#also can I just say arithmetic is the one math thing I actually use in my life

[sometimes-sleeby-octopus]

I made a ninjago bracelet with clay beads

[raven-at-the-writing-desk]

fellow + gidel ssr time fellas

(This bastard took an entire soft pity :(( but hey, I got a Dorm Uniform Jade dupe and finally FINALLY my first Dorm Uniform Floyd on the way, so I ain't too pressed about it.) RISE UP FELLOWIVES NOW’S YOUR TIME

***Character profile, voice lines, Groovy, and vignette spoilers below the cut!!***

First off! His official profile, coffin, and candy (Fox Candy):

(School) Grade/Class: None

Birthday: May 17 (Taurus)

Age: 26

Height: 181 cm

Dominant Hand: Right

Hometown: ???

Club: None

Best Subject: Mathematics (specifically Arithmetic)

Hobby: Watching theater

Dislikes: Saving money

Favorite Food: Apples

Least Favorite Food: Potatoes

Special Skill: Sewing

We finally get confirmation of Fellow’s age! (He had previously said in Playful Land that he was 20-something.)

I love that Fellow’s best subject is math Deuce is jealous/j; it makes so much sense given that his inspiration, Honest John (and Fellow himself) aren’t good at reading. It’s that whole “kids are either good at math or English” stereotype. In Japanese, the phrase 算数 is used. 算数 refers to arithmetic, or very basic math taught in elementary school (adding, subtracting, multiplying, and dividing). That specific phrase explains Fellow’s elementary level of understanding. His dislike being saving money is also related to numbers; he spends the money he has right away to get by in the moment. Fellow doesn’t really have the skill or the luxury of financial planning, he has to focus on the here and now, looking out for both himself and Gidel.

OMG, his favorite and least favorite foods???? 😭 Playful Land has apple (core) flavored candies and popcorn… and again, this is a reference to Honest John and Pinocchio’s first encounter! He takes the kid’s apple and eats it, lol cnsvwiwguwkw Potatoes being his disliked food… Maybe it’s because it’s the “poor” man’s vegetable? Because potatoes are so versatile, keep for a long time, and are filling because of the starch content, Fellow might resort to eating them a lot, so perhaps as a result he got sick of the taste.

What I find most interesting about Fellow’s profile are his listed hobby and special skill. He largely comes off as despicable and a slimy scammer (which he is, don’t get me wrong), but we can see different sides to him in these details—both the inner child that had his dreams trampled but remains hopeful about the future and the big brother figure/guardian to a child. Gidel is actually formally referred to in Fellow’s profile as his (non-blood related) brother, which made my heart melt 🥺 TWST must know I have a thing for beastmen who act shitty but are actually excellent mentors to the children/j

Fellow enjoys watching theater. It’s a way of transporting you away temporarily to new worlds with crazy stories and emotional performances. When words aren’t enough, you sing. And when singing isn’t enough, you dance. It’s an area that’s so full of life and joy, at least from the audience’s perspective. I’m thinking that watching theater must have been a form of escapism for Fellow, especially with how accessible it is (think of like street performances). Watching theater might also serve a dual purpose of teaching Fellow how to come across as amicable and friendly, which says a LOT about his character. He’s resourceful and able to learn from unconventional sources, then is able to apply those skills to real world situations.

Fellow’s special skill being sewing is surprisingly very cute! If you’ve taken one look at his and Gidel’s designs, we may have already spotted some of his handiwork. There’s mismatched fabric patches on their clothes!! The stitches look so clean too. The patterns not matching is probably because Fellow just used whatever scraps he was able to get his hands on, but I also like to imagine that he tried to make the best of the situation by incorporating the mismatched fabrics in a fun way (like the diamonds in his pants).

Next, can we talk about the composition of that GROOVY????

It’s another reference to the same Pinocchio scene! Fellow’s holding his book like Honest John did and it looks like he’s trying to teach Gidel the alphabet from words scratched on the sidewalk. Notice how the C is written backwards too 😂 He even wears glasses like when Honest John was trying hard to act like an intellectual.

