(Short) High Mages

The two men faced each other across an empty plain. Their brows were furrowed in concentration, their eyes, focused and intent. If you paid close attention, you would notice a faint sphere enclosing each of them, swirling and shimmering in different colours.

Every now and then, a ball of fire, or bolt of lightning or whatever, would pop into existence in the air around one of the combatants, speeding for a thinner section of the sphere. But without fail, the other would wave a hand, and the sphere would shift, the fireball smashing to a thousand embers and dispersing harmlessly.

“I’m bored,” the younger combatant announced. He abandoned his dueling stance and straightened, waving off an opportunistic thunderbolt launched by the other. “Battles between High mages are so uneventful. Defence always beats offense. I think we should find another way to settle this.”

“Fine. What do you suggest?”

The young mage pointed into the distance. “If you can take out five of those birds in one try, I’ll acknowledge the loss and turn myself in. But if you can’t, you will return to the Justiciars and say I was too powerful for you.”

The old mage snorted. “Fair enough. You’ll be in the White Court before the day is out, mark my words.” He gazed into the distance, gauging distances and plotting trajectories. A circle of five miniature suns grew in his hands, spinning larger and larger…

… until a spear of rock pierced through his shield and took the old mage straight through the head. His body fell limp, and the five fireballs spun out of control, detonating in an enormous burst of flame and debris.

The young mage turned away from the conflagration. “What an idiot.”

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Efficiency of a Gas Cycle

I was reading up on the physics of refrigeration for … reasons … and while working out some of the math, stumbled on something interesting. It’s almost certainly not novel, but I thought I’d write it out anyway. (It might not even be correct, if I’ve made a careless mistake somewhere. I’m cautiously confident it’s right, though.)

Overview

The gist of my “discovery” is that the (non-standard1) efficiency of refrigeration using a gas cycle is a simple ratio of two temperatures.

I’ll define a gas cycle here for convenience. In a gas cycle, a volume of cold gas is placed in thermal contact with a system in order to extract heat from the system. The gas is then separated, compressed to a high temperature, then placed in contact with a heat sink, rejecting the heat into the sink. The gas is then allowed to expand, cooling down in the process, and the cycle starts again.

More technically, a gas starts off at temperature T_N (N for minimum), and is placed in thermal contact with the system to be cooled, its temperature falling isochorically until equilibrium to T_C (C for cold reservoir). The gas is then compressed adiabatically to a new temperature of T_X (X for maximum), where it is then placed in contact with the heat sink and it cools isochorically until equilibrium to temperature T_H (H for hot reservoir). The gas is then allowed to expand adiabatically, reaching the original temperature T_N.

The quantity I’m interested in is :

\frac{T_{extr}}{T_{rej}}

= \frac{n C_V \Delta T_{extr}}{n C_V \Delta T_{rej}}

= \frac{T_C - T_N}{T_X - T_H}

Obviously, not all four temperatures are free variables, otherwise the efficiency could be anything. So, we need to find the relationship between the four temperatures, and plug it into the expression for efficiency.

The Four Temperatures

The core of the relationship lies in the adiabatic transitions from T_C to T_X and T_H to T_N.

We look first at T_C and T_X.

We start with the equation for adiabatic processes:

TV^{\gamma - 1} = constant

and we know that

V_N = V_C and V_H = V_Xbecause the transition between these states is isochoric.

So, from the adiabatic equation, we proceed to isolate the V terms so that we can later eliminate them with the other equations:

T_X V_X^{\gamma - 1} = T_C V_C ^{\gamma - 1}

\implies (\frac{V_C}{V_X})^{1 - \gamma} = \frac{T_C}{T_X}

\implies \frac{V_C}{V_X} = (\frac{T_C}{T_X})^{\frac{1}{1 - \gamma}}

Similarly, for T_N and T_H we obtain

 \frac{V_N}{V_H} = (\frac{T_N}{T_H})^{\frac{1}{1 - \gamma}}

Substituting into V_N = V_C (isochoric transition):

V_H (\frac{T_N}{T_H})^{\frac{1}{1 - \gamma}} = V_X (\frac{T_C}{T_X})^{\frac{1}{1 - \gamma}}

But since V_H = V_X (isochoric), this simplifies down to:

\frac{T_N}{T_H} = \frac{T_X}{T_C}

which is the desired relationship between the four temperatures.

Efficiency

We pick a variable at random and substitute into the expression for efficiency:

\frac{T_C - T_N}{T_X - T_H}

= \frac{T_C - \frac{T_C T_H}{T_X}}{T_X - T_H}

= \frac{T_C}{T_X} \frac{T_X - T_H}{T_X - T_H}

= \frac{T_C}{T_X}

This is the same expression from the earlier section!!

what sorcery is this??

Conclusion

So somehow the efficiency turns out to be this simple ratio.

It’s not completely out of the blue. The ratio structure of the relationship between the four temperatures hinted that it was describing some underlying parameter, but I didn’t expect the constant to be the efficiency of the transfer.

The expression for efficiency also makes some intuitive sense. You would expect that the efficiency goes down if you pump the gas up to higher temperatures. I didn’t expect it to be such a simple relationship, though.

Maybe there’s actually something really simple going on, but a intuitive physical understanding of this system continues to elude me. But whatever the case, the expression for efficiency is certainly one of the most elegant relationships I’ve seen.


  1. Usually, efficiency is defined as the ratio of useful work extracted (in this case, the energy removed from the system to be cooled) to the work input. However, the value I am interested in is the ratio of heat extracted from the system to be cooled to the heat output to the heat sink. There’s no name for this that I know of, so I just use “efficiency” here. 

Red Like Roses Pt. II (Acoustic/Orchestral)


My cover of Ruby’s second theme song from RWBY.

While the original is made with extremely strong and fast rhythms and electronic instruments, I wanted to make a version which conveys the same emotions, but without a backing rhythm, and using only acoustic instruments. I would have used a female voice but this is the only one I have.

I think it turned out quite well, the weakest link being my vocals which are a little wavery. I may redub this in the future if my vocals improve.

The Watcher

Loosely inspired by: [WP] “A watched pot never boils”, as the old saying goes. Throughout all of history there has always been at least one set of eyes on the ocean. Today, for a split second, everyone looking at the ocean looked away at the exact same time.


His eyes were weary.

The rain beat down on the ruins of a once-majestic castle. It seemed to be hundreds, perhaps thousands of years old. It stood at the top of a cliff, overlooking the ocean. Once in a while, a rock would fall from the crumbling towers, to splash in the water far below.

He sat atop the highest tower, alone. Watching.

He was the last of the Order, now. Gerard had not arrived to serve his watch, and twenty hours had passed since. It had not been unexpected, really; the both of them had been close to eighty years of age. The prospect had loomed over their heads for twenty years. Ever since Michael passed away, the two remaining members of the Order had been forced to keep watch in shifts, neither of them able to leave to spread the cause. Not that it mattered. The Order was viewed as a cult by most of the world. Their teachings were so strange, so alien, that very few were inclined to take them seriously. Their last disciple, forty years ago, had left after less than a week.

