**Introduction:**

The following short story is loosely based on a legend surrounding Hippasus, a Pythagorean philosopher (though he most likely lived about a century after Pythagoras, despite how I wrote the story), who supposedly discovered the existence of irrational numbers, which challenged the Pythagorean belief in a rational world (all numbers can be represented as a ratio of two whole numbers). Legend has it that Hippasus was drowned at sea because his discovery offended the gods, though little is actually known about Hippasus and whether or not he actually discovered irrational numbers at all.

You can read more about what is known about Hippasus here.

**A Celebrated Proof:**

“Hush, hush, please. I know this is exciting, but I need your silence, please,” Pythagoras calls out to his students, all bustling with curiosity about Pythagoras’ announcement. At their teacher’s request though, they calm down, quieting themselves so they can hear what Pythagoras has to say. “Yes, yes, as you know we have recently done a great deal of work with triangles and squares, and we have figured out a method of acquiring the hypotenuse of right triangles, so I would like to have Hippasus here help me demonstrate,” Pythagoras tells his students. Upon hearing this, the students roar out, awestruck at what Pythagoras has just told them. “Please, everyone, I have yet to even get to the proof, please keep it together for just a few more moments. I promise this will be impressively quick.” The students fall quiet again. Pythagoras turns to you and continues, “Let us begin with a simple setup: Say we took three squares and laid them corner to corner, such that two meet in a right angle and the third bridges the other two, creating a right triangle between all the squares. I would like to posit that, in a scenario such as this, the area of the two squares that meet at a right angle combined equals the area of the larger square. This would suggest that for any right triangle with legs *a* and *b*, and hypotenuse *c*, *a*²+*b*² must equal *c*², giving us the value of any side of a right triangle so long as we have the other two.” The students begin buzzing again, but Pythagoras silences them with a simple wave of his hand. “Now, in order to prove this, we must look at a different scenario.” He turns to you.

“Now, Hippasus, consider a triangle with legs *a* and *b* and a hypotenuse, *c*. Look at this square, made up of four triangles, *abc*, laid out edge to edge, such that each side of the outer square is *a* and *b* combined, and the four hypotenuses make a smaller, slightly rotated square in the center. We can see five clear areas contained by the large square: the four triangles and one inner square. We can represent the area of this square a few different ways, but what’s the simplest way?”

You don’t hesitate, everyone in the room knows the answer, “(*a*+*b*)²”.

“Excellent. Because each side is *a*+*b*, squaring that number gives us the area of the square. Simple, but not super helpful on its own. Hippasus, what is another way we could depict the area of this square?”

While a more complicated question, you answer just as quickly, having gone over this proof with your teacher repeatedly, “2ab + c²”.

“Exactly. This takes a little more thought, but we can get here easily. Each triangle has an area of (1/2)*ab*, multiplied by four we get the 2*ab*. Then we just need to add the *c*² from the center square. Now, we know that these two equations, (*a*+*b*)² and 2*ab* +* c*² equal each other, so we’ll set them as such. What should our next step be, Hippasus?”

“Expand out the left side.” Easy.

“Yes. (*a*+*b*)² is not particularly useful to us here, so we must expand it out. When we do that we get *a*² + 2*ab* + *b*² = 2*ab* + *c*². Just one more step, Hippasus.”

“Subtract 2*ab* from both sides.”

“And when we do that, we get the simple equation, *a*² + *b*² = *c*². Plain as day, my friends, any right triangle’s hypotenuse will equal the sum of each of the other side’s squares. You may resume.” Pythagoras’ students fall into an even louder, more celebratory uproar, cheering and marveling at the groundbreaking work of your teacher. They storm up to you two and begin asking you all questions. You yourself find yourself rather dumbfounded by Pythagoras as well. Not only has Pythagoras found such a fundamental and fascinating truth of the world, but for it to have such a simple proof as well! The proof reminds you of Erdos’ idea of the *Book*, a grand book maintained by God that contains the most core proofs in mathematics. You ponder your teacher’s work and can’t help but think that if this doesn’t count as a *Book* proof, you don’t know what would.