And Gidel!!! Pencil and pad of paper in hand, he looks so interested to learn (something which was hinted at in Playful Land). Gideon in the film is also shown with a pen and pad of paper, scribbling down nonsense as Fellow pretends to diagnose Pinocchio.

Gidel glances up at Fellow with an expression of admiration. I love how wholesome their relationship is depicted as, it leaves a warm feeling in the heart.

The framing of this Groovy is very interesting. We have Fellow to our left—a direction has historically been associated with evil (in Italian, the word for left is even sinistra, as if to imply something sinister) and in the darkness. Gidel is the one to our right and in the light. It presents Fellow to us as someone who has given up on his dreams—but not completely, since we see some light touching his hat, gloves, and highest features + he is currently teaching Gidel and still has dreams of opening his own school. Gidel is shown in the light because he’s still a naive child that doesn’t understand how the world works. His dreams haven’t been destroyed yet, and there’s hope for him to have a better life since Fellow is looking after him and instructing him.

CHECK THIS OUT, GIDEL FOLLOWS FELLOW TO CLASS LIKE MARY'S LAMB OR SOMETHING????? Gidel pops out from under the desk or out of/behind Fellow's cape! Gidel also joins Fellow on the homescreen.

Some of Fellow's expressions are so priceless... For example, look at him in Flight! There's an unsure face and a little bead of sweat. (He makes a lot of pathetic accompanying sounds too, lol) Flying takes magic, so he's probably not confident or powerful enough to maintain flight for long stretches of time--though when he does nail it, he looks ultra smug.

ADGKVAVFOOEFIEQOfsl HIS SHOCKED FACE

How uncool, Fellow-san...

His attack sprites are very similar/identical to what we saw in Playful Land--Fellow's just playing for the opposite team now.

Gidel hops into battle to assist, so I guess they count as the first two-character card. It's been a while since I've seen these animations, but I think I can appreciate them a lot more now. I'm noticing new little things like how Fellow adds a bunch of showmanship into his attack, little flashy flourishes and even presenting Gidel with his arms splayed, as if welcoming a star to the spotlight. The attention to detail really is crazy for these.

If you want to read his voice lines in full, you can find an excellent fan translation of them here! I'll just be remarking on things I noticed while combing through the voice lines myself:

First off: bro calls himself Fellow Hones-SAMA???????? OKAY, KING 😭 Love that confidence you got goin' on there...

bifabsiyofbefe Love how he just reads a textbook and then flat-out admits he has no clue what the heck it's saying. Hey, honesty is a virtue.

Ace 💀 He has the balls to play a prank on an adult... I kind of want to know what the prank was, but at the same time I feel like I should be shaking my head and telling him off for doing it in the first place. I do appreciate that Ace being shitty brought out Fellow's true personality there for a second though, I live for it when Fellow gets real steamed and starts shouting that the NRC students are brats or that they should drop out if they have no motivation in school.

The way Fellow automatically clocked that Kalim is way too trusting and would surely be in danger even if he wasn't the one to come for him... Fellow, watch your back. Jamill WILL come for your sketchy ass for what you did back then.

I didn't find anything super interesting in Fellow's comments about Ortho, but I do think it reveals that there is value in him coming to school. It's only at NRC where Fellow can see such a curious thing like Ortho, and that speaks to the value of really going out there and being exposed to different things. That's part of Lilia's own growth arc too, and a large part of why he now spreads that same rhetoric.

Fellow remarks that Ramshackle is "pretty sweet", which means one of two things: either this is the refurbished post-book 6 dorm OR it's still the shabby pre-book 6 dorm, but since Fellow and Gidel have never really had their own stable housing, even run-down old Ramshackle seems like a massive upgrade.

Fellow and Gidel must have been so happy to see that lunch at NRC is free and served buffet style (so there's no limits to how much you can take). On top of that, they got dead chefs from 5 star restaurants staffing the kitchen! Those two really hit the jackpot, I hope they eat well.

