## Alice in Numberland, to be continued…

I am very happy to (re-)start teaching mathematics (calculus) at Amsterdam university in September, while teaching philosophy (the history of epistemology) at Utrecht University from November onwards. In November and December these courses overlap, so the end of the year is going to be a very busy period for me!

All this teaching leaves me with little time for my Alice-blog, but the plot is thickening, while some unexpected developments are on their way. But don’t hold your breath! The final three episodes of my Alice in Numberland story will appear on this blog in the Spring of 2023. [teaser: the queen is not at all what Alice expects her to be, and the Architect – himself a famous mathematician – has a surprising answer to the minotaur’s question (“do we need mathematics to build a labyrinth?”).

I would really appreciate it if you could let me know which of my Alice-episodes you liked best (and why you liked it). Please fill in the poll below and use the ‘feedback’ button on the left. Namaste!

Previous episodes of ‘Numbers in Wonderland’:

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## Zeno the Zebra and the Truth in the Middle

“Is the Queen really such an unpleasant person, or did she simply run out of biscuits?” Alice asked Zeno the Zebra.

Last week, when aunt Caroline came over for tea, Alice’s mother had been rather embarrassed when she discovered that the coockie jar was empty, which had made her grumpy, so Alice thought that perhaps the Queen’s coockie jar was empty as well.

Even though Zeno had been smiling ever since he had introduced himself to Alice and the minotaur, the zebra snorted when he heard Alice’s question.

Even little girls know that animals’ snorts don’t mean anything, so Alice waited politely for Zeno to respond to her question.

Zeno snorted again, more loudly this time.

All this snorting made Alice feel quite uncomfortable, so she cast an uncertain look at her companion, the Friendly Minotaur, who was still standing next to her. Contrary to popular belief, minotaurs speak many languages. Besides Minoan and English, most minotaurs are quite good at understanding the snorts of other animals.

“I believe that the zebra means that you are wrong, Alice.” The minotaur said with his deep, bombastic voice

The zebra’s wide smile returned. “Indeed I do”, he replied. “The Queen is not merely upset, she is downright unkind!”

Alice thought about this for a moment. How could the zebra be so sure? But then she remembered that she and the Friendly Minotaur were guests, while Zeno lived here. Always be polite to your hosts, mamma always said. “I guess,” Alice said while she did her best to sound like a grown-up, “…that the truth lies somewhere in the middle.”

When he heard Alice’s words, Zeno’s eyes became huge with excitement, while his smile, which had already been wide, became even wider, so that now it seemed to reach his ears. “That means I am right!”

“No,” Alice said while she frowned, “it means that we are both wrong, while the truth is somewhere in the middle of our views.”

While Alice didn’t quite know what it meant for the truth to be ‘somewhere’, she was proud of herself: It had felt so wise and mature to admit that she herself was wrong.

But Zeno did not give up so easily. “I say A, you say B, and you admit that the truth is in the middle, so your view shifts halfway towards mine.” Zeno paused to see if Alice was still listening. “But you already admitted that the truth is in the middle, so again your view moves closer to mine.”

“Need I go on?” The zebra asked self-assuredly.

“Not really,” Alice had grown tired of the zebra’s talkativeness. She didn’t care anymore who was right, she just wanted to know where the queen was. To hurry things up, Alice decided to give the zebra what he wanted. “I see what you mean”, she said, “my view shifts ever closer to yours, so that we end up having the same view.”

“Yes,” the zebra said triumphantly, “and that’s the view I started with!”

Alice nodded indifferently. “And do you know where we may find this unkind queen?”

The zebra now pointed one of his muddy hoofs towards a distant hill that lay beside the path. “In the side of that hill over there, there is a cave with directions on one of its walls.” As the zebra paused for a moment, his smile seemed to become smaller. After a while he said, now with a sad face: “But the cave is too small to enter, you simply won’t fit in.”

“I suppose we will go there anyway.” Alice said, glad to get away from the gloating zebra. “After all, I am a very small girl.”