But the fact remained that the Order was the key to the survival of the world. In ages past, during the first Creation War, the multitudes of demons had been first defeated and sealed in a prison of light. The prison did not hold, and evil was let loose upon the world. This time, the Authority sealed them in a prison of fire. But they absorbed the fire, rebelling once again, growing smarter and stronger. The Third Creation War had been hard-fought, and Good had nearly lost. But this time, the demons were sealed in a prison of rock and water, to fetter their movements and quench their fire. And finally, the threat was contained. However, the demons were powerful; given time, their flames would overcome the water that restrained them. And so the Authority made Man, to watch over the demons and ensure that they would never escape.

Once, the Order had been the pinnacle of mankind. They were the bearers of its purpose, the reason for mankind’s existence. But over time this purpose had been forgotten, the Order shunned.

He was the last of them all.

And his eyes were weary.

He had not slept in thirty-two hours. The rain helped. The cold kept him alert, and each drop was a reminder of his duty.

He knew he could not keep this up forever. He was nearing the limits of his endurance, and even if he did not sleep for the rest of his life, he would eventually die of starvation, or old age.

He thought of the people he was protecting. All around the world, people were continuing with their daily routine, unaware of the great terror that would soon befall them. Somewhere out there, children were playing. Somewhere, lovers held each other under the shade of a tree, whispering to each other. Everywhere, people were living.

Mankind was doomed. But he was determined to hold out for as long as he could. Even for just an hour. Even for a minute. Although humanity had forsaken him, he would give all he had just to buy them another minute of blissful ignorance.

His eyes began to close. He knew it was happening, but try as he might, he could not get them to open again. His mind felt… looser. Unfocused.

His vision grew darker. He could see it happening. He could feel it. He hated it. But his mind was splintering from his body. He could almost feel them pulling apart, the loose threads of his consciousness stretching…

…thinning…

…breaking.

His head slumped forward. His eyes were closed.

For a moment, nothing happened.

There was a deep rumbling from below. The ground shook. People stopped what they were doing and glanced at each other, confused and fearful.

And the oceans began to boil.

Equal

The men’s rights movement (MRM) has had a significant amount of bad press. Men’s rights activists (MRAs) are routinely labelled toxic misogynistic nutjobs, and are commonly silenced by feminists, disrupted without the chance to even state their case.

Granted, there are blatant misogynists hiding under the banner of the men’s rights movement. These misogynists are responsible for most of the vitriolic attacks on women and women’s rights. However, these misogynists are just a vocal minority of those calling themselves MRAs.

The majority of MRAs hold substantiated, reasonable views, which are in line with the core ideals of feminism. Moderate MRAs’ views are distinct from those of the fringe MRA misogynists, and should not be treated as equivalent1. Generally speaking, while moderate MRAs agree with core tenets of feminism, they may disagree with specific aspects of its theory or implementation.

To illustrate this, I will enumerate two of the core issues the men’s rights movement raises.

Domestic Violence

Many feminists (or at least, those who influence policymaking) believe that domestic violence can only be committed by men against women, or else they believe that it constitutes the vast majority of cases. As a result, there are very few shelters accepting men compared to women. Shelters for men have been refused funding, and battered men have been ridiculed and blamed for the violence they suffered, or even accused of being the assaulters.

Although many studies show that a similar number of men and women are victims of domestic violence, many cling to the narrative that domestic violence is due to a patriarchal desire of men to exert power over women, and that women are incapable of committing similar violence. The Duluth Model of rehabilitation, the one most widely used in the world, assumes this from the outset, attempting to “re-educate” all men, and only the men, involved in domestic violence cases.

The MRM’s position is that although feminists have done a good job fighting domestic violence with regards to women victims, the flawed narrative around domestic violence discriminates against men2. Male victims are blamed for their partner’s violence, and are not provided safe refuge from their abusers. MRAs argue that this narrative must change, in order to protect all victims of domestic abuse, not just a subset.

False Rape Accusations

Rape is a horrible crime, and rapists should be severely punished. At the same time, being falsely accused of rape is also a terrible experience, whether or not it results in a false conviction. Feminists have pushed for rape law reforms, arguing that alleged rapists’ identities should be revealed, rape victims’ identities protected, and cross-examination of victims limited, in order to protect the victims from reliving their trauma. These policies would certainly increase the probability that rapists are convicted. However, implementing these policies necessarily also increase the probability of a Type I error (false conviction), in the event of a false accusation.

False accusations of rape can be extremely damaging to the accused. Terry Brown was forced to move his family into a tent in the woods, after suffering prolonged abuse by the public, even long after charges were cleared. He says he had rocks thrown at him while walking in the street, and could not go to the shops without being called a rapist. Although true accusations outnumber false ones3, the proportions are not so small that the possibility can be ignored altogether.

This sort of “guilty even though proven innocent” mentality is not isolated. Emma Sulkowicz continued her “Mattress Performance (Carry That Weight)” even after her alleged rapist had been cleared of wrongdoing twice over, and her allegations continued to be supported by many people, among them Senator Kirsten Gillibrand.

The danger of false accusation increases on college campuses. In all colleges in the USA who receive federal funding, a 2011 ruling:

  • reduces the standard of evidence required to only a “preponderance of evidence” (50% confidence), rather than “beyond reasonable doubt” or “clear and convincing evidence”,
  • “strongly discourages” male students being given the right to question their accuser,
  • allows accusers to appeal verdicts, subjecting the accused to double jeopardy.

In the hands of a sufficiently intelligent malicious accuser, even completely innocent people can be found guilty, because they are unable to cross-examine the accuser’s statements, combined with the low standard of evidence. The accused may not even be allowed to know the identity of the accuser, and as a result are unable to gather extenuating evidence. In 2013, an unnamed student at Amherst College was expelled for rape, finally clearing his name in 2015 when he filed a lawsuit demonstrating that his hearings had been unfair and evidence purposefully withheld.

If it is made excessively easy to convict, not only does this mean that a high proportion of innocent people may be found guilty of rape, this may also result in an increase in false accusations. If false accusations become easy to pull off, and there are insufficient checks against it, more people will use them for their own purposes.

Rapists should be brought to justice, but this should not be at the expense of the innocent.

Silencing as a “Standard Response”

You may not agree with the above arguments. Perhaps the statistics I have read are misleading. Perhaps my facts are correct, but I have drawn incorrect conclusions. But are they so wrong, so offensive, so innately destructive that they must at all costs be silenced without even being heard?

It is true that misogynists masquerading as MRAs cause trouble online and in the real world, derailing discussions and disrupting discourse. Wanting to completely shut them out is perfectly understandable, and may even be the only viable course of action.

But, when a professor at the University of Ottawa tried to hold a lecture on men’s issues, what was the rationale behind silencing her? She was not disrupting an existing discussion, merely presenting her views on her own time, in her own venue. Shutting down her lecture was purely out of the desire to prevent her arguments from being heard. 

