Brain calisthenics from the distant past
By John F. Sase, Ph.D.
Gerard J. Senick, senior editor
Julie G. Sase, copyeditor
William A. Gross, researcher
"A good decision is based on knowledge and not on numbers.
Thinking the talking of the soul with itself.
There is no harm in repeating a good thing.
Truth is the beginning of every good to the gods, and of every good to man.
Knowledge without justice ought to be called cunning rather than wisdom."
-Plato, Greek Polymath, circa 428 347, BCE
In our preceding two episodes, we explored some lessons from the distant past. Writing this column, along with working on case materials for clients, generally keeps my (Dr. Sase's) brain in gear. However, too much of good things can lead to getting stuck in mental muck. Therefore, for this episode, I have chosen to share some of the ongoing brain exercises that allow me to transcend the New Normal. In this, what we can call "Fun with a Purpose," I wish to focus on some activities that remain interesting and entertaining while maintaining brain strength and general skills. These are the skills that helped you, the reader, to get into and then to graduate from Law School, to pass the Bar Exam, and finally to establish your career in the profession of Law.
What Does it Take?
According to the American Bar Association (ABA), successful Law students and lawyers take courses that develop the following skills. The top seven skill groups identified include 1) Analytical methods and problem-solving, 2) Critical reading, 3) Professional writing, 4) Oral communication and listening, 5) General research skills, 6) Task organization and management, and 7) Public service and promotion of justice.
In her article "Choosing Your Major for Prelaw," Pre-Law advisor Carol Leach compiled a ranking of the top twenty undergraduate majors that helped many of you to achieve the highest Law School acceptance rate several years ago. The top seven majors include 1) Physics, 2) Philosophy, 3) Biology (specialized), 4) Chemistry, 5) Government, 6) Anthropology, and 7) Economics (huzzah!) (http://lawschoolnumbers.com/application-prep/Choosing-Your-Major-for-Prelaw).
A decade ago, College Consensus reported the seven foremost undergraduate majors and their points above average on the LSAT in their article "Majors with Highest Above Average LSAT Score." These include 1) Physics/Math (+9.0), 2) Economics (+6.4), 3) Philosophy/Theology (+6.4), 4) International Relations (+5.5), 5) Engineering (+5.2), 6) Chemistry (+5.1), and 7) Government (+5.1) (https://www.collegeconsensus.com/features/best-degrees-for-law-school, 2008).
Let us condense these twenty-one observations into what we can call "The Strengths of Analysis and Problem Solving of Philosophical and Theological Issues Using the Powers of Physics, Mathematics, and Economics." Perhaps you may want to share and to discuss these observations with younger members of your own family who may be considering a profession in Law.
As we grow older, our minds get cluttered with the details of our professions and everyday life. Therefore, our minds may continue to thrive through a regular mental exercise of a higher nature and order. My favorite activities include finding, exploring, and understanding ideas from the ancient past from around the world. In Western Thought, what better place to begin than with the Greek philosopher/mathematician Pythagoras (c. 570 c. 495 BCE) and Plato, who provided our opening quote? In our preceding pair of episodes, we considered their work, which reaches back in history and advances through thoughts of the past two-and-a-half millennia. Therefore, let us leap headfirst into Plato and swim to the legendary island of Atlantis.
The Urban Allegories of Plato
In "The Republic," along with Timaeus and Critias, Plato presents us with four urban allegories to help us to ponder the cosmology and nature of the universe and daily living. His first two tales focus on visions of the City of Athens, which he envisions as a hill that divides into simple sections. Plato cuts the mountain in the middle vertically to form two levels. Simultaneously, he splits his mountain site of Athens into two sloping sides.
In this second of Plato's urban allegories, the layout of Magnesia, an actual ancient city, bore little resemblance to that in his story. Plato describes his version as two concentric circles with twelve equally spaced highways that extend radially from a central capital city with each road passing through a market/temple area at the halfway point and continuing to military garrisons at the outer border.
Plato's allegory of the City of Atlantis remains the most challenging of his four tales. He presents Atlantis as a "city of excess" as he provides a copious amount of numeric symbolism in his story. For years, I took the description at face value. However, I gradually began to look deeper, reading the works of authors such as Ernest G. McClain, Francis M. Cornford, and Robert S. Brumbaugh. I did not include Atlantis in my past writings because of the complexity and general lack of relevant economics at the time. However, as I became more enlightened about the math of Pythagoras and his predecessors, I began to find deeper meaning in the Atlantis allegory.
Pythagorean Mathematics
Exploring Pythagorean mathematics turned into a routine of excellent mental exercise that has helped me to clear my brain for other mathematically based work, such as my determinations of economic losses for legal matters. I have found that returning to the lessons of Pythagoras and the mathematical allegories of Plato has helped me to maintain a higher level of vision while strengthening my mental discipline for working with complex numeric databases.
Therefore, I have decided to share these mental exercises with the legal community and others. I have included a series of illustrations in my walk-through of these exercises. I recognize that the newsprints of the visuals may pose difficulties for some readers, so I have posted higher resolution versions online that those interested may download freely from www.saseassociates.com/pdf.html.
