By Roger E. Bissell

Confession #1: I haven’t seriously considered being a mathematician for about 38 years. I’m a professional musician and have been since 9^th grade, when I worked my first paid music job.

Actually, music didn’t become my livelihood until I finished college. That’s when I moved to Nashville, Tennessee and started working at music for a living.

So, even if you don’t count the nine years I used music to support my college education, I have been a professional musician for 34 years now. Long enough to have had second thoughts about this profession.

What might I have been, had I chosen differently? Most likely, either a psychologist, or a philosopher – or a mathematician.

A psychologist, because I have always tried to understand and deal with people’s hang-ups, my own included.

A philosopher, because I have always wondered why the world is in the mess it is, and how to fix it.

The common denominator of these two deep interests is not only understanding the causes of important problems, but also solving them.

But why a mathematician? Not really for any moral cause, nor even for any obviously practical reason.

Oh, sure, I balance my own checkbook and keep the family budget records. I do my own and others’ taxes. But I have no special love for any of these uses of math.

What about the pleasure of solving problems? Well, I do enjoy challenges. And I do get a kick out of using my ingenuity and insight, almost as combat weapons, to defeat “the enemy,” the problem. But this is not the heart of it.

What about the supposed deep connection between math and music? After all, I was a double major my second year in college. Was I drawn by the so-called Pythagorean bridge between the two fields? No way.

I am well aware of the “music of the spheres” and the mathematical ratios in tones and harmonies. But these have nothing to do with why I love either subject.

I love music because it lets me exercise my ingenuity and senses of humor and beauty in creating patterns. I love math for similar reasons: using ingenuity to build patterns.

The way I like to do music is artistic and inventive, creating expressive patterns that didn’t exist before. But math (the way I like to do it) is more like detective work, finding patterns that already exist.

So, if I had been a mathematician, it would have been for the thrill of the chase, the joy of discovering the elusive pattern, the pleasure of capturing the wily theorem. Like a Sherlock Holmes of numbers.

It’s also very similar to science. In a chapter of Mathematics and Plausible Reasoning (1954), Stanford professor George Polya wrote about “Guessing and Scientific Method.” He spoke of:

an aspect of mathematics which is as important as it is rarely mentioned: mathematics appears here as a close relative to the natural sciences, as a sort of “observational science” in which observation and analogy may lead to discoveries.

That’s what I like best about mathematics: the opportunity to be a scientist, to explore and discover what exists. In general, as a matter of fact, I am powerfully drawn to investigating, to finding the hidden, deep truth – whether on the concrete level (tracing “lost” people in the past or present, understanding what makes a person “tick”) or the abstract level (philosophy, psychology, mathematics, etc.).

It’s not just curiosity, wanting knowledge and understanding. It’s more like being a hunter-gatherer, stalking the elusive, hidden fact or essence. My various strengths, such as logic or ingenuity or abstract thinking ability, all seem channeled into my desire to track down something and say, “Gotcha!”

But when did this love first manifest itself? And why didn’t I follow through with it as a career? And how have I dealt with this unrequited love since then?

As far back as I can remember, I’ve always been good at math. I got A’s all through grade school and high school, and I scored at or near the 99^th percentile in the math sections of all the achievement tests I took.

But I haven’t always had a clear sense that there was something important and special about math. That didn’t happen until the 10^th grade, when I took geometry. For the first time, it seemed, my mind had come alive. (Or at least, part of my mind; my writing skills didn’t really ignite until my freshman and sophomore years of college.)

What was so important about geometry? For me, it was learning how to think. Geometry is logical. Not that algebra isn’t. But in geometry, you learn how to reach conclusions that may require many steps of proof.

Why was geometry so special? Because it’s visual. You’re not just manipulating numbers or letters. Instead, you deal with lines, shapes, angles, and figures. Being a doodler, I felt very much at home in geometry.

Trigonometry, which I took the fall of my senior year, was part algebra, part geometry. The main use of proof was in deriving formulas. The same was true for analytical geometry and calculus, the last half of my senior year.

But something really amazing happened during my senior year. (Amazing, in retrospect. At the time, it just seemed like fun.) And it was not part of the high school math curriculum either. I discovered number theory.

Of course, I didn’t know it was number theory. I didn’t realize there was a whole branch of math based on exploring the relationships between numbers. I just stumbled onto it one day.

I had long been intrigued by the fact that a given number’s double and square are almost always different. The only except is when the number is 2, in which case its square and double are both 4.

I decided to make two columns listing the doubles and squares side by side. The first thing I noticed was that for numbers greater than 2, the square was larger. Moreover, the difference between them increased rapidly, as the numbers became larger.

