The Numeral System According to George Bruijsols
— J. Chr. de Vries
I
‘“In general, the numbers of the numerical system are represented using the digits of the decimal system. But does this mean the system is based on that particular numeral system? I don’t think so. We could also use the binary system, though that might not be so convenient for humans. Computers, however, are devoted to it. As you know, my foremost talent lies in the art of reinvention. Regarding this numerical system, I have formulated the following law: In any n-based system, the number ‘n’ does not exist as a digit; it is always represented by the combination of the second digit, ‘1,’ followed by the first digit, ‘0.’ The number ‘10’ in the decimal system has the value ‘10.’ The number ‘8’ in the octal system does not exist as a single digit but is also represented by ‘10.’ The number ‘10’ in the binary system has the value ‘2.’ This law is determined by the first value (and the first digit) of the n-based series: namely, the number ‘0.’
Incidentally, I deliberately speak of a ‘law’ rather than a ‘property.’ A law is a logical consequence, whereas a property is a construction. The redness of a particular rose is not a law but a property. Gravity, on the other hand, is indeed a law. The absence of the digit ‘n’ in an n-based system is a result, not an added construction. I find it essential to make this distinction because I believe the numerical system is not a construction but a universal, lawful principle — not invented but discovered. Of course, the discovery is a human endeavour, but the system lay hidden in potential within the cosmos.
To understand this, I will investigate the number ‘0’ further. It is the starting point of the entire numerical system — the absolute nothingness. However, this ‘nothingness,’ while it is nothing, is not without meaning. It can carry significance. In the case of the numerical system, it represents a mirror. Around the number ‘0,’ the values of the numbers pivot into positive and negative values. The number ‘0’ is the black hole around which these values orbit.
Let us consider the difference between so-called even and odd numbers. Even numbers are divisible by two into two equal parts without a remainder; odd numbers are not. The number ‘0’ is considered even. There is something to be said about this. All even numbers above (or below) zero, when divided by two, are split into two (equal) smaller values. The number ‘0’ is not. For example, 2 gives 1 + 1, but 0 gives 0 + 0. This creates a tension — a small flaw in the definition of even numbers. However, there is another argument for considering zero as an even number. In the sequence of whole numbers — positive, negative, and zero — odd and even numbers alternate. After an even number always comes an odd number, and vice versa. If the number ‘0’ were not even, this property would be disrupted. The numbers ‘-1’ and ‘1’ would then be two consecutive odd numbers without an even number between them. That would also be a flaw. The first imperfection is preferable to the second.
A second digression: prime numbers. Here too, I perceive a flaw in the definition. The first prime number, according to the definition, is the number 2: A prime number is a natural number greater than 1 that has exactly two natural divisors, namely 1 and itself. But my sense of logic tells me that the number 1 could very well be the first prime number. After all, 1 is divisible by 1 and itself. The stipulation ‘greater than 1’ in the definition is a construction, not a law.
A second flaw in this definition lies in the premise that only natural numbers are considered — only positive integers. But all prime numbers are mirrored around the number zero. Why should ‘-13’ not be a prime number? It is divisible only by ‘1’ (or ‘-1’) and itself. That the number zero cannot be a prime number is clear: it is divisible by all numbers (resulting in ‘0’) except itself. In this latter case, the absurd zero division arises.
However, there is yet another reason to argue against considering the number ‘1’ as a prime number. The number ‘1’ is a special case, possessing a quality that sets it apart from all other numbers: it forms the entire sequence of natural numbers. It is the yardstick of the sequence. By adding itself to itself, the number ‘1’ generates the next number, ‘2,’ and then ‘3,’ and so forth, extending infinitely. Without the number ‘1,’ the sequence would not exist. This makes the numbers ‘0’ and ‘1’ unique, distinct from all others. For this reason, the binary system is the most logical.
