#Mean value theorem

[bugmistake]

dude the substitute calculus professor we've had for the past like 3 classes is gonna fuck us up so bad i think

[falseandrealultravival]

"The Mean Value Theorem" in Home Economics (logistics-4)

In mathematics, calculus, there is a famous theorem called the "mean value theorem", but in this case the "mean value theorem" is a theorem in home economics... (a study related to household management). The discovery and proof is, of course, this Rei Morishita. This theorem can also be used to analyze phenomena such as cheating, adultery, divorce, and crime. First, write the axioms and definitions.

Heaven and earth axiom...Men receive energy from heaven, and women receive energy from earth. This is the major principle of the I Ching.

　Definition of S value (Soul value)... an indicator of vitality, including vitality. In principle, it takes values from 100 to -100. Usually not negative. People with a minus sign are "dangerous" (in many ways). Men usually start at 100 and women usually start at 0. (This is a consequence of the axiom of heaven and earth.)　

There are exceptions.

　The definition of S value calculation... Mr. A with S value X and Mr. B with S value Y interact (including sex)

　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　; This is the usual (arithmetic) mean.

Here, before explaining the theorem, an axiom is a truth that anyone can accept, and a definition is a problem

A theorem is a truth that comes out as a result of reasoning from an axiom.

It is something that has great application value.

"The Mean Value Theorem" in Home Economics

(1) In normal family life, the S value does not become negative.

(2) When a woman interacts, the S value of the couple increases. (Conditions of (1))

(3) When a man detains, the S value of the two couples decreases. (Conditions of (1))

(4) If the S value becomes negative, crime may occur.

(5) Men and women with the same S value when they meet will never get married.

家政学における「平均値の定理」

　数学の微分積分学に「平均値の定理」という有名な定理がありますが、この場合の「平均値の定理」は家政学・・・（家庭の運営に関わる学問）における定理です。発見・証明は、もちろん、この森下　礼です。この定理は、浮気、不倫、離婚、犯罪といった現象の分析にも利用可能です。まずは、公理と定義を書きます。 　天地公理・・・男は天から、女は地からエネルギーをもらいます。（中国・四書五経で最重要な文献である易経（えききょう）の大原則。乾坤一擲という言葉がありますが、乾が男・天、坤が女・地を意味します。） 　S値（Soul値）の定義・・・生活力も含めた活力の指標。原則的に１００から−１００までの値を取ります。通常はマイナスにはなりません。マイナス符号をもつ人は｢キケン｣（いろいろな意味で）。男は通常１００からスタート、女は通常０からスタートする。（これは天地公理からの帰結です。）　 例外もあります。 　S値の計算の定義・・・・・S値XのAさんと、S値YのBさんが交流（SEXも含む） 　　　　　　　　　　　　　すると、両者のS値は同じ値の（X+Y）/２となる。これは、普通に言う（算術）平均です。

ここで、定理の説明にイクまえに、公理とは、誰でも認める真理のこと、定義とは、問題 としている対象の意味を確定すること、定理とは、公理から推理した結果出てくる真理で、 応用価値の大きいもののことです。

家政学における「平均値の定理」 (1) 通常の家庭生活においては、S値はマイナスにはならない。 (2)女が交流をする場合は、夫婦２人のS値が上昇する。（(1)の条件下） (3) 男が拘留をする場合は、夫婦２人のS値が下降する。（(1)の条件下） (4)S値がマイナスになると、犯罪が起きる場合がある。 (5) 出会ったときのS値が同じ男女は、決して結婚しない。

[minmin-vs-physics]

HUP, EPR, and Bell’s Theorem

Abstract

An educational document discussing the Heisenberg Uncertainty Principle, the EPR (Einstein- Podolsky-Rosen) Paradox, and Bell’s Theorem, written for an audience without a background in physics, but with their head still screwed on right.

1 Introduction

Ah, quantum mechanics. A bizarre theory which unfortunately describes our physical world exceed- ingly well. Einstein didn’t get it. Bohr didn’t get it. I don’t get it. And soon, you won’t get it either. As the saying goes, the more you know about quantum mechanics, the less you understand it.

I will be skipping around in terms of topics covered in undergraduate quantum mechanics courses to prepare you for the actual beast, Entanglement.

Entanglement, the property of quantum systems to remain correlated even when separated, is a concept which has transformed from a worrisome byproduct of a thought experiment [1] into a cornerstone of quantum mechanics itself. What is a quantum mechanics? Google is your friend, my dear reader. My time with you is limited„ and I cannot teach you the alphabet to make you read Shakespeare. I can only explain what you directly need to understand this article. Anything else shall be your homework, and if I am feeling kind at the end, I will provide a list of accessible resources on learning quantum mechanics the RIGHT way.

As we dive into the frankly confusing world of entanglement, it is vital that you remember one thing– A quantum particle is described by a wave function, Ψ. This wave function is a solution to the Schrodinger equation.

This is what they mean when they say something is both a particle and a wave; It’s behavior can be described by a special kind of wave equation, which we all know and love as the Schrodinger Wave Equation. But that’s not important right now. I’ll explain more if I need to. We need to get to HUP.

2 Heisenberg’s Uncertainty Principle

Formulated by Werner Heisenberg in 1927, the Uncertainty Principle is an indomitable tenet in the field of quantum mechanics. Its premise is simple. The more precisely a particle’s momentum is determined, the less precisely is its position. In one dimension, this can be summarized with the following mathematical statement: ∆x∆p ≤ 2

Here, ∆x is the standard deviation or “spread” of the position x, while ∆p is the standard deviation of the momentum p. As the spread of one quantity decreases, the other must increase in order to maintain the inequality. I will not bother proving the Principle in full, but I have Heisenberg’s original proof in the references.

Is that it?

Ummmmm, no. An important thing to remember about HUP is that it is not exclusive to x and p. HUP applies to any two quantum mechanical operators, A, B, which do not commute with each other i.e. [A, B] = AB − BA = 0. But that’s all mathematical nonsense, Min! What does it really mean?

Fine! I’m only doing this because it will be useful when we get to measurements in the EPR paradox and Bell theorems. In order to understand what “not commuting” means in the physical sense, let’s use our favorites, position and momentum, as an example. In quantum mechanics, xˆ andpˆ are referred to as the position and momentum operators respectively. (Why the little hats? Firstly, they’re cute, and secondly, well, you’ll see.) The whole point of calling them operators is that they act on wave functions. And in the crudest sense possible (please don’t try this at home, folks), hitting an operator on a wave function and taking the expectation value, gives a measurement of the quantum mechanical system.

There is about three semesters of quantum mechanical education I’m waving off right now, but bear with me. When we act the momentum operator on the system, in some sense we extract the momentum. Same thing for position. However, the whole deal about x and p is that they do not commute. So, the order in which you conduct the measurements absolutely does matter. First measuring x and then p would give you a different answer than first measuring p and then x. This is because the very act of measuring a quantum state changes it. That’s right! It changes. This makes all the difference when you consider the standard deviation of a bunch of measurements. If my memory of introductory quantum mechanics serves me right, after about three pages of algebra you arrive at the familiar position-momentum uncertainty principle.

The moral of the story is that the non-commutativity of these operators manifests as a sort of granularity in the accuracy of measurements you can make on a physical system. This granularity is retained between any other kinds of non-commuting measurements you can make!

On second thought, do you really need this? Probably not. But, the algebra of uncertainty principles is a pet project to me. Especially the strangest of them all, the energy-time uncertainty principle. Enough on that! Here’s the main takeaway (other than the actual HUP statement) that you need from this section:

Making a measurement on a state changes its wave function. No exceptions. None. The detached observer is not a reality in the quantum mechanical world.

3 Spin

I realized that the following sections will not make any sense if you don’t at least know what spin is. So, let’s make a short pit-stop at Spin City to learn about this nonsensical physical quantity.

We’re all aware of angular momentum– its the rotational analog of linear momentum (which we talked about the previous section). We all agree that it is a property related to the motion of an object, right? WRONG! Sometime in the 1900s (Seriously, 20th Century Physicists should chill out), it was discovered this angular momentum from motion i.e. “orbital” angular momentum, as it was called in the atomic physics context it was first described, does not account for all the angular momentum of a particle. Long story short, the remaining angular momentum, which is intrinsic to a particle, is now called Spin. Every fundamental particle has a particular value of spin, which, in quantum mechanical jargon, is the eigenvalue of the spin operator.

For understanding the following sections, we really only need to care about spin-1/2 particles, which are lovingly called fermions, and are the building blocks of all ordinary matter. The shining feature of spin-1/2 particles is that their spin can either be +1 or −1 , which is often referred to as spin-up (↑) and spin-down (↓) respectively.

Physically, the up or down comes from whether the measured spin is along the axis it is measured, or opposite to it. Yes, spin is a vector, so it does have three independent components in the three spatial directions, but it is convention to consider the z-component of the spin for calculations and experiments. Any references to up and down in the next sections are along the z-direction.

Oh, and one more thing, spin-0 particles have no intrinsic spin. This will be important when we encounter the EPR Paradox.

4 EPR Paradox

After skipping a whole bunch of most-likely important concepts in the study of quantum mechanics we arrive at the EPR paradox.

The EPR paradox is a thought experiment first described in the groundbreaking paper [1] by Einstein, Podolsky, and Rosen in 1935. Einstein was quite vocally a hater, and the EPR paradox was proposed as evidence that the description of reality provided by quantum mechanics is incomplete. Reality doesn’t care, of course, and the EPR Paradox isn’t really a paradox. In fact, it is the foundation of entanglement– a magnificent, very real feature of reality which spans black holes, quantum computers and even my field of research: Entanglement in elementary particle physics.

In fact, I’m so self-centered that the example we will use to illustrate the EPR paradox is from particle physics. Just kidding, my explanation follows Chapter 12 in Griffiths’ Introduction to Quantum Mechanics, and is a simplified version credited to David Bohm

EPRB Paradox

Suppose a pion (funky particle with spin-0) at rest, decays to an electron and positron which fly off into opposite directions. Since the pion has spin-0, conservation of angular momentum dictates that the electron and positron occupy the following spin configuration.

√(1/2) (|↑↓⟩−|↓↑⟩)

BE NOT AFRAID of the mathematical jumpscare. The fancy bracket |·⟩ is what’s called a “ket”, and is used to denote the state of a quantum system. All the expression says is that either the electron is

spin-up (+1) and the positron is spin-down (−1) or vice-versa, because the total spin of the system 22

must add up to 0. (Since the initial state is spin zero, the system must stay spin-zero even after the decay occurs. That’s what angular momentum conservation is all about.) We don’t know which combination we will get, but it must be one of the above. Measuring the spin of one of the particles will automatically tell us what the spin of the other particle is. This means that the spins of the electron and positron are correlated. In modern terms, such a state is called entangled.

