27: Chapter 27 Wasteland Five-Year Plan

Day 19.

Jiang Lin had been spinning around in the first five pages of Advanced Mathematics for a full seven days.

Seven days ago, using the yield of his Wasteland potato field, he pried open a crack in the door of sequence limits.

The first year was a little over a catty per square meter, the second year was two catties, the third year was two and a half catties, and by the seventh and eighth years, it gradually stabilized.

Those yields were not a straight upward-sloping line; there was acid rain, storms, rotten seeds, scorched seedlings, and inexplicably low-yield years in between, but as long as the time span was stretched out, the subsequent data indeed converged toward a certain stable value.

At that time, he understood for the first time that the so-called sequence limit was not about looking at how chaotic the front was, but whether the back could stabilize.

The front could be chaotic, could fail, and could fluctuate, but as long as from a certain position onward, every subsequent term was locked within your specified error range, it was convergent.

This excited Jiang Lin greatly.

He even felt that he had grasped the temper of college mathematics.

Yet this illusion only lasted until 8:17 this morning.

On the screen, the teacher dropped the chalk onto the blackboard and wrote another line of definition.

[As x approaches x₀, f(x) approaches A.]

Then came that string of familiar yet unfamiliar Greek letters.

ε.

δ.

Jiang Lin looked at the screen and remained silent for a full three minutes.

It was not because it was much longer than the sequence limit, nor because the symbols were overly complex.

Quite the contrary, it looked far too similar to the previous definition.

It was the same arbitrary given ε, the same always being able to find something to cope with it, and the same ultimately making the distance between a certain value and the target value less than ε.

But the more he looked at it, the more wrong it felt.

The N in sequence limits moved onward.

If the 10th term didn't work, then the 100th term would.

If the 100th term didn't work, then the 10,000th term would.

If this sequence truly converged, then no matter how small the error requirement was, as long as you were willing to keep flipping forward, you could always find a starting point; after that starting point, all terms would obediently enter the range.

But function limits were not like that.

In function limits, there were no first, second, or third terms.

x was not a student standing in line marching forward, but something that could approach from the left, approach from the right, or even probe back and forth near the target point.

It did not move onward.

It closed inward.

N was like a temporal threshold, telling you from which day onward the world finally stabilized.

δ, however, was like a spatial fence, telling you that as long as the input was corralled within this sufficiently small range, the output could not escape your designated error.

Staring at that line of definition, Jiang Lin suddenly had the feeling of being pushed back outside the door.

He had flipped forward thinking he understood limits.

Only today did he discover that what he understood was merely a shadow of limits.

Sequence limits were the Wasteland potato field, looking forward year by year.

Function limits, however, were more like him walking from the Stone House toward a puddle.

He hadn't truly stood at the center of the puddle yet, but with every step forward, the soil underfoot grew a bit wetter, the mineral acidity in the air grew a bit heavier, and the dampness carried in the wind became a bit more pronounced.

His position was closing in.

Some kind of result was closing in along with it.

The question was, how to prove this approach was not an illusion?

How to prove it wasn't that his nose had become a bit more sensitive, or that a gust of wind just happened to blow the acidity over, but rather that as long as he kept himself within a sufficiently close range to the center of the puddle, the concentration of that acidity would inevitably approach a certain stable value?

Jiang Lin pressed the spacebar to pause the video.

The Stone House suddenly grew quiet.

Only the wind squeezed in through the crack in the door, gently fluttering the edges of the draft paper.

He didn't rush to copy down the definition.

This time, he knew that copying it down was useless.

The hurdle of sequence limits had already taught him once: being able to recite it didn't mean understanding it; being able to translate the symbols into his own words and press it into his own life was what true understanding meant.

He stood up from in front of the stone table, turning his back to the blank stone wall, like he was giving a lecture to a nonexistent student.

"Come on, classmate Jiang Lin, today we are not going to talk about those damned terms."

After saying this, he coughed once himself.

Having not spoken for too long, his throat felt a bit dry, but he still forced himself to speak aloud.

"Last time, what we talked about was N."

"That was from which term onward."

"Today, we talk about δ."

"This is how close to the target point."

Jiang Lin raised his hand and drew an invisible circle in mid-air.

"Sequence limits ask: Have you gone far enough forward?"

"Function limits ask: Have you gotten close enough?"

"These two things look alike, but they are not the same problem."

When he reached this point, the fog in his brain dissipated slightly.

So he continued lecturing.

"Suppose there is a machine, input x, output y. Now you want to prove that when x gets closer and closer to 2, y gets closer and closer to 5."

"How would a high school student do it?"

"A high school student would say, wouldn't it work just to plug in 1.9, 1.99, and 1.999 and try? Calculating it and seeing it get closer and closer to 5, isn't that a limit?"

Jiang Lin shook his head.

"No."

"Because trying a few numbers can only prove that these few numbers behave, it cannot prove that all x sufficiently close to 2 behave."

"What college mathematics needs to prove is not that a few examples look similar."

"What it needs to prove is: as long as the input is controlled by you into a sufficiently small range, the output is guaranteed to be locked within the target error."

"Therefore, ε is not determined by me."

"ε is the error requirement given to me by someone else first. You can require the output distance from 5 to be less than 0.1, or less than 0.0001, or even smaller."

"What I can choose is δ."

"But this δ is not drawn at random; it must follow ε. The smaller others compress the error requirement, the narrower I must pull in the input range."

"Even more fatally, this range cannot just pick a few well-behaved x's; rather, all x's satisfying the condition must make f(x) fall within the error prescribed by ε."

