Early the following morning, before the frost had melted from shingled roofs and canvas tents, a woman with hauntingly green eyes appeared in the little mountain town of Nug. No one saw her walk into town from either the west or the east, and the frosty trails revealed no footprints, but they assumed she was from the strange monastery deep in the mountains. Indeed, she told a tale of a group of travelers, with a donkey, who had arrived at the monastery in a snowstorm the day before. They had been given bread and sent on their way, and had returned the way they came, to the west, where the weather was mild and snow did not fall in mid-summer.
As soon as he heard this news, the elder priest yelled at the innkeeper until he hurried to make the morning mush. Then he screamed at the stable boy to saddle the one horse that had survived their first crossing of the mountains, and spurred it back into the west.
The younger priest took to the trail on foot. With a sad face, he watched his poor horse disappear into the distance.
The rising summer sun came streaming into the monastery guest house.
Rini stretched his slender arms toward the sky and hopped up to see if Tera needed anything. Both his shoulders and Tera’s back soon hosted little peeping birds. He lifted his arms so more could perch. Tera, however, twitched so they would take flight. All around them, the snow rapidly melted from tree branches and meadow grass.
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Miko was able to kindle the fire from coals, and just as he got it going, a young sister arrived with a tray, her hood back and a small bird on her shoulder. Although she smiled at her guests, she remained silent, and soon sat down and closed her eyes.
“Ilika,” Rini asked as he slowly ate his porridge, “do you know what they’re doing when they sit like that?”
“I believe they’re meditating.”
“I don’t know what that is.”
“Meditation is a way of controlling the state of your mind to avoid unhealthy thoughts, feelings, and attitudes. It’s a very common practice in religious orders.”
“Is it like being asleep?” Mati asked while spooning a bit of porridge to a chipmunk.
“No, just the opposite. They are more aware than we are right now, because all distractions are ignored. The hardest one to ignore is the chatter of our own minds. It can take years to learn to quiet the mind.”
Ilika noticed a variety of reactions to his explanation. Several students seemed interested, but somewhat unnerved by the idea. Toli and Neti both squirmed as if they would rather wash dishes.
“I have taught you all the math that’s useful at this point,” Ilika began as they sat in the sunshine just outside the guest house. “We’ll keep reviewing and practicing, but I need to deepen your understanding of logic.
“You’ve already noticed many things deductive logic can’t handle because it’s based on statements that are true or false, all or nothing. So now we have to learn set theory, and its cousin, quantification theory.”
Several students tried to get their mouths around the big word.
“Many, many things in life can be dealt with in sets, or groups. Any time we put things in a group, it’s because they have something in common. Noni has a set of creatures she calls her flock. The important quality all the members of her flock share is that they’re sheep. She doesn’t want the sheep shearer to accidentally shear her dog, or her donkey, or herself.”
Buna burst out laughing, rolled backwards, and couldn’t stop for several minutes. Everyone else laughed or smiled. The meditating priestess tried to
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suppress her own laughter, without complete success.
Ilika grinned until they regained their composure. “We can also have sets that share something more subtle. Let’s say we have a group called ‘red.’ We could include a red apple, those mushrooms Kibi told me not to eat, a glowing hot coal . . .”
“One of Pica’s paints,” Mati jumped in.
“Miko’s hand after the steam vent,” Neti said with a smirk.
Miko looked at her with a loving snarl.
“The sunset with clouds in the west,” Boro proposed while smiling at Neti and Miko.
“You get the idea,” Ilika said, pulling out a piece of half-used paper. “Now let’s see how sets relate to each other. This rectangle is the whole universe, everything everywhere. Circle R is red things, circle E is things we can eat.”
With something to look at, everyone moved in close.
“Two groups create four regions — things that are red but not edible, things that are edible but not red, things that are both, and things that are neither. Buna, where would a sheep go in this diagram?”
“Um . . . in the E.”
“Right. Neti, a red apple?”
“Between R and E, where they overlap.”
“Good. Sata, your boots?”
“Well . . . if you were really hungry . . . but no, outside the circles.”
“Right . . . assuming you aren’t that hungry. Kibi, Miko’s burned hand?”
“If he had cooked it just a little bit more, it would go in the middle with the apple and we could have had it for lunch!”
Miko pouted as everyone else howled.
“But since he didn’t cook it enough,” Kibi continued with a grin, “it would
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go in the large part of the R.”
Ilika smiled at her. “Let’s try three groups.”
“The groups are green, wet, and dangerous. Algae, Boro?”
“Um . . . it’s a water plant . . . um . . . between the G and the W.”
“Yes. Fire, Misa?”
She almost jumped out of her skin. It was the first time he had called on her in a lesson, but she quickly recovered and looked at the diagram, determination written on her face.
“Green, wet, or dangerous,” Ilika reminded her. “Which of those qualities does fire have?”
“Dangerous!”
Everyone
clapped.