Was there some relationship between them, I wondered? A pattern seemed to leap out at me. For any number, its square was one more than the sum of the next smaller number’s square and double.

An application immediately suggested itself. Suppose you want to square a number. Simply add the square and the double of the next smaller number, plus one. Speed math! If you know the smaller square, of course.

In my excitement, I wondered if there were some general expression for this discovery. There was. Every freshman algebra student has seen it:  ( x + 1 )² = x² + 2x + 1. That realization took some of the wind out of my sails.

Several years later, when reading Ayn Rand’s Introduction to Objectivist Epistemology, I realized the importance of this discovery. There are two paths to some conclusions. Upward from particulars, or induction. And downward from more general knowledge, or deduction.

In the years since, I have inductively discovered other “speed math” techniques. (See “Speed Math.”) I realize I could have deduced them using algebra. But would I have known what I was looking for?

For instance, I discovered that the square of any number minus one is equal to the product of the next larger and next smaller number. I doubt this technique would have occurred to me merely by knowing that x² – 1 = ( x + 1 )( x – 1 ).

Induction just seems to work better for discovery, for forming hypotheses. Deduction, on the other hand, seems more effective at proving those conclusions. I have seen this principle at work in physics, too. (See “Light Speed and the Mass-Energy Equivalence.”

So – confession #2 – while I love deductive thinking and proof, and I find them very necessary in doing math, my real love affair has been with intuitive induction. The raw, unrefined search for patterns is what really turns me on.

But as I said, I was unaware that number theory was an option, something I might specialize in. I just knew that, more than ever, math looked like a fun thing to do. This definitely reinforced my decision to major in math (along with the paucity of scholarship offers from music schools).

My freshman year at Iowa State University was pretty uneventful. Despite a busy schedule of music activities, I chugged through three quarters of calculus with A’s and B’s, and I aced Beginning Symbolic Logic.

I started my sophomore year with a double major – and great expectations. Then all hell broke loose. I met my Waterloo: Abstract Algebra and Intermediate Symbolic Logic.

Part of my trouble was the distraction from music and outside interests. But the main problem was what I call the “reality factor.” My math studies had finally taken me past all the familiar signposts. I was no longer in the real world!

In Abstract Algebra, we no longer had nice friendly problems like: if train A is traveling west at 50 miles an hour and train B, 200 miles away, is traveling east at 75 miles an hour, how long before they collide? (For details, see Atlas Shrugged!)

And in Intermediate Symbolic Logic, we were no longer concerned with proving a conclusion, given certain assumptions, or with analyzing an argument to see if it contained a fallacy.

Instead, to my bewilderment and frustration, I found myself wandering around in alternate systems of arithmetic and logic. Frameworks where 2 x 3 is not necessarily equal to 3 x 2, and where A is not necessarily A. Or so it seemed.

With nothing solid to hold on to, I floundered about like a non-swimmer in the deep end. I prayed for the end of the quarter. It finally came, with C’s in both courses, and my self-esteem and G.P.A. both took a nosedive.

This personal and scholarly disaster was in stark contrast to my glowing successes in music performance, theory and composition. Even two B’s in Advanced Calculus were not enough to dispel the gloom that had fallen over math.

Bottom line: I dropped the math major like a hot potato and finished my B.S. degree with honors in music. (I completed my math minor by eking out a B in something I vaguely remember as “Discrete, Finite Mathematics.”)

Unaware that the mathematical “potato” had sprouts that would someday grow back to the surface, I turned instead to other areas. Philosophy and psychology became my principal intellectual outlets.

But time after time, along the way, my pattern-lust continued to assert itself. Not just in playing jazz and in writing songs and musical arrangements, but in various other ways.

I summarized eight years of income and work frequency figures of my recording career, comparing it with the ups and downs of the economy and the music business. I used this to illustrate points I made to would-be professional trombonists. (See “What They Didn’t Teach Me in Music School.”)

I analyzed 12 years of taxing, spending, and employment figures for the Metro Nashville government and public schools. I used this to argue that taxes and spending should both be drastically cut.

I perused Nashville election and referendum returns for 1976 through 1982. I used this information to assess the effectiveness of targeted campaigning and to determine which precincts to target in the next election.

I compared 10 years of trends in employment and wages in the recording business and the general economy. I used this to urge that recording scale be drastically lowered. (See “Union Scale is Killing Our Work.”)

In each case, I was trying to help solve problems for myself and others in my profession and my community. But more importantly to my ego and my psyche, I was hot on the trail of patterns hidden in the numbers.

My love of mathematics continued to assert itself in these underground sorts of ways. Finally in 1988, an article on a supposed proof of Fermat’s Last Theorem hooked me again, this time for good.