Pythagoras, enjoying the attention and explaining his proof in more detail to those asking, pauses for a moment to loudly exclaim, “Friends, we have made it just one step closer to understanding this rational world! We still have a long way to go and I look forward to exploring it with you!”

Amongst the crowd of students mobbing you, asking questions about how it felt to rehearse this with Pythagoras, what his thought process looked like, and if they could work with you on your next project, Pythagoras finds you and says softly in your ear, just audible over the crowd to you and only you, “When the crowd dies down, I’m bringing a group of students out sailing tonight to celebrate. You should come, I have some phenomenally fine wine for the occasion”. Pythagoras doesn’t even await a response; as soon as he finishes he slips away back into the crowd, positive you’ll attend. And you will.

**Pondering Nature:**

You, Pythagoras, and a handful of other students sit on a moderately sized sailboat with oars to help control direction. The sun has begun to set, leaving the sky and sea-colored in various shades of reds and pinks and yellows. The sea sits still, basking in the sun along with you and the other students. The vessel sways calmly back and forth, as if trying to lull you all into a sleep to accompany the setting sun.

Much like the sea, you students have remained largely quiet, calmed by the sea, you all seem to ponder the discovery you and Pythagoras revealed today. The students all sit, running numbers in their heads, checking values to make sure it works. After a few moments though, Speculus speaks softly: “You know that poem, ‘Sailing to Byzantium’ by William Butler Yeats? Doesn’t it kind of feel like we’re on the same kind of journey as the speaker in that poem?”

“You mean because we’re on a boat,” Sageron asks sardonically, unimpressed by Speculus’s idea.

“Well the speaker in the poem writes about going on an intellectual journey, sailing to ‘the holy city of Byzantium ’, seeking to learn how to sing,” (“Sailing” 15–16) Speculus retorts. “Doesn’t that feel like what we’re doing with Pythagoras, at least in some way?”

“How do you know they wrote about an intellectual journey?” Sageron, still skeptical, asks.

Pythagoras steps in for the first time, offering his thoughts on the topic: “Just think about Byzantium. Sure, initially, given the morbid language of the poem, Byzantium appears as a metaphor for heaven or at least the afterlife, but the founders of Byzantium built the ancient city on fertile ground, and its strategic location to both the Greeks and Romans resulted in a prosperous economy and a long life for the city, so Speculus’ assumption makes sense, at least on the surface. In the introduction to *A Vision*, a philosophy book also written by Yeats, while discussing his work before *The Tower*, Yeats writes that ‘Browning’s Paracelsus did not obtain the secret until he had written his spiritual history at the bidding of his Byzantine teacher’ (*A Vision* 9). Paracelsus, a Swiss physician from the 16th century, rejected traditional education and medicine, seeking education from all over the world, believing that truth exists in all places, and ignoring ideas outside of your own limited culture neglects vast amounts of truth. The multiple mentions to Byzantium compare Paracelsus’ pursuit to the speaker’s, especially considering that *The Tower* was Yeat’s first collection after writing *A Vision*, revealing the speaker’s goal in life: to find eternal truth. By making ‘Sailing to Byzantium’ the opening poem to *The Tower*, the speaker establishes this journey immediately, coloring the entire collection with it.”

The students all mumble amongst themselves about what Pythagoras said. The reference to Paracelsus grabbed their attention in a very particular way. The relationship between student and teacher, with an emphasis on the student’s own journey, appears to have given some form of hope for their own future and their own ability to find such powerful truths. Not to mention, as Sageron pointed out, the connection between sailing in the poem and your own literal sailing feels almost too on the nose, though you have to wonder why Yeats chose to emphasize the sea so extensively. A different, more abstruse part of the poem grabs your particular attention, though. You think of the third stanza, where the speaker pleads with divine sages to “Come from the holy fire, perne in a gyre,/And be the singing-masters of my soul” (“Sailing…” 19–20).