Now, let’s pretend that these particles fly off in opposite directions, until say, they are several light years apart. What would happen if we found the electron and measured its spin to be +1 ? We instantly know that the positron’s spin is −1 . This is obvious. Why are we mad about this?

Naturally, we may think that the electron really was spin-up from the moment it was created and it was only that quantum mechanics did not know until we made a measurement. But by the principles of quantum mechanics, neither particle had a definite spin, until we made a measurement, causing the wave function to “collapse” and instanteously produce the spin of the positron which is lights years away!

The EPR bros were NOT having it. Einstein famously called this phenomenon “spooky action at a distance”. They stated that the quantum mechanical standpoint must be wrong! The electron and positron must have had well-defined spins from their creation, even if quantum mechanics does not know it. Quantum mechanics is not a complete description of reality and there must be some hidden variables which describe a physical system that we do not yet know.

The fundamental assumption guiding the EPR argument is that no information can propagate faster than light. This the principle of locality. In order to appease this, we can say that the wave function collapsed at some finite velocity and is not instantaneous. However, this violates conservation– If we measured the positron spin as well before the information of collapse reached it, there is a 50–50 chance that both particles are spin-up, which means the system has total spin-1. Preposterous! You can mess with anything you want in this universe, but you don’t mess with conservation laws. What do we do now?

Okay, let’s calm down. The theorists may say whatever they want, but experiment doesn’t lie. Experiment tells us that in these cases, spin is perfectly correlated. The wave function collapse is instantaneous. That’s crazy. Call your mom and tell her you want to go home. The EPR Bros are frightening you— Quantum Mechanics is NOT local so it is NOT complete.

...Except. It is. Enter, Bell’s Theorem.

5 Bell’s Theorem

Now, what’s the situation? The EPR gang is not happy. I’m not happy. You’re not happy. Is quantum mechanics wrong? No, silly! EPR said it themselves: they think it’s merely incomplete. So, in order to completely describe a quantum mechanical state, you not only need the wave function Ψ, you also need some unknown, hidden variable λ. Lots of hidden variable theories were proposed after the Einstein-Podolsky-Rosen paper, but none of them ever gained traction. It was still a respectable area of study until 1964, when J.S. Bell proved that any local (Remember locality from the last section?) hidden variable theory is incompatible with quantum mechanics.

I’ll spare you the details of Bell’s work, dear reader. One thought experiment in an essay is gruesome enough. (It is also getting quite late and I still didn’t code my calculations. I have spent far too much time on this already.)

Bell’s proof involves the wonderful use of probability, and the barest assumptions that can be made about local hidden variable theories. Basically, in any local hidden variable theory, the probabilities of various outcomes are related by what’s known as a Bell inequality. If EPR’s conjecture is right, and there really are hidden variables we don’t know about, then any physical system must obey its Bell inequality.

Except, there have been various experiments since the 1960s confirming that Bell’s inequality is indeed violated. This came as a rude shock to scientists as it is not fun to learn that reality is very much nonlocal. It was all fun and games when this was all merely a mathematical artifact, but nonlocality felt like a gateway drug to a much grimmer violation.

Causality

Bell inequality violations, no matter how surprising, are merely wonderful correlations between two sets of otherwise random data. Sure, the measurement of the spin of the electron affects the positron, but it does not cause it in any meaningful way. The person measuring the electron spin cannot use this collapse of the wave function to send a message to the person with the positron, since they don’t control the outcome of the experiment. They can decide whether to measure the electron at all, but the other person only has access to the positron’s spin and cannot tell whether the electron has been measured or not.

Phew! This sort of nonlocal influence does not transmit any energy or information, so it is exempt from the speed of light. Meanwhile, causal influences, those which do transmit information or energy, cannot travel faster than light. According to special relativity, if this was possible then, there are reference frames in which information can propagate backwards through time. And that, my dear reader, is what we call a big nono. Since the EPR paradox does not imply that causality is violated, we can lie uncomfortably on our bed of nonlocal but causal theory of quantum mechanics.

So rest easy, quantum mechanics is weird, but safe. Entanglement is not a fairytale, but also not the boogeyman. It’s probably more scared of you than you of it. Just give it some time. More answers will follow.

What Do I Do Now?

So, you want to know more? Or curl up in a ball and never think about this again? Either is fine. I won’t judge. If your answer is the former, here are some resources to guide you through the thickets of quantum mechanics.

PopSci Sources

1. IDTIMWYTIM: Heisenberg Uncertainty Principle 2. Why did Quantum Entanglement Win the Nobel Prize in Physics? 3. Bell’s Theorem: The Quantum Venn Diagram Paradox

Surely, you’ll get more out of these wonderful science Youtubers than you did from me yapping for four pages. There are a bunch more probably, but you’ll have to find them yourself.

Academic Sources

1. An Introduction to Quantum Mechanics, D.J. Griffiths.

Of course, there are other quantum mechanics textbooks that I like much more than this one. But, this is the least daunting, so I’ll leave it here.

Don’t forget to like and subscribe for more silly academic style papers.

References

[1]  A. Einstein, B. Podolsky and N. Rosen, “Can quantum mechanical description of physical reality be considered complete”? Phys. Rev. 47, 777–780 (1935) doi:10.1103/PhysRev.47.777

[2]  Heisenberg, W. “Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik”. Z. Physik 43, 172–-198 (1927). https://doi.org/10.1007/BF01397280

[3]  D.J. Griffiths, D.F. Schroeter, “Introduction to Quantum Mechanics, Third Edition” Cambridge University Press (2018) 978–1–107–18963–8,

[flagellant]

Re: VIKTOR & ROLF Couture Fall/Winter 2024

I know fashion, especially couture, is meant to be an expressive and exploratory art. I think the exploration of geometry in fashion, silhouettes, and textures here is really neat! But I am unfamiliar with the designer - is there more to those pieces than meets the eye?

Sometimes there is less evident meaning in a collection to those who don't spend a lot of time and study in the industry. Where would you recommend going for more information on current collections?

If you remember these dresses from last year, Viktor & Rolf have been having a very interesting conversation about current fashion trends and the intersection of the cultural zeitgeist. Structuralism is big in fashion right now, arguably the biggest since Antoinette did her shit. A bunch of designers are fucking around with pushing the limits of what is possible to be worn on the body in a very literal sense, treating garments as almost like physics theorems to be solved. Textile sciences, 3d modeling, etc--it's all being used to make dresses that are more fiendishly complex than the last, than arguably we've ever been able to make.

And Viktor & Rolf specifically seem to be having a very purposeful dialogue of the absurdity of it in the face of what beauty standards are "supposed" to be, what high fashion is "supposed" to be. Their works have always relied heavily on pushing those boundaries and the current couture tech boom is just enabling a lot of their weirdness, which is absolutely fantastic to watch.

I would argue that their 2024 FW collection is specifically exploring what it means to be prioritizing those structuralist tendencies over the actual bodies they're being worn on. It's a conversation piece going back to the old hack question: "Who would ever want to wear this?". The harsh geometries make most of the collection entirely infeasible and impractical for actual wear. No celebrity is gonna want to sit at the Met wearing any of these. Their only value is...what, exactly? Where IS the value coming from? Why do we consider Viktor & Rolf as high fashion icons if the dresses they keep making seem to be actively infeasible for even their "intended" purpose as red carpet event shit? What happens to the fashion industry when the ability to make whatever you want no longer requires it to be intended to be worn, to be used a single time in that runway show and then shoved in an archive?

When couture maisons have to pump out hundreds of dresses every two months, when the world is burning down around us, when none of this matters--isn't it refreshing to see some absolutely buckwild shit on a runway? In a world where the spectacle is supposed to distract from the horror, it feels lately like Viktor & Rolf are specifically hamming up the absurdity of it all. They've watched the industry change from when they first started out into becoming something alien and unrecognizable. Fashion design demands Autocad and 3d printing and literally engineering in order to keep ahead of the pack--and in what is supposed to be an industry defined by the inherent humanity needed to design couture, perhaps these strange polygons are a sort of reckoning with that.

[ultimateswag]

𝚂𝚝𝚎𝚟𝚎𝚗 𝙼𝚎𝚎𝚔𝚜- Study Sesh

Paring : Steven Meeks x fem reader

TW : (brief) mentions of biblical studies and Pythagorean Identities.

[this is my first imagines by the way, so please excuse any mistake or poor writing (I tried to not mention any racial aspects, but lmk what I can improve)]

Latin was always a subject you mastered. It got much easier once you began to assimilate Latin with French, which, by the way, you were also pretty good at. One was a dead language, not spoken nowadays, but really useful when doing biblical studies, and then the other was the language of romance, the one people thought of when asked to say the most romantic and sexy language there is.

Though, besides languages and the occasional English, you weren’t really exceptional in any other subject, especially Trigonometry. It's not like you were failing, not at all actually, but you simply couldn't excel it like, by example, literature. That's why you’re pushing yourself to the edge, so you can get better at the calculus assimilated subjects, and that's why you asked Steven Meeks for help.

The two of you have always been friends, talking to each other in the corridors, of course being together in study groups, and also being a part of the same secret society. But apart from those friendly interactions, you didn't stand out as actually being friends, so much so, that few people actually believed that you even knew who Steven was. Being the only girl at Welton preparatory school due to your high intellectual level ( and family history) made you quite popular, so why would you, willingly, hang out with Steven Meeks? That's what they all thought, though, no one dared to say it out loud.

"I still don't understand this." You said, a loud sigh leaving your mouth quickly after as you stared at the problem filled paper in front of you.

Steven let out a small low chuckle, though it was clear he didn't mean it in an offensive way. "This is actually simpler than it looks" he begins, moving the paper a bit closer to him, and scooting lightly closer to you. "Once you understand the concept, it'll all make sense." And to that, you nods, gaze switching from the paper to his face, admiring his soft freckles, beautiful eyes and- and how he begins to explain Pythagorean Identities.

"Alright. So. Pythagorean identities. The most important one is kind of like the golden rule of trig: sin?0 + cos 0 = 1. That's the foundation, everything else branches from this." He explains.

You leaned in, the scent of your sweet raspberry perfume just barely brushing the air between them. "But why does that even work?"

Steven looked at you not just any glance, but the kind of look that says I see you struggling, and I want to help you out of it. "Because it comes from the Pythagorean Theorem." He says, but swing the confusion linger on your face, he grabs a piece of blank paper and a pencil and begins to draw." Picture this: a right triangle inside the unit circle. The hypotenuse is 1. The x-value is cose, the y-value is sine. So when you square both and add them up... it just always equals 1."

You blinked, eyes narrowing slightly in concentration as they switch from the homework, to Steven's drawing. "So it's not just a random formula-they actually mean something."

