“If I were again beginning my studies, I would follow the advice of Plato and start with mathematics.”
Galileo Galilei (1564-1642)
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I was on the verge of enlightenment.
Seated on the hard stone of my humble room with my eyes closed, I cycled the raw energies that permeated the world through the network of meridians originating from my dantian. My mind was cleared of all earthly distractions, that I may better ponder the truths of the universe I now dwelled in.
Inhale.
“A point is that which has no part, and a line is breadthless length.”
Exhale.
Inhale.
“The ends of a line are points, and a straight line is a line which lies evenly with the points on itself.”
Exhale.
“A surface is that which has length and breadth only, and the edges of a surface are lines. A plane surface is a surface which lies evenly with the straight lines on itself…”
I peeked an eye open, letting the word hang in the air.
“… still nothing?”
I sighed, uncrossing my legs from the lotus position I’d settled into for the past few minutes. Man, cultivation was hard work. I hadn’t expected much, but still Euclid’s Elements was worth a try. Maybe I needed something from Plato instead. Heck, wasn’t his name supposedly derived from how wide and swole his chest was? Maybe he was an ancient cultivator himself?
All jokes aside, though. How was it that this failed, when the realisation that ‘if there is life, there must also be death’ was sufficiently profound to bring a xianxia protagonist to a cultivation breakthrough?
Life truly wasn’t fair.
The issue wasn’t a matter of abundance, or so I thought. Yao’s cultivation had stagnated at a bottleneck, and though his memories were hazy on the details, breaking through this limit required either some form of enlightenment or a manual that guided one through the process of progression.
I stood up and stretched my limbs. The stone room was a spartan one with little in the way of amenities or decoration. A straw bedroll in one corner was the only piece of furniture, if you could even consider that as such. Admittedly, Yao could have afforded to purchase a bed, but aside from the fact that most of his sect contribution points were reserved for procuring cultivation resources and his blind hope of saving for a manual, a proper bed wouldn’t have made it past the door of the confined space.
The first night had been a claustrophobic experience, though I suspected that Yao’s memories and past acclimatisation to the enclosed space had calmed my nerves. Now that I’d lived in this stone prison for over a week, the notion of upgrading to better accommodations was quickly rivalling my original motivation of cultivating so that I could study mathematics without having to dread dying of old age.
Yes, I was pretty certain I was singularly unique in that regard.
One need look no further than the walls of my room to know that I hadn’t been idle in my recent mathematical pursuits. I’d fashioned a makeshift compass out of a pair of bundled sticks, and a narrow strip of wood was my rudimentary straightedge. On the left wall relative to the entrance were artful displays; little geometric constructions drawn on paper that I’d created with those tools of antiquity and was particularly pleased with.
On the right wall were… err, works in progress.
This world hadn’t invented post-it notes, and I’d simply purchased a hefty stack of talisman paper of the lowest grade from the sect quartermaster that I could write loose notes on whenever a stray thought arose. These were haphazardly slapped on the wall without rhyme or reason, though if that many had accumulated over the span of just over a week, I probably ought to start tidying them up soon.
Some contained helpful reminders — logarithm and trigonometric tables that I’d worked out approximate values for, various trigonometric identities, lists of results for some of the commonly-encountered integrals and derivatives. Others were open questions that currently occupied my thoughts, and I winced as I spotted a frustrated ‘WHY????’ on one of those yellow talismans and a ‘WTF??!!’ on another.
What even was that curve I’d sketched on there, anyway?
Eh, I’m sure it’d come back to me if it was important. No biggie.
I was in no rush to advance, or so my excuse went. Even at the fourth phase of the Qi Gathering stage, my body was quite literally superhuman compared to the typical earthling. Even if I spent another ten full years doing nothing but math, I should still have enough years left in my life to get right back to cultivating and outracing the clock.
Today was meant to be a day of relaxation, after all the hard work I’d put in the past day. I’d cleared my quota of compulsory sect duties early on, and I didn’t undertake additional ones to earn points the way that Yao had once done. Much of that free time had been spent lying with my chest pressed against the floor all day, trying to figure out just how and why the Gauss-Wentzel Theorem worked in determining the criteria for regular n-gon constructibility.