Trying to parse the meaning of “perne in a gyre” yields little information on its own at all. To begin with, the word “gyre”, while it has a definition, to Yeats, it carried a much more complicated, difficult-to-describe idea. Fascinated by the occult, Yeats worked with his wife, an automatic writer, to communicate with spirits, whom he claims gave him “metaphors for poetry” (*A Vision* 8), the gyre acting as one of the primary ones. Yeats even wrote the entire aforementioned book, *A Vision*, to explain this metaphor and the others surrounding it. Suffice it to say, the brief mention of a gyre in the poem could never fully explain what an entire book struggles to do, though the word does evoke the sensation of spiraling, spinning, or something similar. The other key word in that line, “perne”, has no clear denotative meaning. Likely derived from the word “pern”, which, in some cases can refer to a spool one winds thread onto, it does little more than create a feeling. By pairing two incredibly ambiguous words together in one phrase, Yeats manages to completely remove any imagery or definition from one of the most complex ideas he explores in his poetry and instead inserts nothing but raw feeling. Despite all this, when read along with the rest of the poem, you can’t help but feel that you understand exactly what the speaker asks of those sages.

Your face must have furled itself, lost in thought because Sageron notices and asks why you seem bothered. You explain your train of thought and they look at you in understanding, though you don’t quite know how they could understand such a complex issue so quickly. “I see what you’re saying, Hippasus, I do, but I feel like you may be reaching a little too much.” Clearly, they did not understand. “Are you familiar with the George Orwell essay ‘Politics and the English Language’?” You tell Sageron you don’t know it. “Well, in it Orwell argues against a series of trends he sees in modern political writing and discourse. He first gives a handful of passages that demonstrate the problems and then goes into explaining the problems. Right at the start, he details that the two main problems he sees in modern political writing have to do with ‘staleness of imagery’ and ‘lack of precision’ (Orwell). He goes on to say that ‘The writer either has a meaning and cannot express it, or he inadvertently says something else, or he is almost indifferent as to whether his words mean anything or not’ (Orwell). I think that last point, in particular, applies to your confusion with Yeats. Obviously the speaker in ‘Sailing to Byzantium’ made a choice, presumably intentionally, to ignore what his words meant. Orwell explores this exact idea when he talks about how a good deal of literary criticism comes with an abundance of ‘meaningless words’ (Orwell), and he even explicitly says, ‘That is, the person who uses them has his own private definition but allows his hearer to think he means something quite different’ (Orwell), clearly referring to, at least in some capacity, the practice the speaker in the poem employs. Orwell largely takes issue with this practice because it doesn’t get people anywhere. Perhaps you don’t actually understand the speaker’s intention behind ‘perne in a gyre’ but rather have come with your own meaning for it and have projected that onto the speaker as their intended meaning.”

You respond to Sageron that, while yes, Orwell likely would not have approved of the practice Yeats employed in the poem, you don’t think that Orwell’s argument in that essay applies to your discussion of the poem, because Orwell’s essay centers around political speech, not poetry. “True, of course, the argument can’t apply entirely,” Sageron says. “But I don’t think we can write off the entire essay as only about politics. After all, wouldn’t you say that, when writing a political pamphlet, you need to make every word and every inch of the page count much in the same way you do a poem?”

Indeed, they make a fair comparison. You go on to point out though that you think the two of you have begun to veer off into two slightly different arguments. You want to point out that Yeats’ poem peculiarly suggests that denotative language actually has little importance when it comes to conveying ideas, implying that human comprehension of ideas happens on some level beyond simple language, while Sageron wants to argue that people should pay more attention to the denotative meaning of their words. Sageron seems to have taken the importance of denotative meaning as a given, arguing about how we should use language in practice, while you want to challenge that premise entirely, looking at language more theoretically.

Before Sageron can continue their argument, Speculus chimes in with an idea of their own: “I mean, think about Dr. Seuss, guys.” They look around, realizing they interrupted. “Sorry, I was just thinking that if you look at basically any Dr. Seuss book he constantly used words that mean nothing, and for all sorts of purposes: nouns, adjectives, verbs, he would make up words for anything.” Lerkims, Snuvvs, miff-muffered moof. gruvvulous gloves, snergelly, Bar-ba-loots. Those all come from *The Lorax*, just one book. Speculus makes a fair point; if anyone can prove that we can understand people without the need for language, leave it to Dr. Seuss. When Dr. Seuss wrote about the “snergelly hose” (*The Lorax*) everyone knew what he meant.

