You may remember that the famous knight, Don Quixote, had endeared himself to the peoples of this unnamed Spanish village by saving a popular cat from the grips of a tall tree. Never mind that he was the worse for wear from the experience and it had taken some time for him to recover from his sudden discovery of the law of gravity. Sir Isaac Newton may have gained fame and be credited for the discovery, but his was second hand experience not demonstrated by personal involvement.
So it was in the early morning after the men of the village had left to attend their fields and flocks and the womenfolk were about the business of gossip and such, that one of the girls, of not more than ten and two, assembled the younger boys and girls for the Knight's instruction. Sancho though better of this waste of time but as he noted the large baskets of fruits, breads, meats and smoked fish, he agreed to put Amadis to the task of carrying the load to the nearby shady brook. Don Quixote had dressed himself for the occasion and wore his best (and truth be known) his only pair of pants which one of the ladies of the village had seen fit to clean and repair so that not much of the knight was revealed that should not have been. His boots, sad to say, showed signs of having seen their better days and had not cardboard been placed in the soles, Quixote would have felt the sharp edges of the stones that littered the path to the brook. His breast plate was polished, and his hat which some say looked much like a brass bowl more likely used for the art of bleeding than for adorning such a famous head, sat at a rakish angle (not because of intent but due the ill fit between his pointed head and the round bottom of the bowl; I meant to say hat. The mighty sword strapped to his waist caused not a little trouble as with his shuffling gate, the blade sometimes swung wide and if by chance caught on a weed at the path's edge would on its return like as not became intertwined with his legs, causing him to stumble, and had not he held his oak lance in his good right hand, he would have been sprawled on the path. Had he fallen, the anxious children would have run atop him as they darted back and forth along the way. The strange procession moved along, followed by three village cats with tails upraised, perhaps in tribute to the knight, but more likely because they enjoyed an outing with the children as much as did the children themselves. Finally they arrived at their place in the shade and when all was properly set, Sancho and Amadis sought out a protected area out of ear shot of the happy youngsters who knew not what they were to experience.
Our knight of Exemplary Courage took his place at the base of a mighty cork tree and looked upon his cast of actors who hoped to give the impression that they were students. Removing his helmet, adjusting his mighty sword, wiping his long nose on the sleeve of his shirt and clearing his throat; he began:
"I have here several sheets of paper which have long since passed their use as pages in a book of no importance. In addition I have several bits of charcoal from the fire which retains its good property of making a mark. Each of you will take and do as I instruct; and you will see that not all is as it at first appears to be."
"A bit of magic." exclaimed one. "Nonsense." said another. "We have been gulled, for the promise of a picnic, we've got to listen to the old fuddy duddy who will probably go on until the milk clabbers and the flies spot the meat and cheese." "Hush. He probably will forget what he's doing and go off to fight dragons and leave us here for a dip in the brook." And so the rumblings continued.
"I will tell you a story about a person of the far-away isles inhabited by those who speak English (or some variation of it.) Who have a way with war and the winning of battles. And, if things go not the way they intend, then ^� it's off with your head!" "Shall I tell you more?"
His crowd of students, for that is what we shall call them, became as quiet as mice waiting for the next utterance from this famous knight. Would the tale be of blood and guts, drawing and quartering, perhaps fire at the stake and stoning. It would appear that the morning was to be well spent.
"First I shall tell you of Occam's Razor(a)." Ah, the children thought, the story must be one that cannot be told to the adults of the village. How good it is that the great one should share his tale with us alone. And they gathered closer.
"To first understand the way in which Occam, the William, who was also called Doctor Singularis et Invincibilis, made his contribution to science we need the assistance of the paper and the charcoal which I have brought and will share with you. First, let us tear a strip from the piece - the width of the second joint of your first finger and the length of the page will be adequate. Good, I see you are all with me."
"We make two simple marks on the paper strip an X and then another X. It matters not how close or far apart they be, but put them close to the ends if you will. A line(b) is to be drawn between the two marks. And, now I pose the first puzzle."