He nodded, a little too pleased you were catching on. "Exactly. And the other two identities are built off this one. Like, if you divide everything by cos?0, you get: 1 + tan?0 = sec?0. And if you divide by sin'® instead, it becomes: cot?0 + 1 = cscª0. It's like a family of truths that are all connected."

You bit your lip, scribbling it down, then looked sideways at him. "Do you always talk about math like it's poetry?" You ask with a small smile. Steven, just like you, has always taken a special liking to poetry, and just like everyone else, after just a few classes with Mr. Keating, it became a bigger, more important part of his life, so of course it was no surprise to see him romanticizing trigonometry.

Steven smiled, a faint flush rising in his cheeks. "It makes it easier to understand, does it not?" he tilts his head lightly at you, seeing as you wrote the problem's answer down on the paper, with your perfect calligraphy, while your beautiful, hair fell next to you. Your eyes seemed to shine as you understood each problem, they looked beautiful, you looked so beautiful. At least that's what he thought.

"I suppose it does" A soft laughter is heard leaving your mouth. There was a beat of silence-just the hush of pages turning in the distance and the soft tapping of your pen again. "You know," you said, not looking at him this time, "I usually hate asking for help."

"I know," he replied, glancing at her-noticing the tension in your jaw softening a little. "But I kind of like that you asked me.

Their eyes met. Just for a second. A heartbeat.

“I do too.” She said softly, turning to look at her notes once again. Her cheeks were feeling hotter, she didn’t want to look up.

“Do you need help with anything els-“ Before he could finish his frase, Aurora began speaking once again, mustering up enough courage to glance at him.

“Would you like to hang out? Outside of school?” She asks, cheeks getting hotter and hotter .

“That’d be pretty nice, actually.” Steven’s now held held a crimson tint, although he tried to act like he wasn’t about to jump out of happiness.

“Cool” she says, attempting to act nonchalant, but giving him a shy smile.

“Cool.” He answers back, returning her smile.

[legionofpotatoes]

I had been floating through my apartment these past days suppressing every sense that blared alarms at the gaping hole in the middle of everything. Was pretending she was still snuggled up somewhere out of view. Finally processed it by cleaning up last night. Her toilet, her brushes, her little paw prints on the counter. This shit sucks fellas

I'll self soothe by pointlessly journaling again I guess!

I just don't know how to exist without her at home it feels like. After cremating her I stayed with my partner for a few days and then the first night back home was the emptiest I'd felt in my entire life. Dramatic!

why'd I just make a joke there? It did feel that way. Her smell was everywhere, her stuff was everywhere. Took so many photos of those faint paw prints. I move through the rooms like a ghost. I feel like I am floating because my body is rejecting every attempt at being present. Does this make sense? I don't want to focus on any one moment. Like rolling a boulder up an incline and the boulder is also punching me repeatedly because the boulder is reality itself that I am pushing away.

I am not good at grief and death. This is a lesson that I had to have beaten into me these past three years. I don't deal with it linearly, properly, healthily, or at all like an adult should, I think. I hate how much it shuts me down and turns me into a black hole of energy. Through the haze I can see myself being a nuisance to my loved ones sometimes but I have no heart to push through it. I feel like this is a pretty universal thing. Seen it happen from outside in.

they say grief isn't a finite thing but a process. And I can attest that it definitely transforms over time! But it's also additive? Everything else in life feels balanced in its give and take. But death only compounds. And I often feel like I am running out of places to store it in. It's like a game of tetris each time. Where does this terribly mangled piece fit into my psyche in a way that doesn't unravel everything? Oh it's nowhere? No worries just bolt it on there anyway and the aerodynamics of moving through life will smooth some of the edges eventually. Others not so much.

I have a deeper insecurity that pisses me off in this whole thing. A lot of my worldview and baseline beliefs rest - theoretically, it now seems - on a staunch respect and acceptance of what death is supposed to be. In the natural order I mean. Circle of life and all that. I feel like I look at so much loathsome reactionary sentiment around and trace it back to feelings of inadequacy, powerlessness, and on an even deeper level, a paralyzing fear of death. So, intellectually, my ego rages against it. I chase those themes in stories and philosophical theorems, I pay lip service to the ineffable value of mortality, I worship at the altar of transience, of change, of being a temporary perspective point of a universe that constantly expels and subsumes them to and fro. I like that framework of understanding life and death; I want to inhabit those beliefs. They feel right, primal, divine.

But then death touches my life and I have such idiotic trouble squaring what I feel with those beliefs. There's a lot of physiological trauma associated with it for sure - body recoils at the sudden shift in routines, absence of familiar patterns, lack of crucial stimuli. Maybe that's why? Feels almost like love is a resource I gently tap from multiple sources across the lives that surround me, for nourishment and fuel, and when an ore is lost there's no recourse. I cannot simply truck on and aggressively reallocate remaining channels to maintain bandwidth. Gotta grieve, gotta change somehow.

It's not for them, my partner tells me. And she's right. The cats are at peace, no longer in pain, no longer inhabiting their tiny perspectives of our greater whole. Everyone else too, all the loss I keep recirculating like a demented merry-go-round inside my head, they're okay now. In whatever context that can be defined in, however you spin it - their corporeal suffering is at a confident zero. Their essence, one could say, is in a far less entropic state, peacefully blanketing and guiding the flow instead of being confined to a confusing and traumatic flesh puppet caught in its currents. So yeah I know they don't need my tears, my head understands that, but I think there's something less personal in mourning all this.

Like with chichi specifically when I drill down to it it's the experiences. I collapsed on the floor and wept at the small hairs still visible on the brushing glove because the specificity of that ritual between this particular kitty and her dad, that small gesture of love, is forever lost. The kitty is at peace and her idiot companion will be fine with his drawings and video games, but the experiences that bound them cannot carry on unchanged. That's quite sad. Bodies and lives ending is normal, necessary, and kind. Relationships and experiences, however, were never a predestined vessel born out of stardust. They formed out of thin air, from that magnetic chemistry between unlikely cohorts. Seeing those end abruptly isn't as easy to chalk up to cosmic scales. It's fair to lag and glitch while your brain deals with the sudden loss of those calcified rituals. Maybe that is what's crossing my wires when reconciling personal loss with my highfalutin beliefs.

I sound like a loon. But mostly just reassuring myself that my grief is normal maybe and not as weird and childish as I try to make it seem. I know cats are idiots and pets in general represent a selfish type of unconditional love we like to cultivate, but it's also more than that. It's stupid to even assume otherwise. My favorite artwork of all time is, and don't punch too hard if you've heard this one before, brosio's two earthlings. And it is one hundred percent because of the name. Cause they're both the same thing. Separated not just by time, but biology, sapience, death itself. and yet they're still somehow the same exact thing. Like me and chichi were.

Different eyes and different brains watching and interpreting the same strange reality, and choosing to coalesce in that experience. I don't know how true it is that ancient felines domesticated themselves in order to live alongside people for a steadier supply of food, but there's something so whimsically sacred in that possibility. these utterly alien hairless beasts seem prone to throw food our way, so maybe we'll hitch a ride and see where it takes us.

I hope chichi had a good one with me. She was the sweetest, gentlest, kindest soul in the tiniest, fluffiest body. I knew her mom, was there for her birth, and watched her maintain that peculiar enjoyment of human contact even as she matured and went through her own challenges on the streets (I couldn't rescue her until after I moved into my own apartment). Nothing seemed to blunt her spirit, no expected pavlovian defensiveness or aggression manifested. one day I discovered she had a BB pellet embedded between her shoulder blades, skin already healed over the impact point, trapping the tiny bullet under her epidermis. the vets said it was okay to leave it in.

I hope whoever shot at her is living their worst life. but her response to me touching the pellet was to immediately plonk into my arms for a cuddle. that was her response to most things. she lived to love and be loved. she made pancakes all the time. it was her form of response prioritized even above meowing. she'd be asleep with her paws up and if I quietly whispered her name, her upwards beans would start gently curling and opening in mid-air before a single peep would leave her mouth. I kept her her nails untrimmed due to us living on a higher floor - better safe than sorry - so those kneads were felt keenly. today I would rip my arm off and eat it just to feel one of her paws digging into my knee again. but I like to think the clouds are extra cushiony in the sky since she went up there and started fluffing them up.

she had many litters, and after arriving into my home, she enjoyed a calm half a decade of retirement. eleven years isn't a long time for a cat, but I like to think it's more than most get in her predicament. we had to remove her teefs due to an infection, move to an all-soft-food diet, and take extra care not to run into that silly tongue hanging out as a result. I'm not sure if she would have lived more years or less without me swooping her into my arms, but I like to think it's the former. and that it was her own choices that made that possible.

I loved not just the warmth she spilled into my heart, but the very real being behind her eyes. the interior life of this impossibly small idiot who made it her life's mission to cuddle every living thing. I would sometimes look at her contended face squished against my thigh, gently vibrating from a deceptively strong purr drive, and I would wonder. just how can any form of life possibly hope to aspire to something more than this, something purer or kinder than a simple yearning for quiet togetherness. why greatness? why is goodness not enough?

chichi was very good. she made me better, too. she greeted death as she lived life, with dignity, calm, and relentless biscuits. I'll never forget cradling her head as we drove to the final vet appointment. I was in the left back seat, she was lying down in her bed on the right side, and her body was too weak to move. but her eyes were darting, inquisitive, curious. I lifted her noggin and gently cupped it from behind to guide her gaze upwards. dappled lights danced on the window as we approached the clinic, and I could see them reflect in her eyes like stars. we parked the car, and her side of the door opened.

right as she was about to get picked up for her final on-foot transit, I saw her squeeze out one last pancake. simply built different.

I love you pancake monster. I'll try to be okay and remember you fondly. I would say you'd hate the gauche etsy urn I picked out for your ashes, but you hated nothing. you contained only startdust and love.

I hope to feel you again sometime.

[rbrooksdesign]

"DMT-EDMT Series for December Geometry #8"

This acrylic painting image from the recent past (2023) is yet another perspective on the Butterfly Fractal 1 expression: BF1 --- 1--2 --- (center YELLOW & WHITE) goes up p=2 STEPS (levels) where it reveals the value 2 = x. The LEFT side of the image gives the various multiple of the p value = STEPS, the RIGHT side gives the ID Parameters and their values (x, y, z=Mp, z²=MPS, xy=PN, xy=CR).

At 2p=4 STEPS, we reach 6 = Perfect Number (PN) -- also in WHITE. p=2 ID parameter means 2ᵖ⁻¹ = 2¹ = 2 = x, and, 2ᵖ -1 = 4-1 =3 = z = Mp candidate. (See bottom.)

This image came before the discovery of the Divisor Matrix Table (DMT). It holds the same vital information:

There are two sets of doublings within every "container" that relate to the BF1:

1. First set is the BF1 itself --- 1--2

2. Second set is the second doubling that starts with the ODD 3 to give --- 3--6 ---.

On the DMT, we would see this as:

1--2--4--8...

