigner defined mathematics as the science of
skillful operations with concepts and rules invented just for this purpose.
This is too open-ended. What is skillful operation? What are the concepts and
Rules? Who invented them? What is the purpose? Do all concepts and rules have
to be mathematical? Wigner says: The great mathematician fully, almost ruthlessly,
exploits the domain of permissible reasoning and skirts the impermissible, but
leaves out what is permissible and what is not; leaving scope for manipulation.

Burrowing from M. Polanyi, Wigner says: The
principal point …. is that the mathematician could formulate only a handful of
interesting theorems without defining concepts beyond those contained in the
axioms and that the concepts outside those contained in the axioms are defined
with a view of permitting ingenious logical operations which appeal to our
aesthetic sense both as operations and also in their results of great
generality and simplicity. Wigner admits not only the incompleteness of
mathematics but also its manipulation according to the aesthetic sense of the
operator. He gives the example of complex numbers and burrowing from Hilbert,
admits: Certainly, nothing in our experience suggests the introduction of these
quantities. Indeed, if a mathematician is asked to justify his interest in
complex numbers, he will point, with some indignation, to the many beautiful
theorems in the theory of equations, of power series, and of analytic functions
in general, which owe their origin to the introduction of complex numbers. The
mathematician is not willing to give up his interest in these most beautiful
accomplishments of his genius. A reverse self-fulfilling effect!

Mathematics is the ordered accumulation and
reduction in numbers of the same class (linear or vector) or partially similar
class (non-linear or set) of objects. Number is one of the properties of all
substances by which we differentiate between similars. If there is nothing
similar at here-now, the number associated with the object is one. If there are
similars, the number is many. Our sense organs and measuring instruments are capable
of measuring only one at a time. Thus, many is a collection of successive
one’s. Based on the sequence of perception of such one’s, many can be 2, 3,
4….n. In a fraction, the denominator represents the one’s, out of which some
(numerator) are taken. Zero is the absence of something at here-now that is
known to exist elsewhere (otherwise we will not perceive its absence at all).

Language is the transposition of information to
another system’s CPU or mind by signals or sounds using energy (self-communication
is perception). The transposition may relate to a fixed object/information. It
can be used in different domains and different contexts or require
modifications in prescribed manner depending upon the context. Since
mathematics follows these rules, it is also a language. Mathematics explains
only how much one quantity, whether scalar or vector; accumulate or reduce
linearly or non-linearly in interactions involving similar or partly similar
quantities and not what, why, when, where, or with whom about the objects.
These are subject matters of physics. The interactions are chemistry. There is
no equation for Observer. The enchanting smile on the lips of the beloved is
not the same as geometry of mouth or curvature of lips. Thus, mathematics is
not the sole language of Nature.

Nature prohibits reductionism. Whole is a sum of
its parts and more. Water is more than 2H and O. A triangle is more than three
straight lines. This is natural number theory.
5 has independent perceptual value than 5 ones. If we can purchase a car in € 5k, with € 1k,
we can purchase 1/5 of a car. This may look mathematically valid, but 1/5 of a
car is an undecidable proposition. Hilbert’s problem whether mathematics is
complete (every statement in the language of number theory can be either proved
or disproved) and Gödel’s negative solution arise out of such unnatural
mathematics. Brute force approach is similarly unnatural, though sometimes it
may succeed by chance.

The structure or form principle that comes out of
the evolution of point in time constitutes the entire cosmic manifestation.
According to the mode of evolution; the evolution of the point can take two
forms: whether it is open ended or closed on itself. The open-ended evolution
of the point is called the non-compact line and the close ended evolution of
the point is called the compact line. These two along with other lines or
curvature evolve into two dimensional and three dimensional structures such as
planes and cubes or spheres and their not so symmetric manifestations that
embrace the whole display of visible forms in the Cosmos. Any line, curved or
otherwise, which is not closed on itself, is called a non-compact line or
simply a line. On the other hand, any line that is closed on itself, is called
a compact line, whose symmetric manifestation is a circle. The circles and
other compact structures can be of different types, but a line can only be
straight or curved. Ellipse is a circle, whose center is moving. Its
eccentricity depends upon the velocity of the center, with reference to any
point on the circumference. Thus, the planetary paths are circular, but appear
elliptical due to the relative movements of the Sun and the planets around it.
The parabola and hyperbola are lines, as they are open ended.

Euclid spoke of straight lines on plane surfaces,
as only straight lines could be perpendicular to each other. The nineteenth
century mathematicians used a curved plane to draw a line - two lines
(longitudes) which are perpendicular to the same line (the equator) at the
point of intersection. Had the lines been straight lines, it would not have
projected itself into space in that direction and not curved to end up
intersecting at the north and the south poles. Further, the equator is not a
straight line, but is closed on itself – a circle. Thus, by changing the
initial conditions - nature of lines - we were destined to arrive at a
different conclusion. But using this to disprove the fifth Axiom is nothing
less than fraud with Euclid.

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let noble thoughts come to us from all around