FORMULAS and SIMPLICITY

Formulas in mathematics and physics are often distinguished by their simplicity. And simplicity in a mathematical or physical formula correlates with beauty. Some examples of this simplicity include Euler's formula for polyhedra, Euler's basic formula in complex analysis, the basic inverse factorial series representation of Euler's number, the de Moivre's theorem from complex analysis, the law of cosines, the law of sines, the formula for length in a Cartesian coordinate system, the Bragg law, Newtown's inverse square law, Coulomb's inverse square law, the first and second and third laws of thermodynamics, the formulas for lattice spacings in different crystal systems, the infinite limiting definition of the golden ratio, de Broglie's matter wave hypothesis, the matter wave-light wave hypothesis, the photon hypothesis and so on........It can thus be concluded from these examples that mathematical and physical profundity and beauty comes from simplicity of expression and parsimony in the choice of elements within a physical or mathematical expression. These expressions identified above are just some prominent examples of mathematical truth or physical truth that has an enduring and not ephemeral value. The introduction of too many elements (or factors) within an expression correlates directly with the dimunition of the meaningfulness of the expression. The meaningfulness of mathematical and physical equations tends to wilt away at the edges the farther such expression move from simplicity to complication.........One should always seek after economy and parsimony of expression in the formulation of profound statements.

IRRATIONALITY in MATHEMATICS

Numbers form the elementary fabric in the elaboration of the structure of mathematics. The Pythagorean School in Ancient Greece held the view that numbers must be Gods because they do not change with time. It could be said that the Pythagorean School identified the primacy of numbers in formulating mathematical truths. We learn from school days that in elementary algebra a positive number multiplied by a positive number yields another positive number.........and in mixed multiplications a positive number multiplied by a negative number yields a negative number.......and vice versa.........By symmetry, in the final possible circumstance, one would then expect a negative number multiplied by a negative number to yield another negative number...........But for algebra to be self-consistent we consider or define the product of 2 negative numbers to be a positive number...............this is an example then of where counterintuitive results creep in to the foundations of mathematics..........Another  and related example where counterintuitive results creep in to mathematics is with respect to defining roots of the numbers........for positive numbers all roots are defined by generating the identical multipliers to a given number........but since negative numbers are not formed from multiplying any given set of identical multipliers...............because negative numbers and positive numbers multiply together respectively to yield only positive numbers................we thus have a situation where some elementary mathematical operations on ordinary negative numbers are simply not definable because of the earlier multiplication rules identified at the beginning of this piece. Mathematics gets around this embarrassing situation by defining such operations as leading to imaginary numbers and there is a ver elaborate field of mathematics that has worked out the properties of imaginary numbers..........it is called complex analysis...........And there are many other consequences of the basic rules of numbers and their operations......like 0! = 1! = 1 and 10^0 = 1^0 = ....... = 1 that are uncomfortable and counterintuitive and that lead the wise to conclude that mathematics, at the bottom of the chain of reasoning in a reductio ad absurdum scheme is full of irrationality and this pervasive irrationality acts to wilt away the edges of the rationalists arguments about our world.