3--6-12-24...

5-10-20-40...

with the 1--2 and 3--6 in BOLD.

The pattern holds for ALL BF1-derived "containers."

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Math: Euclid-Euler Theorem = 2ᵖ⁻¹ (2ᵖ -1) = Perfect Number (PN) where 2ᵖ⁻¹ = x = 2¹ =2, and, 2ᵖ -1 = z = 3 = Mersenne Prime (Mp), as xz=PN.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

[experiment-001-izuru-kamukura]

How does one objectively measure boringness, hm? Did Hajime not have his own personality? His own passions? Friends? Family? Dreams? What do YOU have to be able to make that decision?

Let's see. I shall use a one-sample t-test since we do not know the population mean μ nor its standard deviation σ.

We will use a scale of 0-10 ("intolerable"-"endlessly fascinating") and consider 3 to be "boring". I will call this variable "B".

H0: B = 3 Ha: B =/=3

I have collected data from a few of my own talents, which has provided me with a sample of 156. I have calculated the sample mean x̄ to be 2.89, with a sample standard deviation s of 0.56.

The test statistic is then -2.453, and the degrees of freedom is estimated to be 155. With a confidence interval of (2.33, 2.51), it is 95% confident that Hajime Hinata's boredom level exists between 2.33 and 2.51. The p-value is 1.984, and since it is greater than 0.05, we reject the alternate hypothesis.

In that sense, Hajime Hinata is not found to be statistically more interesting than 3, but also not less interesting than 3.

As I am compromised of at least +30 opinions, we can invoke the Central Limit Theorem regardless of the QQ plot presented. As a result, we can presume normality is satisfied.

You could make an argument independence is not satisfied. I do not care. I consider myself an independent vessel for many voices.

[autogeneity]

I think something that was underemphasized in my education was the framing of learning as a process that occurs over time.

I mean, to some extent and more so in certain subjects, you are just looking to accumulate some static information in your brain*. but I think most professors would say they are ideally looking to impart a more nebulous sort of skill in "how to think". like, if you are studying mathematics, you need to know the content, and be able to prove theorems, but you also want to be developing a sort of intuition for thinking mathematically.

[*I said this originally in the draft, but in fact even static information seems to be best acquired through spaced repetition!]

they will tell you to practice problems, of course. but it is framed as a way of getting better at doing the problems, not as a generalisable skill. not like in: you have to move the material through your brain repeatedly in different ways to get its value. not like in: you are trying to forge adaptations, not much differently to building muscle. often like: "it might take some time to understand this [because it is hard and you will struggle with it]" (read: if I'm clever enough I could totally pull it off quickly, skill issue), but not like: time is an actual ingredient in this recipe.

[storiesofsvu]

Fake it til You Make It

Elizabeth Donnelly x reader warnings: language, sass, smut mentioned? covers the 5+1 trope for @prentiss-theorem s bingo!

The Dilemma.

“Heard you took a plea on the Harrison case.” Liz’s sentence was spoken as she sauntered into your office, perching on the side of your desk as you finished scribbling something on the work in front of you.

“Hey now, I drove Buchanan up from the original offer before I accepted.”

“Good girl.” She praised with a wink and you laughed, “Kristen took the other SVU case, correct?”

“Yeah.” You let out a happy sigh, flipping the folder shut and leaning back in your chair.

“Means you’ve got time for an extended lunch then?”

“Only if you’re buying.” You grinned up at her and she scoffed, swatting at your shoulder while she stood from the desk.

“You want your fall bonus or not?” She teased and you rolled your eyes as you grabbed your blazer.

“Fine. I’ll pay. But it’s on you next time.”

“Seems fair.”  Liz shot you a playful smile as you scooped up you bag and followed her out of your office.

Since transferring into SVU you’d had a great relationship with Liz, even if she had been terribly intimidating for the first couple of months. Once you’d proved that you could hold your own against the detectives, that you were a whiz in the courtroom and happened to actually play by the rules, you’d shifted into her good books. Into the books that meant she liked you, that you were someone to be watched because she knew you’d go places. She took you under her wing like she did with a lot of previous A.D.A’s, her office becoming the automatic place that you went when you needed a sounding board, when you got stuck with a case, or couldn’t remember which law journal you saw a specific note in. There were evenings she’d be on her way out of the building only to find you still working away in your office and she’d bribe you out of there with the promise of dinner and drinks, first they were at local lounges, then they upgraded to her Brownstone, where she kept the good scotch.

Nights, lunches and coffee breaks that started out shooting the shit about work, spit balling court strategies based on the defence and presiding judge with Liz as your mentor shifted into ones about office gossip, the little whisperings in the hallways that everyone thought Liz didn’t hear. Then they became a way to escape work for an hour, you learnt about her past, her personal life and she heard stories about your family, friends, days back at law school. You became friends, and you took that with a large badge of honour, Liz was someone you looked up to, someone that you very much valued her opinion of you, having her call you an actual friend was the highlight of your month. She didn’t dole out that label very willingly.

It was on your slow walk back to the office from lunch that day that you paused in your words, stalling in your step as something caught your eye in the window of a shopping mall. Liz barely caught it in time to turn back, grinning at the way you tugged your lip into your mouth as you stared at the dress.

“See something you like?”

“It’s cute…” you muttered, “but I think it’s a little too fancy.”

“Hot date?” She asked and you snorted.

“Liz please, I just spent twenty minutes complaining about Tinder.” You laughed, “my sister’s engagement party is next weekend. I think the vibes are somewhere between brunch with the girlies and garden party, this could work for the wedding though….” Your head tilted for a moment and then you threw you head back and let out a groan, “fuck I hate this.”

“Weddings?”

“More so that it’s my younger sister, my much younger sister. Who has been obsessed with her wedding since she was like, six so you know it’s gonna be a huge, all out affair, no holding back on anything or any expense. I had high hopes for that last Tinder girl so all the invites I replied with a plus one and now I’ll have to suffer through it on my own. Better get ready for a million comments about how it’s lucky I’m a spinster cause there’ll be no money left for my wedding after this.”

“Oh come on.” Liz barked out a laugh, “you know a plus one doesn’t have to be a date, right? You can bring a friend. You said next weekend? Saturday or Sunday?”

“Saturday at four.” You sighed, turning away from the store and falling back into step beside her.

“Well then I guess you’re lucky Lena and I play poker on Sundays.”

“Wait..” Your head shot up to look at her, “you’ll come with me?”

“Sure, why not.” She smiled, linking her elbow with yours, “you said no expense spared and to me that sounds like an open bar and a lot of free food, who am I to say no to that and family drama?”

“Have I told you recently that you’re my favourite boss?”

“You don’t have to.” She grinned, “I already know everything.”

**

The Misunderstanding.

When Saturday rolled around you and Liz were pulling up to the Equinox Hotel at exactly four forty five. Early enough you wouldn’t be scolded for being tardy yet late enough that the rooftop terrace would be swimming with people who could distract your parents now that the party was in full swing. The terrace boasted incredible views of the Hudson River along with the rest of New York, a very well stocked bar, roaming cater waiters with various trays of food and a long table of hot food.

You were able to get to the bar without being noticed, both of you quickly downing a shot of maple whiskey, thanking the bartender for your cocktails before turning to the crowd. Liz’s eyes moved through the crowd, wondering if she was going to run into anyone she knew before she spotted her.

“Incoming.” She nodded over your shoulder, “white sundress, I’m assuming your sister. What was her name again, Jennifer?”

“Jessica.” You muttered and she chuckled at the way you chugged back half your cocktail before Jess was squealing in your ear, wrapping you in half a hug.

“Hiiii! I’m so happy you came!”

“Oh come on, you’re my sister, I wouldn’t miss it.” Smiling, you pressed a kiss to her cheek, “you look beautiful.”

“Thank you.” She beamed, squeezing at your hand, “love your dress, is it new?”

“Yeah, picked it up last week.”

“Cuuuute.” Grinning she then turned to the woman beside you, “who’d you bring?”

“Jess, this is Liz, Liz, Jess.” You gestured between them, watching as they shook hands and said hello.

“Pleasure.” Liz smiled.

“It’s so nice to meet you, y/n never brings anyone to holidays.” Jess giggled, “and I love those earrings, are they Mejuri?”

Liz’s head tilted, a small grin on her lips, “she know her brands and she’s got taste.” Her eyes flicked down to her wrist, “Chopard can do no wrong when it comes to their bracelets.”

“Isn’t it stunning?” She lifted her arm up, her eyes wide, “it was a gift from Colin for tonight.” She let out another giggle, her smile getting even larger if that was possible.

“There you are!” A voice called out and your head shot up to see your parents and Colin making their way over to the small group, “I was beginning to think you weren’t coming.” Your mother clicked her tongue and you let your sigh take form into a smile but before you could speak, Jess was turning around with a gasp. There were so many voices jumping back and fourth as they approached you nearly missed what came out of her mouth next, taking a second for your brain to register it.

“Mom! Come meet y/n’s girlfriend!”

Your eyes flashed wide and you barely had time to glance up to Liz before Colin let out a laugh,

“Girlfriend or sugar mommy?”

“Colin!” Jess hissed, slapping him on the arm.

“Don’t be rude.” Your mother scolded.

“She’s not—” You tried to interject but your father’s hand squeezed your shoulder.

“Sweetheart, we don’t care how you live your life as long as your happy and healthy.”

“Dad!” You tried again but suddenly Liz was squeezing at your hand gently, having completely slipped into the role of fake girlfriend as she stepped closer to you, smiling at your parents.

“I assure you; your daughter is very well taken care of with no need for anything like that. It’s a pleasure to finally meet you, I’ve heard so much already.”

“And to think we haven’t.” Your mother nearly frowned in your direction, “girl certainly likes to keep her secrets. Call me Paula.” She extended her hand,

“Elizabeth.”

“Daniel.” Your dad’s turn was next, firmly shaking Liz’s hand, “how long have the two of you been together?”

“Oh, a few months give or take.” She replied with a small laugh, “you know how it is, sometimes these things are hard to put a solid date on, but we’re happy, and that’s all that really matters.” Her hand slid back into yours, giving it a soft squeeze as she grinned over at you.

“Do I dare ask how you met?” Paula asked, wincing, “because if Colin was right, you can just say a dating app or something.”

“Mom!” You nearly choked on your drink, feeling the heat creeping into your cheeks.

“No, not at all.” Elizabeth laughed, “we work together. And your daughter is one of the brightest and best lawyers we have, she’s got an incredible future ahead of her.” This time you were sure your cheeks were tinged pink, hearing Liz tell you how good of a job you did on a case always made you more proud of yourself, like you were impressing your favourite teacher in school, you knew you were on the right track.

