Mathematical Biography

Introduction

For most of my life I have lead a secret second life as a “mad scientist” reveling in the megalomaniacal contemplation of mathematics and science. This document has several purposes; foremost, is an attempt to come out of the closet and to move forward as a whole person; it provides a base for my research into the psychodynamics of mathematical though, uniting my interest in Yoga, psychology, and mathematics, information which may be of interest to others; regardless of the scientific veracity of my work, as a human, I am on a journey that many other people are on, I wish them to know that they are not as alone as they may think. Understand that this document in large part my own internal mythology; memories of memories that I have used to reinforce my sense of self. It is my hope that the process of writing will serve as a form of spiritual archeology, allowing me to categorize the outer layers of my self and too excavate deeper, bringing the light into the dark.
 
Beginning in 3rd grade, I began realizing that intellectually, I was profoundly different from other people. My vocabulary was growing at an incredible rate, but as other children commented on it, I adjusted my speech to sound like the other children. I had suffered a horrible humiliation the previous year when I had to take second grade over a second time; by third grade I remember having the desire to be the greatest scientist that had ever lived. During my freshman year of high school I was repeatedly shaken by the magnitude of my grasp of mathematics; many times I would read about some new piece or area of mathematics for the first time, only to find that I had given substantial though to the subject and already had a degree of familiarity with it; I was reading a vast amount of mathematics, but I was figuring out things even faster than I could read them. Frequently I would begin reading about a new subject, only to be struck with a vision of the relevant mathematical landscape in considerable detail, which continued reading then confirmed. Having deep insights is wonderful; I was doing things mathematically that I though were beyond the capability of humans and it really bothered me, it made me feel like I wasn’t quite human. The positive side was that I was willing to plunge into the deepest, most freighting mathematical abyss I could conceive of.
 

Exotic Arithmetic

I have a deep love of numbers; more than specific numbers, I am interested in the nature of their most universal and abstract forms; the types of numbers that must exist and the types of numbers that may exit. Arithmetic serves as the basic machinery for creating numbers; questions regarding the possible types of numbers quickly lead to question regarding the possible types of arithmetic. Table 1 displays a list of arithmetic operators; the first three rows contain the classical arithmetic operators, the rows beyond list operators display chaotic properties. Only a very small amount of mathematical literature tangentially relates to tetration. The operators listed below exponentiation in Table 1 are only defined between two positive integers; beyond the fact that successive operators generate even larger numbers that there predecessors, their nature is completely unknown.
 

Arithmetic
Type	Operators	Inverse Operators
	Addition	Subtraction
	Multiplication	Division
	Exponentiation	Root	Logarithm
Chaotic	Tetration
	Pentation
	Hexation
	Heptation
	Octation
	Nonation
	Decation
	…

Table 1
 
The development of quantum mechanics provides a good analogy to the development of arithmetic. Quantum mechanics draws a veil over the nature of world at distances smaller than the atom. Piercing the veil required an understanding of new mathematical techniques as well as the construction of a series of increasingly powerful atom smashers.
Likewise, chaos draws a veil over arithmetic when going beyond exponentiation and piecing this veil also requires new mathematical techniques. Computer hardware and software serve the role of mathematical atom smashers, exposing the workings of tetration, pentation and beyond.
 
I use two different types of arithmetical notation in my research, the spiral notation in Figure 1 and tiered notation in Figure 2. Spiral notation is more visceral whereas tiered notation is more general and supports the inverse operators.
 

Figure 1 Spiral notation for tetration, pentation, and hexation

Figure 2 Tiered notation

 
The most general form of arithmetic using tiered notation is a b c
 
 
 
An example of an exotic type of number is the spinor, which.
 