“I don’t think that Dr. Suess’ use of made-up words necessarily works the same way as Yeats’ though,” Sageron says, pondering Speculus’ point. “Because Dr. Seuss uses made-up words to describe things that he could describe in other ways, his words just fit the rhythm and rhyme. Yeats on the other hand, at least according to Hippasus, makes up words to describe things that existing words cannot. While I suppose Orwell would not necessarily approve of either writer, I don’t think we can equate them either.”

What Sageron says strikes you. You tell them that they seem to have a very limited interpretation of Seuss and that they should consider that maybe Seuss didn’t want to describe anything we can currently know, but instead wanted to appeal to a more base-level feeling within us. A word like “snergelly”, while maybe similar to “winding” or “twisted”, feels more comedic than either of those words and doesn’t quite evoke the same image either. Dr. Seuss has a kind of beauty in that he wrote his books for a young audience with a loose grasp on the English language. Instead of worrying about exact meaning in the way that Orwell describes, he simply makes up his own words to appeal to his audience who wouldn’t fully understand the meaning of his words no matter what. If Seuss can evoke meaning from nothing at all, then do we need language at all? Why does Orwell seek to preserve an institution that writers like Seuss and Yeats prove unnecessary?

“All true, but we cannot forget that Seuss and Yeats largely used existing language to write, inventing new words of their own rarely compared to their use of existing words. Clearly, even they depended on language, and we cannot pretend that it is purely useless. In order to convey meaning it seems that we must have language at least on some level.”

You tell Sageron of the Icelandic band, Sigur Rós, and their interesting approach to language and emotion. You explain that their 2002 album, *( )*, contains no words, and even the album and its songs have no names. Instead, the singer of the band, Jónsi, created a language called Vonlenska (Hopelandic if you were to translate it to English), which consists of nothing more than gibberish vocals meant to sound similar to the Icelandic language. Because of the lack of denotative meaning, listeners have the ability to interpret their own meaning into the song, using the clear emotions of the album as guidelines. Plenty of other music does this as well, either with no vocals or nonsense vocals, so clearly music, among other things, can convey meaning and emotion without the need for specific language. Of course, you clarify, these do not convey specific narratives either, so they don’t perfectly compare to Yeats and Seuss, but the connection is still clear.

“So, perhaps we should say,” Sageron starts, “that while one can convey certain things — such as emotion — without actual language, in order to explain more specific things like narrative, we must use, at least to some extent, language common to the entire audience, though made-up words can still make up parts of a narrative in the interest of emotion, flow, rhyme, or other devices.”

Speculus and Sageron sit back, content with their conclusion, and you can’t necessarily argue with them, though something doesn’t quite sit right with you. You certainly created a system to determine when made-up words and emotive sounds can function, but it feels odd to systematize something as free-flowing and human as expression. Can the system the three of you just designed really apply to all of written and spoken, verbal and nonverbal communication? You feel that Sageron clings too firmly to Pythagoras’ idea of a “rational world”. Yes, perhaps everything in the world can be depicted using fractional numbers, but this doesn’t have to do with numbers, rather language. Though, to Sageron’s point, perhaps one can consider numbers a form of cosmic expression, and thus the rules you use to understand that can also help understand your own forms of expression. You sit back and look out at the sea to ponder. The sun has almost completely set by this point, and the sky is mostly dark, the sea darkening with it. It seems ominous, but still peaceful. Not hostile, just a calm reminder of inevitable darkness.

**Irrationality:**

“Friends, enough with the gloom, today is a momentous occasion that calls for a momentous wine!” Pythagoras raises a bottle of wine up in the air. “I have brought out my finest as a thanks for all your hard work.”

The students meet Pythagoras’ call with praise and adoration. Students respond with things like “We couldn’t help without your guidance”, “None work harder than you”, and “We all have come here to learn, so you have earned the most thanks”.