“Well after that Ivy league education I sure hope so.” Daniel commented, glancing to you, “would hate for that to go to waste. I was beginning to think you were so married to the job you’d never find anyone.”

“Wow.” You muttered and Liz chuckled.

“Well it certainly helps being in the same field. We both know how many hours you have to clock if you want to be successful. That you’re not always going to get the time for those fancy dinner dates by candlelight, sometimes its takeout surrounded by case files in a dim lit office. Isn’t that right darling?”

“Date night is date night.” You shrugged, leaning into her embrace, “doesn’t matter where it is, we’ve gotten the balance figured out perfectly.”

“I’m so glad you’ve finally found someone.” Paula beamed.

“Mom!” Jess interjected, grabbing her hand, “Auntie Carol just got here!”

“Alright, alright.” She laughed, swatting her away before turning back to Liz, “can’t wait to see you at the wedding, we’ll have so much more time to chat. It was wonderful meeting you.”

“You too.”

Quick waves and smiles were shot through the group before your sister was dragging them off elsewhere. Letting out a quiet groan you turned back to the bar, sucking back the rest of your drink.

“I am so sorry.” You muttered.

“It’s alright.” Liz chuckled, flagging down the bartender to get you a refill before her arm loosely wrapped around your waist and she leant in, her lips nearly brushing against your temple when she spoke, “parents love me. Now relax, drink,” she slid the refilled glass into your hands, “enjoy the party with your so called girlfriend.”

Your head tilted to look up at her as your eyes narrowed, replaying the last few minutes in your brain, “you’ve done this before…”

“I need entertainment in my life somehow.” She shrugged and you laughed.

“Are you serious?”

“Why not?” She smiled back at you, “sometimes someone just wants a plus one, sometimes it’s the whole fake girlfriend thing to piss off the family or get them off your case about being single. As long as they’re a friend I really don’t mind. I enjoy getting out of the house and it’s the closest I’m going to get to a date now, I’m divorced and remarried to my job. The only time I have is for a one night event to play pretend.”

“You really are good with parents.”

“That one can be a little trickier,” she commented, sucking back on her drink, “sometimes they freak out over same sex or the age gap, you’ve got supportive ones.”

“I think it’s just gotten to the point where all they care about is that I don’t get eaten by my plethora of cats when I die alone.”

“Do you even have a cat?”

“No. I don’t even like them.”

“Then they’ve got nothing to worry about.” She smirked, “you’ll just stink up your apartment for a week until your downstairs neighbour makes a complaint.”

“Wow,” you burst out a laugh, “thanks.”

“C’mon.” She pinched at your elbow, nodding toward the buffet, “let’s get some food before all the good stuff is gone.”

**

The One with Tequila.

Night clubs were not your thing.

They hadn’t been your thing fifteen years ago when you were much more in your partying era, so they certainly weren’t your thing now that you’d matured. You preferred nights at quiet pubs or dive bars where it was acceptable to wear leggings and an oversized hoodie, even better, you preferred nights in on your couch where you had absolutely no one to impress and you could control the volume. Squeezing yourself into a tiny dress after spending half an hour on your hair and make up just didn’t seem worth it, even if there was bottle service.

You’d been genuinely surprised when Liz said she would come with you, expecting her to tap out of the more wild nights leading up to the big event, but she was there by your side in the limo, and didn’t shy away from sliding into the booth beside you once you were at the club. The bass of the music thumped through the room, coloured lights flashing on the dance floor, there was no choice but to yell over the music while your sister dolled out tequila shots. Even sitting directly next to each other you struggled to hear the other person, causing you to lean right into Liz, your lips nearly brushing the shell of her ear when you spoke. She braced herself when she leant into you, hand softly squeezing your thigh as she spoke and she didn’t bother to remove it after, her thumb brushing against your bare skin.

She laughed along with the jokes and innuendos thrown around the table, indulged in more than a few shots with the group and even shared a few of her own stories. Some of which left you wondering if they actually happened of if she was just that good at making things up. You were disappointed when she was in full support of Jess dragging you out onto the dance floor, shooting daggers at her as you disappeared into the crowd and she was left laughing at the table. She watched with a grin as you loosened up, letting the liquor flow through your veins, dancing alongside your sister and some of her friends. If she was to be completely honest, she really liked getting to see this other side of you, to know for sure that you let your hair down once in a while and weren’t completely consumed by work.

Neither of you had any concept of how long you were out dancing, but Liz knew she had finished another drink and a half by the time the rest of the group returned to the table. You collapsed down next to her with a laugh, reaching for your drink so you could guzzle it down, trying to combat how out of breath you were. Her hand found yours on the table top, fingers interlacing with yours,

“Wasn’t so bad, was it?” She teased, pressing a feather light kiss to your heated cheek and you giggled softly.

“Okay maybe it was kinda fun.”

“See?” With a grin she slid a shot glass toward you and you laughed once more, clinking the glass with her own before you threw back the shots.

“Okay!” Jess’s arm flailed over the table top, smacking at your hand, “now that mom and dad aren’t here, can you fucking tell us the truth?”

“About what?” You asked with a laugh.

“Is she—” she glanced from Liz to you and then turned fully to Liz, “are you her sugar mommy?”

“Jess!” You shrieked, “no!”

“No.” Liz chuckled, sipping at her own drink.

“I so don’t fucking believe you.”

“Why not?!” You asked with a bark of a laugh.

“Uh… she’s older than you, she’s clearly got money and taste, she’s paying your tab tonight, there’s a vibe that she’s the dominant one.”

“Jessica!” Your eyes widened in shock, “she’s my boss! That’s where that ‘vibe’ is coming from.”

“Darling I don’t think you’re helping the matter.” Liz squeezed at your hand with a chuckle before turning to her, “I can choose to spoil my girlfriend where and whenever, tonight seemed like a good opportunity.”

“Lies.” Jess’s eyes narrowed as she stared across at the two of you, “you have way too many sugar mommy vibes.”

Liz laughed, playfully shaking her head, “believe what you want, I guess.”

Luckily for you, a new song came on that made Jess let out a shriek and completely forget about the topic at hand, racing back out onto the dance floor with her friends. Liz squeezed at your wrist, sliding another shot toward you that you thanked her for, swiftly taking it before swallowing and pulling a face.

“I’m drunk.” You stated and she laughed.

“Yes you most certainly are.”

“Sorry.” You grimaced, suddenly realizing you were more intoxicated than you’d meant to get, especially being around your boss.

“No need.” She laughed, “this is the drunkest I’ve been in ages.” She reassured you with a squeeze to your hand and your moment of worry was completely gone from your brain.

“I want pizza.”

“There’s a place around the corner, c’mon.” She tugged you from the booth, the idea of hot greasy food now the only one on her mind, “let’s get out of here before they notice.”

You were incredibly thankful fifteen minutes later when you were leaving the pizza shop with fresh slices, a bottle of water tucked away in your purse, Liz’s arm slipped through yours as the two of you laughed your way through the New York streets. Truly, both of you had a pretty good time tonight, it was something neither of you would have done voluntarily, so drastically different from how you’d spend a Saturday night and that was always a nice break from life.

“I cannot believe Jess is convinced you’re a sugar mommy.” You mumbled over a bite of pizza.

“I’m just shocked your dad was so quick to accept it, he didn’t even seem surprised. Most parents get even the inkling of that type of dynamic and they lose it, thinking there’s some kind of taking advantage going on. Or their upset because they equate it to some form of prostitution, that their kid should be working for money in a real career path.”

You slowed in your steps as your head turned up to look at Liz, your brow furrowing as you took another bite of pizza.

“What?” She asked with a laugh.

“Have… you actually done that before?” You asked and Liz smirked, the lowered inhibitions thanks to the tequila letting her reveal more than she would have on another day.

“Oh come on.” She chuckled, “Casey’s wardrobe was in desperate need of an overhaul and she was raised so strictly catholic she barely understood sexuality, much less her own. I figured why not help her out with both?”

“I knew it!!” You shrieked, fully stopping in your tracks as you turned to face her on the sidewalk, “oh I absolutely knew it! But I was so fucking confused! I thought I was going crazy cause you’d only ever mentioned your ex husband.”

“Just because I married a man once doesn’t mean I’ll do it again.” She grinned, “I didn’t think I was that secretive about it, you should have caught onto at least some hints.”

“I mean I kinda did.” You laughed, “especially when it came to Casey…”

“Oh come on, we weren’t that obvious.”

“Oh, no, of course not.” You smirked, “aside from the constant eye fucking in the courthouse halls.”

“We were not that bad!” She objected, smacking your arm with a playful frown.

“Sure.” You laughed, slipping your arm back into hers as you fell into step beside her again, “just keep telling yourself that.”

Liz walked you home that night, making sure you were safely behind locked doors and promising to drink some water before bed before she flagged down a cab. Upstairs you found yourself replaying moments from the night, interactions between the two of you, stories she’d told you that all were shone in a different light now that your speculations were confirmed true. Maybe there really was a lot you didn’t actually know about when it came to Elizabeth Donnelly.

**

Caught up in the Clouds.

Weddings had never been events that caught your eye. You thought they were overpriced, overhyped, heteronormative celebrations filled with too many relatives and old friends who hadn’t seen each other in years. Emotions were high, stress levels to make sure everything was perfect were through the roof and alcohol only exaggerated every single little issue.

Yet somehow you’d forgotten all of those issues as you sat in a church pew beside Liz, her fingers tangled with yours while her thumb soothed over the back of your hand. There was nothing but love and adoration flowing through the air, winding its way through the crowd of guests before wrapping the happy couple up in a perfect little bow. You had no idea that Jess and Colin were such hopeless romantics, their personally written vows bringing a misting into your eyes. And when a tear finally managed to escape the corner of your eye before you could even think about it, Liz caught it, gently wiping it away with a warm smile. You let out a breath of a laugh, your cheeks heating as you felt a little silly for tearing up at a wedding, a warmth blooming in your chest as Liz returned the small laugh, squeezing at your hand before both of you returned your gazes to the alter.

Cocktail hour got everything moving, people up out of their chairs, flitting around the room as they reconnected with friends they hadn’t seen in ages or got introduced to new ones. Liz never once left your side, her arm securely wrapped around your waist, making sure your champagne flute was never once empty and you always had her backup when trapped with relatives. She was introduced to multiple family members and friends and you swore each time she had some new, horribly romantic story about how the two of you had started dating, who asked who out, your first date, a very over the top story about a candlelit trail in Central Park, surrounded by roses the first time you exchanged I love you’s. With each anecdote (and each glass of bubbly) you could have sworn she was looking at you with more adoration than the last, and that you were returning it, happily nuzzling into her shoulder, feeling her lips press to the top of your head.