Figure 3 A spinor with its tetrational, pentaional, and hexational generalizations
 

1965 — First Awakening

 
Consider the series of squares; 1, 4, 9, 16, and 25, as shown below. To calculate the next term begin by looking at the differences between neighboring terms of the series and writes the difference down on the line below. The process is repeated until a row of zeros is generated. The differences are then added back in by moving from the lower right to the upper right.  This process is referred to as difference calculus and is a discreet form of differential calculus. Below is an example based on squares, below that cubes; I made these discoveries in 3rd and 4th grade respectively. The squares have two no-zero rows because square are second-degree polynomials with only the first and second derivatives as non-zero; this is why the cubes have three non-zero rows. I discovered this technique in third grade while taking an IQ test. There were many problems where a series of numbers were presented and the answer consisted of determining the next number in the series. It seemed that since the number series problems were of the same form that there must be a unified way to solve all of them.
 

1		4		9		16		25		36
	3		5		7		9		11
		2		2		2		2
			0		0		0

 
 

1		8		27		64		125		216
	7		19		37		61		91
		12		18		24		30
			6		6		6
				0		0

 
My father kept the relics of his tour of the Korean War in a large ornate box that our Catholic bible had come in. The box contained a combination padlock that I opened by the age of eight, by dint of hyperactivity; this gives some indication of the amount of mental energy that I was capable of bringing to bear on a single simple item. Also inside the box was the Book, not the Bible, but an Army manual on applied mathematics. The manual contained tables for the areas and volumes of different shapes, trigonometry tables, and most importantly a table of the squares, square roots, cubes, and cube roots of numbers. I used to gaze at the table of powers as if it held all the mystical secrets of the world. I find it amusing that my research into tetration had its roots in my research into squares in the 3rd grade. During this time I made a second discovery about squares; I noticed that 4*6 was one less than 5*5, just as 4*7 is one less that 6*6; I understood that this was a universal property of numbers. I ask many teachers why this was so, but they were unable to answer me. The answer is that (x-1)(x+1) = x2-1. I was discovering algebra on my own in 3rd grade, I could have easily learned it with a little help; this one incident has made me a advocate of school reform.
 
Three incidences helped with my scientific education. Mrs. Barnholt, a neighbor, gave me her son’s old high school chemistry book and a college biochemistry book. These books gave me an acquaintance with mathematics without having to immediately tackle its abstractions. I spent a good deal of time searching, trying to find pictures of the different electron energy states of the elements, as are in current chemistry books. The incredible response that I obtained from a letter that I wrote to the Atomic Energy Commission was mind-boggling. I don’t remember why I wrote the letter but I received a package of half a dozen booklets on particle physics and a thick white paper summarizing the physics of the previous year. These were well written booklets presenting the latest information in physics; there is no way to describe how much that meant to me. The lesson is that a small act of kindness to a child can have a profound effect on the child’s development and direction in life; I like that. My father actually bought me several mathematics workbooks; it was the only time in my life that he had done anything like this; it may have given me a bit more of an emotional boost towards mathematics.
 
In 4th grade, I correctly concluded that special relativity was an unusual form of rotation from plotting the equations of special relativity on graph paper. I also felt that the mathematical basis of quantum mechanics needed to be fixed when I read about the problems of infinities associated with the practice of renormalization; this is interesting since this had been an important area of my mathematical research and part of the reason I have invested so much time in it. Over a two-year period of time I gained a solid layman’s understanding of science and mathematics but by the end of 4th grade I ended my intense study of science; I realized that I desperately needed to understand calculus to continue on.
 
During 3rd grade, my teacher Mrs. Vogel had an ongoing arithmetic competition in speed and accuracy with an owl broach the winner wore; the owl went back and forth between my friend Julie and myself. It can be seen in my case that nurture was a very large factor in my mathematical development.

1970 — Calculus

The summer after 7th grade, I taught myself algebra, matrix theory, and basic differential and integral calculus; I learned algebra in two weeks, sans trigonometry; I had to memorize the formulas for the solution to quadratic equations and the sum of geometric progression, everything else I could derive on my own. Although I didn’t have a good grasp of the calculus, I spent 8th grade unsuccessfully trying to determine the formula for the volume of a sphere of arbitrary dimensions.
 