The binary system consists solely of zeroes and ones, which transform these digits into a kind of switch — something is either true or false. This property makes the system so fitting for computers. Its absolute consistency is perfect and, therefore, sublime. Yet, even this exceptional system leaves a loose end: the mystery of prime numbers becomes no more transparent, and the alternation between even and odd numbers remains similarly elusive.
No matter how we twist or turn it, the seemingly perfectly logical system of integers always reveals some flaws. This is deeply dissatisfying. The system proclaims the illusion of infallibility, and consequently, that all operations or processes carried out using it are likewise flawless — provided they are applied correctly, of course. You might call it a comforting illusion in an imperfect world.
Something that appears perfect, and indeed is almost so, I find unbearable. Almost. Not quite. That, I find appalling.”
“Is that the real reason you ended your life?”
“Of course not. You know that perfectly well.”
“Yes, dear Taunis, I am aware that the immediate cause was something else, but perhaps this is the so-called underlying reason?”
“You are conflating a few matters, as usual. The numerical system is a discovered phenomenon; in other words, it was given to us. Something that is given cannot be refused; that would be unacceptable and, thus, immoral. In my case, something was taken away, which is the opposite. Such an act can be rejected or even undone. That is everyone’s right. To be deprived of something is a form of dying, if you will allow me this hyperbole. Death may be answered with death.”’
— § —
II
‘“Good heavens!” Anita raised both hands in a disapproving gesture. “That’s not okay! What a twist! Completely over the line!”
“Pardon?” I looked at her, bewildered. “I did tell you it was a dream! I can’t help what I dream, can I? Or should I not have shared it with you? If that’s the case, my apologies!”
It was a year after Taunis Haas’s passing when I travelled to Nuremberg to visit his grave. 1) That’s where I met Anita Strödil. After a brief ceremony — laying down a bouquet of flowers and saying a few words — we sought out a café to get better acquainted. She had been born in the United States but had moved to her parents’ hometown, Nuremberg, about ten years ago — the city where Haas had lived and worked. They had been colleagues at the local university, which meant she knew a thing or two about his troubles. She’d changed her surname from Stroedil to Strödil.
“I don’t know whether or not you’re responsible for what you dream, but it’s absurd to claim you’re speaking to someone who’s deceased.” Anita shook her head at me and took a sip of her wine. “Dreams are overrated anyway. They’re nothing more than the brain transferring data from its RAM to its hard drive. People make far too much of them — life analyses, predictions of the future — all nonsense!”
This wasn’t a topic I felt like delving into just now, and certainly not with her.
To steer the conversation back on track — and to silence her for a bit — I continued explaining the context of my dream. “That I dreamt of Taunis isn’t so strange, considering the reason for my visit to this city. But there’s another element that may have played a role — or at least I assume so; I’m no expert on dreams.” She looked at me with a thoughtful expression but, thankfully, said nothing further.
“A week ago, I had an interesting conversation with George Bruijsols, an acquaintance of mine in France, where I live. We occasionally meet for lunch, and our discussions usually revolve around topics like infinity, mathematics, and other speculative matters.” Anita stared into her glass. It was empty, so I poured more wine. She nodded with a faint smile.
I refilled my own glass, took a sip, and continued. “The story that came from Taunis’s mouth in my dream largely originated from George. Maybe not in those exact words, but the gist of it came from him. I must have projected George onto Taunis. Taunis also loved numerical mysticism, so all things considered, it wasn’t that strange.”
Anita interrupted me now. “But if the story from your dream actually came from Bruijsols, why didn’t you just say so straight away? You could have simply said you dreamt of Taunis but that the dream was essentially a paraphrase of your conversation with your friend. That bizarre detour into that morbid dialogue with Taunis — you could have left that out entirely.”
She seemed to blame me for including the question about Taunis’s chosen death, as if out of some morbid curiosity. Was she right? I had to admit, I wasn’t sure. I needed a moment to think. “Maybe you have a point,” I said, mostly to buy myself some time.