You were thankful when dinner was finally announced and you were guided to your seats, food taking focus before all attention was back on the newlyweds. The two of you had been playing into things so well, surrounded by the looming feeling of love, happiness and forever that things simply felt natural, they felt right. Everyone in the room thought you were a couple, there was no reason to not play into that. Liz’s hand gently playing with the baby hairs at the back of your neck while you listened to speeches, your hand resting on her thigh as you leant in to whisper a joke, the shared quiet laughter as your eyes sparkled.

Wrapped up in your own little bubble you even managed to get out on the dance floor, first it was bopping along to top forties, scream singing along to favourites from your younger years, indulging both the kids and the grandparents with things like the chicken dance. Then the lights dimmed, the playlist shifting to something slower and instead of stepping off the dance floor, Liz grabbed your hand, twirling you under her arm before wrapping it around your waist and pulling you close. She murmured something about it not being that easy to get away from her, that she deserved at least once dance tonight and you laughed softly,

“Never would have pegged you for the dancing type.”

“There’s plenty you don’t know about me yet” She muttered back with a grin and you playfully shook your head at her, relaxing into her embrace as you glided around the dance floor.

It was during one of those dances, held close enough to her to feel her heartbeat, where she could feel your breath on her skin when you spoke, that it felt like you were in an alternate universe. That this weird fake dating thing was actually real, that you were happily in love with the other person and looking forward to spending your future together. It was on pure instinct that you turned to kiss her cheek, only, she’d had the same instinct in the same moment, her lips landing on yours and your body tensed for the briefest second before you relaxed against her. You felt the warmth of her hand coming to cup your cheek and you let out a soft sigh into the kiss, your lips curving into a small smile as you pulled away.

Before either of you could even think about saying anything the DJ blasted an air horn, the lights flashing while they turned up brighter and he announced the bouquet toss. You got whisked away an instant later on sister duty, your task making sure the maid of honour got safely into a cab without Jess noticing, knowing she’d be pissed her MOH was belligerently drunk halfway through the night. You thought you’d be free, but then discovered Jess had spilt a few drops of red wine on her dress and they still had a handful of professional photos to get through. By the time you returned to the table where Liz was now deep in a conversation with your mother, you were carrying your shoes, the buzzed feeling from the bubbly very worn off.

“You look exhausted.” Liz greeted with a small laugh.

“I feel exhausted.” You replied.

“Well, let’s get you home.” She stood from the table, her arm winding around your waist, “I’ll call a car.”

You briefly heard something from your mother about seeing Liz again before the three of you managed to say goodnight and you finally escaped the reception. You were half asleep on Liz’s shoulder by the time the car pulled up in front of your building and she was gently nudging you awake, watching until you were safely inside before she pulled away from the curb.

**

Playing it Cool.

After the wedding it was as if everything had been some kind of fever dream, that the weird little bubble of your fake dating had been simply that. Your parents were ridiculously distracted by Jess and Colin, squealing over honeymoon photos, eager to hear any and all stories from their lavish vacation that you were finally left in silence and at peace. Your new so called girlfriend practically forgotten from their minds.

The first Monday at the office after the wedding you’d been a little apprehensive about how to act but both of you seamlessly slipped back into your appropriate roles and nothing else was said on the matter. You picked up cases, brought the perps to justice and did your job. Liz would pop up in your office just as often as before, commending you on your work, bribing you to take a break with lunch and you went about life just as it had been before, even if you now did know a little more about the other person. Months went by while things returned to normal, your usual bantering, late night gossip fests and the like.

“What?” She asked you one day, watching the way you were stabbing at your plate of pasta and you let out a huff, leaning back in your seat.

“I know I said you were free of this whole fake dating thing after the wedding but apparently Jess is the honeymoon baby type.” You risked a glance up to her, “I figured I had until Christmas and could bullshit my way around an amicable break up by then but she’s nearly seven months next week and is insisting that you come with me to the baby shower.”

“Is this one of those absolutely ridiculous garden parties covered in half pink and half blue decorations until they pop a balloon and everyone screams and cries over the reveal?” She deadpanned and you laughed.

“No. They’re not finding out the gender, this is just your run of the mill mocktails for her, cocktails for everyone else, finger foods and sandwiches and gifts while everyone weirdly demands to touch her belly.”

“Yeah, I was never one for things like that either.” She took a sip of her wine and you nearly frowned, “but I’d hate for you to have to suffer through it alone, so of course I’ll come.”

“You are a literal life saver.”

“I know.” She grinned across at you, causing you to playfully roll your eyes before she flagged down a server to settle the bill.

Four days later the two of you found yourselves in your parents back yard, a gift bag with a handful of onesies, a cute card and a wad of cash being passed off to Jess and Colin. Liz’s hand slipped into yours with ease as she chatted with your family, accepting a drink from your mother before pressing a gentle kiss to your temple. You were able to avoid some of the silly games Jess had planned, escaping off to a small table in the corner with your mother as she watched your father play photographer for the day.

“Oh sweetheart I can’t wait to see you all excited when it’s your turn for this.” Paula smiled and you practically choked on your drink, barking out a laugh.

“Mom, please. Kids are on my absolute do not want list.”

“You say that now.” She chided, “I mean have you even asked Elizabeth about it?”

“Oh I do think I’m rather passed that point.” Liz replied with a chuckle, “and even if I wasn’t, I agree with her. Kids aren’t something I see in my future, I’ve got enough of a maternal role at work, wrangling all those baby prosecutors, I don’t need to come home at the end of the day to keep doing the same thing.”

“Yeah.” You sighed, swigging at your drink, “and they’re messy, and loud… and always sticky and covered in snot. Why would I want that anywhere near me? I’d much rather be able to travel, be the rich mysterious fun Aunt. Like, do I want to have to shell out millions for a chance at a good preschool, or do I want to be able to buy a yacht?” Liz laughed, squeezing at your hand.

“You’ve been hanging out with Calhoun too much.” She teased and you swatted at her arm, shaking your head.

“Well if there aren’t any grandbabies coming my way, can I at least expect a wedding?”

“Mom…” you groaned, earning a chuckle from Liz, who picked up your left hand in hers.

“You know, I have been telling her that it looks a little bland without a diamond.” She kissed the back of your hand and your mother practically burst with glee, “we should take you to get your nails done next week.”

“Liz…” You warned with a glare and she laughed softly, her gaze drifting over to the rest of the party as your mother let out a happy sigh.

It wasn’t until you were getting into the car at the end of the afternoon that it finally came up again.

“I can’t believe you.” You laughed, “here I am, trying to figure out how to fake a break up by Christmas and you go bragging to my mom about rings!”

“What can I say, I get bored, I like to meddle.”

“Clearly.” You shoved at her shoulder, “fucking shit disturber. They’re gonna be expecting a wedding next year now.”

“Oh they’ll be distracted by that thought for all of two months until that baby’s born and then the crotch goblin will be the only thing they think of for another three years, you’re fine.” She waved you off and your eyes narrowed in her direction, glaring at her.

“You owe me a new set of nails.”

**

The Italicized ‘Oh’.

Exhausted was an understatement.

After working your ass off to try and get justice for a victim who could no longer speak up for themselves you’d been torn apart, chewed up and spat out in the courtroom. You were left embarrassed, mad at yourself and the system and incredibly disappointed, all you wanted was to curl up into a ball and disappear. The only saving grace right now was that it happened to be Friday, so at least that was possible for a couple of days, hopefully everyone would forget by the time Monday rolled around.

You’d been home long enough to wipe off your make up, tug your hair up off your neck and change into sweats, flicking the tv on for some background noise while you wandered into the kitchen to see what you had in the liquor cabinet when there was a knock on the door. Your shoulders drooped as you let out a huff, padding over to the door, not totally surprised when you discovered Liz on the other side.

“Hey.” You greeted with a sigh, stepping back from the door so she could step into your apartment, following you back into the kitchen.

“It’s not like you to take off early.” She commented.

“Hmm?” You barely glanced over your shoulder, attention refocussed on deciding between wine or whiskey.

“Swung by your office, thought I’d find you there. I wanted to chat.”

“Liz, please, spare me the lecture.” You grumbled, opting for the whiskey, grabbing a glass from the cupboard.

“You had a rough go; I wasn’t going to.” She chuckled, “it’s not about work.”

“Okay.” You poured out a glass of liquor, gesturing to see if she wanted one as well before pouring out a second when she nodded slightly. “What then?”

“Well,” Liz huffed softly, feeling her heart thump in her chest, “for the better part of a year we’ve been pretending to date and I don’t want to pretend anymore.”

“Great.” You let out a sad laugh, “a terrible week, garbage day, I lose a case and now my fake girlfriend’s dumping me. You really know how to kick a girl when she’s down.” You took a swig of your drink, doing your best to keep your emotions at bay.

“Sweetheart…” She plucked the drink from your hand, “you’re not hearing me.” The glass came to rest on the counter before her finger curled under your chin, frowning at the misting in your eyes. “The part I don’t want to do anymore is the fake part.”

“What?” Your breath caught in your throat as your brow furrowed, still not totally understanding where she was going with this.

“I didn’t really realize how much I was enjoying it until there were no more excuses to get to hold your hand, somewhere along the way of pretending to be in love with you I actually started to fall.” Her thumb gently stroked across your cheek and you instinctively leant into the touch, “I went by your office to ask if I could take you to dinner, on an actual date this time. When you weren’t there I realized you were probably sulking and I didn’t like the idea of you having to do that on your own, so…” She gestured to the counter behind her and that was when you finally realized she hadn’t shown up empty handed. A take out bag with a bottle of nice wine sticking out the top sat beside a bouquet of lilies.

“Liz…” Your eyes moved between the counter and her, shaking your head in near disbelief, “I…”

“If we’re not on the same page, I’ll go.” She shrugged, “leave you to wallow in peace. I just couldn’t stay silent any longer.”

“No!” You gasped, grabbing her arm, “I… god.” You laughed, “I’ve been thinking of, and trying not to keep thinking about that fucking kiss since the wedding. I’ve been waiting for my sister to send out some stupid invitation to another ridiculous and annoying event that I’d be forced into to have some sort of excuse to play pretend again.”

“Well…” smiling she took your hands in hers, squeezing softly, “then how about we don’t pretend any longer?

“I think I’d really like that.” You murmured back, closing the gap between you as one of her hands cupped your cheek before her lips met yours in a tender kiss. You couldn’t help the grin your lips curved up into, warmth blooming through your chest at the feeling her smiling against your lips.

Breaking the kiss she smiled softly at you, her thumb stroking at your cheek before she left a feather light kiss on the tip of your nose, “now, I know you had a bad day but you need something beyond liquor for dinner. How do you feel about Thai?”

“Sounds perfect.”