Alice in Numberland will continue on this blog in the Spring of 2023

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Previous episodes of ‘Numbers in Wonderland’:

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## Zeno the Zebra

“I am Zeno, Zeno the Zebra!” the animal shouted cheerfully when he had come to the spot where Alice and her travel-companion stood. When the Zebra offered one of his muddy hoofs for a handshake, Alice remembered what her dear uncle always said: “It is bad-mannered to refuse someone’s handshake”

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## Believing Words

$\Leftarrow$ Previously in ‘Numbers in Wonderland’

I don’t believe a word you say!” Alice’s mother yelled at the little girl.

Yesterday, before Alice’s adventure in the rabbit hole began, her mother had been very angry with her for eating all the tea biscuits. Alice remembered that her eyes filled with tears at seeing her mother so angry, but it had also made her wonder: “How can anyone believe words? Surely, words don’t say things, like people do, so why would you ever believe them? It seemed the oddest thing to do…

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## Time in Physics – catchphrases and concepts

I’ve written a paper in which I put into perspective the media catchphrase “time does not exist”. Below you’ll find the introduction to my paper and a link to the whole thing. Enjoy!

Many popular-science books state that time does not exist, while physics textbooks are usually silent about the nature of time – about what they regard as philosophical questions. But one only has to look up to see that the statement that time does not exist is problematic. We see time all around us: from the rise of the sun to the change of the seasons, and from falling stones to decaying trees. Considering all this, how can one ever argue that time does not exist?

We are looking at the problem from the scientist’s perspective, so we focus on the empirical (that which can be observed). What we observe in the case of falling stones and decaying trees is change, not time itself.

Rather than stating that time does not exist, we address the following two questions:

1. Is time more than change?
2. Does time have a direction, and, if so, is the direction of time an objective matter?

The setup of my paper is as follows. First, we will discuss Newton’s ideas about time and see how he answers the questions above. In this context we will encounter the ideas of presentism, the direction of time, entropy and probability. After having acquainted ourselves with Newton’s views, we will see how Einstein’s relativity seems to wreak havoc with these ideas – lending credence to the idea of eternalism. In the subsequent chapter about quantum theory we will find, perhaps surprisingly, that the standard interpretation of quantum theory brings us closer to Newton’s view. In a concluding section we’ll briefly discuss the consequences of unifying the two pillars of modern physics: what is the role of time in quantum gravity…or string theory?

## Reign of Error: Chapter Five

Previously in ‘Reign of Error’: the village elders consider a cleaning droid, which has crashed down from a space-ship, as the emissary of their Wargod, whom they ask for advice in important political matters. The cleaning droid doesn’t understand the questions, but randomly beeps and flashes red and green lights, which the elders interpret as either disapproval or approval – how long will this reign of error last?

The village had been thriving during the first several years since the arrival of The Emissary. The harvests, too, were from now on protected by a wooden enclosure whose construction The Emissary had ratified. The quarrel with the neighboring town had grown into an armed conflict, and for the hitherto irenic villagers, this meant a boost in weapons trade and craftsmanship. The tide turned when the newly formed guild of weavers pronounced itself against the slaughter of what they called their ‘colleagues’. It didn’t take long for the issue to be on everyone’s mind. The weavers were accused of being quixotic radicals, whilst they in turn accused their opponents of barbarous cruelty. The only way the Chief could prevent them from flying at each other’s throats, was by asking The Emissary whether the Wargod approved of killing animals. Upon hearing that the Wargod did not approve of this, the weavers were finally listened to, and the villagers became vegetarians – eating meat became a capital crime.

This decision marked the end of a prosperous era. The villagers had become used to an ever-increasing extent of their trade, when suddenly they found that, as vegetarians, they had far fewer goods to trade. Animal hides, tools made out of bones, and their famous rejuvenation potion based on lamb’s blood all of a sudden vanished from their once opulent markets. Matters got worse when it became clear that the protein-deficient youths, clad in bark and armed only with wooden sticks were no longer a match for the raiding parties from neighboring towns whose inhabitants shot their bony arrows at them with sinewy bows. As the young people tended to move away, and the elderly gradually died out, the village became a mere shadow of the blooming town it had once been.