AVUSDGVUADOVIAISDBIDAS THE DIALOGUE IMPLYING FELLOW CASUALLY BYPASSED THE SCHOOL'S BARRIER AND OTHER SECURITY MEASURES... So Chenya-core of him, really. Fellow may not have magical might, but he's seriously street smart to have found a way in like he has.

Small detail but I appreciate how Fellow has lines which call attention to Gidel. It doesn't just remind us that Gidel is there too, but it also demonstrates to us that Fellow actively tries to include him in the conversation despite Gidel's muteness (a condition which may lead others to outright ignoring him or talking down to him).

LAST THING (though it's not in MysteryShopTL's linked post): in his birthday greeting to the player, Fellow says that both you and him don't have talent for magic, so you should get along. I didn't think the game would acknowledge the player and Fellow's similarity in that sense, so it was very nice to be proven wrong.

And to finish off this post (which ended up being way more massive than I thought it would be), a quick summary of the vignettes!! If you want to read them in full, please check out MysteryShopTLs’ post!

In vignette 1, Fellow and Gidel are putting on a street performance in Silk City. Fellow collects fees from the onlookers and then tries to milk more out of them by spinning a story about how Gidel is a puppet that can walk without strings. Buuut Gidel moves like a normal living being and sneezes, which ruins the ruse and leads to the crowd getting mad at them. The duo run off, but Fellow reveals that while the locals were looking at Gidel, he used magic to steal some of their precious metals and jewelry. In the next vignette, Fellow and Gidel have moved on to Fairest City. It's said that they live a nomadic lifestyle and hop from place to place, never staying for too long in any one location because word of their scams may spread and cause a situation where they cannot reasonably make money through their lies. (Cute detail: Fellow listens to Gidel's suggestions on where they should go next and even praises Gidel's smarts.) This time Fellow's trying to auction off a magestone that he claims will allow anyone who holds it to use magic. The people of Fairest City don't believe him and give him the cold shoulder, which upsets Fellow (since he really hates it when others look down on him). He ends up using his UM to get his audience to be more pliant and manages to sell the magestone for a pretty penny. At the end of this vignette, Fellow drops a line about how he and Gidel are so free, how they can do whatever they want since they have nothing holding them back. I really love that thought~

AND IF YOU THOUGHT VIGNETTES 1 AND 2 WERE FUNNY HAHA TEEHEE CUTE, WAIT UNTIL YOU SEE VIGNETTE 3 💀 VIGNETTE 3 FELT LIKE IT WAS A TARGETTED SNIPE ON MY HEART

The setting is Sunrise City! Fellow and Gidel are being chased off by an angry person they tried to rob. It looks like they're unsuccessful today and will be going hungry. Gidel tries opening a random can of OIL in search of food, so Fellow scolds him and tells him to leave it be. Apparently Gidel does this a lot when he's hungry (just grabbing random shit and trying to eat it), even though Fellow has tried teaching him how to read. THIS IS WHAT THE CONTEXT OF THE GROOVY IS, FELLOW SQUATS DOWN (like we literally see his 2D model lowering) AND DRAWS IT ON THE GROUND FOR GIDEL TO SEE. O is for orange, I is for ice-cream, and L is for laugh. Fellow realizes that L is the only non-food word, but he couldn't come up with anything else. I wonder if like... this is some common game they do to distract from hunger. They have to imagine the food they could have but can't. And L being "laugh"? That can't be a coincidence. Fellow could have used any other L word as an example, even if he couldn't come up with a food that starts with L. It makes me think he picked "laugh" on purpose in an effort to lift Gidel's spirits and to try and distract from their circumstances.

