By John F. Sase
Gerard J. Senick, general editor
Julie Gale Sase, copyeditor
“Wise men talk because they have something to say, fools because they have to say something. Be kind, for everyone you meet is fighting a hard battle.”
—Plato, ancient Greek philosopher
“Advertising is to a genuine article what manure is to land—it largely increases the product.”
—P.T. Barnum, 19th Century American entrepreneur
Last month, we began to explore Allegorical Economics by delving into the source of all economic understanding—ourselves as human storytellers. Economists and Attorneys “tell stories” in both the classroom and the courtroom. In their respective cases, attorneys need to condense the salient facts of client backgrounds in such a way that they evoke understanding and empathy from jurors. This month, we address the issue of using numbers in the stories that we tell in the courtroom and the classroom.
Joseph Campbell, “The Power of Myth” (Doubleday, 1988)
Having taught more than 10,000 students of Economics and Business and serving as a Forensic Economist on approximately 700 litigation cases, I (Dr. Sase) have learned to keep math-laden examples as simple as possible without losing the meaning and intent of the underlying economic story. I accomplish this by using “easy” numbers. For example, instead of using a number such as $357.84, I have found that the rounded value of $360 appears more comprehensible for students, especially when the story contains numbers and calculations throughout.
I find that limiting the math (whenever appropriate) to elementary addition, subtraction, multiplication, and division tends to provide both understanding and a universal appeal. However, the simplicity in the calculations, as well as the numbers themselves, is not enough. The interrelation of the values provides the necessary ingredient for following the mathematical progression in a story.
Plato understood this concept of simplicity and clarity well. He developed allegories by using the mathematics that his predecessor Pythagoras brought back to Greece. As a young man, Pythagoras traveled from his home on the Aegean island of Samos to Alexandria and Giza, Egypt. He devoted two decades of his life to being an initiate at the Temple School at Giza. Following the Babylonian invasion of Egypt and the burning of the school, Pythagoras, fellow initiates, and teachers were captured and taken to Babylon. During their years of captivity, they assimilated knowledge from Babylonia as well as from more ancient civilizations.
Finally, Pythagoras was released from Babylonia and returned home to the Isle of Samos at the age of 56. He devoted the remainder of his life to teaching in the Mediterranean region, applying his knowledge of Mathematics to the fields of Art, Music, and the Sciences. One still can observe the influence of his mathematical methods throughout these fields, where we find a basis in the sums, products, and dividends of the powers of “2” and “3.”
Before we begin our tutorial example, we will consider the methods of Pythagoras in greater detail. Within the century after his death in approximately 495 BCE, the mathematical relationships that he introduced found their way into the allegories composed by Plato, especially those concerning the cities of Atlantis, Athens, and Magnesia. These stories continue to resonate with Economists, Mathematicians, Musicians, and others in modern times.
The City of Atlantis remains a mystery in respect to its possible location and layout. According to Plato, Neptune, the Greek god of the seas, founded Atlantis as a utopian city-state in which a trio of concentric rings of land separated by water surrounded a central temple-island. However, the citizens of Atlantis became greedy and failed to appease the gods. For their vanity, divine powers supposedly destroyed Atlantis with fire and earthquakes.
Contrary to popular belief, Plato created elaborate descriptions of the structure of and behavior in Atlantis as an allegorical tale rather than a factual one. He appears to use the Atlantean story in order to present a cosmology based upon the interrelationship of numbers.
The other allegories of Plato about the ancient Greek cities of Athens and Magnesia help to support this current view. The real cities of his time bore little resemblance to those in his stories, which use the locales and basic details of these cities as springboards into mathematically-based morality tales.
Four centuries later, Saint John, the New Testament Theologian, constructed a similar allegory while living on the Aegean Isle of Patmos, which is twenty-one miles from Pythagoras’s birthplace of Samos. This writing of John has emerged as the text known throughout Christianity as the Book of Revelations. His text, which reflects colorful apocalyptic images, contains many of the same numerical constructs and relationships used by Plato as based upon the works of Pythagoras.
The ancient tools of Pythagoras have been brought forward in western civilization through other writings of the past two millennia. In the following tutorial, we explore the application of these tools to develop a tale about the growth of financial assets through interest rates and inflation rates. Here, Economics interfaces with Philosophy, which interfaces with the study of Law.
The Challenge
In my practice as an Economist, I am called upon to explain advanced Economic measurements by using graphs like the Bid-Rent Profile, which I have presented below. For this research, I rely upon Negative-Exponential, Quadratic, and Cubic functions, which sometimes need to be doubly integrated for three-dimensional spaces such as cities. Simply by uttering the preceding sentence, I would expect to lose the attention of half of my audience or readership. However, I have discovered a way to illustrate higher forms of mathematical measurement by using simple arithmetic—a little trick that I learned from the ancient Greek polymaths Pythagoras and Plato.
The Fibonacci Series
Many scientists regard the Italian Mathematician Fibonacci (aka Leonardo of Pisa or son of Bonacci) as the greatest European in his field during the Middle Ages. Born in Pisa, Italy, during the late 12th Century, Fibonacci received a North African education under the Moors, largely due to the professional focus of his father, a customs officer and trader in the Mediterranean Region. As a young adult, Fibonacci travelled extensively around the Mediterranean coast. Many scholars believe that he met with many of the Arabic merchants and learned of their systems of doing Mathematics, which incorporate the teaching of Plato, Pythagoras, and earlier scholars, during his business trips. From these experiences, Fibonacci began to understand the many advantages provided by the more-ancient system of Mathematics evidenced in the structure of the Giza complex in Egypt and the many temples in India and beyond.
