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This is a story written in 1979 ,

THE BEGINNIING

A Short Novel by

Paul S. Prueitt

February, 1979

Copyright 1979, Dallas Texas

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"You cay that you know very little about mathematics but you are educated and hold several degrees: one in Art, on in Rhetoric, and one in History. This interests me very much. Shall we walk together for a little while in this evening light and talk about a topic or two from the history of mathematics? I have been thinking, as of late, about the whole numbers and the Pythagorean concept of the harmony of the Spheres."

So Fahlso and his evening visitor walked along that path made of flat rock and leading towards the vegetable and flower gardens. Her name is Niya. They had known each other for only a short time, but during this time they had grown to enjoy each other's presence. Both had sensed a complementing of something that was fundamentally a part of their basic being; something that had been for many years very empty.

Fahlso reflected: "What was this intimacy of life: this thing of Oneness and of Twoness? The Pythagoreans taught us that one and two were, in some respects, not really numbers -- this is, since one generated all of the whole numbers and two differentiated the even numbers from the odd numbers. Why is One the generator? Why not Two? Is the world in essence monistic, or it it fundamentally dualistic, one part not being complete without the other part?"

Fahlso had come to two answers to this question. He concluded that questions about the nature of the world could not be dichotomized without coming face to face with the nature of the mind, but (and contradictorily) the mind might make use of the mental notions of mathematics to discover some aspects about the nature of the world. Two classes of examples now occupied his own mind. These where the class of mathematical models of the Pythagorean and Chaldean schools in the six hundred years surrounding the Manifestation of the Buddha and the class of modern mathematical models of the "same" universe. In the last weeks, he had found that his conversations with Niya provided many interesting perceptions and was very happy to have seen her Volkswagen making a dust trial on the road this day to his farm."

Niya reflected: "What was this intimacy that Fahlso and she had felt, and might feel again? Were Fahlso and she two people who could become as one person.. if but for a short while. The interest was on several levels. Niya wanted to know about these concepts of number and quantity, which seemed to lie at the basis of an external world. She remembered back to their first meeting. It was at the university coffee shop, whose quaint little gathering places, so popular during the late '60s, now once again part of the campus environment. "

At the first meeting, Fahlso was speaking to a group of students about the evolution of mathematical concepts. What had first impressed Niya was the way in which he simplified philosophy and mathematics. "It is most rewarding," he said, "to find a simple answer to a complex question, for continuing in this fashion, one's overall perceptions about the world increasingly approaches unity." Niya wanted to learn more about mathematics and how it tied in with history, so she had sought Fahlso out to question him further on these things.

Returning to the present moment, she said: "How is it that our world seems to have so much order to it? I feel at peace walking here among the long lines of strawberries. Do you hear the breeze rustle the fall corn stalks? The rows of corn, Fahlso, how may rows are there?"

Fahlso shifted his gate, walking more slowly. "Oh, now let me remember. I planted five squares. In each square, I planted ninety rows of corn, each stalk about five inches apart and each row being about ninety feet long. Irrigation trenches are between each row and the distance between the middle of one of these to the middle of an adjacent one is about one foot. The rows of corn, then, are in five squares of about eighty-one hundred square feet each. The number nine times nine itself is, I believe, a perfect type of quantity. It represents to me a type of unity since nine times nice is eighty-one, and the digits of eight and one add up to be nine again."

Niya could now see the other fields of corn beyond the first one. Paths and flower gardens filled in the spaces in between. . Looking back at Fahlso, she felt at a meaning that these fields must signify to her new friend. They were like the colors of a painting or the rhythm of a poem. The planning pleased her. She had always felt that the human role, on this Mother Earth, was to beautify and somehow bring to perfection the nature of the physical world surrounding them.

She picked up Fahlso's thought. "Five groups of eighty one hundred is 40,500 square feet. The numbers of the coefficients of the powers of are here -- four, zero, five and then two zeros. These digits do add to nine, as does any sum of groups of eighty-one. You believe that there is a symmetry here, and that this symmetry is an important one in the structuring of the world.."