"In other words, for any ε comes first, there exists a δ that comes after, and finally it must manage all x's that are sufficiently close to 2 but not equal to 2."

"Once this order is wrong, the proof collapses."

"That is to say, ε is the target error, and δ is the input control range."

"In sequence limits, what I look for is N."

"In function limits, what I look for is δ."

"The former is from which term onward everything behaves, and the latter is how close to the target point onward everything behaves."

At this moment, the logical chain clicked together seamlessly.

High school mathematics was about calculating results, while college mathematics was about mastering rules.

Jiang Lin turned around, plopped back down in front of the desk built from marble slabs, grabbed his pen, and heavily wrote down two words next to the example problem that had once left him completely clueless.

Done!

A sense of accomplishment ten times more refreshing than solving a Gaokao final-answer bottleneck question instantly washed away all the exhaustion and self-doubt accumulated over the past dozen days.

He finally dug the first shovel of earth on this Wasteland called calculus, seeing the dampness beneath that wasn't entirely dead yet.

However, this joy did not last long.

Jiang Lin picked up the e-ink screen and glanced at the total page count of this Advanced Mathematics textbook: 428 pages.

And this was merely Advanced Mathematics (Volume 1).

Afterwards, there was Volume 2, Linear Algebra, as well as General Physics mechanics, thermodynamics, electromagnetism...

From the start of the fourth Wasteland until now, he had spent nearly twenty days back and forth, and had only pried open a corner of the limit language in the first five pages.

At this speed, let alone fifty years, he might not even finish grinding through the sophomore Physics curriculum in a hundred years.

The smile on Jiang Lin's face slowly faded.

He looked at his fingers, which had begun to grow rough from expanding the Stone House and doing farm work, and then looked at the Fourth Wasteland Study Master Outline pasted on the wall.

Passion and sudden enlightenment were unreliable.

A person could not forever rely on a pumped-up surge of adrenaline to fight against such a massive and profound knowledge system.

"I need a little discipline."

Jiang Lin muttered to himself.

He knew all too well the inherent flaws of human beings as a species.

On Earth, there were class bells, evening self-study sessions, monthly exam rankings, and Old Liu patrolling outside the window.

Those were the anchor points that constituted the social coordinates of Jiang Lin, a high school senior.

But in the Wasteland, none of these existed.

There was no calendar here, no submission deadline, and no one to check his homework.

If he didn't want to study calculus today, he could completely lie in his sleeping bag and sleep for a whole day, or sit at the door of the Stone House and space out for a whole day.

The Wasteland wouldn't urge him.

But absolute freedom often led to absolute depravity.

If a person remained for a long time in a vacuum environment without external constraints or positive feedback, his perception of time would distort, his willpower would be chewed up bit by bit by boredom and loneliness, and he would finally turn into a walking corpse wandering across this red land.

Looking at the gradually darkening Wasteland sky, Jiang Lin took the draft paper with [Done] written on it and flipped it to the back.

[First Edition: Wasteland Five-Year Plan and Daily Schedule]

Early morning 6:00 - 8:00: Farmland and survival maintenance.

No matter how late he derived formulas the night before, he must wake up at six in the morning.

This was not just for staying alive, but even more so to pull his brain—which had been floating in abstract mathematics all night—back down to the ground every morning through solid physical labor.

Those two hundred square meters of farmland were his only attachment in this world.

He had to go check if the potato seedlings were withered, dig up some clay that still carried moisture at the bottom of the dried-up riverbed, test the pH of the groundwater, and turn over the red soil that had caked as hard as rock.

The vibration of splitting the mud open with a shovel and the sweat flowing out of his body were also the best medicine to combat loneliness.

Morning 8:00 - 12:00: Core difficulty tackling (currently Advanced Mathematics).

After breakfast was when his mental energy was most abundant, which had to be used to gnaw on the hardest bones.

He could not avoid the important and dwell on the trivial, nor could he skip over to do calculation problems just because the ε-δ language was hard to understand.

Even if he only understood one page of a book in four hours, he must honestly sit in front of the stone table.

Afternoon 13:00 - 17:00: Major Physics category advancement (currently General Physics).

Jumping out of the purely abstract mathematical world and entering the physical world.

This was a process of switching brains.

In the evening, there were two hours of free activity time.

It could be used for spacing out, repairing the drafty gaps in the Stone House, or scavenging in the ruins.

Finally, and most importantly.

Evening 19:00 - 21:00: Review and time capsule.

Human memory had a half-life.

He had no doubt that if he understood calculus today, ten years later when he learned Quantum Mechanics and needed to use this tool, his brain would definitely be left with nothing but a mush of paste.

He couldn't rely on his brain; he had to rely on an external brain.

He set an ironclad rule for himself: every time he finished studying a chapter, no matter how tired or sleepy he was, he had to take out a precious piece of paper draft paper and write a page of Instruction Manual for My Future Self.

The content format of the manual was fixed.

What was the core essence of this chapter?

(It must be written in human language, piling up formulas was forbidden.)

Before studying this chapter, what were the prerequisite knowledge points that must be mastered?

When studying this chapter, where were the biggest pits he stepped into and the places where he got stuck the longest?

If he forgot in the future, which page and which example problem should the first step fall back to?

The concept book solved whether one could understand today.

The time capsule solved whether one could find it back ten years later.

One was a tool, one was a signpost.

He could not study just for today's eighteen-year-old Jiang Lin.

He also had to leave a path back for thirty-year-old, forty-year-old, or even sixty-year-old Jiang Lin.