Ways of Seeing
- Exploring Modern Metaphysics -
Mathematics - Relationships within the Physical World
The Real numbers (such as lengths between 1 and 2 on a tape measure) and logic lie at the foundation of mathematics:
* The Counting numbers (1,2,3,…) come obviously and naturally once you have made a distinction of any kind (I have an apple and picked another, so now I have two apples)
* The Rational numbers follow easily (I have 3 apples and eat half of one, leaving 2 ½ apples) and consists of all Counting number quotients, their negatives, and the number '0'
* It is easy to provide a one-to-one relationship between the Counting numbers and the Rational numbers, showing that (counter-intuitively) they are “equally” and infinitely numerous
* It can be proven that there are many numbers (such as π and the square root of 2) which are not Rational (can not be expressed as the quotient of Counting numbers) and are called the Irrational numbers; note that the Rational and Irrational numbers combined comprise the Real numbers
* Although it can be easily shown that between every two Irrational numbers there is a Rational number (and vice versa), it has been proven that there are “more” (a higher level of infinity) Irrational numbers than there are Rational numbers
* Cantor’s famous unsolved “continuum hypothesis” asks whether there is a collection of any kind that is more numerous than the Counting numbers and less numerous than the Real numbers
* Interestingly, Gödel has shown that a complete and consistent set of axioms (rules) for all of mathematics is impossible, and further, that some mathematical statements that are true are not provable
Real numbers and their operators (+, -, x, /, √, f, ∑, ∞, d, ∂, ∫, etc.) when related algebraically can provide striking and exacting insight into present and future interactions within the physical world of which we are a part:
* There are the normal circumference, area and volume formulas, such as the easily proved Pythagorean Theorem for determination of the length of the missing side of a right triangle: a2 + b2 = c2
* There are three basic formulas upon which all the world's electrical and electronic systems have been built and maintained over the years involving power, current, voltage, resistance, frequency, inductance and capacitance: P = IE, E = IR, F = 1 / 2π√LC. (For example, by substituting E from the second formula into the first we get P = I2R, showing that the power loss P in an electrical transmission line of resistance R increases as the square of the current I, and so from the first formula the voltage E is maximized on the line for maximum power transmission at the smallest possible loss-causing current.)
* It is easy to prove the fundamental theorem of arithmetic which states that every Counting number greater than 1 can be written as a unique product of prime numbers, that is, numbers which have exactly two distinct Counting number divisors; this is at the core of modern data encryption algorithms
* Matter can be converted to pure energy, with the speed of light squared as the conversion factor, per Einstein’s special theory of relativity: E = Mc2
* The movement of matter through space and time can be described and predicted per Einstein’s general theory of relativity: Guv + Λguv = Tuv 8πG / c4 (Λ is the Cosmological Constant which may help explain dark energy, dark matter, and the increasing expansion of our current universe.)
* Heisenberg has shown that the measurement of the position and movement of a particle can not together be determined below Planck’s constant = 10-34 Joule Seconds
* There are some deep, broad and “beautiful” representations of relationships within the physical world such as Euler’s eiπ + 1 = 0, and the energy of a particle being defined by Plank's Constant, the speed of light and wave length E = ħc / λ
* And, a formula has been derived empirically which shows the relationship between some of the fundamental physical constants (Plank's Constant, speed of light, Pi, Permittivity of free space, and elementary charge) which define (in part) how our particular universe works: ħc 4πϵ0 / e2 = 137.03599911
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