Back in 1637, Pierre de Fermat claimed to have proved that the only integral solution to the equation xⁿ + yⁿ = zⁿ was n = 2. Since he did not record his proof, mathematicians have struggled in vain ever since to prove his claim.

I first heard of this theory about 43 years ago back in freshman algebra. We were studying its Pythagorean form, the equation x² + y² = z². Our teacher challenged us (facetiously) to find an integral solution where n > 2.

When I returned to the problem in 1988, I took a different approach. Rather than trying to prove that it didn’t work for n > 2, I tried to find out what it is about the equation that lets it work for n = 2.

I systematically laid out a number of cases and looked for patterns. As a result, I made the completely unexpected discovery of a method of generating what are called “Pythagorean triples” – sets of three whole numbers that satisfy the Pythagorean equation (e.g., 3, 4, and 5). I inductively discovered the method first, then I eventually figured out how to prove it deductively.

At about the same time I did this, I discovered George Polya’s books, and his discussions of “methods of guessing at mathematical truths and solutions” seemed to speak directly to me and to what I was doing – indeed, had been doing for many years. He said that a truly creative mathematician is a good guesser first and a good prover afterward.

That is certainly the pattern of my process of discovery – very much the way my mind works. My “psycho-epistemology,” as Ayn Rand would call it.

And so far, it seems that my method is an original, distinct alternative to the standard method for generating triples. In some ways, it works better. (See “A New Procedure for Generating Pythagorean Triples.”)

When I wrote the first draft of my “confessions” in 1992, I was attempting to prepare a paper on my method to submit to mathematical journals, and sending copies of it to select people with a mathematical background for their comments. I was of the firm belief that, if I had not unknowingly “reinvented the wheel,” my discovery would turn out to be a significant contribution to mathematics.

Since then, however, my progress on the Pythagorean paper has stalled, thanks to fruitless struggles with using mathematical text-writing programs. I have decided to go ahead and do up a version using MS Word, using fractional superscripts in lieu of radicals for square and other roots, offering the above-linked abbreviated version as a way of putting it on-record, in case it is not accepted by a journal.

One other discovery or insight I have had in more recent years came about during a seminar held over the Internet by the Institute for Objectivist Studies (now The Objectivist Center) during the 1995-96 academic year. The question was posed: what do zero exponents refer to in reality?

ven the mathematics majors in the seminar seemed stumped. But it occurred to me that exponents don’t refer to numbers of things in the world, but instead to operations that we do mentally.

Beginning with the unit 1, I reasoned that if an exponent of 2 (i.e., some number squared) meant that the number 1 is to be multiplied by that number twice, then a zero exponent (i.e., some number to the zero power) meant that the number 1 is to be multiplied by that number no times. In other words, any number to the zero power is 1.

Since then, I have written up this insight, along with other hopefully helpful comments, and posted them as companion papers on my website. (See “What's the Deal with X-to-the-Zero-Power?? A Brief Note on the Non-Fiction of Zero Exponents” and “Sequel: What’s the Deal with Fractional Exponents?”)

Also, somewhat to my chagrin, I have in recent months heard Leonard Peikoff, in his lectures on induction, glowingly praise one of his students for having suggested to him the correct interpretation of zero exponents. It’s certainly possible that the fellow made his discovery independently, but my website and the zero exponent paper have been up for over 5 years now, and people in Objectivist circles are well aware of it. (Perhaps I’ll hear soon that Peikoff’s student has also stumbled onto a new method for generating Pythagorean triples!) I guess I should be content to know that an insight that apparently didn’t exist before I had it in 1995 is now “making the rounds.”

More recently, I have considered changing careers from music to high school math teaching. I took a general qualifying exam (C-BEST) and began preparing for three more specific qualifying exams. However, the material is so vast and my free time so limited, I have decided not to pursue the certification. Instead, I have decided to continue studying philosophy of mathematics and number theory and to explore at my leisure this fascinating world.

In addition, I have sketched out a paper called “Pathways to Discovery.” In it, I will discuss induction and deduction. Illustrations will include material from some of the essays linked from the present essay.

In particular, I will compare “rational” mathematics, using deduction-demonstrative reasoning-proof with “empirical” mathematics, using induction-plausible reasoning-guessing. Both can be ways of discovering new knowledge, but it seems to me that the latter is much more likely to be productive.

Will I blaze any new trails in mathematics? It remains to be seen. But consider this blurb on the back of Ogilvy and Anderson’s Excursions in Number Theory:

It [the theory of numbers] is also a popular topic among amateur mathematicians (who have made many contributions to the field) because of its accessibility: it does not require knowledge of higher mathematics…No special training is needed – just high school mathematics, a fondness for figures and an inquisitive mind.

You rang? I must confess – and that’s confession #3 – that I am very encouraged!