1971 — The Great Awaking

I was introduced to the Revolution in 1971, as a freshman in High school. We fought against bourgeois dress codes, and in particular, male student’s right to wear long hair. My nemesis was a gym teacher by the name of Squid. Well… Squid probably wasn’t his legal name, but it was well earned. Squid looked like a teddy bear with a crew cut. He taught swimming, although he couldn’t swim himself. Squid felt compelled to verbally attach me in the classroom for my long hair and wrong thinking. This was somewhat problematic for him as he was quite mentally slow for a gym teacher. I was able to verbally dance him around anywhere I wanted him to go on stage in front of a large number of my peers, repeatedly. It was almost perfect. Almost. Squid also happened to teach a second activity, wrestling. It just so happened that the star quarterback of the football team was in my gym class. I guess Squid finally had a perceptive thought and wondered how I would do against the quarterback who had twenty-five pounds on me. I did damn good for a while, going 2-3 against the quarterback. An ill-considered maneuver on my part ended with a broken right arm. Having a broken arm, my good arm, changed the direction of my life. The prime impact was that it redirected my hyperactivity into the mental realm. I began to spend a great deal of time reading and thinking about mathematics.
 
A piece of mathematics I did at this point in time shows my mathematical orientation. Many people are familiar with Pascal’s triangle (see Table 2) which gives the coefficients of the different powers of (1+x)n . I wondered what the number was halfway between the two 1’s in the first row; I seem to remember that the answer was something odd like the square root of pi. Virtually all my work in mathematics has been oriented towards deriving continuous extensions of discreet systems, as with Pascal’s triangle.
 

				1		1
			1		2		1
		1		3		3		1
	1		4		6		4		1
1		5		10		10		5		1

Table 2 Pascal's triangle
A year after my work with Pascal’s triangle I became deeply involved with Yoga; the fundamental principle being that all is one, that there is no true duality. I had obviously taken this to heart regarding mathematics even before I understood what Yoga was; while intellectually I am not ignorant of mathematically discreet systems, emotionally I feel that they don’t actually exist, that there is always a deeper continuous structure; this is humorous considering that I have made my living in the bastion of discreet mathematics, computer science. The reason I have spent an extravagant amount of time trying to define tetration for complex numbers is I am incapable of believing that tetration may only be defined for integers. A second example of my orientation towards continuous mathematics is that I gave considerable thought to what is now referred to as fractals; objects that exist, not in one or two dimensions, but in a number of dimensions that can’t be expressed in terms of integers. The coasts of islands provide an example of objects whose dimensions are between one and two, more than a line but less than a plane. I never made any progress; in fact when I read Mandelbrot’s definition of a fractal, it took a while to understand his methodology.
 
The primary research in my life began when I questioned whether higher analogs of i, the square root of negative one, and the complex number plane might be necessary in advanced arithmetic. Researching through mathematics books, found that the complex numbers were closed under addition, multiplication, and exponentiation; in other words, no mixture of complex numbers, addition, multiplication, and exponentiation led to anything beyond complex numbers. An example of the lack of closure will help show what closure is. The real number line is closed under addition and multiplication; it’s a happy little universe unto itself. The inclusion of exponentiation opens a crack in this universe; 1 squared is 1, -1 squared is also 1. Only a number of with a magnitude of one can be squared to produce a number with a magnitude of one, yet 1 and –1 are not candidates. This necessitated going beyond the real number line and discovering the complex number plane, which contains numbers with a magnitude of one besides 1 and –1.
 
