How A Handful Of Pebbles Contained The Universe, Part 1.

December 2, 2016 Leave a comment

For thousands and thousands of years, people had only two words for quantities:  “one” and “many”.  Even ancient languages reveal this fact.  When people saw an aggregate of stones, they referred to them as “many”.  And when they picked up a single stone they used the word “one”.  “I have one stone not many stones”, they would say.  It must have seemed obvious to them.  Little did they know that by using these terms they were blocking off the awesome universe which surrounded them.

At some mysterious point in time, someone must have picked up the stones individually and wondered how to designate them.  Perhaps, one person wanted to exchange his “many” for another person’s “many” in a business matter.  How could each determine that he wasn’t cheated except by counting.  Stones, then, were no longer simply a collection of “many”, but had special identities or designations.  Thus, from such primitive beginnings the world of number was born.  The simplest numbers uncovered were the “counting numbers” or “natural numbers”.  The numbers 2, 3, 4, 5, 6, 7, 8, 9 came into being, which followed 1.

The Babylonians and Egyptians needed numbers to measure fields and the shapes of the fields necessitated some rudimentary elements of geometry for buying and selling properties.  However, it was the ancient Greeks that studied number as something that had its own existence independent of human needs.  Indeed, to Pythagoras, all things were numbers.  Men and women were numbers and he assumed there were only ten heavenly bodies, since 10 had special significance.  In fact, Pythagoreans worshipped the “tetrakys”, a triangle composed of four rows of dots.  The first row had one dot, the second two dots, the third three dots and the fourth four dots.  When the four rows were added, the sacred 10 was the result.  Everything appeared to represent perfect balance and harmony until…  Yes, even in this well-constructed world, irrationality stuck out it’s ugly snout.  Someone constructed a square with sides one.  When a diagonal was added, the length of the diagonal had to be the square root of 2.  Pythagoras’s own theorem led to the unhappy result.  The irrational could be dealt with later.  But there was still one important number missing:  0.

The Greeks never could find 0, and this fact imposed strict limits on what they could do with numbers.  For zero, we have to go to another country, India.  The Indians had long had a concept of nullity.  It came from their philosophy.  It came from their religion.  When “0” joined the counting numbers, a major step was put in place for solving equations, the construction of the Cartestian plane, and, in today’s world, the binary system which is the basis of computer circuitry.

In the Middle Ages, the first algebraic equations were born, arising from Arabic countries.  The mysterious x and y of algebra represented an abstract way of thinking hitherto unknown in mathematics.  The Greeks may have been the philosophers of number, but the Arabs were not only philosophers, but active participants in extending the range of number to greater practical and theoretical heights.  However, algebra and geometry were still separated.  It required a major step to bring them together.

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The Journey Continues…

January 23, 2014 Leave a comment

“Assume 1 exists.  Now prove 2 exists.”–Thomas D. Hedden’s cynical concept of mathematics

Numbers always fascinated me.  I remember going down to the Rogue River bed to construct numbers out of stones.  I liked particularly the shape of 4.  In fact, when I turn the TV off, I always leave it on channel 4.  4 has seemed to me the “perfect” number.  And when I was a child, I wanted to see 4 and feel the stones that formed it.  3 has been a troublesome number; I had difficulty forming 3s in my writing and the oddness of the number disturbed me.  The mirror twins of 6 and 9 were also troubling.  I enjoyed looking at 1, but felt I didn’t understand it.  0 was a special number, but my multiplication cards somehow excluded it.  I did my times tables from 1×1 to 12×12.  There was no 13.  I’ve felt there is something malevolent about 13, and perhaps the makers of the multiplication tables did as well.  To this day, I mark all my checks that end in 13 VOID!, and shred them.  My accountant doesn’t like the way I handle checks, but how can you deal with such an irrational person as I?

The cabins at Diamond Lake fascinated me.  I would follow one cabin marked 12, and try to construct the whole number sequence.  I remember looking through bushes, circling trees and hills, in an effort to complete the sequence.  78 record albums were also a source for numbers.  In my grandparents’ home at Amesbury near Griffith Park, I recall seeing records of South Pacific on the floor.  I saw 5, 9, 10, 4, but the others were missing.  I was quite disappointed.  However, I made up for the loss when Grandma bought me the complete Columbia album of South Pacific many years later.  Motels, of course, often proved an exercise in futility, because some would start with 100 or some other number.  Still, I considered the number of the motel I stayed at quite special and have the key with its number to one of them!

Multiplication by 0 was intriguing;  What is President Obama x 0?  But division by 0 was even more bewildering.  Common sense tells us that if we divide an object by nothing, we are not dividing, so the result should be the object unchanged.  Look!  I will divide this chocolate cake by nothing.  See this knife!  I will hold it up in the air.  Now everyone, dig in!  But in mathematics, if 5/0=5, then through cross multiplication, 5×0 should equal 5, but that contradicts the way we originally defined multiplication by 0.  What can we do?  Simple! We’ll take the easy way out and say that division by 0 is undefined.  I must admit that multiplying a chocolate cake by 0 is something I cannot fathom!

Xs and Ys hurt my eyes.  Ys and Zs no more please!– childhood verse about algebra

I remember algebra only too well.  The subject inspired my first story, The Tale of the Brilliant Xaquenta Qualzifaz Xitg and the Birth of Algebra.  Xaqenta falls in love with Yakshwe Reginald Yorkes and they meet at a special place called the origin.  At that point, they decide to form a family.  There was a communicative law, so that all members could recognize one another better and an associative law to improve family relations.  The eternal optimist, a small man on stilts called absolute value, was never far away.  But the story faded into fractal dust, and so it has remained.  At the time, I decided there were other worlds to explore. And so, armed with my protractor and a straight edge, I was ready for whatever shapes and symbols I might encounter.

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