In the midst of chaos
“Is this the entire file?” the analyst asked.
“Yes,” Anna replied, “it’s all in there.” It was her first visit to a Lyapunov office, which promised to ‘analyse past life choices to live a better future’.
The analyst checked the data. “Will you be able to supply us with more info?”
“Like what?” Anna couldn’t imagine anything was missing. Her file contained all crucial life events that she’d recorded meticulously, mulled over, cried over and now turned over to this man — a professional, no doubt, but still a stranger. She felt a sting of panic as she saw him casually browsing her entire history of love, betrayal and loss. What more could he possibly want?
“The weather at some specific dates or how crowded it was at key locations might help us.”
She nodded. If that was all it took!
“Fine, then we can proceed. What’s your question for us?”
“I’d like to know how stable the history is,” Anna said.
“Could you rephrase that, please? Just to make sure we understand your request.”
“Like, how much variation in some of the actions would’ve changed the outcome, and how drastically,” she said.
“But that varies moment by moment.”
“I guess so,” Anna said. Wasn’t he supposed to be the expert?
“So, you want us to compute the degree of chaoticity for this entire history. Correct?”
“Yes, please,” she replied, as she recognized terms from the brochure. “Can you do that?”
“Sure, you can pick up your results tomorrow at 10 a.m.. If you place your RFID on the counter, we’ll subtract your credit right away,” the analyst said routinely.
For Anna none of this was routine, of course. Her heart fluttered as she heard the beep that confirmed her payment. The analyst passed her a ticket with the order number. She couldn’t believe answers were so easy to come by once you knew how to phrase your questions — and where to ask them.
The next day, Anna returned to the office with an energetic flair, even though she’d hardly slept. She expected to meet the same analyst, but a woman was on duty. Anna gave her the ticket.
Without looking up, the analyst said: “I haven’t done it.”
“Oh, that’s alright,” Anna replied, always ready to accept an apology, even when none was offered.
“I could do the computation while you wait, but I won’t,” the analyst declared, as though that was the end of their conversation.
“Why not, is it too hard?”
“No, it’s easy enough.”
“Is anything missing? I did answer all the questions as well as I could.”
She didn’t respond. Just when Anna started to worry that the analyst might be feeling unwell, she said: “We’re specialized in computing Lyapunov exponents of customers’ life histories. You know what that means, right?”
“You estimate the degree of chaoticity,” parroted Anna from the documentation that she’d been close-reading all night.
“Suppose I tell you the Lyapunov exponent for your history was high...”
“I knew it,” Anna replied.
“Hold your horses, I don’t know the result, that’s why I said ‘suppose’.”
Anna felt embarrassed. She tried to forget the supposed piece of information as quickly as she’d accepted it, which was curiously hard to do.
“So, suppose the analysis showed that the measure of chaoticity for your history was high. Then what?”
“It would’ve taken the slightest change to any of the events to come out completely differently.” Anna sounded excited, as though she still hadn’t understood the hypothetical nature of the question.
“Yes, that’s what it would mean. Now suppose the Lyapunov exponent was low.”
“Little would it have mattered what anyone would’ve done.”
“So, what’re you trying to tell me?” Anna felt especially stupid today. Where was the man who’d agreed to do the computation for her? He’d promised! And she’d already paid him, too. This business was an ordinary scam and she’d fallen for it. Tears sprang to her eyes.
“See, that’s why I won’t answer your question.”
Anna clenched her jaw. Crying in front of this unbudging woman was the last thing she wanted to do.
“We’re here to analyse the past,” the analyst continued. “We’re excellent at it, but we can’t provide what you’re after, so I’ll refund you in full.”
“But all I want are some answers,” Anna pleaded her case. “An analysis of my life’s history, as you call it.”
“What you’re really after is something else though, isn’t it? You want to change the past. That’s why you’re so eager to have me say that a small change would’ve made all the difference. Perhaps listening to a different song or leaving a little later than you did would’ve spared you much heartache. Wouldn’t that be wonderful? But even if such little changes had sufficed, changing history isn’t on our menu. And, in any case, what you need is something different still.”
The analyst swiped for the refund.
“No, please keep the money and give me my result,” Anna said, quickly withdrawing her RFID from the counter. But it was too late, the register beeped.
“It’s something money can’t buy,” the analyst said, “but I do hope this refund will set you on a path where you’ll be able to find it.”
Anna looked clueless.
“Forgiveness, I mean.”
“I’ve forgiven him already,” Anna blurted out.
“Not just for others, but for yourself.”
She must’ve done the calculation and extrapolated it, Anna thought as she left the office. She must’ve seen that this was for the best.
Sylvia Wenmackers is a professor in philosophy of science at KU Leuven, Belgium. She likes paradoxes, infinitesimal probabilities and chocolates.
She tweets as @SylviaFysica.
The story behind the story
As a teenager in the nineties, I enjoyed taking psychedelic trips into the depths of fractals. It all started after chancing upon a copy of James Gleick’s Chaos at our local library. A professor of physics would later tell me that the way Gleick had written about chaos theory was ‘almost poetry’. This was not meant as a compliment, but I remember fondly the coloured pages at the centre of the book. Because I regretted not owning these pictures, I bought a blue and turquoise poster of a detail of the Mandelbrot set to put on the ceiling of my attic room. Each swirl contained galaxies filled with countless marvels.
One night, I sat beneath the Mandelbrot poster holding a set of dried yarrow stalks that I had picked earlier that year in the wilderness outside. I separated the stalks according to a procedure outlined in a second-hand copy of a translation of the Book of Changes, better known as the I Ching. I had not yet discovered the fine line that divides science from divination, and happily crossed it to compose a hexagram that would answer my heart’s question.
Only later, when I studied physics, did I learn to suspect magic, although I never managed to condemn its poetry. In fact, by studying chance events mathematically, I came to understand that the tools for divination are not arbitrary. Small changes in initial position and velocity that we don’t control change the way a tossed coin lands, and whether yarrow stalks fall parallel or crossing each other. The way such objects fall is sensitive to the way things were at the beginning.
Our eyes don’t show us what really matters about such situations, but computers can depict it. We can program them to show us the possible paths in a phase space. Some paths converge, smoothing out initial differences; others diverge, blowing up unnoticeable differences to undeniable consequences. Lyapunov exponents quantify how fast trajectories separate and can be used as an estimate for unpredictability. Sometimes the weather is easier to predict than on other days. Likewise, life is uneventful on some days, whereas it throws us curveballs on others. And sometimes we are the ones who cause the storm. If only we could tell which is which.