Paradigm Sage: the hidden structure of rules and invariants

Paradigm hefted the ornately wrapped present in front of him. Flipping open the tag he saw it was from Understanding, an interesting twist given that this was Understanding’s birthday party. He glanced up at the old guy and suppressed a smile, certain it was because of his influence that presents were being passed in a non-traditional direction.

He unwrapped the thick box and found inside a doll. No wait, it was a doll shaped box made from cardboard. Its outline was that of a doll, and he now saw a lid, painted and in relief. He carefully removed the lid, setting it upside down, and found a second cardboard figure inside, smaller but otherwise identical. Immediately he perceived that there would be another, smaller doll inside this one, and wondered how far this would be taken. He removed this lid too, placing it upside down inside the first one, intrigued that some impulse had compelled him to place the original lid upside down in the first place. Several openings later, he had a stack of nested lids ascending, and a sequence of nested containers descending.

Before continuing, Paradigm took a moment to reflect on this strange gift, and to survey the assembly. It seemed that all whom Understanding had mentored were in attendance. Curiosity was flitting from spot to spot trying to find out what everyone else had received and in general keeping things lively and interesting. She could be exhausting. Logic was counting points off one-by-one in some argument with Proof, and Mathematics was skeptically toying with one of Understanding’s new computation engines. Everyone was having a good time. Well, almost everyone. Dogma had a strange look about him, not angry really, but stressed, or maybe flustered. He looked uncomfortable. Dogma had been around a long time and was usually a very stabilizing influence – Paradigm always felt uneasy around him.

He returned to opening his present, then paused. He wanted to guess how it was going to end. Looking for clues outside of the nested boxes he finally noticed that Understanding had not only signed the label, but had also penned a short phrase on it, “Begin at the center.”

Ah, a cryptic clue, the best kind. Paradigm loved incongruities. He was persuaded that with enough cogitation anything could become clear, and it was surprising how often chaos would give way to pattern. He removed one more lid and this time found a doll that looked different. It was bulging, obviously stuffed with something, with crinkly paper extruding from underneath its lid.

Before he could open it, Reason plopped down next to him and peered into his nested boxes, “What’s this?” he queried. “It’s not a puzzle” responded Paradigm, “and it’s not a game. I’m thinking that Understanding is playing with me nonetheless – maybe it’s a metaphor?” Paradigm looked up at Reason. Reason returned the look, but then peered again at the gift. After a moment’s silence, he said, “I get it. This gift wouldn’t work for anyone but you.” Paradigm nodded, then added, “Understanding proffered a clue, ‘Begin at the center.’ Does that match what you think you’ve figured out?”

“Absolutely. Work on it, I’ll check back with you later.” With a bounce, Reason left, mingling seamlessly with the crowd. Paradigm decided he’d been challenged. This was going to be good. Before returning to his deliberations, he surveyed the group one more time. There was lightness and celebration and stimulating conversations. Paradigm, as the latest addition to Understanding’s group was humbled. He couldn’t imagine a finer assemblage. He vowed to be worth the honor of membership. Only Dogma was pensive.

Dogma looked at the present from Understanding and then fingered his own present hidden in his cloak. This whole present thing still had him bothered. Presents were to be given to the guest of honor, not received from him. That was the right way to do things. He had worked on his present for a long time in anticipation of this event, but then the announcement arrived that Understanding would be giving presents, not receiving them. He had been sure it was a misprint; then he arrived and, to his horror, watched Understanding pass out present after present.

He couldn’t give his present now, not here, that wouldn’t be right; it went against the flow. He fidgeted. The flow was in the wrong direction, but to go in the right direction would be to go against the flow. Neither option was right. How could neither of the only two options be right; how could both be wrong? He’d have to find another opportunity to offer it.

Dogma had been by Understanding’s side for a long time. At each new discovery, Dogma would faithfully categorize it and put the new thing in its proper place. The places formed a pattern. Any place without a thing was a new discovery to be made, a predictable discovery. Stability was comforting and Dogma was proud of the foundation that his work provided. It had led to many new and fruitful questions and was continuing to produce interesting research. Surely there was no better way.

There had been a catastrophe once. He shuddered at the recollection. An unexpected discovery had been producing facts at a furious pace and finding a proper place for all of them got way behind. There didn’t seem to be places for some of these new things. It was a time of great turmoil. It was the first and only time he lost faith. Finally, a breakthrough occurred. An entirely new place-scheme seemingly came out of nowhere, and after much work, there was again a pattern for the places, and some were empty, so new discoveries beckoned. Exciting. But it was odd for a while, because many previous things had to be moved from their old places to new ones. Things shouldn’t be able to fit in more than one place. Eventually, he got used to the new places and now they seemed almost natural. The Great Catastrophe usually only haunted him in his dreams.

The new place-scheme had been proposed from outside the group, by Paradigm. It was then, not even that long ago, that Understanding decided to mentor Paradigm – one more addition to the group. Since then, Paradigm had little new to offer, just nodding as each new discovery took its proper place. Dogma didn’t want to seem prejudiced, but it wasn’t clear what he added to the group’s efforts. It was no secret that Dogma was uncomfortable with Paradigm. Their few conversations were just exercises in frustration, like they were speaking different languages. He glanced in Paradigm’s direction. Sure enough, he was lost in thought, Dogma muttered under his breath without even realizing he had vocalized, “Lost in his own world again.”

His reverie was interrupted by the sound of a doorbell. Understanding got up to answer it, and Dogma made a complete survey of the assemblage. The Bickering Twins were late, as usual.

Paradigm looked up at the sound and noted that Understanding was walking over to answer it. He had thought everyone was here, but a quick look at Dogma revealed that he had missed a detail, again. Dogma was methodically scanning the attendees no doubt ticking off who was here and who wasn’t. Paradigm looked the other way; it would be faster just to see who showed up. He was not disappointed, at that moment Understanding returned followed by Theory and Experiment still engrossed in whatever debate they had brought with them. Paradigm smiled, get those two together and two things were guaranteed to happen, sparks and progress. He found them invigorating.

Returning to his nested boxes, Paradigm finally opened the last one. Inside was a piece of paper, smooth in the middle, but crumpled around the edges to fit inside the doll’s shape. The center of the smooth part consisted of beautiful and simple figures, surrounded by a flurry of weird and complicated designs extending into the crumpled border. The border was a jumble of lines, many hidden by the folds in the paper. No patterns here, just chaos. The visible image was dissonant, but there was going to be a pattern; he was sure of it. He carefully smoothed the paper out and found that it just fit in the next larger box. Now with the folds gone, the complicated figures adjacent to but around the smooth center took on simple yet very different forms. Being able to see the complete images resolved the dissonance. Paradigm laughed. What a beautiful metaphor for his one big contribution. He stared at his mentor fondly.

Reason, attracted by Paradigm’s laugh, hustled back over. Without a word, Paradigm showed him the spread-out paper, fitting within the second smallest box with scant room to spare. Beaming he said, “It’s a metaphor for my penchant for thinking out of the box.” Reason nodded, but then queried, “what is the significance of the rest of the boxes then?” Paradigm froze, he’d actually forgotten about them, so focused was he on the inner two boxes. Paradigm stared at them for a long time, first at the two in the center, then out to the biggest box, and finally scanning back to the box with the flattened paper. He looked up at Reason, “Oh, no, you’re always in a box.”

From across the way, Understanding looked up and noted the tableau between Reason and Paradigm. His smile and silent nod went unnoticed.