Aaaah, as I was saying! Fellow gets upset that he doesn't know as much as your average 26-year old would since he never went to school. Gidel seems to sense his frustrations and reassures him with a pat, which reenergizes Fellow. He says he'll try to find some food, so Gidel should focus on making a fire. While gathering wood to burn, they come across a job posting by a shady rich man that Fellow and Gidel supposedly did another job for in the past. Fellow suggests that they check out the job and if they don't like it then they can leave. ADSKJBBSLDIADBLUBAB These are the events leading up to Playful Land... meaning that Fellow’s showmanship is wasn’t something he developed at the amusement park, but as a general coping and survival mechanism to get by day-to-day.

I uh. May or may not have cried a little at Fellow and Gidel's really wholesome interaction 😭 I MEAN YEAH OF COURSE I'M A SUCKER FOR BIG BROTHER CHARACTERS (and we certainly see that in how Fellow scolds Gidel and looks out for his wellbeing, both in the vignettes and in Playful Land) but also???????? ? ? ? ? ?? ?????? ? ? ? ?? I love Love LOVE how Gidel is shown to be supportive of Fellow as well. Fellow as the older person, the adult, and the able-bodied one of the duo is pulling most of the weight when it comes to getting resources and handling communication. However, Gidel plays an important role in their dynamic as well. He's the heart and the emotional support that Fellow needs when he's down in the dumps and being hard on himself. Gidel not only serves as a "reason" for Fellow to work hard (to support a child), but he also gives Fellow motivation and hope that tomorrow can be another day. YOU CAN REALLY TELL HOW MUCH THESE TWO CARE AND LOOK OUT FOR ONE ANOTHER OTL

OOOOOOOoooOOooOOGGHHHH MY HEART *clutches it* I CAN'T TAKE THIS ANYMORE, I CAN'T HANDLE THE ONII-SAMA OF IT ALL

[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.

[oliviax727]

Physics Friday #3: No Seriously, why is 1+1 = 2? (and what a real number really is)

Refer to this link if you're confused as to what this is all about.

If you were wondering where my part 2 to the Dark Energy vs. Dark Matter post is, it'll come next week. I just wanted to divert for a bit and stick my head into mathematics. I generally won't do two parts back to back.

Preamble

Education Level: Middle School (Y6 - 8)

Topic: Logic and Construction (Mathematics)

Introduction: 1 + 1 = 2 because I said so

What is 1+1?

Why does it equal 2?

How can we say such a simple thing without falling into the depths of chaotic mathematical thinking?

What is a number?

What does it mean to be real?

Many people are asking this ...

Well, really to answer those questions directly. Mathematics, unlike a lot of other subjects, is founded on the principles of hard logic. Definitions and statements that derive new definitions and statements. Truth follows from more truth.

But in order to have true statements, some of those statements must given i.e. we just have to assume or declare they're correct. Otherwise we wouldn't have true statements to begin with!

Consider the logical statement "The sun is a star".

In order to prove that this statement is true, we need to:

Define the existence of an object called "the sun"

Define what a "star" is

Define what it means for an object to be "is" another object

We could then come up with these statements:

The sun exists

A star is a bright burning ball of gas

An object is something else when that object has the traits of that something else

But then we are faced with a problem: how do we know that the sun exists? Well, we can see it of course!

But this doesn't apply to maths - after all, can you see the number 1? Like, can you see the concept of the number 1?

The answer is that we have to just accept some statements as simply true, no questions asked. These statements are called axioms.

In any mathematical system, we have a set of rules, or axioms, that dictate how our system works.

In most cases, we say that 1 + 1 = 2 by definition. That the number 2 is purely defined by 1 + 1. Any properties it has, like 2 representing an amount of objects (cardinality), or 2 coming after 1 (ordinality) is merely coincidental, an aspect of the system itself, or entirely irrelevant.

Real Numbers

Let's start off with how we can play with these numbers, using the Reals and an example.

A real number is real simple. Here's some examples:

2

16

2/3

-8

-9091/2311

0.0404583439484328423490 ....

Pi

It's basically any number that you've dealt with before: decimals, fractions, integers, and the like.

But how did we get to this stage? Like how can we define the real numbers to mean a specific thing?