Were the protesters so insecure that they felt the need to parentalistically insulate the community from all alternate arguments? Did they think that current feminist ideology is flawless? Just because the protesters refuse to listen to others, doesn’t mean that their professed view is likely to be right. Silencing opposing viewpoints passes over opportunities to discover and correct flaws in beliefs. Because of this, if those protesters were in the habit of silencing opposing viewpoints, then, all else being equal, their arguments are in fact more likely to be wrong.

Furthermore, even if opponents are known to be wrong, silencing opponents will not change their minds, nor that of anyone who wants to listen to them. The combination of the Streisand effect and the underdog effect will only entrench their beliefs, and perhaps draw more people to their cause.

If progress is to be made, and the divide between feminists and the men’s rights movement healed, silencing must cease to be the knee-jerk response to rational dissent. Feminists and men’s rights activists have similar ideals, and working together would save a significant amount of time and blood pressure. But before that can happen, they must first be willing to listen to each other.


  1. This is often misconstrued as a “no true Scotsman” fallacy, which it is not. There are usually fringe, radical elements attaching themselves to any given cause. Just as “kill all men” radical feminists are not representative of feminism, and ISIS terrorists are not representative of Islam, so too fringe misogynists are not representative of the men’s rights movement. 
  2. Not all feminists agree with the narrative. Nevertheless, this is the narrative employed in most government-sanctioned programs and the narrative believed by a significant fraction of the population. 
  3. Most reliable sources quote values between 2% to 10%, although some estimates go up to 17% or even 65%

Apprentice

He was an adventurer. An explorer. An archaeologist with a knack for finding ancient secrets, retrieving them — and coming back, alive.

And he needed an assistant.

I was but a lowly fruit-seller in the town marketplace when Ulric came calling. And I knew that this was the moment that would change my life.

So here I am, creeping through the entrance hall to the crypt of Aygairo. It is famous in the surrounding villages, not only for the storied treasures hidden within, but also for its lethal traps. Many a bold youth has made the trek into the mountains to try their hand at Aygairo’s puzzles. A thousand years ago, he had been a rich king, but at the same time cunning and highly intelligent. The few explorers who survive return without their original fervour, and usually at least one limb. I am apprehensive, but hopeful, for I have something they did not. I have Ulric.

Ulric pauses, and holds up his hand. “Stop.” I halt immediately, a chill running down my spine. I had not seen any danger.

“Do you see these diamond tiles on the floor? These are pressure traps. Some will already have been triggered by animals or past explorers, but no sense taking chances. Step between them, or you won’t be long for life.” Ulric points, and I can vaguely see the diamond tiles in the half-light of his torch. I can also see some larger mounds and long, whitish objects lining the sides of the passageway. The stench along the hallway suddenly makes sense.

Ulric strides ahead, seeming to pay almost no attention to the ground beneath his feet. But I can see the subtle positioning of his feet, angling away from each trap. I do my best to follow in his footsteps, but I am clumsy, accidentally stepping on a few. Thankfully, they seemed to have already been triggered, and do not move.

The passage ahead zigzags, but the pressure traps are still present. I am getting better at navigating them, which is just as well, because the stench is diminishing, and I doubt that many of the traps at this point have already been neutralised.

At long last, we reach a door, and the end of the pressure traps. Rather, Ulric reaches a door, while I struggle to catch up. He kneels down and examines the stone mechanisms covering the door, while I make my way through the last few traps. Finally, I pass them. Exhausted, I lean against the walls, without thinking.

I feel the wall move under my weight. Gears grind and shift. Time slows down.

Oh shit.

Ulric’s quick reflexes save me. Before I had even grasped the situation, Ulric had already leapt up, and thrown us both to the ground. I lie there, stunned, as crossbow bolts fly over our heads.

When things have quietened down, he pulls me to my feet.

“Seriously kid, try to think a little before you get the both of us killed.” He shrugs. “But I guess that’s not too bad. I’ve done worse.” He returns to his examination of the door.

I can’t help but notice that while I am sweaty, dirty, and exhausted, Ulric is none of them. His leather armour is clean and polished, and his cloak straight and smooth. His hair is immaculate, and his face displays no fear, no hesitation, only a detached concentration on the task at hand.

“Ah,” Ulric says. He begins to adjust the mechanisms, pulling vines from some gears, and repositioning others. “What do you know of the history of King Aygairo?”

“He was a king long ago,” I reply. “He and his brothers ruled the world, and Aygairo was king of this region.”

“Mostly correct. They didn’t rule the world, just the continent of Aviurre. The brothers divided up the land after their conquest, and Aygairo, as the fourth and youngest brother, obtained the poorest land in the centre, far from the sea. As luck would have it, however, Aygairo discovered an enormous deposit of gold and precious metals right here in these mountains. He shared it with his brothers, and the sudden windfall helped all of Aviurre prosper.”

“I see.”

“So, when Aygairo was on his deathbed, he decided that he would share his wealth once again. However, his brothers had passed before him, so he decided he would share it with any of their worthy descendants. And so…”

Ulric smiles, and from his satchel, he pulls out a golden cube. It is covered in intricate symbols, and on one face, has deep ridges carved into it. And on each face, there is a single symbol overlaying everything else.

“The Avire family crest,” I say, surprised. “So you’re the worthy descendant?”

“No,” he says, laughing. “I took this from King Ayheaor’s burial site two years ago.”

Ulric fits the cube into part of the door’s mechanism, and turns it slowly, like a key. There is a low whirring as mechanisms turn and grind. The door does not open, but the wall to our side slides away, revealing a hidden passageway.

“This is a secret passageway, only for descendants of Avire, which bypasses most of the traps here. It was hinted at in the tablets I found, but it only just fell into place when I studied the door.” Ulric laughs. “I’m not really who it was intended for, but this is just so much easier, you know?”

We follow the passageway, which is safe for the most part, other than the occasional arrow trap, rockfall trap, or unexpected flight of descending stairs.

The passage widens, and opens into a large chamber.

“The final chamber,” Ulric says, holding up his torch. In the dim light, I see a golden door at the other end of the room. Even if there is no treasure behind it, the door itself would be worth several hundred fortunes. I feel my heartbeat quicken. We are close.

Ulric notices my excitement. “Not so fast,” he says. “The pathway is trapped.” I look closely, and see that he is right. Beginning halfway across the chamber, the floor is entirely made of traps. Unlike the entrance hall, where there were still spaces between traps, the diamonds here are completely tiled together. There is no safe way forward.

“Don’t worry. It’s actually not difficult to pass this chamber, according to the tablets. All you need to do is go over there and pull that lever.” Ulric indicates a lever on the right side of the room. “I’ll let you do it. Go on.”

I jog over. Near the lever, the floor changes from solid stone to wooden beams with gaps between them. If I lose my footing here, I could fall to my death. Yet, Ulric was right; this task is still surprisingly simple compared to all the other traps in the chamber.

I reach the lever, give Ulric a thumbs-up, and pull

motion

confusion

impact

Pain. Dizziness. Darkness.

