The Mirror of Atropos

— J. Chr. de Vries

For Samuel V.

At the beginning of the afternoon on a warm late summer day, I encountered George Bruijsols on the terrace of L’Auberge la Source du Périgord, a simple hotel-restaurant one kilometer outside Le Coux. The restaurant offered a choice of two menus, simple yet very tasty. And rightfully so, as they had been serving the same menus for decades without changing a single dish. 


George was a retired Flemish librarian, and we hadn’t spoken in a while. We had agreed to discuss the phenomenon of infinity over lunch. I had picked up George in my 25-year-old Peugeot from the covered market square in Le Bugue. His eyes had noticeably deteriorated in recent years, and he really shouldn’t be driving anymore. However, it’s not always possible to avoid it when you need to run errands for instance. But in this case, the lame could lead the blind — I had twisted my left knee slightly, luckily the Peugeot was an automatic. We both ordered a Kir Royal as an aperitif. 

“The subject you want to address is actually recursive,” George said as he took a few of those typical French savory snacks that the waiter had placed on the table along with the Kir. “The theory about it is infinite,” he clarified. 

“So the question is whether infinity generates its own infinity,” I said. “In other words, does something like a ‘degree of infinity’ exist, something measurable? That would imply the existence of different sizes of infinity. And then, what would that measure be?”


And so began our conversation about a subject whose essence is unimaginable but yet has the allure of a black hole.

“Do you still remember Leo de Hoogt?” George asked after we studied the menu and made our choices. He ordered one menu (Croustillant de Cabécou — Confit de canard — Crème brûlée), and I the other (Terrine de foie gras canard du chef — Magret de canard — Les fromages), both served with Bergerac wine, a carafe for each of us — red for him, white for me.
 “Only from your stories. He had a peculiar theory about the reversibility of words and how it ultimately always leads to death, if I remember correctly. Like: ‘droom [dream] becomes ‘moord’ [murder], ‘leven’ [life] becomes ‘nevel’ [mist], but ‘dood’ [death] remains ‘dood’.”

“Yes, if you take Dutch words, in English, you can still do it with ‘live’ and ‘evil’, but then it quickly ends. In German, you can do it with ‘Leben’ and ‘Nebel’, but in French, you only get as far as ‘trop’ and ‘port’, but I don’t see death in that. Unless you drink too much port, of course…” George loves corny jokes.

“Too much port leads to safe harbor.”

“Wise guy!” he retorted. “But I didn’t bring it up for some silly theory.”

“He also had a peculiar theory about whole numbers, if I remember this correctly. Or rather a question, but it has been a while since you told me about it, so I don’t remember much. Something about addition and multiplication, I think.”

“Oh sure, yes that’s right! I had completely forgotten about it. But wait, it will come back to me.” He took a sip of his Kir, and from the furrow of his eyebrows, I could tell he was delving into his memories.

“It had to do with even and odd numbers,” he continued. “If you add an even number to another even number, it results in an even number. If you add an even number to an odd number, it becomes odd. Odd plus odd becomes even again. The same applies to multiplication. Positive times positive is positive, negative times negative is also positive, but negative times positive is negative. So, there are twice as many even or positive outcomes as there are odd or negative ones. Even and positive was, for him, a metaphor for death, so this number theory supported his word theory. Everything dies.”

“But he wasn’t so off his rocker to believe that there are twice as many even numbers as odd numbers, no?”

”Non, il n’est pas si seul dans sa tête,” George replied. “But he did investigate the question of whether there are more even numbers than odd numbers, and his conclusion was that there is ultimately one more even number than odd numbers, namely the number ‘zero’. That number must be even because it is placed between two odd numbers, right in the center.”

“But there are infinitely many even numbers and infinitely many odd numbers, so does that one which seems to be extra really matter? Infinity is infinity,” I objected. “I mean, you’re not suggesting that there are infinit plus one even numbers, right?”

“Well…,” George took the last sip of his Kir, “the key question then is whether there exists a degree of infinity, whether one form of infinity can be greater than another form. For example, comparing all natural numbers to all whole numbers.”

“I would think that there are twice as many whole numbers as natural numbers, except for that ‘zero’, which is unique.”