Liz wrapped an arm around your waist, picking up the take out bag while you grabbed the glasses off the counter and the two of you made your way into the living room to settle on the couch. It was there that you finally let your guard down, all the walls you’d kept up to maintain the professional relationship being torn down piece by piece as you relaxed into each other. Your mood lifted as Liz recounted stupid stories to make you laugh, watching you with an adoring smile as you began to come out of your pit of disappointment, only interrupting to steal kisses or brush a piece of mussed up hair off your face.

By the end of the evening, limbs were tangled together, a blanket half tossed over the both of you while a movie played on the television. Liz’s fingers gently combed through your hair and you let out a happy hum, grabbing her free hand to leave a kiss on the back of it. You couldn’t help but smile, feeling a little bit silly at the entire thing, sometimes what you needed the most really was what was sitting right in front of you the entire time.

___________________

@svulife-rl l @anya-casablanca @mickey-gomez @naturalxselection @yesterdaysgone @momlifebehard @alexusonfire @alexxavicry @rainbowelshrhian @daddy-heather-dunbar @evilregal2002 @alcabots @ladysc @dextur @disneyfan624 @kdaghay @svushots @onmykneesformarvel @kalixxh @kmc1989 @irishavengersassemble @louderfortheback @swimmingstudentchaos891 @baubeautyandthegeek

[tomomiisasleep]

notes on Ianthe

just casually re-reading GtN Ianthe moments

here she kinda bested Palamedes and Harrow in picking out information out of bone fragments, which is very impressive considering the specialities of the 69 necros.

it seems that when it comes to necromantic "theory", she really is the best of the generation, this deduced from her grand feat of figuring out the Eightfold all on her own.

here Babs kinda jabs at her for being jealous of Corona's ability to gain attention. hmm. I have mixed feelings about this. I mean Babs knows her pretty well and I would believe that she actually does care about whether or not she could show off, which is ugggghhhhhh PAIN I don't wanna think about the fact that it may have hurt her entire life to co-exist with Corona and let her draw in all the attention but she does it anyway because Corona is the only person that really loves her and knows her, aaaaaaa and she started at six. SIX. WHAT does that mean. Did she even give it much thought at first? or were they still in the same twin us vs. world mindset at the time?

anyways I always thought that their layered relationship is delicious(in a bad way but in a good bad way)

1. Ianthe seems to be only good at theorems while Corona is the perfect heir, thus Corona holds power over Ianthe.

2. Corona is not actually a necro and depends entirely on Ianthe to put on the act, thus Ianthe holds power over Corona.

3. Corona is the main and often sole source of recognition Ianthe receives, likely causing her to value Corona more and others less over time, thus Corona holds power over Ianthe. (This may be the reason Tamsyn deems Corona the "worse" because she is skilled in manipulation, not necessary with malign intent)

4.(Not entirely sure about this but probably) Giving Corona power is Ianthe's "goal" in life(I've seen this in other posts) which she deems highest priority and worth any cost(as expressed in The Unwanted Guest). She may measure her self-worth by Corona's success, which like, means that their relationship is not "one controls the other" but "-I need you. -I need you."

BUT! That last one depends on which of them decides on what "Corona's success" entails. The one who has the final say in what their goal is, is the twin who controls the other. And. There are a few pieces of evidence I've gathered that Corona is in charge. HOWEVER there is the evidence that Corona wants Ianthe to eat her. and she refused. ugh maybe they'd be better off becoming a merged soul. Maybe that's the real reason Pal bested her in The Unwanted Guest. She is overtaken by regret that they missed the chance to become one. Now they'd never be truly the two of them together, because even if they merge there'd always be Babs in there third-wheeling

but I understand that despite all that, Ianthe does want to be recognized outside of being "one of the twins". It's natural. and she does have a very hard to ignore crush on Harrow.

And maybe I'll scratch all the stuff above later when I read their interactions(plus the Corona side-story! I've been saving that for when I get mental from alectopause) because I'm just speaking from memory and memory deceives.

ok I'll shut up about Tridentarii stuff now, what I actually wanted to say that Ianthe gives off different vibes in GtN and HtN. In GtN she's just this huge evil nerd, and in HtN she's gross-hot sexy-bitch. And it's funny how different she is from Gideon's and Harrow's pov, like all the fanfic I read in which she parties a lot and plays with girls a lot are all mostly drawing from Harrow's pov, and Harrow is a repressed nun who has only ever met 1 girl her age in her life and has a bone fetish(especially for Ianthe) and is obsessed with her scent. Though I enjoy reading all fan-interpretations of her, I think her canon lifestyle might actually be closer to how Gideon sees her(in GtN. in NtN they become bros which I love btw tower princes 4ever), which is wall-flowering in the corners and reading trashy romance novels while Corona deals with the socializing.(I kinda believe that people can be categoriesed into "reads about dating" and "dates" and Ianthe belongs to the first because she gets excited over a sexy makeover

she also spends a horrifying amount of time studying (because she's as good with theory as Pal who "made his life into a war" for Dulcie, and Harrow who studies with the weight of 200 dead kids on her consciousness.

ok Ianthe's such a hot nerd I'm gonna faint I have become obsessed with her aaaaaaaaa she's like everything that drives me crazy in a hot tall glass of skank(a skank for knowledge

[palikaraki]

AFFIRMATIONS FOR MONDAY

θεμα Α definition from the pre calculus chapter🙏🙏NO THEOREM PROOFS🙏🙏nothing too long🙏🙏three lines maximum 🙏🙏

θεμα Α normal swsto lathos🙏🙏

θεμα Β normal function analysis🙏🙏 and graph design🙏🙏🙏🙏something pretty🙏🙏and symmetrical🙏🙏no weird things🙏🙏

θεμα Γ simple mean value theorem🙏🙏 and rate of change🙏🙏NO ABSOLUTE VALUES🙏🙏🙏that you have to solve and take a gajillion different cases and functions🙏🙏🙏🙏no absolute values🙏🙏🙏only the sith deal in absolutes🙏🙏🙏

θεμα Δ NO(!!!!!) THIRD OR FOURTH DERIVATIVE (!!!!) 🙏🙏simple things🙏🙏 simple problems with simple solutions🙏🙏 NO third or fourth derivative.🙏🙏no fuckass data hidden in the panhellenics council's ass that you have to decipher in 5939292 lines to be able to start solving the problem🙏🙏simple definite integral🙏🙏 without weirdass x0 x1 x2 variants🙏regular numbers🙏🙏🙏🙏one two three🙏🙏four five🙏🙏🙏maybe even six🙏🙏🙏2+2=1🙏🙏🙏1+1=2🙏🙏

normal mathematics🙏🙏nothing too weird🙏🙏or strange🙏🙏or peculiar🙏🙏like 2 affirm☝☝reblog 2 manifest🙏🙏🙏

[max1461]

I believe that the following philosophical argument in favor of the second order Peano axioms as ultimately "correct" works:

We know from Gödel that no effectively definable formal system can capture the full behavior of the "true" natural numbers. That is, it's impossible, as finitistic beings, to give a formal definition which precisely characterizes the standard natural numbers. We will always "leave out some details" in the definition, among these the Gödel sentence in the given system and so on.

This makes the meaning of the phrase "the standard natural numbers" itself philosophically problematic. In the context of a given meta-theory (say ZFC), we can take the standard naturals to be some particular meta-theoretic construction (say, the von Neumann ordinals). In this context, the incompleteness theorems as internalized in the meta-theory say that no effectively definable formal system as internalized in the meta-theory can prove all the true facts about our chosen standard model. But of course this doesn't save us, because the incompleteness theorems "on the outside" of the meta-theory say that it can't prove everything there is to know about the "true" external standard model of the naturals, whatever it is.

Of course this last part is possibly bullshit and may rely on some kind of Platonism to make sense. So to be a conservative as possible one should stick to just asserting the meta-theory-internal version of the incompleteness theorems. After that you can, if you want, let them inspire by implication a sort of fog of uncertainty in the reader about what fucked up epistemic shit is going on "outside" the meta-theory, even though that perhaps does not make sense (or perhaps it does...). Of course you can make "outside the meta-theory" make sense by internalizing the meta-theory in a meta-meta-theory, but then you just get the same situation one level up.

So, ok, the point is that you are never going to be able to write down a formal system that unambiguously defines what you mean by "the true standard model of the naturals", such that exactly the statements which can be derived from this system (=definition) are exactly the true ones. Which sucks! That's lame, because math is supposed to involved being precise about what we mean by shit.

There are a couple of ways out. One is to just take some effectively definable formal system like first order PA and say "this is what we mean by the naturals, we mean the shit that can be proved from this. Yeah that leaves a lot of stuff hanging, a lot of statements about arithmetic of-ambiguous-truth-value, but whatever". Because, you know, PA is not categorical, so it has many inequivalent models. Or you can say "I will take second order PA as internalized in ZFC (so basically, the von Neumann ordinals) as my definition of the naturals". Which I think is more powerful(?) but still suffers from the same problem when you look at it "from the outside" of ZFC. Actually, you can do that for any (expressive enough) meta-theory M, you can put second-order PA inside it and take that as your naturals.

With the stage set, a brief digression:

I think that, informally, we should all be able to agree on the following about the "true" set of natural numbers, if such a thing can be said to exist (and imo it sort of must, because it's implicitly invoked in a meta-way when we define formal systems to begin with, and so on):

1. The number 0 is a natural number 2. If n is a natural number, then the successor of n (that is, n+1) is also a natural number 3. If m and n are two natural numbers and they have the same successor (that is, n+1 = m+1), then m = n 4. There is no natural number whose successor is 0 5. If P is some property which might or might not hold of a natural number, and we know that P holds of 0, and we furthermore know that whenever P holds of one number it must hold for the next number, then we know that P must hold for every natural number

Some people are philosophical uncomfortable with the last one, but I think it's intuitively undeniable. Like imagine a fucking... guy hopping from one number to the next, and he never stops. Can you pick a number he never gets to? No you fucking can't. You believe in induction.

So, ok, back to models and shit: both first order and second order PA try to formalize this intuition, and the key way that they differ is in terms of what a "property" (mentioned in (5)) is. First order PA says that a "property" is a first order formula. This is very powerful because we can effectively define the set of first order formulas over a given language. They are finite objects and we can work with them direction. From this flows all the nice properties of first order logic, like completeness and so on. But this effectively definability also makes it susceptible to the incompleteness theorems, and so first order PA ends up "leaving stuff out".

Second order PA defers the notion of a "property" to the meta-theory. It basically says "a property is whatever you think it is, big guy ;)" to ZFC or whatever theory it's being formulated in. ZFC thinks a property is a ZFC-set. Meta theory M thinks a property is an M-set. And second order PA as formalized in M agrees. Mathematically this makes second order PA harder to study as an object in itself. But philosophically I think it's kind of desirable?