I recalled reading a small except about the general Ackerman function, a function that mapped the positive integers to arithmetic operators. One property of positive integers is that adding one to them creates a new larger positive integer. This is mirrored by the recursive or repeated use of an arithmetic operator to create a new “bigger” arithmetic operator, an operator that generates larger numbers than its predecessor. Recursive addition is multiplication and recursive multiplication is exponentiation; but this process of recursive recursion can continue, leading to operators such as tetration, pentation, hexation, and beyond. This infinite collection of arithmetic operators is referred to as First Order Arithmetic. The general Ackerman function defines how to calculate any First Order Arithmetic expression involving integers; it can be implemented as a very small program in many computer languages. The general Ackerman function has three inputs, two for the numbers to be operated on and one for the arithmetic operator. Ackerman created a function that could not be represented by First Order Arithmetic by setting all three input variables of the general Ackerman function to the same number; this function is referred to simply as the Ackerman function and can be shown to grow more rapidly than any possible expression using First Order Arithmetic.
 
The Ackerman function leads to the possibility that something beyond complex numbers might be necessitated; but that it would have to be necessitated by one of the arithmetic operators beyond exponentiation. The problem was that the Ackerman function was defined only for integers, where things are boring and safe; I expected closure to break down in more unusual places. The more I tried to find any reference to the realm beyond exponentiation, the more I “felt” that there just wasn’t anything out there in the literature. Ancient maps denote the ends of the world with annotations indicating, “There be monsters!”; that is what the Ackerman function is, a sign proclaiming, “There be monsters!” at the edge of the mathematical world.
 
This is the point at which I began my research into tetration, recursive exponentiation, with the primary goal of being able to define tetration between two complex numbers. I spent vast amounts of time through the 1970’s researching tetration to no avail. The more I worked, the more my totem image for my work became digging through a vast, beautiful, extremely thick wall of diamond. Tetration was like some exotic alien metal out of Star Trek; absolutely, perfectly unaffected by any known force applied to it. There was no rational reason for me to work on tetration; it was a case of unrequited love; I had reason to walk away from my research in tetration a dozen times, and I did. I was completely compelled by a vision of transcendental beauty; it was like seeing the face of God in the distance. In fact, I actually did “see” something; an ocean of large molecules of arithmetic; the atoms were numbers and the operators were the molecular bonds. I suspect that many mathematicians unconsciously hack their visual cortex so that they “see” their mathematics. Many of my mathematical epiphanies come in the form of visual phenomena, I see the operators of modern algebra as small colored disks the size of a very large dot.

1973 — Fame

Reviewing my SAT scores the summer after my sophomore year in high school, my counselor recommended that I seek entrance Bradley University, which I did. Although I majored in physics department, I was “discovered” by Dr. McGaughy, the chairman of the Mathematics department. Dr. McGaughy was someone I respected as both a person and a mathematician; he listened to my ideas about tetration with interest and respect. Entering college early have minimal impact on my mathematical studied but had a profound impact on my mathematical self respect.
 

1978  — AFTAC

I enlisted into the Air Force in 1978 and was offered a position as a Scientific Measurement Technician working for AFTAC. This was an elite position requiring the highest scores on the AFQT and a Top Secret Clearance. AFTAC was tasked with monitoring the 1963 Comprehensive Nuclear Test Ban Treaty. Tech school lasted nine months and covered monitoring nuclear events with satellite systems for atmospheric and outer space events, hydrophone networks for underwater events and seismic systems for underground event. The basic principles of computer hardware and software were also covered. I completed tech school with the highest test scores in the history of the outfit.

In early June I bought my first computer, a TI 58 Programmable Calculator, for about $105, with my first paycheck. The TI 58 supported the addition, multiplication, and exponentiation of complex numbers; I probably ran tens of thousand if not hundreds of thousands of calculation on the TI 58 and manually plotted the results on hundreds of sheets of graph paper. The five-year period after obtaining my first computer was when I did the foundational research in tetration. Countless plots showed that tetration’s behavior was quite erratic; I could see why nothing was written on the subject. Chaos theory was emerging in this time frame; at first I saw similarities between the odd behavior of tetration and chaos, as time progressed I slowly began to realize that tetration was an actual example of chaos. This explained why three hundred years after the development of logarithms, the next arithmetic operator, tetration, had never been developed; significant research into chaotic systems without using computers is nightmarish, if not impossible. Chaotic systems contain an infinite amount of detail; only computers can store and manipulate the great amounts of information needed to even crudely model chaotic systems.