“That’s what you would think because we tend to think everything based on our empirical reality. But it’s not the case. Let’s look at that ‘special case’, that ‘zero’. If we take the sequence of whole numbers, ‘zero’ lies in the middle. But you could also view it as a matter of perspective. The distance between two consecutive whole numbers is always ‘1’. Let’s say we move the ‘zero’ to the extreme negative number and call that number ‘negative infinity’, symbolized as ‘-∞’,” — he took out an old receipt from his jacket pocket and started noting a series of symbols. — “Look, we can rewrite that most negative number as ‘negative infinity plus zero,’ so: ‘-∞+0’. The next number would be ‘-∞+1,’ then ‘-∞+2,’ ‘-∞+3,’ and so on, up to the extreme positive number. If we then omit the symbols ‘-∞+’, we are left with the sequence of natural numbers, ‘0, 1, 2, 3…’ and so on. Conclusion: there are the same number of whole numbers as there are natural numbers.”


Fortunately, the waiter arrived with an amuse-bouche, which gave me some extra time to think. “But what if we write that extreme positive number in your system as ‘-∞+∞,’ so ‘negative infinity plus infinity’? Would the sum then not equal to ‘zero’?”
 George chuckled, “Very clever boy!” He stuffed the amuse-bouche, a piece of walnut bread with pâté and an olive, into his mouth in one go. “Infinity presents us with unsuspected paradoxes. We can also perform the exercise differently. Wait, my paper…”
 He began writing some series again, this time a portion of the sequence of whole numbers around the ‘zero’: … -4, -3, -2, -1, 0, 1, 2, 3, 4… “If we assign each number a value from the sequence of natural numbers, for example, assigning the value ‘0’ to the number ‘0’, the value ‘1’ to the number ‘1’, the value ‘2’ to the number ‘-1’, the value ‘3’ to the number ‘2’, the value ‘4’ to the number ‘-2’, and the value ‘5’ to the number ‘3’, and so on. Then we see the following pattern,” he noted some more and showed it to me.
 “Voilà, alternative method:”


	8	6	4	2	0	1	3	5	7
…	-4	-3	-2	-1	0	1	2	3	4	…

“What stands out now is that the positive numbers (excluding ‘zero’) which are the natural numbers, are replaced by all odd values, and the negative numbers by all even values. All integers are thus represented by all natural numbers. Conclusion: there are as many natural numbers as there are integers, there is no difference in the degree of infinity between the two sequences.”

“No words come out of my mouth.”

“That works out well then,” said George, as the waiter arrived with both appetizers.


As we enjoyed our appetizers, I reflected on the complicated matters we had discussed so far. The numbers were swirling in my head, but I didn’t have time to keep worrying about it further, as George continued his argument with great enthusiasm.
 “We need to address a distinction in order to understand this, namely the distinction between the ‘countable’ and the ‘uncountable’ sets.”
 He explained that both the sequence of integers and the one of natural numbers are ‘countable sets’, and their measures of infinity are equal. However, the set of real numbers, the set of numbers after the decimal point, is ‘uncountable’. There are infinitely more real numbers than there are integers because between any two numbers in the latter sequence, there are infinitely many numbers with values after the decimal point. For example, between ‘1’ and ‘2,’ there exist values such as ‘1.000[…]1, 1.000[…]2, 1.000[…]3’, and so on. It takes an eternity to move from ‘1’ to ‘2’.
It reminded me of Zeno.
 “Like Achilles and the Tortoise,” I suggested. That turned out to be a foolish remark. George patiently explained that Zeno’s paradox is invalid since Zeno believed the sum of an infinite number of values to be infinite also, but that’s wrong. The distance the tortoise has to cover is a finite distance. The same goes for the distance from ‘1.000[…]1′ to ‘2’.”

“This is all way too mathematical for me, so let’s shift the topic a bit.” After the waiter cleared our plates and cutlery, I said, “You asked about Leo de Hoogt earlier, but you haven’t told me the reason for that yet.”

“Ah, indeed. That was because of the limited menu options in this otherwise excellent restaurant. It reminded me of an invitation I received from him years ago. He was still living in Mas de Cause, a lieu-dit above the village Daglan. He had a house from which you had a sublime view of the Céou valley, the river that flows through Daglan.”


The waiter came by and noticed our almost empty carafes. He asked if we wanted some more wine. We gladly accepted. George continued his story, “You gaze into the abyss and see mountains emerge in the distance. On one of those mountains, you can spot the domes of an observatory, where you can study the infinity of space. Anyway, he invited me to an imaginary dinner at a special restaurant called L’Auberge Chez Hilbert. I accepted the invitation, and while we sat outside on his terrace, overlooking the sublime abyss, he described the menu. In contrast to the restaurant where we are currently enjoying our lunch, the choice at Chez Hilbert was unlimited. There were an infinite number of different appetizers, main courses, and desserts to choose from. Not only that, but you could also eat them all because they were imaginary, so you would never feel full. That immediately posed us an elementary problem because if we had to consume all the appetizers first, we would never get to the main courses, let alone the desserts. All the dishes disappear into infinity; with each new appetizer, we push the main courses and desserts further into the future. So in the end I left his place with an empty stomach.”