First of all because, at a basic level, "property" seems like a much more fundamental notion to me than "natural number", and one I am much more willing to accept an intuition based definition of. Like, I don't know what you mean if you say "the true natural numbers". That seems pretty wishy-washy! But if you say "the real-world, ordinary definition of 'a property'", I can kinda be like "yeah, properties of things. I know how to reason about those!". And then second order PA, because it's categorical, will tell me "great: since you know what a property is, here's what a natural number is". And that's something I can work with.

This was overly long-winded I think. But in other words, what I am basically advocating for is conceptualizing second order PA as a function from "notions of property" to "notions of the natural numbers". And because models of PA are unique up to isomorphism (in whatever (sufficiently powerful) meta-theory you formalize it in, not "from the outside" of course) this means you can take up SOPA as your definition of the natural numbers and then "lug it around with you" into whatever different foundational system or meta-theory you fancy. And when you lug it into the real world, where "properties" mean actual properties of things, you get the real, true natural numbers.

This is all purely philosophizing of course. But I think this is about the situation.

[bubbloquacious]

Effortpost registry

Kernels and Injectivity; 19 jun 2025

The Topology Game; 2 apr 2025

Graphs as presheaves 4: coverages; 13 feb 2025

The general linear group as a Hopf algebra; 31 oct 2024

Zariski topologies; 14 oct 2024

On integer multiplication and endomorphism algebras; 2 sep 2024

Recommendations for learning category theory; 28 mar 2024

The hairy ball theorem and stably free modules; 11 feb 2024

Topological connectedness and generalized paths; 24 nov 2023

Graphs as presheaves 3: subobject classifiers; 19 oct 2023

Effortpost registry; 18 oct 2023

Graphs as presheaves 2: limits and colimits; 11 oct 2023

Hydrogen bomb vs. coughing baby: graphs and the Yoneda embedding; 7 oct 2023

Extending the D ⊣ U ⊣ I adjunction sequence; 23 sep 2023

The Riemann rearrangement theorem and net convergence; 18 sep 2023

Thoughts on the axiom of choice; 18 feb 2023

Topological spaces and simple graphs as neighbourhood spaces; 15 feb 2023

What is a space?; 10 jan 2023

The exponential function applied to sets; 24 dec 2022

On nilpotent eigenvalues; 23 dec 2022

But IS the empty space connected?; 11 nov 2022

Monads monads monads; 8 nov 2022

Calculating what the triangle identities mean for a bunch of adjunctions and being amazed when it works every time; 7 nov 2022

Defining the Lebesgue integral as a net limit; 27 jul 2022

Rambles about describable sets; 28 oct 2021

Functions with cycling derivatives; 30 aug 2021

Why the rationals have zero length; 31 may 2021

An infinite cardinal valued random variable; 30 may 2021

A field-based functor; 20 mar 2021

Generalized sides; 13 mar 2021

Rambles about metric convexity; 22 feb 2021

Wiggle function convergence; 28 jan 2021

Rambles about infinity; 5 sep 2020

Generalized golf; 24 jun 2020

Rambles about continuousifying series; 10 may 2020

Rambles about being closed under exponentiation; 7 may 2020

Rambles about the groups that come with fields; 3 may 2020

A compilation of donutified functions; 17 mar 2020

Rambles about arithmetic functions; 24 jan 2020

Graphing real functions on a torus >:); 29 nov 2019

[ahuactl]

on mathematical models

the question “is math invented or discovered?” remains to be the most contentious questions in mathematics.

a significant portion of people including mathematicians believe that mathematics holds objective truth insofar as their models, theorems and equations tell the truth about reality in which the Subject: their proposition exists independently of them. this position is familiar among those in the Platonist or Realist traditions of mathematics. common knowledge states that any proposition such as  P: 1 + 1  = 2  is true in which ever universe it is stated in such that its truth is not constructed upon the confines of our society and more strongly, not in our minds or cognitions.

i will present my argument here on why i disagree with this view. first, propositions like P: 1 + 1  = 2 are certainly not true in every construction of a mathematical system. for example, in boolean algebra and nonstandard number systems. even non-Platonist mathematicians would quickly disagree with this as they would claim that the Platonist view holds that the truth of a proposition exists not solely but with the pair (P, S) whereas P is the proposition and S is the system in which it is considered to be in. on such note, this leads me to my next argument.

from the development of set theory in the 19th century, it was discovered by mathematicians that mathematics as a field was built on shaky logical foundations (re: Russel’s Paradox). this thus necessitated the construction of a new structure which must possess the power to rebuild mathematics. this new foundation came with the introduction of ZFC. it is an attempt to systematically rebuild the previously known and accepted results of mathematics while working to eliminate the logical inconsistencies of previous naive set theory. from how it was made, ZFC does not posit the truth of any propositions insofar that it exists as starting axioms in which rules of inference can be applied. as with any theories of logic, its axioms must be decided and chosen. it could be noted that these axioms were posteriori in a sense that they were built to solve the issues such as Russel’s Paradox. it can then be argued that the process by in which these axioms were chosen is contingent upon the values of the persons that chose it as well as the developed culture of mathematics. in fact, it is by no means that ZFC is only the possible theory of mathematics (e.g. Model Theory, Category Theory). no matter the “objective” metaphysical existence of mathematical objects, by the way mathematics was built and practiced, these starting axioms still remain a choice.

to reiterate, the way mathematics has been agreed to be used by mathematicians, whichever system set in place still remains a choice. this reveals its use as pragmatic: the system, that is, ZFC was adapted because it works. this naturally extends out from its foundations to concrete mathematical objects studied in abstract algebra, analysis, etc. for example, a group is not just a set with specific rules with its binary operation; the construction of the properties that denote a Group came a posteriori in a way that it was discovered that the set of properties in the first place proved to be a useful abstraction to study. 

on a related note, i would argue that the existence of abstract mathematical objects actualize themselves by our need and desire for the world to conform to standard rules as appeal to our intuition and senses. the truth value of P: 1 + 1  = 2 in (P, S) in an arithmetic system S signifies an essence in which it “makes sense” that combining abstract objects  result to a set in which it succeeds singular objects by the value of each one of those objects. the value of (P, S) does not reveal a Platonic truth of our universe but instead a declaration of common sensical deductions. in such, a mathematical statement is a “model” the same way a bachelor is “unmarried”.

on usage:

regardless of the metaphysical existence of mathematical objects, it, as a system, is undoubtedly one of the finest works of humankind. i advise that one ought not to use this powerful tool to make claims of absolute truth. in a sense, the existence of mathematical objects remain irrelevant to how it ought be used. one should not at all invoke mathematics to justify repressive systems and to cause grave harm to humanity. as a tool, it must not be used for evil. one must always remain cautious of individuals who use mathematics as arguments by certainty. these individuals who swear by the sharp sword of mathematics is most likely using it to silence thought and keep one from questioning. always look at the mathematics; an equation is not just a set of symbols that denote some property; it is a theory of the world and this theory must be met with most criticisms when it is being used to make claims for human policies. look at the mathematics, specifically which assumptions were made by the ones who present them: it must reveal the true character of such persons.

[what-about-zaladane]

Entropy, Intuition, and The Art & Science of Shuffling A Tarot Deck (Or: How Many Times Should You Shuffle It, Mathematically?)

This post is specifically about riffle shuffling -- not overhand or pile shuffles.

tl;dr: If you care about randomness and making sure all the cards have a fair chance of coming up, 9-10 times.

But how much should you really care about randomness?

Let's back up.

Mathematicians Gilbert, Shannon and Reed developed a model of randomness for riffle shuffling (i.e. the snappy type of shuffling you do with a playing card deck). They found that the first few shuffles of a deck of playing cards only rearrange the order of the cards a little…but by the 7th shuffle, the deck's order is almost indistinguishable from random.

You might ask -- but a tarot deck has 26 more cards! Does that matter?

Yes, it does.

Each added card increases the permutation space of the deck. You go from 52! (a number with 68 digits) to 78! (a number with ~115 digits). The number of possible orderings explodes -- as does the time and effort it takes to "mix" those orderings.

The Math

If you do a perfect riffle shuffle, the deck becomes increasingly mixed. The GSR Theorem uses total variation distance from uniform randomness to measure just how mixed it is. Total Variation Distance is a number between 0 and 1 that tells you how different two probability distributions are. The two distributions we are comparing are:

The probability distribution of the deck after n shuffles

A perfectly random deck, where every single card has an equal chance of being in every position.

What do different values of TVD mean?

If TVD = 1 -> the two distributions are completely different

If TVD = 0 -> the two distributions are essentially the same

Imagine you’re drawing the top card of a tarot deck after shuffling.

If the deck isn’t well shuffled yet, certain cards are more likely to be on top.

If the deck is perfectly shuffled, every card has exactly a 1 in 78 chance to show up.

Total Variation Distance measures:

"How far are we from that perfect 1-in-78-for-every-card situation?"

TVD is basically asking: how unfair is this shuffle still?

If you shuffle enough times, you'll get close to 0.

TVD Convergence Chart

Here’s what the convergence of TVD looks like for 52- and 78-card decks:

You can see that the first few shuffles don't bring us much closer to a totally "fair" deck -- but once we hit 4 or 5 shuffles, we start to converge to 0 rather quickly!

Back to the Shuffling

GSR modeled the distribution of permutations using the random riffle shuffle model, which assumes:

You cut the deck into two halves with a binomial split

You then interleave the cards in all possible ways

(Btw: This gives you a Markov chain on the symmetric group S_n, where n is the number of cards. If you just flinched at that sentence -- don't worry about this! Forget you read it! It doesn't come up again in the post.)

What about a 78-card Deck?

According to Bayer & Diaconis (1992) in “Trailing the Dovetail Shuffle to its Lair”:

The number of riffle shuffles needed to mix a deck of n cards is approximately:

So 9 to 10 riffle shuffles are needed to randomize a 78-card tarot deck.

Okay But…How Important is Randomness in Tarot, Anyways?

Honestly, it depends on you.

If you believe in pure chance, then randomness is the point. You want the cards to speak without being nudged by muscle memory or old orderings.

But if you believe in "divine" order, then even imperfect shuffles are sacred. Every shuffle is a divination. Every card is a mirror.

So shuffle 3 times. Shuffle 9. Shuffle until the deck feels right.

But just keep in mind: mathematically, 10 riffle shuffles are optimal if you want to approach full chaos. And chaos is very good at telling the truth.

What Do I Do Personally?

You'll notice that the graph of TVD converges pretty fast after about 6-7 shuffles. Personally, when I'm feeling the need to completely reset the deck, I shuffle 10 times. Any other time, I shuffle 8. I'd love to tell you that there's some fantastic reason for this, but it's mostly because I'm Chinese-American and I'd probably do everything eight times if I could. :D