Most of my recent efforts were of a similar vein: I vaguely remembered a theorem statement, but knew nothing of how it was proven. All I hoped was that it was provable using the tools that I had, since recreating more complex mathematical techniques wasn’t something I was confident I could do at present.
“Today’s focus,” I said to myself, staring at the wall with a slip of talisman paper held in my hand. “Cultivation.”
I stuck the yellow sheet on the far wall – far being a relative term – and wrote that single word down. Then I pasted another talisman to its lift, raising my brush to pen down my thoughts.
“First order of business — cultivation process.” I mulled over the word. “How does one cultivate, really?”
And wasn’t that the question? Yao’s memories were generally unhelpful, though stemming more from the selfishness of others than lack of effort on his part — as an orphan who’d joined the sect as an outer disciple, he had little in the way of guidance or resources, his journey to enlightenment fully dependent on execution of sect duties to earn contribution points. Alas, he knew little about how to progress from here on, since the inferior manual he’d scrimped and saved for had been meant only for the initiation of the Qi Gathering stage, detailing the barest minimum of how to draw qi in and feed it into the dantian.
Part of it was due to his lacklustre inherent talent. He’d just barely met the threshold for joining the sect – the ability to cultivate qi at all – and the inner disciple who’d administered his testing had judged him to have mixed spiritual roots of abysmal quality and purity. Over his years in the sect, he’d since learned that it meant the rate of his natural cultivation was far slower than others blessed with better spiritual roots, and could hardly hope to draw in more esoteric forms of qi to cultivate with.
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Accordingly, the only use of qi he’d been able to achieve was bolstering his own strength, and only with great effort. While the more talented members among the outer disciples were fully at ease in the fluent usage of qi, utilising it in the form of footwork techniques and martial styles, my current abilities were far more mundane.
“The goal of the Qi Gathering stage is the formation and growth of the nascent dantian via the cycling of qi through the meridians,” I muttered, scribbling that down. “Once the dantian reaches a certain critical mass — for lack of better terminology — it becomes amenable to additional alteration via the imbuement of elemental qi to be projected and manifested. At this point, the cultivator formally enters the second realm: Foundation Establishment.”
I didn’t feel any different from my earlier parroting of Euclid while gathering and coaxing qi to condense the way that Yao had. There was a chance that this was meant to be a slow process and a mere few minutes would hardly be noticeable at all, but surely there had to be more to cultivation than repeating the same set of processes at every stage. In earlier experimentation, I’d tried to push the qi that I drew in through my meridians in different patterns of flow, but likewise didn’t yield any result of note save growing self-awareness of my own impatience.
Come to think of it, that was probably a contributing factor to why I’d spent most of my time on rediscovering mathematics rather than working on cultivation.
“Postulate: every cultivation stage has a distinct goal.” I stared at the statement, appending it to one corner. Then, for good measure, I listed down the stages I currently knew of. Each had five phases, and it was generally agreed upon that each successive phase and stage was harder to break through than the previous one.
Qi Gathering.
Foundation Establishment.
Core Condensation.
Golden Core.
Spirit Expansion.
Nascent Soul.
Soul Formation.
There were probably others more, but those were the stuff of legend. As a mere neophyte in the art of cultivation, I wasn’t too fussed about whether or not my set was complete.
The goal of Qi Gathering was self explanatory. Trivial, one might even be tempted to say.
How on Earth was I meant to keep gathering it, though? Yao had hit the bottleneck in the middle phase of Qi Gathering, and try as he (and I) might, the dantian simply refused to accept additional input. It felt… rigid, for lack of a better term, every trickle of qi that I fed in merely brushing against the condensed sphere of supernatural energy.
The trick was in the method, that much was clear, since cultivation manuals were the gems sought by those at every stage. There were ranks to them, such that some were strictly greater than others even within the same stage. Yao had dreamed of saving enough points to acquire the sect’s Whirling Tiger Sword manual that outer sect members could obtain, that described a set of breathing exercises and swordplay aimed at both cycling qi and preparing one’s dantian for the future cultivation of sword qi.