I groan, and open my eyes. I am lying on cold stone.

Where am I?

I try to sit up, but my right shin screams in protest. I look, and it is mangled. Halfway down, the bone has been bent to almost a right angle, and bone is protruding through the skin. Blood flows freely. I retch.

I hear footsteps from above, and find the energy to look up. “Help,” I croak.

Ulric looks down at me.

“As I thought,” he says, speaking more to himself than to me. “This was a trap of temptation.”

I don’t care what kind of trap it is. “Help me. My leg is broken.”

Ulric pauses. “I can’t. There is no way back up.”

I stare at him, confused.

“No easy way, at least. You see, the way this crypt was designed was that plunderers would be given a glimpse of the treasure to sap their spirit. Although pulling the lever would indeed disable the traps, they would be dropped into another series of traps. To get to the treasure, they will need to find their way back to the treasure room a second time.”

“So you’re coming down and we’ll find our way back? Together?”

“Well… I could, I suppose.” Ulric flashes me a terrible, terrible grin. “But this is just so much easier, you know?”

I sit in shock as Ulric began to walk away. I hear metallic clangs and creaks as Ulric opens the golden door and slams it shut again.

Then there is silence, and I am slowly bleeding out on a cold, stone floor.

And I decide to take matters into my own hands.

I am the apprentice of Ulric Lanhart. Although he has abandoned me to die, I will survive. I will escape. I will train. And I will have my revenge.

With newfound determination, I drag my screaming body toward the nearest wall.

I grab a small outcrop, grunting as I pull myself upright.

I feel the wall move under my weight. Gears grind and shift.

Oh shit.


Inspired by: [WP] “I didn’t hire you because you would be valuable to our team. I hired you to DIE.”


A/N: In retrospect, not my best work. Written without planning, it turned out long-winded and unfocused:

  • Awkward transition from past to present tense at the beginning.
  • Initial plans for the gold cube key artifact were not used in the end, making it redundant.
  • Backstory of Aygairo et al. not properly thought out nor utilised.
  • Rapport between MC and Ulric not established sufficiently to make the betrayal effective.

Stagnation and Chaos

Sometimes I get the feeling that I’m stagnating. I wake up, go to my computer, study some, read some, write some, and maybe watch some videos. But my creativity feels stunted, and my writing uninspired. I achieve nothing and I go to sleep, to start over the next day.

What an absolute waste of time. Not only is it boring to be stuck like that, but nothing gets done, and there is no foreseeable exit from the cycle — it doesn’t seem like my life is within my control.

Two weeks ago, I had been competing against one of my brilliant to-be classmates in a programming competition, and I learnt a lot over the course of the week-long competition. But after that, I began to stagnate. I made one song cover in the first three days, and after that, nothing much. I’d fallen into the cycle.

Until yesterday. I met with a friend, went out for lunch and had some conversation, listened to music… Nothing of consequence, but somehow it managed to shake me out of the cycle. So here I am, today, breaking out of the cycle and making something new.

And I think I can explain why, and reliably break out of cycles of stagnation in the future.

Dynamical Model of the Mind

The mind is complex. We have myriad thoughts about a plethora of subjects, and similarly with our emotions. However, the configurations are bounded — there are not an infinity of subjects to think about, nor an infinite number of mental states. This follows from the finite size of the brain and the finite maximum information density of matter. Certain thoughts also tend to lead into certain other thoughts, reasonably predictably — thinking about an apple tends to lead only into thoughts about a very small number of related concepts such as “red”, “tree”, or “Isaac Newton” and is very unlikely to bring up thoughts about “river”, “fragmentation”,”municipality” or the near-infinite multitudes of things that have nothing to do with “apple”.

This naturally (to me, at least — due to my physics background) leads to a mental model of the mind as a dynamical system, where the state of the mind is represented as a single point in a n-dimensional space, where n is the number of distinct concepts that the mind in question could be thinking about. This dynamical system should probably be modelled as chaotic, due to the highly nonlinear and unpredictable nature of the brain. The dimensions in mindspace can probably also be modelled as continuous, due to the aforementioned nature of thoughts to link only to close-by thoughts, leading to some degree of adjacency. A mind, given a certain initial position (which corresponds to an initial configuration of thoughts), will then tend to move along trajectories in mindspace according to what thoughts are likely to be formed next from the current set of thoughts.

Attractors in Mindspace

Boredom or stagnation, then, occurs when the mind falls into an attractor within mindspace. It begins to cycle, to orbit or otherwise stay within a certain trajectory or subspace within mindspace. There may be some variation between orbits, but ultimately the mind’s trajectory is locked within the attractor — there does not exist, or exists very few, trajectories that the mind can take within mindspace that will lead it out of the attractor. In the absence of sufficiently large external input, the mind is unlikely to exit the attractor. It will be stuck and will repeat the same or similar thoughts until it is either able to exit, or the mind ceases to exist.

This is a familiar phenomenon with people who are addicted to a certain thing, for example computer gaming. The game is a sufficiently attractive prospect that the mind is dragged into it. The mind may leave due to the exigencies of food and water, but ultimately thoughts will flow back to the game.

And that’s fine. People can be perfectly happy living their lives while their minds only inhabit small set of thoughts. But I, for one, would not. I would hate to be there. While I might be happy in the short term, in the long term I would still be there — there is no potential for growth, for development. There is no meaning for me to find in an attractor.

Exiting Attractors

How, then, to exit an attractor? The model of the mind I am using is a dynamical system, and in a dynamical system, the evolution of the point’s position in time is dependent on the equations of the system, which in this case would be the environment and surroundings of the mind. For example, the thing drawing gaming addicts away from the game would be the body sending signals of hunger to the mind, each of which will tug the mind-point towards the “eat” thought. In essence, an “eat” attractor is created by hunger signals, until the hunger is sated at which point the “eat” attractor disappears. To exit the attractor, the equations of the mind must be changed to create a new, more attractive attractor, or to weaken the existing undesired attractor, or both.

One way that this may happen is boredom. Sometimes the mind notices that it is travelling the same old path, causing it to become bored. The mind’s trajectory will change slightly, because boredom is part of the mind’s state. Eventually, sufficient boredom may build up, leading the mind close enough to the fringes of the attractor until a path out is found. It can also be imagined as the attractor becoming shallower, until finally it becomes convex.

This is why people like variety. By injecting a small degree of variety, people are able to reduce boredom while having the same enjoyment they once did. For example, Bobby Fischer invented Chess960 in an attempt to switch up the chess metagame, in part because he was bored of seeing the same old openings every time (also because mediocre Russians were memorising openings and performing disproportionately well). Games like Pokemon also change game mechanics or introduce new ones, to revitalise players’ interest.

Deliberately Exiting Attractors

However, for a person trying to avoid stagnation, waiting for boredom to run its course may take too long, sometimes on the order of weeks, months, or years. How can a person speed things up?