"The village sage and the idiot will both tell you that the line you have drawn, if it be straight, is the shortest distance between these two marks (or points as the great men of science call them.) Those very same men of science will agree with your village members."
"Try as you might, can you find another way between the two points that is of shorter distance. No. We have what is called an axiom - that is a self evident proposition which is accepted as true without proof. That is; it is obvious."
"Our friend, William the Ockham, another name under which he is known, teaches that it is always best to reduce problems to the simplest terms, hence the word, razor. In other words cut to the quick or heart of the matter. He says to avoid complexity at all expense. Ignore the mumble - jumble of the high priest. Seek out the truth. Alas, that is not what most of the learned ones would have you believe."
"But, my students, the greatest teachings of Ockham are quite different. To solve a problem, he insist that you understand that which may affect your logic and reasoning, and here is where our simple piece of paper comes in. All the wise men will tell you that the shortest distance between two points is a straight line. They are wrong!"
The students looked to their paper, to one another and to the knight. They thought; the villagers are right, Don Quixote is crazy. He has proven himself so by his very word and deed.
"My students now take your charcoal and draw a line from each X to the very edge of the paper. Now, take the paper by the edge and match each end to the other so that your lines join. Behold! You have found a shorter distance between the two Xs. Yes it is true. By your ability, you have now demonstrated that there is in fact a shorter distance between the two points than the men of science thought possible."
"He's right." said one of the lasses who seemed to be brighter than most.
"There's more." said the knight.
"With a bit of spittle and a pinch of flour(c) which I took from the kitchen this morn, join the two ends and hold them until they are dry. You have formed what is called a circle or a cylinder since it has a wall. Your cylinder need not be perfectly round. What you are to observe is yet another distance between the two spots or Xs. While you cannot draw a line across the inside of your cylinder, you can clearly see that the distance between the Xs is now different from that which you observed in just the past two experiments.
"Look, if I push in on the side of the cylinder, the spots get closer or farther apart." cried Jose. "I can make the cylinder round or like an egg or whatever."
Quixote was amazed. He had intended to demonstrate this but the child had stolen his thunder. But standing straight and tall, or at least best as one in his physical condition could, he went on. "What do you suppose is the shortest distance between our two spots?"
The tiniest little girl gave the answer. "If I press the paper together the spots are as close as they can be."
The "class" agreed!
"That's a very good answer and certainly not to be expected when we started out with our sheet of paper. And it is just what Occam would have liked you to discover. Remember, when you study something, keep the questions as simple as possible but always look out for the unexpected. In this case, the distance between the two spots is about two thicknesses of our sheet of paper"
"Now I am going to surprise you with a special treat. For this you must follow my instructions very carefully. We will make another strip of paper from the old book page. Instead of sticking the ends together as we did before, we will do something different. For this we will make two "X"s on the edge of one side of the paper and two "O"s on the other side in about the same place. Carefully, twist the paper so that an "X" and an "O" match up. Good. Now put some spit on the joint and let it set for a moment."
What in the world was he up to? No one had a clue. While some of the papers would not stick, more than a few did and so they were ready for the next stage.
"Support the strip so that you can draw a line on it." And, he demonstrated how it was intended to be. Since some papers had fallen apart, those with an intact loop showed the rest how it was done.
"My line went all the way around. It's on both sides of the paper and it just keeps on going. I don't know why but it's strange." said the oldest boy in the group.
"You really don't have to understand, just to be aware that something interesting is going on. Again we return to William and trust in his advice. Keep it simple but always be on the lookout for something that will alter the outcome. If you look closely, you will see that if you put an "X" on the charcoal line on one side then continue along the line until it is underneath your first "X" and add a second "X", you can see that the shortest distance between the two "X"s is the thickness of a single sheet of paper."
"So what have we learned? Nothing is as it first appears to be. You must be careful to look for the unusual and not be surprised. Then try to explain the unexpected."
"There is another trick to this paper. I have borrowed a pair of scissors from the good dame who is providing for the care of my Squire and me. Look what happens when I cut along the line that you have drawn."
"You now have one very long loop." said the young lass who had assembled this group for Don Quixote this morning.
"You have done very well. Now take your charcoal and draw another line along the surface."