“Let’s not make a drama out of this,” Anita said. Her smile was genuine. “Go on about those Bruijsols, I’m curious.”
“I do want to think further about that point you mentioned earlier — it’s nagging me, but I can’t quite put my finger on it yet. I’ll finish the story about George first; maybe that will give me some insight as to why I felt compelled to bring it up.” I took another sip of my wine, returned her smile, and continued my story.
“George was working on an extraordinarily intriguing yet peculiar project stemming from his theory — if that’s the right word — about the n-base system. He said he found the concept of ‘n-basis’ in numerical systems problematic. He felt it was redundant, that it undermined the logic of the system rather than clarifying it. He referenced Ockham’s principle — a system should be as simple as possible.” I paused to see if Anita was still following me. She nodded attentively. “George wanted to thoroughly revise the notation of numerical systems, such that every value had its own symbol. For example, in the decimal system, the number ‘10’ would need its own symbol. Every number would thereby also be a digit; all numbers and digits, and thus their values, would merge inseparably. The rule of n-basis would thus become obsolete. ‘To die,’ he said.”
“That’s an impossible task,” Anita exclaimed, “there’s an infinite number of numbers! How did he think he could manage that?”
“Well,” I said with a wry grin, “I’ll explain that in a moment.”
“Okay,” said Anita, “I’ll wait…”
“The starting point was the number zero, for which he chose the symbol of a black disk, meant to represent a ‘black hole.’ The number ‘1’ was a vertical line, much like in the decimal system, but with a short horizontal stroke added at the bottom. For negative values, that stroke was placed at the top. All positive numbers rested on the stroke; all negative numbers hung from it. Positive numbers all pointed upward, negative ones downward. The number ‘0’ naturally had no stroke.”
“Aha, so the number ‘-1’ would look like an uppercase ‘T,’ and the number ‘1’ would be the inverse of that?”
“Exactly, that’s correct. The numbers ‘0,’ ‘1,’ and ‘-1’ were the first he devised. He then thought further about the symbol for ‘2.’ He initially considered the Roman numeral system, so a ‘2’ would consist of two vertical lines on a short horizontal base. And ‘-2’ would be its inverse. By extension, the ‘3’ would contain three vertical lines. But that’s where it ended. The number ‘4’ needed to have a single unique symbol. He settled on a diamond shape, with a vertical stroke at one of the points, either above or below, depending on its position relative to zero. The number ‘5’ was analogously represented as a pentagram, with the point upward or downward, and the number ‘6’ as a hexagon.”
“This does anything but clarify matters!” Anita exclaimed, nearly choking on a sip of wine. “This way, you’re actually undermining the numerical system! If there’s no overarching system to differentiate between them, you remove any possible abstraction. It’s like that story about the man with an absolute memory — ‘Funes,’ by that Argentinian writer Borges.”
“Yes, that’s true, but George quickly realized that himself. After designing symbols for the first seven numbers, he hit a wall. For the number ‘7,’ he could still use a heptagon, but ‘8’ posed a problem.”
“He could’ve just used the figure-eight we already use. Nine seems trickier to me. But anyway, I’m interrupting you…”
“That’s a very astute observation. Indeed, he hit a real impasse with the number ‘9,’ but that eventually led to a solution — a brilliant insight, if you ask me. He suddenly realized that ‘9’ is the square of the number ‘3,’ which gave him his first breakthrough: he needed to incorporate the concept of squares into his system. Then he understood that he could incorporate the entire binary system into his framework. He could implement all powers of ‘2’ in one go. And furthermore, all squares of the number ‘3.’ The number ‘4’ was already embedded in the set of squares of the number ‘2.’ This led him to his second breakthrough: he had to base his system on prime numbers. He would only need to implement the squares of the prime numbers, and therefore only the prime numbers themselves. The squares would naturally follow.”