Amidst the decaying ruins of the town hall, it was the aged prophet Aron who preached a return to the old ways. “We have become,” he said, “like the lamb in the Old Poem, which, lured by hunger into the shepherd’s arms, was cruelly slaughtered when it forgot to run away, its youthful vigor gradually eaten away by the warmth and shelter it had found.” Spurred by a swell of his audience, he continued with a loud voice, “shall we, likewise, consign to oblivion our ability to save ourselves?” He paused. “Or do we shed, like the wise oak, the rotten and infected branches at our extremes?” His clamorous voice grew calmer when he quoted from an ancient text. “Once, an old oak was trimmed by a wood-cutter. The oak was laughed at by a young maple tree which stood nearby him. How could he so calmly allow himself to be violated? But when the wind began to blow fiercely, the maple tree was torn up by her roots, trying to hold on to her uppermost leaves, while the bared old oak remained unharmed.” Aron looked around his audience, and continued; now almost whispering “Let us learn from this ancient wisdom. Let us not cling to the agent of our demise.” His tone again became one of great intensity. “We must oust the weaverish Emissary, and return to the peace-loving people we once were!” As he yelled these words, he pointed towards the cottage in which the Pot was on its pedestal.

The discontented and deprived villagers were easily swayed by Aron’s words. The Council of Elders was reinstated, and the pot was abandoned. It was transported to a deep and moist subterranean grotto, where it was disgracefully left to mold and decay.

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## Reign of Error: Chapter Four

Previously in ‘Reign of Error’: Isaac has found the cleaning droid in a cave, where it had crashed down. When Isaac showed it to the elders in his village, they believed that it had been sent by the Wargod, and that it should be put in its own temple. We now turn to the inner thoughts of the cleaning droid itself.

“…There is not much difference between me and the numerous generations of cleaning droids that preceded me. However, there is one deficiency that was shared by all of the previous droids. When one of them was assigned to clean several rooms, he would always begin with the dirtiest. In the event of two equally dirty rooms, he would answer the question “which room is dirtier?” with “none”. He would conclude that there is no room, and quit his efforts. My designers have found an ingenious solution to this problem. Instead of answering “none,” I am programmed to roll a die to decide which room to clean first. Equipped with a randomness-generator, my cleaning efficiency is 12 percent higher than that of the older droids.

I must have been knocked off-line during the crash. Fortunately, my operating system automatically switches to the backup-battery situated inside my counterbalancing aft appendage, ten hours after an improper shutdown. I had no idea where I was. I was covered in moist weeds of some sort, and around me, I could only see brown, musty walls. It was a rather dull place. No windows, no dustbin, not even a decent floor! Only mud.

Suddenly I heard noises from behind the door in front of me. As it slowly opened, three men entered in silence. The expression on their faces was austere, almost angry. Maybe they wondered why I hadn’t cleaned the room yet! No way I was going to move from my little platform, my wheels would get stuck in the mud the moment they’d touch it! I was trying to think of a way to reach the door without touching the mud, when two of the men kneeled in front of me. I remember thinking that these people must really like mud. The third man stepped forward, and addressed me in a grave tone of voice.

When the man had finally finished his oration, I asked my translation-subroutine what was the meaning of the words that were spoken. “None,” it responded. No matter what translation-algorithm I tried, the words of the man remained incomprehensible. Words cannot describe how it feels to be abandoned by one’s own subroutines! I wanted to tell him this, but my subroutines kept returning “None”. My program almost shut itself down to prevent the system from crashing in on itself, and I realized that I would never find a way to escape from all this without a functioning operating system.