The mind is not an isolated system; interaction with objects or other minds can change the mind’s position within mindspace. For example, a writer may find their creative well running dry. They cannot think of any new topics or ideas. They may decide to take a holiday, meet new people, read a book, or otherwise jump to a new position in mindspace, from which they can start anew with fresh ideas. They may even encounter new concepts they have not considered before, which will have the effect of adding completely new dimensions to their mindspace.

So to avoid boredom and stagnation, do something new. Find some new input that will jumpstart your thoughts out of the attractor.

Reading a piece of writing, for example, will help. In fact, writing something is essentially recording down a position or set of positions in mindspace in a manner such that the position can be located by another mind. By reading it, the mind will be able to move to that point, hopefully exiting the attractor.

Still Bounded

There is a problem, however.

Mindspace is finite. No matter what you do, your mind is ultimately stuck within the bounds of mindspace. Even if you are able to escape small attractors, your mind is still necessarily bound within this “final attractor” that is the entirety of mindspace. You may be able to expand mindspace, but it is not infinitely expandable, so given enough time there will always be a point in mindspace visited again and again and again (pigeonhole). Even within a normal amount of time, most of our lives may be being lived in a large attractor, though different people will have different periods of reoccurrence.

This has unfortunate implications for people searching for the meaning of life. People tend to think of themselves as free, but assuming that the mind is physical (i.e. souls do not exist — a very reasonable assumption to me), then any living being is certainly bounded by the limited information density of space. Any pursuit therefore must end. There is nothing that can be done for all eternity without repeating oneself. Anyone achieving immortality had better find some way to entertain themselves, because they’re going to be oh so very bored.

But as a mere mortal, I don’t think exploring every point in mindspace is an option for me. So I am happy to settle for being restricted to an attractor… so long as it is functionally infinite.

Balls and Balances

Abstract

I explain the solution to a common, moderately difficult logic puzzle, and propose a novel variation with maximised difficulty. The wording of this variation, without solution, can be found in this1 footnote. The (obfuscated) solution is available in this2 footnote. The method is provided in the main text.

 


About two weeks ago, I attempted a set of logic puzzles that I came across on the Internet. I solved the first few, but there was one that I did not complete (mainly because my bus reached its stop). After getting off the bus, I forgot about the puzzle and did not think further on it.

Until yesterday. I had some free time, and found my mind wandering over various things, and for some reason thought about the puzzle. Since I had nothing better to do, I set about to solving it.

The puzzle went something like this:

“You have a bag of 12 balls, of which one is defective. The balls are all identical, except for the defective ball, which weighs slightly less or more; you do not know which. You have a scale which can tell you which of two sets of balls weigh more, or if they are the same weight. What is the minimum number of uses of the scale you need to identify the defective ball with certainty?”

I will go on to explain my thoughts and my solution to this puzzle, and raise related puzzles. If you want to attempt the puzzle by yourself, now is the time to do so.


Done?

Having done vaguely similar puzzles before, I immediately framed the question in terms of information. This is what I thought:

“There are twelve balls, therefore the location of the defective ball is described by three-point-something binary bits of information. Each comparison provides one bit of information. Ergo, four comparisons should be the minimum required.”

I then set out to find the exact algorithm that would find the ball, and immediately found something wrong.

I imagined weighing six of the balls against six of the others. The scale would tip to one side, but the problem is that this doesn’t provide any information as to the location of the defective ball. I wouldn’t know which side the defective ball was, since I didn’t know whether it was supposed to be lighter or heavier.

This seemed to indicate that one additional bit of information would be required to identify whether the defective ball was heavier or lighter, making a total of five bits.

At this point, alarm bells started going off in my head. Five comparisons seemed to be far too high a number for this problem. There also seemed to be a significant amount of wasted information from the comparison operations. I was also wondering why twelve balls were given, rather than sixteen, which I imagined would eliminate information redundancy which should strictly increase the difficulty of the puzzle.

Modified Solution

After some thought, I realised that I had gotten two things wrong:

  1. The location of the defective ball is not described by one in twelve possibilities, instead it is one in twenty-four, accounting for whether it is heavier or lighter.
  2. The comparison does not give one binary bit of information, instead it gives one ternary bit: in addition to telling which side is heavier, both sides could be equal.

The approach to obtain a solution becomes clear: I need to find an algorithm that distributes the twenty-four possibilities as flatly as possible in a ternary outcome tree, minimising the maximum depth of the tree. Since twenty-four is two-point-something ternary bits, I should expect a solution to use no more than three comparisons.

The Weighing Operation

Before we proceed, we must establish what exactly happens in a single weighing operation.

Suppose I weigh one ball against one ball, and the scale shows the left side is heavier. Out of the four possibilities (Ball 1 is heavier, Ball 1 is lighter, Ball 2 is heavier, Ball 2 is lighter), I have eliminated half of them, and these two remain: Ball 1 is heavier, Ball 2 is lighter.

I shall abbreviate these as follows:

  • 1HL means that neither of “Ball 1 is heavier” or “Ball 1 is lighter” have been eliminated.
  • 5H means that “Ball 5 is lighter” has been eliminated, and “Ball 5 is heavier” remains.
  • 12- means that both possibilities have been eliminated, and 12 is certainly not the defective ball.

So weighing balls 1HL vs 2HL gives:

  • Case 1: 1 is heavier
    • 1H, 2L
  • Case 2: 2 is heavier
    • 1L, 2H
  • Case 3: Equal weight
    • 1-, 2-

However, the above is not complete, failing to take into account the implications on the other balls. The complete list would be:

  • Case 1: 1 is heavier (1 > 2)
    • 1H, 2L, (3-12)-
  • Case 2: 2 is heavier (1 < 2)
    • 1L 2H, (3-12)-
  • Case 3: Equal weight (1 = 2)
    • 1-, 2-, (3-12)HL

Thus, within each of Case 1 and 2 there are 2 remaining possibilities, and within Case 3 there are 20. If we want to minimise the depth of the tree, we need to even out the spread of possibilities between each of the 3 cases.

The First Step

It turns out that for the first comparison, weighing 4 balls against 4 balls gives the best spread:

  • Case 1: (1-4) > (5-8)
    • (1-4)H, (5-8)L, (9-12)-   [8 possibilities]
  • Case 2: (1-4) < (5-8)
    • (1-4)L, (5-8)H, (9-12)-   [8 possibilities]
  • Case 3: (1-4) = (5-8)
    • (1-8)-, (9-12)HL                [8 possibilities]

Notice that Cases 1 and 2 are effectively identical, with 4 balls H and 4 balls L, therefore any algorithm that proceeds to solve Case 1 will also apply identically to Case 2.

Solving Case 1

Cases 1 can be solved quite easily. We want to split the 8 possibilities evenly into 3 cases of 3, 3, 2, such that only one more weighing is required.

To do this, I chose to weigh Left: 1H, 2H, 5L vs Right: 3H, 4H, 6L, leaving 7L and 8L aside.