“Sure, didn’t I hear this before?” I scoffed, “but haven’t we already discussed this extensively? This is about those number sequences.”

“You’re right, but you asked me why I mentioned Leo. There are of course an infinite number of stories related to the concept of infinity. By the way, that brings me to another topic, although it’s certainly related: the phenomenon of mirrors.”


After the waiter had served our main courses, George began his argumentation: “You’re surely familiar with the stories of Borges, which sometimes have infinity as their subject. The most famous ones are ‘The Library of Babel’ and, above all, ‘The Aleph’. But I suddenly thought of another rather hermetic story: ‘Tlön, Uqbar, Orbis Tertius’. Right at the beginning of that story, he mentions an imaginary encyclopedia in combination with a mirror. The encyclopedia is called The Anglo-American Cyclopaedia from 1917, a so-called ‘careless reprint of the Encyclopaedia Britannica from 1902’. That is, of course, a typical Borgesian play with reality, but that’s not relevant now. It is about a statement on the phenomenon of ‘mirrors’ from that so-called encyclopedia, which would be found under the entry ‘Uqbar’: A heretic from Uqbar declared that mirrors, like sexual intercourse, are repugnant because they have the property of infinitely expanding the number of people. — I’m paraphrasing this from memory; I don’t have the exact text by Borges at hand, and it’s undoubtedly more beautifully crafted from a literary standpoint. In other words, mirrors behave like Leo’s restaurant, Chez Hilbert.”

“Yes, I remember this well! But it’s been a long time since I read it. I vaguely recall something about objects doubling themselves and then doubling again to the nth degree. There was a term for that, but I can’t remember it.”

“‘Hrön’, plural ‘hrönir’. Yes, of course, that doubling certainly touches upon our subject. But initially, I was referring to those mirrors.”

“Mirrors indeed are doubling things, but don’t stories actually do the same? They conjure up a mirrored world, a fictional world.”

“Haha, Leo’s play on words and his restaurant as ‘The Mirror of Atropos’. Yes, I can agree with that. Then the Library of Babel has a copy of itself. And the Aleph is also mirrored within itself. Language itself becomes a mirror.”

“So, infinity is a product of language,” I concluded. “But then, what does this mean for science? It is also linguistic.”

“At the very least, technology has the property of multiplication,” said George. “Think of lenses, photographs, and films; they double reality by a factor that is potentially infinite. But even printing already had this ability, and nowadays we have computers with word processors that can multiply a text indefinitely. A handwritten text can be worth a fortune because it is unique, while a photocopy of it doesn’t hold a comparable value. Compare a handwritten score by Beethoven with a digital copy on the internet.”

“An error made in a handwritten text is nothing more than a mistake, but an error generated by a computer becomes an elevated principle. So, ultimately it comes down to human action, which is singular. We see this in music as well when we compare live performances to recorded ones. However, perhaps it goes even further: electronic music, music generated by tape or computer, one could call in a sense ‘lifeless,’ residing behind the Mirror of Atropos. It can be judged as beautiful music, just like photos and films can be breathtaking, but there is a vital element missing; quite literally. I realize that this sounds quite conservative, if not even reactionary, but does that make it any less true or, at the very least, worthwhile?”

“Criticism of technology is often labeled as reactionary because it is seen as criticism of progress, and in the end of the future. It is felt as a betrayal of life. But you also mentioned that there is an aesthetic power in art where technology plays a fundamental role, elevating your criticism above banal or religious rejection of any form of modernity. The underlying question, then, is whether every form of technology, which apparently is generic in nature as it leads to a multiplicity of the original form, is condemned. Then the invention of the wheel and agriculture would already be the beginning of our woes. But, and this is the crux of the matter, so is the invention of language. Everything starts with language, our capacity for imagination, art, religion, the monetary system, our visions. Viewed in this light, the ending of Tlön, I mean the postscript of 1947 where the decline of our sciences and languages is described, and their replacement by ‘Tlön,’ is not a prediction but a report of the irreversible project called ‘homo sapiens’. Then we are doomed. This thesis screams for an antithesis…”

His last sentence painfully made me realize that despite his kind words, he perceived me as an entrenched reactionary. Fortunately, this is a story, so it is not I, the writer, who is a burnt-out fossil, but the protagonist of the story. Nowadays, you have to explain this, because hardly anyone can read anymore, corrupted as they are by the comfort of technology, which leads to everything being understood as a literal object, devoid of irony. Thankfully, our dishes were now being removed.