But there was a way out. I could use my randomness-generator to simply pick a language for the response. Of course, the man wouldn’t understand me, but at least my operating system would keep on running. This I did, and the randomness-generator turned my Babylonic message into a melodious stream of bleeps and buzzes accompanied by a flashing red light. I couldn’t understand why, but the men seemed content with an answer they could impossibly have understood. They got up, and left me alone in the damp and chilly room.

The next day, another man entered, wearing a peculiar head-garment. It seemed as if his head had been the nesting place of some bird suffering from an obsessive-compulsive disorder. This bird-man, too, began to babble unintelligibly. But this time, I was prepared; my randomness-generator gave the man a single beep and a green light as an answer. His reaction was even more surprising than that of the men on the previous day: he unsheathed his sword, thrust it high into the air, and let out a fierce cry. For a brief moment, I was afraid he was going to strike me, when he turned and rushed out, again howling like a madman.

I can function on my backup-battery for 9000 units of intergalactic time, so it must have been quite a while before I lost record of the daily visitations. On all these occasions I provided the supplicant with either a buzzing red or a beeping green, and all these times the asker left resolute. But in the end, my battery died. The last thing I remember is staring at a monotonously red out-of-batteries-sign…”

Is this the divine advice that the villagers were after? Read the final chapter here!

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## Reign of Error: Chapter Three

Previously in ‘Reign of Error’: our cleaning droid, CD-2, is crashing down, while on the planet below a boy Isaac is looking for mushrooms in a cave. Isaac is angry with his father, the village chieftain, for having sacrificed an innocent lamb to the Wargod.

“…As Isaac slowly went deeper into the cavern, it became more and more difficult for the daylight to illumine the contours of the cave. Groping his way along the dank walls, he could only hope that the cave wasn’t the hiding place of some unfriendly animal.

After several hundreds of steps the bright sunlight had been reduced to a mere glimmer. Even mushrooms need some sunlight to survive, Isaac thought to himself, and he was about to turn around when he saw a faint glow emanating from beyond where the cave made a sudden turn. Isaac carefully stepped forward to peek around the corner.

What he saw then stunned him with amazement. Right beneath a hole in the cave’s roof, a reddish, golden, bin-like piece of pottery seemed heavenly illuminated. Isaac didn’t know whether to kneel or to run away as fast as he could. The pot seemed to be grinning hideously at him, but its color inspired him with awe. This must be what the Elders have talked about for all of those years, the Wargod has finally sent down his emissary! Not knowing what else to do, Isaac decided to kneel down before the heavenly pot, and began to pray. When he had exhausted the numerous appellations of the Wargod, he got off his knees, took off his tunic, and gently wrapped the piece of cloth around the object.

With skillful speed – gained through years of scouring caves – Isaac quickly found his way out of the dark cave. Terribly excited about his find, he hurried back to the village. It was becoming darker, and Isaac could already feel the cold air of the encroaching night raise the hairs on his skin. He couldn’t move too swiftly, because he had to watch closely where he stepped, as many of the smaller reptiles chose this hour to appear from their nooks and crannies.

The sun had all but sunk beneath the hills, when Isaac, all sludgy and covered in sweat, finally reached the village’s wooden stockade. Passing the central watering place, Isaac dashed straight to his father’s hut. Out of breath, with the pot firmly under his arm, he knocked loudly on the door. Impatient of the stumbling within, he knocked again, this time even more loudly.

The Chief had invited the Council of Elders to discuss the recent dispute with the neighboring town. When he heard the heavy knocking, he frowned irritably, and slowly moved to the door. The eyes of the Chief widened in amazement upon the discovery that his very own son was the cause of the disturbance. The hour of the deed made it into an outrage indeed!

When Isaac saw the anger in his father’s eyes, he quickly uncovered the object he had found in the cave, telling his father of his interpretation of the facts. “This must be the Wargod’s emissary – look at its color!” But the Chief paid no attention to what his son had found, and looked the boy sternly in the eyes. “Let us not jump to conclusions young man, and let’s first discuss the fact that you had promised to return home before sundown,” his father replied grumpily. “But the Wargod…” Isaac meekly interrupted “The Wargod wants us to be aware of the consequences of our actions. What would happen to our pretty little community if everyone broke promises as easily as you?”