It's important to have such rigorous definitions in mathematics, because without them, we won't be able to generate new theorems about how our world works.

The real numbers are known as a complete ordered field. What that means is it has three properties:

A field describes a particular set of numbers with some simple arithmetic laws attached to them

A ordered set is one which as a notion of order

A complete field has no gaps

The Field axioms are as follows. A field is a set of numbers that/where:

Contains two non-equal numbers, 0 and 1

Has a definition for the + and × operators

For any number a:

- a + 0 = 0

- a × 1 = 1

- There exists a number (-a) such that: a + (-a) = 0

- There exists a number 1/a such that: a × 1/a = 1, unless a = 0

For any numbers a, b, and c:

- a + b = b + a

- a × b = b × a

- (a + b) + c = a + (b + c)

- (a × b) × c = a × (b × c)

- a × (b + c) = (a × b) + (a × c)

(Note: I dunno how to format bullet lists properly pls help)

Pretty simple eh? Well there are actually quite a lot of things that are fields. For example the set of all rational numbers (fractions) are a field.

There's also the order axioms. An ordered set is a set of numbers that/where:

Has a definition of something being less than another or a < b

For any numbers a, b, and c:

- If a < b then a + c < b + c

- If a < b and b < c then a < c

- Either a = b or a < b or b < a exclusively

An example of one of these ordered sets is the integers!

Lastly we have the completeness theorem. The completeness theorem is a bit more complicated, and it might be worthwhile to spend a whole topic on it:

Say I were to define a new operation within this set. For example f(x) = a + b + x

A complete set, no matter the definition of the operator, would always evaluate to a number that remained within the set as long as no rules of the set were broken.

i.e. x can be any number, and f(x) can be any operation involving x. But if x and f(x) can be defined entirely by what we had originally, then f(x) will always equal a valid number given that we don't divide by zero.

The rational numbers, for example, is not complete. Here's a small proof:

Define the operator a^2 := a × a

Define the operator sqrt(a) as being sqrt(a)^2 = a

There does not exist a rational number that equals sqrt(2)

Therefore the rationals are not complete

It turns out that the real numbers is the only complete ordered field in existence. That by setting just these axioms, we can have a unique set of numbers.

So how does 1 + 1 come into this? Well, 2 is defined as being 1 + 1. And 1 + 2 = 3, and 1 + 3 = 4 ...

Here's an example proof for 2 + 2 = 4, the bane of all who know about Gregory Orsen's 1894:

2 + 2 = (1 + 1) + (1 + 1) = (1 + 1 + (1 + 1)) = 1 + 1 + 2 = 1 + 3 = 4

Note that these axioms leave out some rather important identities, like:

Any number times 0 is 0

0 = -0

0 < 1

-1 < 0

a < b implies 1/a > 1/b

But the whole point is that we don't need these statements to be axioms! We can prove these from the ones we already have alone!

Set Theory, Peano, & Recursive Addition

There are, of course, other ways to construct mathematical frameworks.

The real number axioms are an example of constructing a system by having a set of rules and then proving afterward that these rules produce a unique set of numbers.

But what if we wanted to go more general, and have numbers not defined by axioms, but have the axioms describe more general maths.

Well, there are several ways in which we can do this:

Set Theory Construction

Lambda Calculus Construction

Surreal Numbers

I'll mention only set theory. A set is something I've used before. What a set essentially is, is just a collection of things.

We can use sets to define numbers, for example:

0 := { } (i.e. the set containing nothing) 1 := { 0 } (i.e. the set containing, the set containing nothing) 2 := { 0, 1 } (i.e. the set containing, the set containing nothing, and the set containing the set containing nothing)

With this, we have numbers! It also comes with the added benefit of:

"The number of elements in a set corresponds with what each number means linguistically in terms of amount".

But what does this even do? Like what about addition?

Well, we can use what's known as a recursive definition to help us figure out what addition is. But first we need the notion of a successor.