So, without further ado, let us walk through the brain exercises that have served me well over a few decades. Let us begin with the relevant basics of Pythagorean math that will lead us to our results. It remains debatable as to whether or not Pythagoras developed these ideas himself or learned them from others. Conservatively, the current discussion among academic philosophers suggests that he absorbed much of his basics as an initiate and student at the Temple School at Giza. Pythagoras learned much of his advanced math from the ancient priesthood of Giza as well as from fellow students who had journeyed to Egypt from India, Babylon, and other centers of ancient knowledge. See Leon Crickmore, "A Possible Mesopotamian Origin for Plato's World Soul," Semantic Scholar, 2016 and Eugene Afonasin and Anna Afonasina, "Pythagoras Traveling East: An Image of a Sage in Late Antiquity." Archai 27, 2019.
In our previous episode, we explored elements of Pythagorean-Plato analysis, which included the less-complicated model of the City of Magnesia that served in our thematic discussion of sufficient affluence within a sustainable economy.
Exercise One
We need to consider the Pythagorean method of creating a progression of ratios that are apparent in the conceptualization of the City of Atlantis. This element builds upon the mathematical products using the exponential powers of 2 and 3. The most straightforward explanation of this construction relies upon drawings of three successive spirals that allow us to set up a simple table of products and ratios.
For simplicity of reference from the smallest to the largest, let us refer to these three spirals as the Curled Snake, the Blooming Plant, and the Human Eye with an eyelid. For assembly, we start with a line of thirteen equidistant points. Leaving the centermost point untouched, we use the remaining dozen points for our construction. When grouped, they appear as three concentric spirals.
We use this composition to generate the specific numeric sequence that provides a progression of exponential powers for the numbers 2 and 3. We begin our construction at the point just to the left of the center to create the inner spiral of the Snake. We draw our Snake from point one across-right to point two, back across-left to three, and, finally, across-right to position four.
We construct the Plant by adding a second point four left of position three of the Snake. Crossing to point five at the right, we circle back to six on the left. Finally, we travel to point seven on the right.
We complete our construction by drawing the Human Eye. From a second number seven left of point six, we crossover right to eight and circle back to nine. We complete the Eye by traveling to point ten on the right.
Hurray! We have successfully finished the first exercise of our Pythagorean-Plato series. Now we possess the primary sequence of the exponential powers of 2 that we need to develop our Pythagorean progression of 13 points.
Exercise Two
Let us move along to our second exercise in which we assemble a T-Chart containing the powers of the numbers 2 and 3. Our first exercise gave us the exponents of the number 2. From left to right in row A that forms the bar of our T-Chart, we have 29, 27, 26, 24, 23, 21, 20, 22, 24, 25, 27, 28, and 210. After we calculate these thirteen values for the number 2, we will multiply each one by 30, which of course, equals the value one as it would for any number taken to the zero power. In the column at the center, the number 2 remains at the exponent zero. However, the corresponding exponents of 3 increase as we descend column B. As a result, we get 30, 31, 32, 33, 34, 35, and 36.
Completing this second exercise, we calculate each of the values of 2i*3j to arrive at the amounts needed for our third exercise.
Exercise Three
Our third exercise involves simple division that most of our readers learned in elementary school. I am guessing that many of us have either children or grandchildren in school (which may be distance learning at home for a while longer). These exercises may offer some intergenerational fun within your family circle.
The quotients of this exercise will provide the basic building blocks for the construction of the Atlantis Allegory developed by Plato. The critical feature produced that differentiates this model from the other three urban allegories of Plato is known as the Pythagorean Comma, which is the gap that exists between the two middlemost values.
Let us form two diagonal sets of boxes that extend from the box containing the number 1 at the top of column B. We have divided each of the values in column B by the respective values along the left side of row A on the left-hand side of the table. For example, the number 3 divided by 2 equals 1.5, and 729 divided by 512 equals 1.424. Similarly, on the left-hand side of the table, we divide each of the values along the right side of row A by the respective values in column B. For example, the number 4 divided by 3 equals 1.333 and 1024 divided by 729 equals 1.405.
Now we have a payoff from our first three exercises. By copying the values from the two diagonals of our T-Chart to a standard 360-degree compass, we get a set of equitably spaced radii. However, the closely paired radii pointing toward the six o'clock position (180 degrees) creates a split pair. The narrow gap of this pair provides the entryway to the Atlantis of Plato. This gap, the Pythagorean Comma, forms the navigable channel from the outer boundary of Atlantis to its central island.
Exercise Four
Let us move onward to our fourth exercise. Whether or not a real City of Atlantis ever existed, the allegorical description by Plato has influenced most of the drawings made of Atlantis throughout modern history. However, Ernest G. McClain, a music professor at the City University of New York to whom we referred earlier, discovered that the Atlantis of Plato contains a remnant of an older form of tuning musical instruments by intervals rather than by our modern standard of equal temperament. Having tuned pianos and guitars while playing classical, rock, jazz, folk, and blues music throughout my lifetime, I understood what McClain described. Traveling full circle on this matter, I decided to confer with Professor McClain by phone before he passed. I came to rely upon his analysis, which used microtonality derived from the inferred mathematics as presented by Plato across what appears to be thirteen octaves (in which the numerical values double at each octave).
Math Is Music / Music Is Math
In Plato's tale of Atlantis, 121 different tones appear across thirteen octaves, with no two octaves being the same. However, some very unusual numerical patterns appear throughout his tale. Given that Plato considered math as music and music as math, this route presents a worthwhile journey.
I started by converting the microtones established by McClain to degrees around a compass delineated into three rings and the central island of Atlantis. Also, I noted the incremental octaves expressed by McClain.
(Continued) ...
