How to Count to Two
A personal memoir, by Nathaniel Hellerstein
with appendix, illustrations and bibliography
about my three-decade struggle
with George Spencer-Brown’s Modulator
I fondly recall when I finally figured out the Brownian Modulator. George Spencer-Brown displayed that circuit at the end of his book, “Laws of Form”. (See the Bibliography.) He said that it counts to two, and that it uses “imaginary truth values”. His circuit, as printed, was a tangle of wires and NOR gates. I couldn’t make head or tail of it. I simulated the thing by hand, and verified that it does indeed count to two - that is, it toggles an output from On to Off, or back again, when the input toggles twice. But that didn’t tell me much about how it works, or how imaginary truth values are involved.
I was a teenager when I first read “Laws of Form”. I liked how elegant his form algebra was, and how efficiently it derived boolean logic. Near the end of his book, George Spencer-Brown pointed beyond his form algebra by introducing imaginary truth values. These are self-enclosing forms which endlessly oscillate between the marked and unmarked states. They correspond to the “Liar” paradox: “this sentence is false”. I was satisfied by the book’s form algebra, intrigued by its imaginary truth values, and baffled by its modulator circuit.
Many years passed. I finished high school, then college, then I went to graduate school at U.C. Berkeley, to study Mathematical Logic.
I thought that the Transfinite Cardinals courses that I took there were my real work, and the Algebra and Trigonometry and Statistics and Calculus courses that I helped teach there were make-work. But when I graduated from Cal, I learned that nobody would pay me a cent for what I knew about transfinite cardinals. That was because I knew absolutely nothing about transfinite cardinals, despite years of study.
But I also learned that I could make a living by teaching algebra and trigonometry and statistics and calculus. I knew how to do that because the professors at Cal never taught anybody anything; all of the teaching work was left to the teaching assistants, like me.
So at the University of California at Berkeley, my real work was make-work, and my make-work was real work.
I knew nothing about transfinite cardinals because infinities beyond infinity make no sense. Consider Beth-sub-six: 6ב. It’s the cardinality of the set of all subsets, of the set of all subsets, of the set of all subsets, of the set of all subsets, of the set of all subsets, of the set of all subsets, of the set of all integers. Now that’s curled up tight! Then consider Beth-sub-Beth-sub-six: 6בב. That’s like Beth-sub-six, but with the phrase “of the set of all subsets” repeated Beth-sub-six times!
Ridiculous! When has a mathematician ever used Beth-sub-Beth-sub-six to solve an equation? When has a physicist ever used Beth-sub-Beth-sub-six to predict a trajectory? When has an engineer ever used Beth-sub-Beth-sub-six to build a machine?
Transfinite arithmetic bored me. For any transfinites Aleph and Beth, Aleph plus Beth equals Aleph times Beth, which equals the larger of the two. How dull! Now consider two raised to the power Aleph-nought. It’s bigger than Aleph-nought, but how much bigger? It could be almost anything; set theory can’t decide. You call that an arithmetic?
The transfinites go on and on because of Cantor’s Theorem, which says that for every infinity there’s an even greater infinity. In particular, the continuum cannot be counted; for if there were such a counting, then a certain “anti-diagonal bit” would be as true as it is false: a paradox. This is impossible in boolean logic; therefore no such counting exists...
... in boolean logic. But what if there’s a logic in which a bit can be as true as it is false? How about a logic including George Spencer-Brown’s imaginary truth values?
Louis Kauffman sent me a paper, one that he co-authored with Francisco Varela, about “wave form logic”. (See the Bibliography.) Wave-form logic contains Spencer-Brown’s imaginary truth values, which are as true as they are false. This gave me an escape from Cantor’s mind-numbing transfinite cardinal hierarchy; for in wave-form logic, Cantor’s antidiagonal proof fails. The anti-diagonal bit is merely imaginary-valued. So the continuum can be counted, and the transfinite hierarchy is unnecessary. Hear, O Logicians: the Infinite is one!