“Good. The ocean, Sata?”
“Between wet and dangerous.”
Ilika gave them each several examples until he was sure they had the idea.
“We never put anything in the middle!” Buna protested.
“Hmm . . . green, wet, and dangerous . . . can anyone think of anything?”
Everyone searched their minds during a long silence.
“A big, hungry turtle!” Mati said with a huge smile.
As mid-day approached, Rini helped Mati brush the donkey, then sat down near the priestess, in the same posture, and closed his eyes. A squirrel climbed onto his leg, making him giggle for a moment, but he soon managed to relax.
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Ilika noticed, but continued combing Kibi’s hair silently.
The lunch tray arrived, and the sister who had been sitting took the breakfast tray and the silver piece back to the monastery. Rini opened his eyes and rejoined the group.
After more logic practice and an hour reading dramatic scenes in The Adventures of Godi and Tima, Ilika left plenty of free time before dinner.
Miko and Toli grabbed their sun hats and dashed off to explore the rocks and snowfields on the mountain above.
Rini and Kibi sat down with the priestess and closed their eyes.
The rest wandered about the meadow, drinking icy-cold water, or talking to Tera and the little native creatures. When Sata and Mati started practicing the deep voice of the Mountain King from the story book, their audience of small creatures quickly vanished. Tera just looked at her people and shook her head.
About an hour later, Boro and Mati quietly settled in with the meditators, and Neti and Buna went off to explore.
Sata sat down on a boulder with Ilika and they talked about the solar wind and other topics of mutual interest. After a while, Sata fell silent and scrunched her face. “I don’t understand how a ship’s captain could have learned all this stuff. We’ve had captains and officers from ships at our inn, and they’re hardly more educated then their crews. The only difference was, they had more money and better clothes.”
Ilika smiled at the eleven-year-old beside him, rapidly learning, growing, and experiencing life beyond her years. “You’ll find out soon. The stuff I’ve learned — and am teaching you — is a lot more important where I come from.
But if you’d prefer, we could just talk about things that other ship captains talk about . . .”
“You mean ale and wenches? No thank you! That would be as boring as washing tables!”
Ilika
chuckled.
At dinner, some students wanted to share their geological discoveries of little caves and gleaming crystals on the mountainside. Others wanted to share their psychological discoveries of old memories and hidden fears from
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meditating. Ilika, who had not taken part in either activity, made sure everyone got a chance to speak. Toli had trouble paying attention when the meditators were talking, but an elbow from Buna corrected his attitude quickly. A pair of rabbits showed up, along with the usual squirrels and chipmunks, and they didn’t care who talked about what, as long as they got a handout.
“Thank you all for sharing what you discovered,” Ilika said as the meal was ending. “There are many levels to reality, and the physical and mental realms are very different, but we have to know about both of them. We live in a physical world, but we can only know and understand it through our minds.
If we ignore either one, we can be easily fooled.”
After reviewing a number of topics, from latitude and longitude, to the causes of hypoxia, they all settled into evening free time. Finding no more food, the rabbits departed as the sky began to turn pink.
Ilika noticed Miko, Sata, Buna, and Misa with the meditators. Only Toli and Neti seemed determined to avoid it.
The temperature plunged well below freezing that night, a dry and windless cold that was easy to bear. Their stockpile of wood made for a quick morning fire, and little creatures gathered even before the tray arrived. Hot tea, fresh biscuits, butter, and stewed fruit soon warmed them on the inside.
“Two of the rules of quantification theory are simple and easy to use,” Ilika began, “but the other two are tricky and can fool you. We’ll start with a simple one.”
“The universal instantiation,” he continued. “Everything, x, has quality A.
Therefore, any one thing, y, also has quality A.”
“Everything is cold,” Mati said with a shiver, “therefore Mati is cold!”
Boro chuckled. “That fire sure looks hot to me!”
She gave him a friendly punch in the shoulder.
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“Isn’t it true that there are always exceptions to everything?” Rini asked.
“Yes,” Ilika said. “Mati’s logic was valid, but in the real world we live in, it’s hard to find absolutes like ‘everything’ or ‘nothing.’ Most things are relative.”
“Everything in the real world is relative. Therefore, cold in the real world is relative.”
All those in the circle looked at Sata, the youngest student of all, with amazement. Ilika even noticed the meditating sister open her eyes for a moment, as if curious to see who had spoken.
“It feels funny to be up here when Toli is down there meditating,” Rini said as he stood on a rock outcropping on the southeastern face of the mountain a good five hundred feet above the guest house.
“I think Toli would have done anything,” Ilika said, “when the sister about his age took him by the hand — wash dishes, stand on his head, anything.”
Rini chuckled. “Did you see Buna’s face?”
“She was very quick to take his other hand, wasn’t she?”
“Yeah. Then Neti couldn’t stand being the only one not doing it.” Rini was silent for a minute. “This is so beautiful. Thanks for making me come up here.”