1979- Seismic Analysis

Consider the following problem. How do you detect an underground nuclear explosion by analyzing the seismic data from a world wide system of sites, with each site having it's own unique sensor array. In 1979 I was assigned to AFTAC Headquarters at Patrick AFB to TGS, the geophysical research division’s seismic department. I was trained to analyze seismic data visually by looking at seismographs on microfiche, identify the arrival of the different waves of a seismic event and then to input that information into a software program that computed the epicenter, depth and seismic magnitude of the event. I worked with a team of seismic analysts reviewing every seismic event in Asia looking for underground nuclear events. AFTAC was a world class scientific institution connected to DARPA with technology many years ahead of what was available in the public domain. I used a state of the art IBM 360 mainframe with the latest terminals with alphanumeric and video capabilities, a number of seismic analysis programs including a sophisticated signal processing program.

1980- Systems Engineer

AFTAC had a major initiative for a next generation all digital, integrated worldwide seismic network with software to store and process the seismic signals, be able to identify and classify seismic events and most importantly be able to detect nuclear seismic events. What was the next step in pushing the technological envelope in detecting underground nuclear events. One concern was that a foreign power could set up an underground nuclear test site in a geologically unstable region that typically produced a number of strong earthquakes and then explode the nuclear device during an earthquake in order to hide the nuclear event. A good seismic analyst can visually separate out multiple seismic events occurring at the same time, but what if the signal for a nuclear seismic event was hidden in signal of a much more powerful natural earthquake.

In 1980 I began to work full time as a FORTRAN programmer writing software to plot seismic signals as orthogonal drawing and the inputting of seismic data from paper tape readers into a digital waveform editor. I shared my office with the Geophysics division's IT personel and a captain working on the design for the next generation digital worldwide seismic network. Many meeting took place in our office and I got involved as a systems engineer in the old understanding of the term. I had a multidisciplinary background in Electrical Engineering, mathematics, physics, computer science and I was a seismologist trained to detect nuclear events. I helped communicate ideas from people in one specialty to another. I studied AI, Artificial Intelligence, but spent much more time on what is now considered Machine Learning. I took particular trouble to master the underlying mathematics relevant to Machine Learning at that time; signal processing, probability, statistics, matrix algebra and control theory.

1986- Complex Systems Theory

The first half of the Eighties saw the introduction of chaos theory to the scientific community at large. I was particularly captivated by Stephen Wolfram’s work with Cellular Automta. In 1986 I moved to Champaign, Illinois only to discover that he was at the new Center for Complex Systems Research. I had several of the most interesting and important conversations I have ever had with anyone. Wolfram was interested in my research into extending tetration to the complex numbers. Since tetration displays chaos, defining tetration for the complex numbers would be the first example of an exact continuous solution for a chaotic equation.

I’ve used the term “chaos theory”, but Wolfram pointed out that there is no chaos theory as a single discipline. Dynamics is not only a concept in physics, but a branch of mathematics. Wolfram commented that dynamics was fragmented into the study of many distinct dynamical systems. He questioned how physics could be unified when the underlying mathematics of dynamics wasn’t unified. Discrete dynamical systems like Cellular Automta and the Mandelbrot set do a great job of modeling chaotic and fractal systems through iteration and recursion. Many systems in physics are chaotic to some degree while many systems in biology are fractal. But hundreds of years of classical mathematics has been invested into continuous dynamical systems. This is in part due to physics appearing to be fundamentally continuous. Also, the laws of physics are most commonly represented as partial differential equations which typically have continuous solutions.

There are two accepted mathematical foundations for expressing any dynamical system in physics. They can be expressed as a partial differential equation or as an iterated function. Unfortunately this separation leads to dynamical models that are either good at representing chaotic behavior or are accessible via classical mathematics. If my research in defining tetration for complex numbers was correct then it would be an example of a dynamical system without this separation. This in turn could lead to insight into the continuous or fractional iteration of functions in general..