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.