That was far beyond the scope of my meagre finances, however. If I was going to get anywhere anytime soon, it would have to be through self-invention.
“How does a manual even work?”
Forget even the mechanics of unleashing a technique. Assuming that the previous postulate was true, the fact that one could progress through the same cultivation stage using methods of different ranks and types meant that there were multiple valid paths that all satisfied the same condition. Yet, the method used did matter, because a cultivation technique of higher quality definitively allowed one to begin at a higher starting point at the next stage.
Hmm…
“For simplicity’s sake, suppose we denote a cultivation method X as a function that maps an element x in S1 to y in S2, where Sk denotes sequential cultivation stages k,” I said, then paused. “Hmm. Then for some X1 is a higher quality method than X2 acting on x, y1 is strictly greater than y2.”
Did it make sense? Not really, too many terms were too loosely defined. Still, it was helping put my thoughts into order.
“Is that statement true for any arbitrary element x in S? Or does this also depend on preceding cultivation methods? In which case, is it the composition of functions that is critical, rather than a single function in itself?” I wrote a few more notes down. “Because incompatibility between cultivation methods and subsequent qi deviation does seem plausible…”
And wasn’t that worrying? Could I inadvertently and irreparably screw something up just by cultivating the wrong way?
“It’s still too crude for my liking, though… I mean, one could engage in multiple techniques at the same stage to accelerate their advancement, so long as they are mutually compatible.” My eyes narrowed. “Should this be modelled as a function of functions, then? A functional?”
Yes, I knew it was the antiquated definition for the term. Still, eighteenth century math was better than no math at all.
And if one spoke of functionals, one couldn’t avoid the obvious classical comparison to —
“Minimization and maximisation of path lengths? Calculus of variations?” I mused. “Could cultivation methods constrained by boundary conditions of start and end points be optimised?”
Would that even work, ignoring the copious number of assumptions underpinning this model? I was only sparingly aware of classic problems like minimising path lengths or actions. Not for the first time, I cursed my lack of grounding in mathematics and never properly taking the time and effort to dive into the subject back on Earth, where resources were plentiful and readily available.
I put that aside for now, leaving a talisman post-it bearing the word ‘Functionals???’ on the wall.
“Qi Gathering. Not all methods are equal, and some are strictly better than others.” I returned to my earlier topic of focus. “Effectiveness of a given method may be partially dependent on the method one applied during the preceding phases of the Qi Gathering stage, and other intrinsic factors associated with each cultivator.”
Spiritual roots, for one, was a major determinant. Though he’d refused to see it, it was plainly obvious that Yao had been written off for any serious consideration of advancement since the day he entered the sect with his poor roots.
Roots.
Hehe.
If mathematical cultivation was real, perhaps I might now have an nth root spiritual root that bolstered my ability to cultivate nth roots?
No! Stay on topic!
But I couldn’t. As soon as that stray thought crossed my mind, I knew the battle was lost.
“Hmph!” I said, spontaneously striking an absurd (hehe; surd!) pose with one palm held before me. “Junior, you dare? I, Ling Yao, have cultivated the Square Root Mantra! Face my Periodic Continued Fraction technique!”
I lashed out, summoning my strength in some bizarre cross between shadow-boxing and interpretive dance. My eyes widened in mock surprise, and I feigned three steps backward.
Sadly, that three steps was almost enough for my back to press against the wall behind me. I definitely needed a better room.
“I-impossible! How could you have survived the famed technique of Grandmaster Euler and Lagrange!” I monologued. Wow, I could really get into this. “No matter. For I have also cultivated the Heaven-Defying Yin-Yang Negative Root Scripture! Behold: the Principal Value of the Complex Square Root technique!”
It was a difficult technique to execute. I did some vague impression of halving the angle of an arbitrary complex number in polar coordinate form to the origin, coupled with a scaling of the absolute value of the original number.
Needless to say, mathematical charade was not one of my talents.
I dashed over to the side, ready to pretend to be the peanut gallery of observers watching a fight play out, a cry of ‘No way, it’s the Principal Value of the Complex Square Root technique!’ and ‘Good! Good! Good!’ already on my lips.
Then came a knocking upon my door, and I froze.