"It stays on one side - one side only." The boy who had been very quiet to this point, turned his loop as he would, but saw no way that the line could be made to blacken the other side of his paper.
Then the Knight errant called them into a close circle so that they could see what would happen when he made another cut with his scissors. He took the boy's loop and carefully cut along the charcoal line. When finished he handed the paper to the small girl.
"There's two loops but I can't separate them." was the puzzled comment. "Why?"
"That I can't explain but if I had a magic carpet I could take you hundreds of years into the future and a scientist by name of Mobius could explain it very well. He studied this art which we can call origami, and the strange structures which we have been making have been given his name, a Mobius Loop(d). Alas, I have no magic carpet so you will have to trust me."
An as if on signal, Amadis began braying for he was accustomed to a mid-day feed by his master who was fast asleep under the shade of a fig. This reminded Quixote that his charges should be fed as well and they went about spreading a most generous lunch before returning to the village.
As they sat for their lunch, the famous knight entertained his students with a simple rope trick. The purpose being to demonstrate, as he said, "there's two ways to skin a cat." Which brings us back to the beginning of our story.
*****
(a)The argument often made as to why Occam's Razor (aka Ockham) is important, is that simplicity is best in all matters. [Stated in different ways: Entities should not be multiplied unnecessarily. We must not assume the existence of any more things than necessary. Plurality should not be posited without necessity.] While the knowing ones use their own buzz words, the "KISS" principle says it best: Keep It Simple, Stupid.
Bayes is thought by some to be the father of statistics because of his probability theorem, but Occam deserves this title for introducing "simplicity" into the scientific method.
(b) For the children who remained after his "class", Don Quixote explained that a line was but a series of tiny dots where the charcoal had rubbed off onto the raised points on the paper's surface. This "fact" is often taught today and some call these points "pixels", perhaps after tiny elves or pixies who reside in very small places.
(c) The great Don did not know that spittle, as he called it, is simply saliva. It contains a protein, amylase which is a digestive enzyme, that "chews up", that is to say digest, starches. By adding the bit of saliva to the flour which is mostly starch, he began a process of reducing the long starch chains to shorter ones to make a better glue. Starch paste continues to be a favored glue for binding of archival materials, such as books, in the twentieth first century.
(d) Occam never intended to have his axiom put to such a test, after all he lived some four hundred years before August Ferdinand Mobius was born. The simplicity of Occam's Razor is that in most instances it is just as true today as it was when he (and others who preceded him) stated it in various forms. It is important in experimental design.
However, before designing "simple" experiments or in making assumptions in logic, it is wise to consider the environment in which you work and how seemingly insignificant "facts" or changes can lead to vastly different results and conclusions from those envisioned at the start.
Case in point is a newly "discovered" surface of glass or plastic which kills bacteria. Scientists at MIT and Tufts University demonstrated that by covalently binding a bacteriostat to the surface, anywhere from 94 to 99% of bacteria that come into contact with the surface are killed. [Proc. Natl. Acad. Sci. USA, 98, 5981 (2001). Not to diminish the importance of their finding but before the shouts of "Eureka" die down; this was discovered long ago, under different circumstances. (Here we go with the environmental considerations.) It seems that researchers, whose names I long ago forgot, were sure that a plastic cutting board would be more sanitary than a wooden one and set out to prove it. Alas, the wood board actually was better! And why was this? Seems wood has within its surface, lignin. Lignin contains polyphenols that are bacteriostatic.
That's science! (And a good reason to revisit the literature from time to time, if you happen to be a researcher at MIT, Tufts or another school as well.)
****
Chaos, Gleick, James, Penquin, 1987
Many a Slip.., James, Gregory, Language Center, Hong Kong University,
2001
Principles of Logic, Aikins, Herbert Austin, Henry Holt Co., 1902
"If we could first know where we are, and whither we are tending, we could
better judge what to do and how to do it." Abraham Lincoln as quoted in
Speaking Frankly (A personal history of the Second World War and
President Roosevelt's role) Byrnes, James F., Harper & Brothers Publishers,
New York, 1947
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