  • Case 1a: Left > Right
    • 1H, 2H, 6L
  • Case 1b: Left < Right
    • 3H, 4H, 5L
  • Case 1c: Left = Right
    • 7L, 8L

After this, the solution is trivial. If we have 1H, 2H and 6L, we just weight 1H vs 2H to see which is heavier; if equal, then 6L is the solution.

This solves Case 1, and therefore Case 2.

Solving Case 3

Case 3 is a little more difficult. It is not obvious (to me, at least, at the time) what needs to be done to split the 8 possibilities into cases of 3, 3, and 2.

However, a solution exists, by borrowing a ball (1-) that has already had its possibilities eliminated.

We weigh Left: 9HL, 10HL vs Right: 11HL, 1-.

  • Case 3a: Left > Right
    • 9H, 10H, 11L
  • Case 3b:
    • 9L, 10L, 11H
  • Case 3c:
    • 12HL (Solved)

Cases 3a and 3b is the same situation found in Case 1: weighing a pair with the same polarity will yield the result. As for Case 3c, it is already solved, although we do not know whether the defective ball is heavier or lighter. However, if we want to find out, we can simply weigh it with 1-.

Thus Case 3 is solved, and thereby the puzzle.

Variation #1: 13 balls, 26 possibilities

Even though I obtained the solution, one question remained in my mind. Why did the puzzle author specify 12 balls? 3 weight comparisons are, in theory, sufficient to distinguish 13 balls: 3 ternary bits can distinguish 27 possibilities, and 13 balls only present 26 possibilities.

As it turns out, a solution is possible.

With 13 balls, it is not immediately possible to split the 26 possibilities into 3 cases of less than 9 each. Weighing 4v4 results in 8,8,10, and 5v5 results in 10,10,6. If there are more than 9 possibilities in one case, then two weighings, each of which provide one ternary bit of information, cannot distinguish every subcase.

However, if we have access to another ball 14-, which is known to be non-defective, we can weigh 5v5 with 14- on one side. This splits the cases into 9,9,8, perfect for our purposes.

As before, Case 1 and 2 are identical, and Case 3 is the same as in the original puzzle. So we need only solve Case 1.

Case 1 is not difficult to solve. It contains 5 balls of one polarity and 4 balls of the other. Let’s take the case where we have (1-5)H and (6-9)L.

We weigh Left: 1H, 2H, 6L vs Right: 3H, 4H, 7L, leaving 5H, 8L and 9L aside.

  • Case 1a: Left > Right
    • 1H, 2H, 7L
  • Case 1b: Left < Right
    • 3H, 4H, 6L
  • Case 1c: Left = Right
    • 5H, 8L, 9L

In all three cases, to solve we need only weigh two balls with the same polarity to distinguish between the three remaining possibilities, solving the puzzle.

We can see that there are 3 possible reasons why the puzzle author didn’t include the thirteenth ball:

  • They didn’t want to involve the use of an additional ball. (However, this could have been overcome by making the setting of the question in a “ball factory”, making more balls readily available. This would have the possibly desirable side effect of increasing the difficulty of the question.
  • They didn’t think to use an additional ball. They tried 13, couldn’t make it to work, and reduced it to 12.
  • They wanted to reduce hints at the solution. 12 is divisible by many factors, thereby not hinting at any particular method to the solution. (Although 13 is prime and similarly doesn’t hint at a solution, 13 immediately discourages thinking about the question in a binary fashion, which is the mistake I made and which by my best guess would be the mistake most people would make.)

Variation #2: 27 possibilities

What if we wanted to have 27 possibilities? We could include another half-ball: 14H. This ball only has one possibility enabled.

As it turns out, this is also possible. The solution is left as an exercise to the reader.

Variation #3: More than 27?

My initial mischaracterisation of the problem as having 12 possible states was flawed, but not unjustified. Although there are 24 states, the answer only distinguishes between twelve balls; the puzzle does not require us to state whether the defective ball is heavier or lighter. Therefore, there might exist some optimisations of the solution where the H and L cases of some balls are grouped together. By grouping these cases, we save the information required to distinguish the H and L cases.

I argue that we can only optimise one ball.

Looking at the weighing operation, so long as a ball is weighed on the scale, at least one of its cases will necessarily be eliminated. If it falls, its L case is eliminated, if it rises, its H case is eliminated, if balanced, both are eliminated.

However, to make use of grouping for optimisation, both H and L cases must remain open. If we have eliminated one of the cases, then we have already expended the information that we want to save by combining the cases.

Combining the above two facts, we deduce that we can only combine cases on a ball we have not weighed before.

This is important because it also guarantees that only one ball can be optimised by combining cases. If we have two balls we can optimise on, then we cannot have weighed either of them. But by symmetry, we also cannot distinguish the two, and therefore we do not know which is the solution. Therefore, if the question is to be solved, only one ball can have both cases remaining.

As we saw in “Solving Case 3”, case 3c, optimisation is possible, as we grouped together cases 12H and 12L. So, theoretically, we are able to solve a puzzle with 14 balls.

Let’s see if this is in fact possible:

We follow the same procedure in Variation #1, weighing 5v5 balls where one of the balls is taken from elsewhere and known to be not defective. This splits cases into 9,9,10. Case 1 and 2 are identical to Variation #1 and can be solved by the same method. The crux is in Case 3, where we seem to have insufficient information.

In Case 3, we have balls (10-14)HL. The solution in fact follows the same lines as Variation 1:  we weigh Left: 10HL, 11HL vs Right: 12HL, 1-,  with 13HL and 14HL set aside.

  • Case 3a: Left > Right
    • 10H, 11H, 12L
  • Case 3b:
    • 10L, 11L, 12H
  • Case 3c:
    • 13HL, 14HL

Cases 3a and 3b are trivial, following the same method outlined previously. Case 3c is the focus, containing 4 possibilities where all others contain 3.

To solve, we weigh 13HL vs 1-.

  • Case 3ci: 13HL > 1-
    • 13H (Solved)
  • Case 3cii: 13HL < 1-
    • 13L (Solved)
  • Case 3ciii: 13HL = 1-
    • 14HL (Solved)

And thereby, 14 balls can be solved.

As the 14 ball variant is solvable, and I have shown above that no more information can be saved by this method, I believe that 14 balls is the maximum that can be distinguished in this type of problem.

This leads me to propose Variant #3 of this problem:

You have 10 bags, each containing N balls.  All balls are identical, with the sole exception of exactly one ball in bag #4. You do not know which ball is the defective ball. It looks identical, but weighs slightly less or more, you do not know which. You have a scale which can tell you which of two sets of balls weigh more, or if they are the same weight. What is the maximum value of N, such that you can certainly identify the defective ball within 3 uses of the scale?

I believe that this variant is the hardest possible variation of this problem. Not only does it require all the reasoning of the original variation, it also requires the reader to:

  • realise that the question phrasing allows access to additional balls of known weight, and apply this knowledge.
  • realise that although 3 uses of the scale can technically distinguish only 27 possibilities, the nature of the question allows 28

I expect that most people will be able to reach an answer of 12. Only some will reach 13, and only the most careful will realise that 14 is possible. I suspect that if I were to pose this question to myself, I might not even reach 12, due to my initial binary misconception; I might even guess 8.