“Of course, of course, that’s what we have this school for. For you to grow, me to teach, and for all of us to think,” Pythagoras says, even more proudly.

“Are you confident that your proof works? Have we tested it?” another student asks.

Pythagoras’ arm drops slightly, and he leans back, somewhat surprised to hear such uncertainty. “Why of course it works. Go on, pass me any pair of numbers and we can quickly tell the equation’s veracity.”

“3, 4, 5”

“9 + 16 = 25, of course.”

“5, 12, 13”

“25 + 144 = 169. Come on, give me a challenge.”

“What about 1 and 1?” Speculus asks.

“Well, the equation would dictate that 1 + 1 equals the square root of 2”

“But what is the square root of two? How can we test it?” At this point, the students begin exchanging whispers, uncertain about the square root of two.

“Students, hush. The equation must work. Hippasus, would you find the value of the square root of 2 for me? This should help clear up any confusion.”

Pythagoras hands you some writing utensils to use to find the square root of 2. You stare at the page in confusion, unsure of where to start.

“What’s the holdup? You know where to start. Simply set the square root of 2 equal to a ratio of two numbers, you know it needs to be a ratio after all.” You set the square root of two equal to *a*/*b*.

“Good. You know that the square root of two must be a ratio of two numbers, so we can assume that *a*/*b* is reduced, meaning that they have no common factors.” From here you know that you need to get rid of the square root, as that is your problem so you square both sides, giving you 2=*a*²/*b*².

“Right, you need to get rid of that pesky square root. That’s causing our problems so get rid of it. Keep going.” Next, you multiply both sides by *b*² to isolate *a* and solve for it. You have 2*b*²=*a*².

“Precisely. Look! You’ve already gained some valuable information. And so quickly, too. Tell them what you’ve found.” You tell all the students that *a* must be an even number because its square equals a multiple of two, and the root of an even square must itself be even.

“What a wonder! We can already narrow *a *down to an even number. Additionally, that allows us to rewrite *a*². Go on Hippasus, replace *a*² with the new value.” You rewrite the equation as 2*b*²=4*c*², replacing *a* with 2*c*, to represent an even number.

“There you go. Go on, don’t let me distract you.” Next, you divide both sides by two to solve for *b* and simplify the equation, getting *b*²=2*c*².

“Good. Hopefully, we should gain some more information from this as well. What do you see?” You announce that, by the same logic as before, *b* must also be even.

“Good. That makes sense. Keep going” You pause for a moment, unsure of what to do with the new information.

“What’s the holdup? We need this value.” You clarify with Pythagoras that *a*/*b* is a reduced fraction.

“Yes, what about it?” You tell Pythagoras that if *a*/*b* is a reduced fraction, then *a* and *b* cannot both be even numbers, because they would have a common factor of 2.

“I see, yes. That does seem like a bit of a contradiction. You must have made a mistake. Redo it for me. I’ll watch closer.” You walk through the same steps as before, not seeing any place where you made a mistake. You come to the same conclusion.

Pythagoras seems unfazed. “Do it again, we’ll see the error certainly this time.”

You tell Pythagoras that you don’t think you made an error, and Pythagoras leans back, looking at you now, and not your work. “I don’t need your input here, Hippasus. Please do the work so we can see where you made an error.” At this point that students have begun whispering amongst themselves again, the air of uncertainty returning to the boat now lit by little more than lamplight. They seem to be discussing your work, asking why you all can’t just accept the square root of two as an answer, and some urge you back to work, telling you to listen to Pythagoras.

Speculus chimes in: “Could it be that the initial assumption that the square root of two must equal the ratio of two numbers is incorrect?”

This sends the boat off the edge. Students’ whispers turn into full discussions, with questions, accusations, threats, and, above all else, fear. If your school’s work is all based on the rational nature of the world, does this mean that all your work means nothing? Can some of it be saved? Did Pythagoras know this to begin with?