"Some individuals." Fahlso replied, "have taught that mathematics tells us about the nature of the world. But here, within mathematics itself, is symmetry of great importance. As you observe, five preserve nine, and any other whole number will preserve nine also."

"Yes, I see that," Niya agreed, " but, how does this relate the whole numbers to the physical world?"

"It might not relate them at all. However, there is seen to be a relationship between our concept of the whole number and our concept of the physical world. It is this second consideration that makes most sense in the number theory of Plato. " Fahlso leaned down and picked two ripe red strawberries, giving one to Niya.

Niya, finding a chance to contribute to the direction of their conversation, said, " This is the later Plato. If I remember my Greek history correctly, Plato's friend Archytas rescued Plato from the Greek tyrant of Syracuse, Dionysuis." "Now", Niya pauses to think on an issue, "Archytas was the ruler of a part of Sicily and was a Pythagorean. This history is a fair interpretation due to the close proximity of Sicily to the centers of Pythagorean learning existing in Italy around the year 388 B.C."

"It was during that year," Fahlso took is turn, having finished the strawberry, "that Plato began a transition from concepts taught by Socrates to concepts used by Empedocles in his training of pre-alchemical scholars. This new concepts where embodied in the work of Plotinus some six hundred years later."

Niya reflected for a moment. "In the Socratic dialogs, Socrates was at first antagonistic against mathematics in general and theoretical mathematics in particular. Later in his life, Socrates is reported by Plato to have said to Theaetetus and Theodorus that mathematics was the mirror of the human mind." She looked at the wonderful shape and color of the strawberry still in her hand. "Since we do not have any of Socrates' own writings, we may suppose that Plato (in his dialogs) gradually softens Socrates's notion towards number theory."

She looked out across the valley floor at the foothills. "But would Plato wish us to understand that the symmetry of nine exists in the mathematics of which we ourselves have born, or to understand this symmetry to exist in the world that lies external to what we can imagine . . . ." She paused, " . . . to the world we can see but not quite fully understand/" Niya had a look of puzzlement. She then noticed the sweet taste of a strawberry and realized that that her strawberry was no longer hold in her hand.

"Do you not feel order both within your being and external to it?" Fahlso ventured to remark. Then he added, "Yes, I know, there is disorder also. The rains of the last week have mildewed some of the strawberry plant's lower leaves."

There was silence for a while. Both were within a separate self, alone. A moment of private reflections and Niya stooped to the ground to repair places in the irrigation ditch, here and there, where the rains has washed away some of Fahlso's carefully planned work.

"Maybe we could work a little while on the garden and talk about the early mathematics and its history and those people whom you have called the Pythagoreans." Light gleamed from Niya's eyes as she saw Fahlso pleased and already moving towards the rock tool shed that stood at the edge of the fields. He took two garden hoes from their storage places.

As they began clearing the irrigation ditches and pushing the loose rich soil into rows, Niya could hear Fahlso begin to hum or chant something in Persian. She too was caught up in the moment. She remembered Baba Ram Das' book, "Be Here Now." Much of what was good about her past was here in the present. She loved talking to this man who seemed in a constant state of enjoyment in his private world of reflections and contemplation. Sometimes, when he was talking, he would be lost in thought. Niya smiled to herself as she became aware that Fahlso had finished his chant and had begun a narrative about his farm.

"In this valley I have planted five squares of corn, so that I might have corn to grind into meal. As you see, one square is in the middle, and there are four others marking the corners of a larger square. Five was significant to the Pythagoreans because this number represented the marriage of the first female number, being two, to the first male number. Three was called the first male number since the number One was not thought of as being a true number." A central theme of their conversations came into view. He paused and looked over some of his gardening work. "One was considered to be very different from all other numbers since by adding one to itself over and over again, we may construct any whole number."

He continued," The Pythagoreans knew about the rational numbers -- that is the set of numbers that may be expressed as a/b, where a and b are whole numbers. However, they felt that it was the whole number that mirrored some fundamental characteristic of the world. The planets themselves were thought of as whirling around the Earth in paths that trace out perfect circles. They knew that decreasing the length of a vibrating cord by a ratio expressible in terms of whole numbers would change the pitch from one harmonic sound to another. This is where the term. 'The Harmony of the Spheres', can from. For, the Pythagoreans deduced, as the planets whirled about the Earth they emitted harmonic sounds. The Pythagorean worldview was simple and elegant." Now a second theme can into view."