I wrote my doctoral dissertation about wave-form logic, which I called diamond logic. (See the Appendix and the Bibliography.) Many years after I graduated, Louis Kauffman was editor at a mathematics textbook company; he approved publication of my first book about diamond logic. (See the Bibliography.) It has made me some money, but not much. I write mostly for other reasons.
By then I had settled into part-time teaching at two colleges. These jobs are secure because both colleges pay me wages so low that they can’t afford to fire me. The community college pays for my medical and dental insurance to keep me healthy enough to be exploited.
The pay, though modest, is fair. They pretend to pay me, so I pretend to work. Teaching algebra and trigonometry doesn’t feel like labor to me. My job is to stand in front of a room full of people and tell them things that are totally obvious to me. Then I ask them questions whose answers I know full well; but the rules of the game say that I don’t have to tell them, they have to tell me. How passive-aggressive!
I show them how obvious it is, and eventually they calm down enough to think as simplistically as a mathematician. Then they agree that it’s obvious, and they prove that it’s obvious by writing down the obvious answers to my obvious questions. For them, the hardest part of the job is the physical labor of writing those answers legibly. For me, it’s the physical labor of reading those answers, marking them, tallying the marks, and recording the tallies.
As soon as they prove that what I told them is obvious, they leave. And for that I get paid! Teaching is the sweetest scam ever.
It helps that I’m in a strong union. Dear reader, I recommend that you, too, join a union.
Years passed. I taught, I wrote. I met Sherri, we fell in love, we moved in together, and we got married.
Louis Kauffman published another book of mine, this time about three-valued paradox logic. (See the Bibliography.) Then I turned my attention back to George Spencer-Brown’s Modulator.
I fondly recall when I finally figured out the Brownian Modulator. I was sitting sideways on a blue couch set beside a wide window. The couch occupied most of a small room, open to the office and the kitchen. The room was bathed in sky-light, so Sherri and I called it the ‘Blue Room’. I lay under a warm blue blanket. I held a clipboard on my lap. I scribbled on it with manic glee. The Blue Room’s floor was littered with crumpled balls of paper.
On that clipboard I pulled apart George Spencer-Brown’s tangle of wires and NOR gates. When I untangled it, an elegant design emerged; two crossed loops inside a circle. To that design I applied “diffraction”, an analytic technique from diamond logic. When diffracted against itself, the modulator acquired an even more elegant design, also circular; the “Brownian Rotor”. (See Illustrations 1, 2, 3.)
This too counts to two, because it’s a rotor. Its circle holds four paradox values, two I’s across from two J’s. When the input toggles once, the I’s and J’s turn a quarter-way around the circle; two toggles turn the I’s and J’s halfway around the circle; and so on. The I’s and J’s chase each other around the circle, bearing phase information. The output equals antipodal values juxtaposed: “I but J”, which equals true; or “J but I”, which equals false. Two flips of the input swaps the I’s and J’s; this flips the output from true to false, or from false to true.
I grinned, I scribbled, I pulled page after page off my clipboard. Some pages I set by my feet, the rest I crumpled and tossed aside. I wrote, I rewrote, and the Brownian Rotor emerged from chaos.
Sherri approached me, smiling. She bore a tray with a mug of hot tea. She set the tray down on the Blue Room’s floor, within my reach.
She said, “Happy?”
I said, “Yes.”
Decades later, I still recall that luminous moment of Yes.
More years passed. Sherri gave birth to Hannah. Hannah learned how to crawl, then walk, then talk. Then Sherri and I bought a house.
Louis Kauffman agreed to publish a second edition of my diamond-logic book, this time including the Brownian Rotor material. (See the Bibliography.) By then I had also diffracted the Kauffman Modulator, which Louis Kauffman had discovered. Diffracting the Kauffman Modulator produces the Kauffman Rotor, which also counts to two by pushing I’s and J’s around a circle. (See Illustrations 3, 4, 5.)
Edition two required lots of re-writing. I put diffraction, modulators, and rotors into a new chapter.
I still recall when I showed the manuscript to Hannah. I told her the new chapter’s title: “How to Count to Two”.
Hannah said, “One, two!”
It took her a second to do what took me three decades to figure out how to do. That’s faster by a factor of almost a billion.
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