“Students, please be quiet,” Pythagoras says assertively, raising his voice, but not in celebration this time. “Speculus, we must not jump to conclusions so readily. We would not be mathematicians if we did so. We must explore all ideas to their full extent before we make such assumptions. Now, Hippasus, run through it again. Slowly this time. I must see where there could be errors in your logic.”

You run through the proof again, going slowly, and following Pythagoras’ directions. You reach the same result. Pythagoras looks at your work, thinking. You tell Pythagoras that you agree with Speculus, and that the only part of the proof you don’t fully understand is the assumption that the square root of 2 must equal the ratio of two numbers. You ask what proof Pythagoras has for that.

“That is a deeply complicated proof, Hippasus, we don’t have time for that now. Please, let me think this through.” You tell Pythagoras that you would like to explore the idea of the square root of two being irrational, and what other numbers might also be irrational. This enrages Pythagoras more than anything you’ve said yet.

“Hippasus, you must drop this line of questioning immediately! It simply will not lead anywhere!” You tell Pythagoras that if he doesn’t want to learn more about this, you would be happy to go to another gymnasium where they would be thrilled to learn of such a new and interesting find. You say that you think this could lead somewhere big, and that irrational numbers could explain a lot of the phenomena in the world your school has yet to explain properly.

Pythagoras seems close to his limit. He lifts you up and pushes you against the side of the boat. Your head now leans over the edge, the breeze from the sea chilling your face. So close to the water, the night settles in on you, and you feel colder than you’ve felt in a long time. Your head tilts to the side and you get a glimpse of the deep, black-blue water, threatening to swallow you whole and never spit you out.

“Hippasus, I will not allow you to spread such falsehoods across the world. I will not allow you to sully the name of this school with your stupidity. I will not allow you to ruin *my name*!” He pushes you a little further over the edge.

“Pythagoras, I’m sure they didn’t mean anything by it,” Speculus says softly, their voice shaky. The other students have all fallen silent. “Let’s just drop the subject, I’m sure we’re all tired.”

Pythagoras’ head twists back sharply, his eyes burning holes in the chest of Speculus. “Speculus, say no more! You needn’t put yourself over the water the way Hippasus has.” Speculus slinks back a little into the crowd, dejected, not ready to oppose Pythagoras the way you have. You see Sageron grab their arm and begin angrily speaking to them, clearly dissatisfied with their dissension with Pythagoras.

You think back to Erdos’ idea of *The Book*. Yes, Pythagoras’ proof earlier certainly was simple, but what you just did proved the existence of numbers that no fraction can ever represent. And using such simple math as well. In just a few steps using simple arithmetic, you have proved one of the most fascinating discoveries in the world. You know that you have found the first proof in Erdos’ *Book*.

Seeing Pythagoras’ harsh opposition to your idea pulls something out of you. Somewhere deep within you, you feel correct in a way you never have before. Pythagoras’ assumption of a rational world must be wrong and you will not accept such dogma. You tell Pythagoras just this, harshly criticizing his rigidity, saying that you cannot be his student should he reject such a plain and simple truth.

“I should say the same to you, Hippasus.” Pythagoras pushes forcefully, and your body tips over the boat headfirst. You feel yourself in the air for a brief moment before your neck slaps against the water. Suddenly, the water’s cold threat has actualized and you can’t even think to look up at your former teacher, the chill of the water occupying your body and mind completely. You can only splash around, keeping yourself afloat for a moment before your limbs freeze up and you feel yourself start sinking, as your head lowers into the water, any remaining light you had gets completely drowned alongside you.

Back on the boat, Pythagoras sits down and puts the still-closed bottle of wine away. “Friends, let’s head back. Speculus was right: it’s been a long day. We’re all tired. In the morning let’s all agree this was all a dream.”

**Beyond Comprehension:**

You can feel yourself sinking quickly, the water pushing up against you, though you feel no weights on your body. The force gently pulls your entire body down, further and further, the light fading rapidly as you fall. You can see the faint light at the surface, in the direction you’ve looked since your baptism and last rites began. The light from the moon grows dimmer and dimmer, consolidating from a broad spot down to a point, like a circle reverting back to the zeroth dimension.