“That’s what teachers are for. Meditation is obviously easy for you.”
They were both silent awhile, gazing out over mountain peaks and valleys.
Rini suddenly smiled. “Something wonderful is about to happen, isn’t it?”
“You feel it?”
Rini nodded. “But I don’t know what it is.”
“Kibi senses it too,” Ilika said with sparkling eyes. “I think we’ll find out soon.”
“The universal generalization is very, very tricky. Some things have quality A, therefore all things do.”
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Most of the students frowned with suspicion.
Seeing this, Ilika smiled. “Let’s say every donkey we’ve ever seen is blue.
We might be tempted to say all donkeys are blue.”
“How could you be sure?” Boro questioned.
“That’s the problem. Sometimes the very definition of the thing makes it a reasonably small number, and you can check them all.”
“Like all the magic bracelets in the kingdom,” Buna proposed, “can tell the altitude, the temperature, and the humidity.”
“How do you know?” Miko challenged. “Have you searched the whole kingdom for magic bracelets? Maybe the healer at Port Town has one in a box somewhere.”
Buna became red-faced for a moment. “Okay! All the magic bracelets on Ilika’s arms!”
Several looked at Ilika to be sure.
He pulled up his tunic sleeves so they could see he had only one. “Buna just discovered how difficult it is to be sure we’ve checked them all. What if, like with most things, we can’t?”
“If we find one that’s different,” Mati said, “like Tera, who isn’t blue, then we can’t do it, we can’t generalize.”
“Right, Mati. If we find even one counter-example, we cannot make a universal generalization.”
“I get it!” Neti suddenly jumped in excitedly. “If we say ‘all’ or ‘none,’
that’s hard to prove because they’re absolutes!”
Ilika nodded vigorously.
“But what if we’re really, really, really sure,” Rini asked, “but can’t check them all?”
“Here’s the rule,” Ilika answered. “If we have very strong reason to believe that any randomly-selected item has the quality, AND no reason to believe otherwise, AND no counter-example, then we can make a tentative universal generalization.”
Boro nodded with a look of satisfaction.
That afternoon, the priestess sitting silently against the wall of the guest
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house could sense that all eleven guests had joined her in meditation. The young sister who had reached out to Toli was gone, but Buna was right there to take her place, hand in hand with Toli, eyes closed.
Neti squirmed until Miko took her hand. Just as she relaxed, something small and furry curled up in her lap and fell asleep.
Deep Learning Notes
What clues might make us think the woman with hauntingly green eyes was from the monastery? What clues might make us think she was not?
The diagrams Ilika drew to explain logical operations on sets are usually called Venn diagrams, after John Venn, an English logician. If the circles overlap to create all possible intersections, as they did in the two examples, then the number of logical regions created is 2 to power of the number of circles. Two circles create 4 logical regions, 3 create 8 regions, 4 create 16
regions, etc.
In this chapter, we can see a sorting process among the students because the idea of meditation often brings up fear in people. During meditation, the usual distraction of life are eliminated, so the memories, fears, hopes, and fantasies that we all have come bubbling up, screaming for attention. An experienced meditator can deal with them. The beginner can be overwhelmed at first. Meditation, like any adventure, if best done with a friend, and in small chunks, especially at first. When learning to camp, we don’t start with month-long backpacks. Similarly, a new meditator will find 5 minutes challenging.
A dry, windless cold is easy to bear because our bodies, with the help of clothes, create a layer of warm air close to the skin. Rain or wind breaks this layer and allows the cold to get to our skin. Cold, even just cool weather, with rain falling or wind blowing, can be more dangerous, and require better clothes (raincoat, windbreaker), than much colder still, dry weather.
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The Universal Instantiation is logically valid, but not often useful, as Mati discovered. There are universally true statements by definition. “All birds have feathers” works because we define a bird as a creature with feathers, so it is not possible to have a bird without them. But without using the “by definition” trick, most things in our universe have exceptions.
“Everything in the real world is relative, therefore cold in the real world is relative.” What do you think of Sata’s statement? Does it help if we define
“real world” as “physical world”?
What emotions did Toli and Buna most likely feel when a young priestess took Toli by the hand and invited him to meditate with her?
What was it about Neti’s personality that made her try meditating when she was the only one not doing it?
Ilika and Rini, together on the mountainside, were able to share an intuition, the sense that something wonderful was about to happen. Which of the other students would have been able to share in that conversation?
As Ilika’s students discovered, the Universal Generalization is much more in tune with the relative nature of most physical reality, and so it is much trickier to use. It is basically a statement of the scientific method. To quote from the story: “If we have very strong reason to believe that any randomly-selected item has the quality, AND no reason to believe otherwise, AND no counter-example, then we can make a tentative universal generalization.”
What was symbolized by a furry creature curling up in Neti’s lap to sleep as she meditated?