Summary

I have solved the most common variation of this problem, and created several variations with increased difficulty, including one with maximal difficulty. I hope that this post has shed light on how to approach and solve this general type of logic puzzle, as well as on the nature of information.


  1. You have 10 bags, each containing N balls.  All balls are identical, with the sole exception of exactly one ball in bag #4. You do not know which ball is the defective ball. It looks identical, but weighs slightly less or more, you do not know which. You have a scale which can tell you which of two sets of balls weigh more, or if they are the same weight. What is the maximum value of N, such that you can certainly identify the defective ball within 3 uses of the scale? 
  2. Solution: sqrt(14*3-6)*3-log(34+47)/log(5-2)
    Solution is obfuscated so that glancing at it will not spoil the question. When you are ready to check your answer, paste it into google or work it out yourself. 

Queen of the Underworld

Inspired by: [WP] in a world where you get superpowers for doing good deeds, write the story of a super villain.

Revised 14 January 2017.


They pulled the hood off my head.

I blinked at the sudden light, disoriented. I was tied to a chair, figures surrounding me. Silhouettes. Agents.

“Agent Spencer. How nice to meet you again.”

His face was grim. “You’ve lost, Natalia. Or should I say, ‘Persephone’.”

“You got me,” I shrugged. “I suppose I couldn’t evade you forever.”

“You should note that your powers are restrained, and that you are surrounded by twelve Empowered agents, excluding myself. Any attempted resistance will be short, futile and suicidal.”

I looked around. “I can see that, thank you. So what brings you all here to this merry little party?”

“WHY?!” Spencer exploded. “You were our very best! Our beacon of hope! Why would you do this to humanity? Seven years. Seven years the world has lived in terror! The Masters granted you power to help your fight against the Horde, not to lead them! Why would you do this? ANSWER ME!”

His face, ruddy with exertion, was inches from mine. I looked into his eyes. What intensity. What foolishness. What ignorance.

I smiled sweetly. “Maybe I just felt like being a bad girl for once,” I lied.

Spencer straightened up, glaring at me, red-hot iron under a thin veneer of professionalism. “If that is true, then how were you able to keep your powers?”

I kept a blank look on my face. “I beg your pardon?”

“Don’t bother stalling for time, Persephone. All our powers are proportional to intent for good. You should have lost them when you turned from the light. How did you keep your powers?

I feigned ignorance. “I don’t know. A lucky accident?”

Spencer scowled. “As if. Try again.”

I maintained the best innocent silence I could.

Spencer tried a different angle. “You are now in Division HQ, and will probably be here for the rest of your life. We were lucky that your veggie powers are somewhat less lethal-”

Veggie powers? How insulting.

“-but if we don’t find the loophole, and the next supervillian turns out to be fire-type, he’s going to attack HQ sooner or later. Let’s see how much your plants can do for you then. It is in your best interests to help us close the loophole.”

Not convincing. I shrugged.

Spencer waited, but finally understood that I wouldn’t be cooperating. He sighed and turned to leave. “As you will.”

As he reached the door, he stopped. “By the way, the Masters will be bringing the Lodestone here to revoke your powers. The escort will arrive tomorrow. On account of our former friendship, I would advise you to prepare yourself mentally. I have heard that it can be, shall we say… unpleasant.”

They were revoking my powers? Captivity I could escape, torture I could endure, but confiscating my powers would set back – no, it would completely undo the plans I had set in motion seven years ago.

“Wait,” I called to Spencer, though I still wasn’t sure what I wanted to tell him. “Come here.”

He approached suspiciously. By the time he stood before me, I had made up my mind. “I give in. I’ll tell you how I kept my powers.”

He smiled. “I knew you would come around. I should have played the revocation card earlier.”

The idiot actually believed he had convinced me to talk. Well, he had, but not for any reason known to him.

“On one condition.” I added.

His eyes narrowed. “At my sole discretion.”

“You send your goons out and turn off all the recording devices here.”

He narrowed his eyes, considering. “Alright,” he said, finally. He waved to his men. “Form a perimeter outside.”

He touched a switch on his headset as his men filed out of the room. I saw a green light on his earpiece wink out.

“And the one under the table. I have worked here before, in case you’ve forgotten.”

Spencer scowled. “Fine, fine. We all know how smart you are, Persephone.” He switched it off. “Now talk.”

“Well, to start off, I’m not really a supervillain.”

He snorted. “Save that for the jury, Persephone. I’m only interested in how you kept your powers.”

“That’s what I’m trying to tell you, you dunce. I really have good intentions. My end goa-”

Good intentions? Tell that to the ones in Vert whose homes you wrecked.”

“Ah, but I didn’t kill any of them, did I? And the Masters repaired the damage.”

“Pure luck. Half the buildings were on the very verge of collapse. If they had, any number of them might have died.”

My expression darkened. “Luck, was it? Say, what time is it?”

Spencer hesitated in confusion, then glanced at his wrist. “Three fif-” He stopped abruptly. In the space of two syllables, a small green tendril had grown around the metal band of his wristwatch. Roots grew, spread, burrowed, consumed.

The watch clattered on the floor.

Spencer was pale. “You are under five layers of restraint,” he said, eyeing the runed bands spiraling across my chest and arms.

“With my level of power and control, you can rest assured that if I wanted to kill the Verds, I could have done so easily. Same goes for killing you, too, restraints or otherwise. But I’m not doing it. Now do you believe that I mean well?”

“I should be calling in-”

“Then you’re a fool. You’ve known me for twenty years, since we were trainees together. You know I’m not evil. You know I’m not crazy. You do, however, know that I am and have always been much smarter than you. So which do you think is more likely  – I suddenly turned from your best buddy into your worst enemy, and became the first person in all history to retain her powers while evil, and entirely failed to kill anyone at all in seven years of tenure? Or, am I the same person I’ve always been, just that I know something you don’t yet understand?”

Spencer ground his teeth.

“Don’t assume that I’m going to believe you.”

“I lived at the Crystal Palace for two years, as a bodyguard to Master Elyn. I saw how the Masters lived. I saw their powers. I saw them use the Lodestone, when Elyn’s sons were Empowered at their coming-of-age. You remember how they recited the ritual to Empower us? You remember the part where they say ‘bestowed for good, strengthened by good, and preserved by good’? Those parts were missing for Elyn’s sons. Missing! And nobody said a thing!”

“I don’t see anything wrong with that.”

“You dont?! They were Empowered unconditionally, Spencer. All the Masters were Empowered unconditionally! While the common people have to be pure of heart to maximise their powers, the Masters get it for free and forever! How do you think they repelled the Horde at Lumhart? How do you think they rebuilt Vert in a day? Did you think they were all more well-intentioned than every other person in the world? One of the Masters might be, maybe five, perhaps ten, but definitely not all of them!”

“So what? Isn’t it good that they use their powers to protect us?”