You open your mouth to cry for help, yet produce nothing more than a smattering of bubbles, with no clear sound coming out. Even if you could scream, who would help you anyway?

Eventually, you lose track of how long you’ve spent in the water. Maybe it’s only been seconds, possibly minutes, maybe longer. You simply feel the pressure of the water squeezing you tighter and tighter, smothering your entire body. You close your eyes and open them again, to ensure that you can still see something, however faint. When you open them again, you begin to hear someone speak to you. The sound does not go into your ears but vibrates through your whole body. The words connect with your heart as much as they do your eardrums.

“Hippasus, Hippasus. Flying up to the realm of gods.”

Calm down now, your mind is still but a fawn

Learned its first truth, more than that Pythagoras

Not but a slight nugget, only a scratch on mathematics”

Your mind fixates on what happened, and you can’t make sense of it. What you did, what got you here, or even simply what “here” is escape you. Though you said no words, you sense that the voice has heard you, and understands your confusion.

“You found a way to open a door without even turning the knob. Now you need to determine how you will step through,” they say.

You ask them about the door they mention, unsure of what they mean.

“A hole through your world, opening up to broader horizons and more possibilities.”

You ask them what’s on the other side of the door.

“Slow down, Hippasus. Take it one step at a time. The path you opened may still close on Icarus’ face yet again.”

Their words remind of other mathematical oddities, similar to what you accomplished; in the same way that you found numbers that exist beyond humanity’s means of representing numbers, you think about four-dimensional geometry, and how it fundamentally exists beyond human comprehension. Just like how people can use the square root of 2 without actually knowing its value, mathematicians can discuss the properties of four-dimensional objects such as a tesseract without even beginning to understand what one would actually look like in practice. Mathematicians can only create 2D and 3D representations of what it might *sort of* look like.

You look back up at the receding moonlight, just a faint sliver at this point. The pressure of the water presses you in, compacting you. Your body has begun to ache under the now intense force. You still feel something pulling your body down indefinitely, just as incessant as when it started.

Your brain, finally catching up, distracts you from this line of thinking and you ask the voice who they are.

“Think about what you did, Hippasus. Can you understand?”

You ask what you did.

“What did you prove, Hippasus? Irrationality.”

Irrationality?

“Irrationality, Hippasus. You know its definition. You know it applies to me.”

Irrationality.

“Yes, Your Irrationality clarified me”

You ask if they can help you understand, still unsure of what they’re trying to say.

“One day you may become Irrationality yourself. You may sail away and dissolve.”

Their words remind you of Yeats’ journey, the journey you discussed with Speculus and Sageron. You wonder if Yeats became irrationality in the way they describe. Was that his intention? Yeats embarked on an intellectual journey, but his plea points to something deeper than simple knowledge. He pleaded with powers greater than humans.

The moon has now disappeared. You don’t know where you are. You can see nothing. The water pressure pains you now. Your aching body has turned to desperation. You struggle to focus on much else at all. As the force pulls you down further, you can feel the pressure increasing with every passing moment.

You suddenly think of the events that led you here. You ask the voice if they can explain what Pythagoras did.

“Pythagoras put his life before truth. His own status. Have you seen this before? Will you see this again? Pythagoras’ decision will plague your kind, as it has plagued you up to this point.”

The word “plague” strikes you.

“Yes, plague. However you may take it, Pythagoras’ sin will remain with you as long as you are.”

You ask if you can stop the plague they speak of.

“You? You already have for yourself. Ignatius cured his, but can you create a vaccine for vainglory?”

You both fall into silence, and you feel as though you look the disembodied voice in its eyes. You nod your head in acknowledgment of its presence. Suddenly the force pulling you down yields and you feel yourself grow stationary. The water pressure disappears and your body takes a deep breath, free and open. You have gone from feeling the weight of the world to feeling nothing at all. You float, suspended in nothing. You see nothing. You feel nothing. Soon your body begins to dissolve away into the water. You are nothing. You can only think. The voice speaks to you:

“Come now, you have things to do.”