Fahlso looked out over the valley, his eyes following the line that a distant path made along the foothills. The path appeared and disappeared only to become one integrated whole within the mind's eyes. He looked back at the rich soil freshly turned by his garden hoe. "This view was destroyed by their discovery of the irrational numbers." Niya could see that his eyes gazed at something quite different from what he was aware of in his mental states. He continued, "I sometimes feel that they might have overcome the irrational if they had understood the modern view of the nature of their number concept."

Fahlso looked at his companion. The wind was blowing gently form the south. It pulled at this, and then tugged at his other one. It mixed everything up into something else. Each moment came into being, then the moment left again. He became aware again of Niya's presence and the questions they were considering.

"Well, after all, oneness is a most difficult notion to understand." Niya collected her thoughts. "I share your belief. I see, in these Greeks of long ago, an understanding that most modern people do not have today. I feel that I might, in fact, come to understand the essence of Greek number theory. But, I do not because I seldom have cause for thinking on that level. Do you sense what I mean?" She questioned Fahlso and set her mind for the up-coming discussion.

"Yes, I do sense what you mean," Fahlso replied. "I also have thought about this evolution of human thought. I wonder if the Greeks saw and felt very much differently than I do." He smiled, looking into those incredibly deep blue eyes that were sparkling in near burst of love and joy.

Niya was aware of his appreciation of her, and felt some need to place the day into perspective. "I have not begin to understood who you are nor how we are together. But our discussions have somehow changed who I am to myself. You have led me to believe that you and I, and all of the other being living now, do think through a different media -- our feelings are slightly of a different nature than those of the ancient ones due to the structure of time's expression being different from the structure of time as experienced by them."

Niya straightened up and leaned on her garden hoe. "Yet some truths are more enduring than others. These concepts of number and quantity are an example. The Greek concepts of number and length still have reasonable truth within them. I am impressed by this reasonableness, and I have the thought that these long lasting reasonable truths are the essence of what are called the Platonic Forms. What do you think, Fahlso? When Plato and his friends where discussing mathematics, for themselves, did they not feel that the concepts of mathematics were independent of time and human thoughts?"

"I don't rightly know." Fahlso shifted the positioning of his own hoe so that another set of his lean muscles would mechanically shape the little canal running along side the strawberry plants. "I suppose that Plato would have been aware of that concept from the East which seems to have penetrated the Greek world from an origin in early Buddhism. How is it that the Tao has been translated into modern thought?"

Niya became aware of Fahlso's shift of thought from Western thought to that of the East. She made a mental note to return to her concept of the Platonic Forms and made a quick reply to this new question. "You mean that all created things must change, for this is the nature of created things."

"Un huh." The strawberry fields were finished, so Fahlso looked out across the valley and thought for a moment. After a pause of some several minutes, he looked at Niya with gentleness. "Help me pull the biggest carrots out of the ground. This way we will have carrots to make juice, and the little carrots left in the ground will grow more quickly."

"O.K." she said. Niya first took Fahlso's hoe and with her own leaned them both against the rock tool shed. It was built of that some native rock that could be found in the foothills surrounding them. She then moved next to Fahlso to gather the carrots.

After some few more moments, Niya questioned him again on the Greek concept of the whole number.

"How is the Greek concept of mathematics different from the modern one? " She wanted to know if the basic concept of numbers had changed any in the last three thousand years.

As he had done so many times in the past, Fahlso thought about the concept of a concept. Heidegger had called it the notion of an object, where here the object was the notion of a concept. Wittgenstein, Whitehead, Russell and Godel finally worked out the theory of infinite regression in its symbolic form. The set composed of all sets -- what is the cardinality of this set? The studies seem to conclude that knowledge is in its very nature either inconsistent or it is incomplete. This is to say that either one will contradict one's own belief system or else one will fail to define some of the fundamental characteristics inherent within the belief system. And if we wish to seek truth, then consistency and completeness are very important.

