I Know What Science Doesn’t

Chapter 005

"Transform, substitute, or discretize directly until the computational load for computer simulation drops significantly."

The direction was crystal clear.

Daniel Hayes began his analysis in earnest, his pen scratching rapidly across the paper without pause. An entire page filled with nothing but the first transformation and discrete expression of the equation.

Claire Whitaker sat beside him, watching with growing doubt. She shook her head. "This won't work. I've tried this approach before. It just gets more and more complicated as you go..."

"Let's see." Daniel didn't even look up as he flipped to a fresh sheet and continued with the next step.

Then the next step. And the next.

Over half an hour later, pages covered in dense notation sprawled across the desk.

He paused briefly, murmuring to himself, "If I transform again, will it start converging?"

Claire pressed her lips together. "I think it'll just get more complicated..."

"Just watch," Daniel kept writing, using three full pages just to complete a single expression. Then he continued onward.

Claire tried to follow along carefully. She found some steps difficult to grasp, but as a mathematics PhD student specializing in partial differential equations, she could at least understand the general framework. What she couldn't figure out was Daniel's thought process. What was the point of this expanding analysis? She'd already tried this route. It didn't work at all.

After watching for a while longer, she rubbed her temples and glanced around. By the time she looked back down, she'd already lost the thread of his reasoning.

That was mathematics for you. One moment of distraction and the entire logical chain could slip away.

After another ten minutes or so, Daniel finally looked up, his face bright with excitement. "Just as I thought. It's converging!"

"Is it?" Claire hadn't caught it.

"You didn't notice?" Daniel explained eagerly, gesturing at the pages. "Some of these terms can be merged and simplified. Others allow for extremely close approximations to be substituted in. Transform it a few more times and the computational load will drop dramatically, maybe even exponentially. At that point, we're done."

Claire stared at him, mouth half open, unable to form words. Continue with transformations this complex? Simplify them? Reduce the computational load exponentially? That couldn't be possible, could it?

Daniel didn't elaborate further. He wasn't going to continue the analysis right now anyway. Instead, he gathered up the mountain of scribbled pages from the desk and handed them all to Claire. "It's all here. Take a look yourself. I need to get back to my own work."

"Oh, right," Claire accepted the papers, stuffed them into her bag, and left wearing a dazed expression.

Daniel stretched his arms overhead with a long, satisfied groan. He stepped outside for a quick walk to clear his head, and when he returned, he dove into his own equation problem.

The earlier session had yielded valuable insights. When he'd been working on his own research before, he hadn't approached it from a mathematical angle at all. His focus had been purely on using transformations and numerical substitutions to obtain more approximate solutions. That was the standard method.

The approach Claire had described helped expand his Intellect and pointed him toward a promising direction for solving the problem. The method should be generalizable to similar types of equations. But the specifics still required more work.

He'd solved Claire's equation problem, sure, but that was just one equation. A different equation would require starting the analysis from scratch. The key issue involved something critical: substitution transformations.

During the analysis process, he'd used approximate substitution transformations multiple times. But these weren't standard methods. They were conclusions he'd arrived at directly through Truth Perception.

That was the problem. He could rely on Truth Perception to complete the approximate simplification process for equations. Other people obviously couldn't.

So he needed to study several more equations, accumulate a large collection of correct approximate substitution transformation forms, and examine them all together to identify patterns. Work backward from the results to deduce the underlying principles and process.

"It's a bit complex, but it should be doable, right?" Daniel thought it over and threw himself into focused research.

Meanwhile, Claire returned to her apartment in a fog. Something felt off, but she couldn't put her finger on it. Even standing at her front door, she still hadn't figured out why she'd just picked up the papers and left like that. Her original plan had been to spend the entire afternoon in the study room doing serious research.

She stepped inside and spotted Sylvia Monroe lounging on the sofa. Suddenly, Claire found her excuse. She pulled the stack of symbol-filled pages from her bag and thrust them forward eagerly. "Guess who I ran into today!"

Sylvia took the papers and glanced at them briefly before setting them on the table. Her expression was flat. "You really think someone from the law school can understand this?"

"I'm not asking you to read them! I'm asking you to guess who I ran into!"

"You're this excited... did you meet some hot guy? A celebrity? Or maybe a cute freshman? Didn't you mention last time that there was this really good-looking underclassman..."

"What are you even talking about?" Claire rolled her eyes in exasperation. "I ran into Daniel Hayes!"

"Daniel Hayes?"

"The guy who came to look at the apartment yesterday! Now I don't have to worry anymore. He's definitely a PhD student. Just look at these. How could anyone who isn't a math PhD write something like this? Look closely... partial differential equation transformation and discretization..."

Sylvia pressed her hand firmly against her forehead and cut her off quickly. "Stop, stop, stop! Please! And also, didn't he say he's from the physics department? A PhD student about to start, right? So he's a year behind you!"

Claire froze. Suddenly she realized what felt so strange. A physics PhD student who hadn't even started yet was teaching a mathematics PhD student a year ahead of him how to analyze partial differential equations? How absurd was that?

And more than that, during the analysis process, there had been parts she couldn't even understand. She felt a wave of deflation wash over her. With a long sigh, she said nothing more and picked up the stack of papers to examine them carefully.

She'd been able to follow the earlier sections, but certain steps still eluded her. She studied them patiently, only to find that the parts she hadn't understood before still made no sense now.

"How can he just substitute like this?"

"Why does this work?"

"How did he calculate this? But when I check the result, there's no problem..."

Claire realized she simply couldn't comprehend it. She clutched at her hair and thought hard, but no matter how much she tried, she couldn't figure it out. She couldn't even fully understand the first transformation.

"No way! I have to figure this out!" Forget about research, forget about anything else. A physics student about to start his program had done mathematical analysis that she, a mathematics PhD student, couldn't understand? Fine, she could accept that much, but she still had to understand the work. Even if it turned out to be wrong, that would be something! Not understanding incorrect work was perfectly normal.

The more Claire thought about it, the more frustrated she became. She shoved the papers back into her bag, grabbed it, and headed out the door again. She made her way to the Department of Physics building and found her advisor, Gideon Pierce.

Gideon was a man approaching fifty with a receding hairline and thick-lensed glasses perched on his nose, the very picture of a seasoned professor. He held the papers and studied them for quite some time before speaking. "He said if you continue the substitution transformations, the equation will keep converging and the computational load will drop significantly?"

"Yes, that is what he said," Claire nodded.

"Looking at the overall structure, that appears to be correct," Gideon confirmed. "By the final section of the analysis, the equation does show a convergence trend."

"I understood that part." Claire nodded, then pointed to a specific spot on the second page. "The issue is here. And there are several more instances like this. Every overall transformation includes them. How are these substitutions made? Why use this particular form? Where does it come from? I can't understand it at all. But I've verified it. The results check out."

Gideon stared intently at the indicated section, then simply picked up a pen and began working through the verification himself. After a thorough calculation, he set down his pen. He adjusted his glasses, raised his head, and sat in thought for a long moment before finally speaking.

"This transformation is definitely correct, but..." He paused. "It's not just that you can't understand it. I can't understand it either. Leave this with me. I'll study it more carefully. By the way, who did this work?"

Claire replied, "Daniel Hayes. A physics PhD student. I think he said he's starting in March."

"Physics department? And he hasn't even started yet?"

"That's right."

Gideon opened and closed his mouth, then nodded thoughtfully. "You can go now. I'll take another look at this."