“Voulez-vous les desserts?” the waiter asked. I asked him to wait a bit.
“Bien sûr, dix minutes?” I nodded kindly.

“Do you know the anecdote about Cantor?” George asked as we sipped our wine.

“Not that I recall, no,” I replied.

“When Cantor had passed away, he arrived at the gates of heaven, where St. Peter was on duty. Cantor asked if he could be admitted. St. Peter furrowed his brow and scratched his beard. ‘That depends,’ he finally said, ‘on answering the following question: Is God infinite?’ Cantor pondered for a moment and chose a diplomatic response: ‘Indeed, God is infinite, but not measurable.’ St. Peter nodded in satisfaction, but something tickled Cantor’s brain, and he couldn’t help adding a remark: ‘But you, on the other hand, are not infinite, and yet measurable.’ St. Peter flew into a rage and banished Cantor straight to hell.

Later, Cantor found himself in the antechamber of hell, with Satan guarding two enormous copper doors. ‘You must choose which door to enter,’ commanded Satan, ‘the left or the right one.’ Cantor looked worriedly at both doors and asked: ‘What is the difference between what lies behind each door?’ Satan, not one to be unkind, was willing to explain: ‘Behind the left door, you will find all the whole numbers, and you must write them all down.’ That seemed like a pointless pastime to Cantor, so he asked: ‘And what lies behind that right door?’ Satan grinned at him and said: ‘Behind that door, you will find all real numbers. Same task, you must record all the numbers.’ Satan pointed at both doors and said, ‘Choose.’
 Cantor didn’t hesitate one second, and said: ‘Then I will choose the right door because the number of numbers behind it is infinitely greater than that behind the left door. God is infinite, so at the time I finish, I will be infinitely older than God, and thus God will be dead. And so will you. It’s just a matter of being patient.’

After politely laughing, the waiter arrived with our desserts. George accompanied his with a glass of Monbazillac, I chose a glass of red port. We enjoyed them in silence, our subject thoroughly exhausted, both of us having had enough of the infinite.

After having enjoyed our dessert, we settled the bill, and I asked George if he preferred me to drop him off at the market square in Le Bugue or if he wanted me to drive him home. He nodded, grateful to avoid a two-kilometer walk. It had been a while since I visited his house, and I didn’t have the route memorized. My little Peugeot didn’t have a GPS, so I had to rely on George’s directions. It was only now that I truly understood how much his eyesight had deteriorated; his instructions were of little help. After an hour of wandering through unfamiliar backroads, we finally arrived at his house — or at least a house that looked strikingly similar to it, but seemed to have undergone some changes. I wondered if it was the right house or if my memory was playing tricks on me.


George stepped out of the car and assured me that it was indeed his house, even though in a new form.
 I stepped out as well, and to my surprise, the house now had a name: ‘Chez Hilbert’, at least I didn’t recall it ever had one. “What happened here?” I asked. “It looks exactly like the house I have seen before, but it also seems to be entirely new. Like it is a copy of the former one, a hrön!” Then I noticed a display case built beneath the nameplate on the wall. I walked towards it and noticed a menu card, which bewilderingly changed its dishes continuously. The descriptions of the appetizers changed every three seconds, the main courses every seven seconds, and the desserts every thirteen seconds. When I asked George if those descriptions were on a loop, he shook his head and laughed.

“No, no loop,” he said, “the texts are generated by a computer program, ensuring that no dish will ever be repeated.”

“And yes, you’re right” he continued after I repeated my question about the house, “the house has been rebuilt from the ground up, making use of new materials. It can indeed be regarded as a hrön, an eleventh-degree hrön, of which the lines are purer than that of the original version. Or perhaps it’s an ur, an object born out of hope.”

“Ah,” I said, pointing to the name of the house on the façade, “something becomes clear to me now: Leo de Hoogt isn’t the original figure of which this house is the hrön. It’s the other way around — Leo de Hoogt is a hrön of you. You are the primordial version.” The remaining question was what all of this made me, the writer.

J. Chr. de Vries — Bonnemort, February 23, 2023