The arithmetic sequence that we commonly call the Fibonacci Series begins with the numbers “zero” and “one” (0, 1). In many languages throughout the ages, philosophers have explained that these numbers form the foundation of the universe and that the root of all of creation is binary, as represented in the values of zero and one. We interpret this pair of symbols through the concepts of “no and yes,” “off and on,” “nothing and all,” and other binary relationships. However, this pair of numbers forms the seed of a series from which we can derive a universe of mathematical relationships. Starting with this binary pair makes the entire process of expansion transparent.
The sequence emerges through the sum of the two largest numbers in the sequence. We start by adding together our binary pair of numbers to obtain a third number that is also a “one”: 0 + 1 = 1. Now we have the pair of two “ones” that is necessary for further expansion of the series. We continue to sum the pair of largest numbers, which sit furthest to the right. At this point, we have a set of three numbers (0, 1, 1). By taking the sum of the second and third values, we obtain the fourth number: 1 + 1 = 2 and the expanded set of four values (0, 1, 1, 2). Again, we sum the two largest numbers located furthest to the right to obtain the fifth number 1 + 2 = 3 and the enlarged set (0, 1, 1, 2, 3). From here, the process is ongoing, such that 2 + 3 = 5 and the set (0, 1, 1, 2, 3, 5) is produced. Next, we have 3 + 5 = 8 for the set (0, 1, 1, 2, 3, 5, 8) and then 5 + 8 = 13 in the set (0, 1, 1, 2, 3, 5, 8, 13). Finally, for our immediate purposes, we take the sum of the two values furthest to the right, such that 8 + 13 = 21, producing the set (0, 1, 1, 2, 3, 5, 8, 13, 21). Hey! We are done.
The preceding graph includes the full set through 21 (0, 1, 1, 2, 3, 5, 8, 13, 21). Alternately, the function represented by the Fibonacci Series herein is a Cubic Function in which y = 0 + 1.29x - .45 x2 + .08x3 with an R2 (i.e., goodness of fit) greater than 99%. In certain cases, we may delete the first two values (0, 1) and move the remaining values leftward so that the y-intercept may equal 1. The function represented by this altered series would be a Quadratic Function for which y = 1 - .57 x + .63x2 with R2 = 99%. This Fibonacci Series can be expanded easily, depending on need. Expansion requires adding 13 and 21 from the set (0, 1, 1, 2, 3, 5, 8, 13, 21) in order to get the next number of 34. Then, 21 and 34 can be added to render the next value and so on toward infinity.
Working with the Fibonacci Series
By arranging these nine values on a square, we produce a curved line that we use to illustrate practical events. We can rotate and flip our line to get a total of sixteen basic lines that can be truncated or expanded as needed. These variations allow us to model an airplane taking off and then accelerating steeply upward, a toboggan sliding down a run, and a car driving up a steep hill that gradually levels off near the top. In business, we also can approximate the change in Average Fixed Cost: as the quantity produced increases, the Average Fixed Cost decreases. Likewise, we can model Variable Costs, which rise at an increasing rate as production increases.
The following composite diagram contains the sixteen rotated and flipped variations of the Fibonacci Series. In addition, the diagram includes four straight lines that may be used to represent Supply and Demand, to discuss the differences within the spectrum of elasticity, and to address other topics in the field of Economics. The four straight lines reflect the ratios of the two binary values that form the base of the Fibonacci Series. The two diagonals are generated by the ratio of 1:1 and the horizontal and vertical lines are generated by the ratios of 0:1 and 1:0, respectively. With suitable expansion or contraction, this set of twenty lines within the square can be used to illustrate all of the relationships found in Economic Principles.
In the following example, we bring the Fibonacci Series into the world of contemporary publishing. The example models book sales as they increase at an accelerating rate over a short period of time. The truncated Series used in the graph above provides a simple illustration of how the sales of hot best-sellers take off quickly when “controversial” marketing techniques are applied. The rapid increase of copies sold reflects promotion and sales strategies that effectively use “controversy” as a marketing tool. Both the concept and the measurement tool are simple. Hopefully, the question of WHY appears obvious (cha-ching).
Takeaway
We hope that by using Economics and mathematical tools such as the Series by Fibonacci and the earlier findings of Pythagoras and Plato, attorneys may dig through the muck within their cases more easily. The ancients may help to find and to explain some simple truth and possible answers to questions that arise during disciplined investigation.
We wish our readership of the Legal News a safe and enjoyable holiday season and a healthy and prosperous New Year!
————————
Dr. John F. Sase (www.saseassociates.com) teaches Economics at Wayne State University and has practiced Forensic and Investigative Economics for twenty years. He graduated from the University of Detroit Jesuit High School and a bachelauriate in humanities/social science from Justin Morrill College at Michigan State. He earned a combined MA in Economics and an MBA at the University of Detroit, followed by a Ph.D. in Economics from Wayne State University.
Gerard J. Senick (www.senick-editing.com) works a freelance writer, editor, and musician. After earning his degree in English at the University of Detroit, he worked as a supervisory editor at Gale Research Company (now Cengage) for more than twenty years. Currently, he edits books and articles for publication.