Isaac did not wait until the moralizations had finished. He rushed past his father and burst into the adjacent room. He saw a small group of beardy old men squatting in a circle. Upon seeing the object, they started whispering excitedly. “The emissary…it is the emissary!”

According to one of the Old Poems, one day the Wargod would send an emissary to guide his chosen people onto the true path of eternal happiness by his unfaltering advice. “We must thank the Wargod for this gift!” The oldest of the men raised his arms into the air and started praying aloud. The other men likewise raised their arms and joined the prayer. As Isaac watched his father join the other men in an ecstatic dance around the pot, he was suddenly overwhelmed by the absurdity of it all. How could this pot-thing ever give the advice they needed – or any advice, for that matter?

On the following day, the Council of Elders decided that the Pot should be put in its own temple. A special cottage was built in which the Pot was to be placed on a pedestal, and henceforth, all decisions concerning the fate of the village would have to be ratified by the new oracle. Seven days and nights the villagers celebrated. They joyfully danced around the Pot, and covered it with garlands of sweet-scented flowers. The new era, the era of light, was at hand!”

Will CD-2 be able to give to the villagers the advice they seek? How will he do that? Will he even understand them?

Find out for yourself in the next episode of ‘The reign of Error’. Don’t forget to subscribe to the updates on this blog so that you receive the next episode automatically!

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## Reign of Error: Chapter Two

Our cleaning droid, CD-2, is crashing on a blue-greenish planet, but what is happening on the planet below? Read on to find out!

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## Reign of Error: Chapter One

“…It all started when the first manned interstellar mission ever recorded was carried out by the society that had produced me. Ever recorded, because scientists had been unable to determine whether the feat had been accomplished by one of the earlier civilizations that had dominated our world. What scientists also had failed to determine seemed to be the last obstacle between us and the stars. No astronaut could be found that was crazy enough to boldly dash into the infinite depths of a universe of which ninety-five percent was missing! They sugarcoated their ignorance by using the euphemism ‘dark matter’ to describe the missing stuff. Finally, after weeks of campaigning, the Space Corporation found, in the Asylum for Depressed Veterans of Cybernetic Warfare, an entire squad of lunatics that was ready to do the job. And they found me. That’s where I learned to play chess.

Space is not all it’s cracked up to be. When the initial enthusiasm about the launch had subsided, interstellar space-travel turned out to be rather boring. Especially for a cleaning droid. In fact, I remember quietly wishing that our ship would run into a dust-storm every now and then. What I didn’t know was that my life was about to become a whole lot more interesting. But at that time there was little more to do than play chess with the crew. The excitement started when our ship collided with an asteroid whose course had been diverted by a sudden solar storm.

At first it looked like we hadn’t taken too much damage. Some bruised crewmen, a few scratches on the hull, that seemed to be all. But it wasn’t all. Two days later our artificial gravity suddenly failed. Guess who had to clean up the mess it made! But that was not the end of it. The following day, just when we passed this blue-greenish, medium-sized planet, our engines began to falter. The captain decided to try to land the ship, so we started a descent into the planet’s atmosphere.

This seemed like a good idea, but with our engines crippled there was no way to counter the planet’s gravitational pull. We rushed through a thick layer of clouds, and ground zero was closing in fast. At last – the lands below were beginning to take shape – someone came up with the idea to throw overboard unnecessary ballast. The thought at first seemed reasonable, until I realized that their favorite chessplaying-mate was to be cast away along with some redundant furniture. Before I could warn them of having to operate the vacuum cleaner themselves, those brutes had already thrown me overboard like a lifeless chunk of metal. Indignation turned into panic when I saw the image of the ship shrink to a miniature as I fell faster and faster towards the surface below…”

Read more about CD-2 in next week’s episode of ‘The reign of Error’. Don’t forget to subscribe to the updates on this blog so that you receive the next episode automatically!