How much of this could he tell Niya? It would depend, of course, on how long she would be studying at the university, or maybe on how long the two would be friends. As he focused his attention, Fahlso became aware that Life, the impact of Life, confronted every part of his being. At each point, in his mind, an infinite regression slit off and vanished in the composite of his awareness. This infiniteness was somehow fundamental.

"Plato knew of an algebraic identy that gives him integer Pythagorean triples a, b, c, such that a^2 + b^2 = c^2, " Fahlso said absent mindedly, as though he were lecturing to a classroom of his students. "You will notice that if one puts 1 into his formula, it being (2n)^2 + (n^2 - 1) = (n+1)^2, where n is an element of the natural numbers 1, 2, 3, . . ; then the second term on the right hand side is 0. The Pythagoreans delighted in this fact, for it again showed to them the uniqueness of the number one. "

Fahlso reflected on the notion of how things come to be regarded as evidence for something else. His awareness turned to recall the discussion of two of his friends about the modern version of reductionist science.

Niya had stopped thinning the carrot plants. With the flat part of her hand she smoothed out an area to write on. Just as the Greeks had done 2500 years ago, she was using a stick to figure the first three triples. She then drew three triangles, one inside the other. The smaller one measured four lengths of a small stick by three by five. The middle one measures eight by six by ten. The third measured fifteen by eight by seventeen.

"Did the Greeks realize that there are an infinite number of triangles such as these?" Niya asked.

"I think they would have said that there are considerably more Pythagorean triangles than there are grains of sand in all the universe. They knew of Archimedes' calculation of the amount of crystals of sand it would take to fill the universe. It was 10^61, if I remember correctly." Fahlso smiled. "However, the Pythagoreans believed that the whole numbers were sufficient in themselves. This belief led them to avoid the concept of an actual infinite. Hence, the paradoxes of Zeno: 'there can be no motion because in order to get anywhere, one first has to go half way and then half way and the half way, etc. Thus there can be no motion, because no one can ever fully get anywhere.'"

Niya listened quietly while Fahlso used the blade of his hunting knife to uncover a carrot that only a moment before he had pulled at too hard, breaking off the top stem.

"Different cultures had approached an understanding of the infinite, but maybe no one really understood the concept of the infinite until Cantor defined a whole bunch of different infinite numbers. In talking about the cardinality of a set, the number of elements that are in the set, we find that different infinite sets have very different properties. There is the same number of points, for example, in the line segment between zero and one as there are in the line segment between zero and two. However, there are fewer natural whole numbers than there are points in any line segment, no matter how small it is. The puzzles continue since one would at first suppose that there are twice as many whole numbers as there are even numbers; but, in fact, the cardinality of both these sets is the same."

Niya at first thought that she understood Fahlso, but then the question of the infinite overcame her and she realized that she did not have enough experience to think with. As she reflected back about her understanding of history, she remembered the turmoil in scientific circles during the last of the 1800s and the beginning of this century.

"So historically we have two fundamental changes in the over-all concept of the number. The ancient Greeks and Babylonians tried to avoid concepts about the ordinary infinity of the whole number. During the Middle Ages this ordinary infinite was gradually accepted, along with, if I remember correctly, the concept of zero. Then, more recently, the mathematicians discovered that there were more than one type of infinite. These types lead to what is called the transfinite numbers." She looked at Fahlso. "Is this right?"

Fahlso started to say something about the historical development of modern mathematical analysis, but held back. "Sometimes enough has been said. The moment is full, and we should be happy with no more." He stooped down to gather carrots together and place them into a basket. Something akin to his fascination with mathematics was moving in his heart.

Niya quickly gathered her carrots and cradled them with her left arm. The reached out and touched the other's hand.

"The magic in the air seems very real." Fahlso felt a slight pressure form Niya's grasp.

The sun was almost down and a crescent moon was rising on the Eastern horizon. "I love new beginnings that start when the crescent moon." Niya said softly, almost to herself.