I ordered another bottle of wine, catching a look of approval from Anita. “Introducing the squares had an additional effect,” I continued after we clinked glasses. “But first, I need to circle back to the issue of even and odd numbers.” She made rolling gestures with her hands, signaling me to keep going. By now, a rosy flush had spread across her cheeks, and I found her increasingly attractive — in vino amor…
“The distinction between even and odd numbers simply indicates that even numbers can be split into two equal parts, which is convenient if you’re with another person and want to divide a freshly stolen batch of apples equally. If it’s an even number, that’s perfect. If it’s odd, not so much — you’ll have one left over, and then you’ll have to fight over it. So, if you and a partner are stealing apples, you know you should aim for an even number. But if there are three of you, this even-odd division doesn’t help at all. You’d need a similar system tailored to three thieves — the even numbers would then be divisible by three, while the odd ones would not. The odd set would now be twice as large:
odd: 1 2 4 5 7 8 10 11
even: 3 6 9 12
“All numbers, if we base them on the prime numbers, can thus be defined in an ‘even’ and ‘odd’ type according to those primes. This makes any number of thieves feasible. Win-win!”
“Phew,” sighed Anita. “That’s indeed brilliant! But at the same time, it’s not without problems, because there are infinitely many prime numbers — not to mention the issue of so-called ‘negative primes.’ His project, despite everything, still seems impossible. Or am I mistaken?”
“You’re absolutely right,” I said with a wide smile, which she immediately returned. The initial awkwardness of our conversation was beginning to dissolve into a pleasant camaraderie. “He recognized that himself but decided not to abandon it.”
“So he just kept going?” Anita looked at me, surprised. “I mean, it’s a wildly ambitious idea, and that madness is part of such a project. Madness doesn’t let itself be stopped easily. But it’s against better judgment.”
The conclusion of reason, Anita liked to keep things sensible. “I now understand why I asked Taunis that question; it ties into this,” I replied. “In my dream, Taunis — or rather, me in the guise of Taunis — added something that didn’t come from George. It was that remark about the distinction between ‘lawfulness’ and ‘property.’ That lawfulness is a given, something we can discover, while a property is a construction. I suspect George sensed this unconsciously. We never discussed it. He couldn’t let go of the idea of discovering a perfectly lawful system. To him, the current numeric system had just a few too many ‘loose ends,’ which made the system essentially a construct. He couldn’t live with that — and he was determined to stay alive. Continuing his work on this new notation system is what keeps him going.”
“Did George come up with this so-called ‘law’ of the positional numeral system himself, or did he get it from somewhere else?” Anita asked suddenly.
“He told me he read about it in a text concerning a mysterious medieval scholar named Sigwind Primus the Chaste. Sigwind had formulated a theory about prime numbers, known only through a treatise by one Matthiam Scuëde, a Danish mathematician from the 19th century. In it, he spoke of the so-called ‘Sieve of Sigwind.’ 2) The term ’n-based system’ is mentioned there.”
“That strengthened George’s sense of a ‘discovery’ and the idea that there’s an inherent lawfulness,” Anita said thoughtfully.
“Unconsciously, I’d say. Either way, Scuëde claimed the theory originated from Sigwind. But that’s unverifiable since Sigwind’s manuscript has been lost — assuming, of course, that the Dane didn’t invent the whole thing. What is certain is that George didn’t invent the law himself. His project, by the way, doesn’t stand alone, but that’s another story.” 3)
“So, if I understand correctly, your comment about Taunis’ death refers to George’s manic project, which was his way of choosing life.” Anita stared into her glass for a moment, reflecting. “Am I correct in concluding that Taunis and George are, in fact, two versions of the same figure — namely, you?” she asked a little later.
“That’s possible, but that would mean dreams are more than a hardware-based processing mechanism. There’s a psychological aspect involved then. Unless you believe psychology is merely an interpretation of a chemical or electronic operating system. In that case, free will wouldn’t exist.”