“You don’t have a clue, do you? They are controlling us, Spencer! Do you realise that any time they wanted, they could destroy the Horde? The Lodestone can grant powers to each and every person in the land, the Masters could provide food, shelter, technology, weapons to every person on the planet! Instead, they use the threat of the Horde to keep people in line, and grant only just enough power to a certain few, the “good people” who promise only to fight the Horde! The Masters are afraid, Spencer. They don’t want to risk Empowered commoners usurping the position they have enjoyed for centuries. The Masters present themselves as saviours, yet allow people to die to the Horde, for their own selfish gain. They sit in their ivory tower, defended by their powers – powers that they have locked away from the rest of the world.”

Spencer was unmoved. “If what your are saying is true, then why didn’t you tell us? Why didn’t you fight them?”

“I wasn’t powerful enough. We aren’t powerful enough. The Masters would just crush us like ants under their heel. We must wait, as I have waited. All these years, I have played the villain, gradually showing more and more of my strength, to force the Masters to Empower more and more common people in response. Only when there are enough Empowered can we reveal our knowledge to the people. Only with all our strength can we overthrow the Masters. Right now, I am only planting the seeds of rebellion. The time is not ripe for the harvest.”

“You mean to claim the Lodestone for yourself.”

“Yes, and no. I mean to claim it, but not for myself. I want to use it for everyone. The Lodestone’s potential is infinite. Potential that is, right now, untapped. The world can, and should, be made better by it. When the time is ripe, when there are enough Empowered, I will reveal everything, and the Empowered will overthrow the Masters, and we will change this system. We will change the world.”

Spencer looked at me, brows furrowed. Considering. “You’re talking really big for someone currently restrained and imprisoned. How are you going to do that?”

“Well, the Masters have been Empowering fewer and fewer people recently. I feel my impact is wearing off. Given time, I could easily escape to continue the fight, but I was wondering if there was something else that could be done… something that would make the Masters sit up and take notice.” I looked up at him.

Spencer narrowed his eyes. “You want me to join you.”

I smiled.

“And you let yourself be captured so that you could talk to me. To bring me to your side.”

I had actually been captured due to inattention and a few moments of blinding stupidity, but I wasn’t about to let him know that. I broadened my smile.

Spencer took a few steps backward, dropped to one knee and punched the ground. The metal floor tore like paper, and a sandstorm raged. I saw that the doors to the room was now blocked with a thick wall of rock, and there was a hole in the ground, a flight of earthen stairs descending into inky darkness.

Spencer tore off my restraints, throwing the runed straps aside. I stood up. Flowers bloomed in the thin layer of soil covering the ground, as I stretched my muscles and mind.

Spencer waved me toward the stairs. “After you, Natalia,” he said.

Knowledge

I argue that knowledge, as the word is currently used, is a flawed concept and is not useful. It is merely a feature of language which does not occur in reality.

Throughout history, many formal definitions of knowledge have been proposed, attempting to formalise its linguistic use. The traditional definition of knowledge as justified true belief has already been shown to be incorrect by the Gettier problems. For completeness, I will raise one Gettier-type counterexample here. Farmer A walks past a field. He sees a cow in the field. He forms a justified, true belief that there is a cow in the field. Unbeknownst to him, the cow he sees is a cardboard cutout. However, a cow does actually exist in the field, but is hidden in a ditch, unseen by Farmer A. Farmer A’s belief that there is a cow in the field is therefore true and justified, but we would not say that “Farmer A knows that there is a cow in the field”. This illustrates the incompleteness of true justified belief as a definition of knowledge.

Further, I argue that even definitions that attempt to take the Gettier problems into account are unsatisfactory.

Take for example Nozick’s definition of knowledge:

S knows that p if and only if:
(1) p is true.
(2) S believes that p.
(3) If p weren’t true, S wouldn’t believe that p.
(4) If p were true, S would believe that p and not-(S believes that not-p).

Knowledge under this definition is guaranteed to be true, seemingly solving the problem. However, I contend that criteria (3) and (4) cannot be fulfilled. Using the farmer and cow example, how would the farmer ensure that there is only one cow in the field? Suppose he searches every square metre of the field. There could be a cow in an underground bunker below him, but still considered in the field. Or, there could be a cow walking silently behind the farmer wherever he was walking in the field. More generally, it is always possible to construct a counterexample where p is not true but S believes p is true (or vice versa). Taken to the extreme, this can take the form of a vatted brain, fed signals at all times identical to that which the brain of a farmer searching a field would experience. Because of the above, I contend that (3) and (4) can never be fulfilled with certainty. (See footnote 1)

Due to this, I argue that any binary definition for knowledge (where something is either not known, or known with absolute certainty) is unsatisfactory. This covers all definitions I have seen. I argue that any binary definition that does not require absolute certainty will have a Gettier-type counterexample, exploiting whatever area is left unverified. Yet, any definitions that do require absolute certainty will find certainty impossible to fulfil, thus all statements will be unknown, making the definition useless.

This could be reworded succintly as:

Nothing can be determined with absolute certainty.

The degree of certainty required for beliefs to be considered knowledge could be set as not absolute certainty, resulting in knowledge that could be wrong, which is unacceptable. Or, it could be set as absolute certainty, and nothing can ever be known, which is unacceptable.

Unfortunately, the way the concept of knowledge is used in language requires that its definition be binary, that knowledge be absolutely certain. It is impossible to say that “Person A knows Statement X is true” while Statement X has a chance of being false. This means that any attempts to define knowledge while conforming to its linguistic use must fail.

I thus argue that knowledge, as the concept is currently used, is not a useful concept. Instead, we should speak of beliefs which have a probability of being true. I propose that belief of a statement be on a continuous spectrum strictly between 0 (statement is certainly false) and 1 (statement is certainly true). The exact values 0 and 1 are unobtainable. The number value of belief is the perceived probability of the statement being true, as determined by Bayesian logic applied to available observations of evidence. (See footnote 2)

With this definition, the farmer in the example may estimate a probability of 0.95 that there is a cow in the field, based on past experience. It is unlikely, after all, that a fake cow would be present. It can then be said that the farmer believes, with 95% probability, that there is a cow in the field.

In summary, I feel that a definition of knowledge is not possible, nor is it useful. Certain knowledge does not exist, only uncertain beliefs.

Footnote 1: In this piece, I argue that beliefs cannot be known with absolute certainty. I realise that this is not strictly true, but have omitted this for simplicity. To the best of my knowledge, it is not true only in the cases of statements that are true by definition, and statements about present observations. I do not consider past observations certain because memories can be lost or mistaken.

Examples:
“Cats are animals.” True by definition.
“I am currently observing visual signals (which my mind interprets as a cat).” A present observation.
“I think, therefore I am.” A present meta-observation.

Footnote 2: In fact, from a Bayesian perspective, the failure of binary definitions of knowledge is a direct consequence of Cromwell’s rule. The linguistic requirement that knowledge be absolutely certain is unreasonable from a rational, Bayesian viewpoint.


Also posted on /r/philosophy.