The world for the most part offers niche occupations for the average citizen, who among all others finds the most contentment living upon it. I live in a country which rewards a university education with a public service position. Now, is that actually counterproductive or not? It turns out, as recent research indicates, that only fifteen percent of the founders of the Fortune 400 companies have an MBA. Face it, life is geared to benefit the average. Many people never really grasp a deeper mathematics beyond the simple concepts involving operations on fractions. That is the point at which they begin to acquire the traits of skepticism. That's grade three arithmetic, by the way.
I am convinced that the roots of skepticism lie for the most part in the inability to grasp concepts that are essentially over one's head. Mention the concept of the imaginary number i for example, and most people will immediately close their minds, cross their legs and fold their arms in a harrumph of incredulity. The square root of minus one, indeed! And yet, the imaginary number exists! It is a real thing after all. To prove it, I will explain it to you. Reading comprehension skills are mandatory if you wish to continue with this blog.
In certain scientific applications, in my example I will use electronics since it encompasses all sciences, it is found useful to use imaginary numbers, particularly with respect to solving electronic circuits. I will begin by discussing the j operator (it is the same as i in non-electronic applications but in that discipline i denotes electric current (amperes), so we use j). The j operator is used to denote rotation of 90° in the counterclockwise direction on the 2-dimensional X-Y matrix. For example, on the X-Y graph a line a units long can be operated on by the operator j to become ja, a line of the same length as before but rotated 90° in the counterclockwise direction to lie on the Y axis. Any quantity operated on by -j will rotate through 90° in the clockwise direction. The quantity j(ja) is written j²a, and j(j(ja)) is written as j³a. So j²a becomes -a (2 90° counterclockwise rotations). Things become interesting when we analyse the situation where j-ing a twice in succession brings it to the same point as a single operation with a minus sign therefore j²=-1 and we can therefore conclude that j=√-1. j³ must equal j(-1) or -j, and j^4 must equal j²•j² = (-1)(-1) = +1. Remember, this is all about rotating around the X-Y axis.
So far, so good. In mathematics, the square root of a negative number is known as an imaginary number. Its terminology is misleading because in dealing with some scientific applications imaginary numbers become real. In order to avoid difficulty in dealing with square roots of negative numbers we consider that every imaginary number can be expressed as the product of a positive number and the square root of -1, for example the √-25 is √-1•√25 = √-1•5. We can then write this expression as j5. The term complex number refers to an expression wherein an imaginary number is united to a real number by a plus or minus sign. 3-j4 is a complex number.
Even if you have understood all of this, you are still inclined to say "so what?" If you haven't understood any of this you are a happy camper and no doubt a skeptic. Go in peace. And if you, the former, wish to dispel any trace of ambiguity with respect to the practical purpose of the imaginary number and are interested in knowing more then continue by reading this short .PDF document (temporarily unavailable). The Adobe Reader is required. The paper was composed using the LaTex engineering and mathematics markup language, used for publishing those clever vector diagrams and formulae with all those neat symbols. Study it, understand it, and go to the head of the class.
I am writing this not merely to impress you, or to acquire a modicum of credibility, although I do hope to have accomplished some of both, but to point out that a lack of understanding of anything should not be allowed to permit the introduction of doubt into that which is real. Oh, and also just because somebody has a university degree doesn't mean that they're better than you.
Friday, January 18, 2008
Trust not the skeptic..
Who is right and who is wrong?
The greatest thinker of them all, Albert Einstein, admitted one time to "The greatest blunder of my life". In nineteen twenty-nine, the noted British Astronomer Edwin Hubble observed that the Universe was expanding, thereby rendering Einstein's Static Universe Theory obsolete. Some time before, a young student had approached him, asking him, "..but Dr. Einstein, wouldn't gravity tend to pull all the objects in a static Universe together?". That required some reformulation, so the good doctor came up with what he termed the "cosmological constant" which, when incorporated into his relativity equations, introduced an anti-gravity component, thereby satisfying the requirements for a static Universe.
There is an important lesson here. Mathematics is not the holy grail. Mathematics can prove the impossible. Therefore mathematics is a potentially misleading and dangerous tool if misused.
So string theory will always be theory and should be taken with a grain of salt (as this very entertaining slideshow demonstrates), for it is impossible to prove, completely untestable, and therefore a figment of the imagination. It falls into the category of interesting mathematical proposition. For my part, I will not even consider it, because it falls outside of the realm of logic and reason. As I've said, mathematics can prove the impossible.
We are all entitled to be wrong. I once believed that the expansion of the Universe occurs in cycles, accelerating until this expansion attains the speed of light which creates a boundary of infinite mass which explains dark matter all around, then beginning the creation process beyond this boundary once again, a different kind of colossal heartbeat.
I am certain now that this is wrong. The Universe is expanding at a constant rate. The observation that it is accelerating is but an illusion which can be proved with a simple demonstration, to wit:
Take an elastic band and snip it open and size it to exactly six inches. Draw the universal symbol for the sun, a point with a circle around it, at the two- inch mark. Draw a point to represent a near star at the one-inch mark and another to represent a far star at the five-inch mark. Now stretch the rubber band until it is twelve inches in length. You will see in the time it took the near star to move one inch farther away from the sun that the far star has moved three inches farther away. The observation is that the farther something is away from our point of reference in this Universe, the faster it is moving away, even though expansion is occurring at a constant rate. And because this Universe is dynamic - ie. it is always expanding, the illusion, based on actual observation, is that this expansion is accelerating. The scientific community currently believes in a misleading perception of what is really happening, a misconstruence if you will, but we know better don't we?
In a Universe which is expanding at a constant rate, there is no requirement and no reason to believe in a cycle of contraction and expansion. Paul Frampton is simply wrong.