“Exactly the other way around.” Anita gave me a slightly instructive look. “I do think that nature is fundamentally based on process-oriented systems, and that includes our brain because we are also a product of nature. But humans have no choice but to interpret those systems — whether it’s the numerical system, psychology, economics, dreams, visions, relationships, language, the law — or where it all comes together: art. Interpretation is all we have. That ability to interpret is what gives us our freedom. It allows us to make choices. But our freedom is limited. We are only capable of influencing natural processes to a very limited degree. Of course, we can build dikes, invent airplanes, or mess up the climate — but that’s essentially the same as making a calculation error in the numerical system. Just as that system isn’t altered by a mistake, natural processes don’t adapt to our interventions. Gravity carries on regardless, no matter how fast we fly in our jets. And, yes, breaking something is remarkably easy. We can change our natural living environment, but not its underlying form. The organic development of nature is subject to change; the Earth of today isn’t the same as the Earth of a hundred thousand years ago. But I believe the structure that drives that development remains unchanged on a fundamental level. Perhaps in a billion years, the Earth will rotate slightly faster or slower, altering gravity — but the principle of gravity will endure.”
“Is that good news or bad news?”
“I don’t have an ethical or aesthetic judgment about it; it is what it is.”’
— § —
— JCdV, Bonnemort, December 31, 2024
Postscript
For some time, at least a year, my dreams have been changing. I’m not even sure if the term ‘dream’ fully captures it. You could call them ‘lucid,’ but also a hybrid form between dreaming and wakefulness, a kind of ‘half-sleep.’ This state of half-sleep usually appears in the middle of the night, around three o’clock. I am then under the impression that I’m thinking about something, but when I wake up, I believe I’ve dreamed it all. But that’s not entirely true because it was indeed about rational and logical thoughts, and I believe dreams aren’t rational; in (my) dreams, at least, there is no dialectic, as far as I know. I’m no expert in this field.
The above text describes two of my ‘half-dreams,’ which came to me on the same night. If ‘coming’ is even an adequate description. You could also say I constructed them. Both are connected, indicating a construction.
The night of the half-dream was more than a week after my stay in Nuremberg, and about two weeks after my lunch with Bruijsols. After that small ceremony at Haas’s grave, I did indeed visit a café with Anita Strödil. I met Strödil there for the first time, and this is — at least so far — the only time I’ve met her. She was there with her friend Esther Woszec, who had also known Haas, mostly through her sister Zoē. But that’s another story. 1) Esther I had met once before, but she didn’t want to join us at the bar.
It was a much more businesslike conversation than the dream suggested. She smiled faintly once, maybe twice, and her smile did indeed make her attractive, but there was in no way a flirtation between us. In my dream, this was different; I think I even became a little in love with her. Is it possible to fall in love with a dreamed character? Maybe. I think infatuation is always a projection. In both half-dreams, projection was the primary theme.
Anita was, of course, right: the half-dreams around Taunis, George (and also herself) are projections of mine. Projections are all we have to grasp the laws of nature and the cosmos — even literally: to grasp and understand. Hence, our dreams and visions. And art. Without our projections, for example, of gravity, no Pegasus, no Da Vinci’s ‘Air Screw,’ no Alberto Santos-Dumont, the first (but almost forgotten) inventor of the airplane, no rocket to the moon. Projections keep us alive.
As for prime numbers, I predict that despite our projections, no overarching formula will be found, as the gaps between consecutive primes will continue to grow infinitely, with no predictable regularity to be discovered.
George’s project also has a flaw: the number of even and odd numbers is not equal, as the number ‘0’ creates an extra even number. I didn’t tell him this because if he became aware of it, he might not only give up his project but also his life. This is the real reason for my question to Taunis.
— Bonnemort, January 11, 2025
1) See: The Common Ground
2) See: The Fateful One
3) See: The Mirror of Atropos