Peano arithmetic, that is, arithmetic with integers, can be constructed from set theory by defining the immediate successor of a number:

S(n) = { n itself and every internal object within n }

We could then use this to redefine our numbers as:

0 := { } 1 := S(0) 2 := S(1)

This is very similar to our 1 + n example back in the real numbers.

From this, we can define what addition is using our recursive action:

For any numbers a and c a + S(c) := if c ≠ 0 then S(a) + c otherwise S(a)

This definition is recursive, as it contains itself. But in order to stop us from going infinitely into the negatives, we must stop the process when c reaches zero.

Here's two examples of our definition

1 + 1 = 1 + S(0) = S(1) = 2

2 + 2 = 2 + S(1) = S(2) + S(0) = 3 + S(0) = S(3) = 4

And thus we have that 1 + 1 = 2!

Conclusion

At last, we have reached the end. Congratulations, if you read this all the way through, you have read an entire tumblr post (and a long one that is) on why we can say that 1 + 1 = 2. This is a very broad topic that I have barely scraped the surface on. Here's some other interesting related subjects:

David Hilbert's formulation of mathematics

Peano Arithmetic

Lambda Calculus

Fields, Ordered Sets, and Completeness

Real Analysis

Zermelo-Frankel Set Theory

As always, feedback is very appreciated! I'm an astronomer, not a mathematician. A lot of this stuff I was taught in my first year of university. And I hope you enjoyed reading this. Feel free to follow if you like seeing stuff in the realm of physics, astronomy, mathematics, and computer science.

[dyspunktional-leviathan]

Tl;dr:

I struggle with basic arithmetics.

But not enough that my school would let me not do math.

So I had to do math in school, straining my entire organism for that.

That is really bad, actually.

I don’t even need math in my life at all.

/End tl;dr

_______

I struggle with basic arithmetics.

I do not actually need math in my life. At all. I am lucky to live in the time and the environment where I can always have a calculator do it for me.

The fact that I had to do math in school is bad, actually, and not just for the general school reasons.

I was forced to strain myself on something my brain just fundamentally doesn’t do.

That something not even being remotely anything I would need in life.

But even if it was only one of those two things!

I want to focus on the former most of all here in particular though.

I have chronic migraines, and at least in the later years, when I was homeschooling, my headache would immediately get much worse when I’d just try to think in the direction of math. I don’t know if it was like that when I was public-schooling, there were too many sources of headache worsening.

I don’t remember if I had much math problems before middle school, though I always disliked math, but in middle school, it was very visible in my math work that my numbers were constantly wrong in the most basic ways.

It’s easier to do math on paper than in my mind, writing it out and using column math and long division *is an aid*, I can say I struggle with basic arithmetics even if in school I struggled less (using aids; also still struggled), I certainly do acknowledge those who struggle more than me and the ableism they face

But, so, my struggles with math were not enough for them to let me not do it, just enough for them to force me to struggle more trying to correct my mistakes, and as a whole I still had to do the thing that my brain just doesn’t do, and it was needlessly straining the entire me.

[the-firebird69]

Taking of Pelham 123 - trailer

There's some confusion you can see Billy O'hare which looks like John Travolta you also see Tony Soprano was actually Tommy F. We're very confused and I think David Boyer is John Travolta and we know he is and he's a mac proper and it's craziness and he's taking hostages trades the hostages for this guy who we think is trump in disguise and it is about Pelham investments but it used to be a grant company for students to learn and that's how it started it was a Pell Grant and it was short for Pelham and it was for students who are doing exceedingly well and they kept our son from that money started giving it to Brian who's an idiot and they hate him for it I said You are a stupid sack of **** and it went out of you like a sieve and we have some sort of barbarian carpenter beer drinker. And he's hanging out with Hulk Hogan with the beer in his hand shrugging saying I don't get it hulk Hogan and that's what they're saying he turned out to be a little bit rough so he looks up at Hulk Hogan and says I don't want this ****. And BJA is astonished but was very rude to our son put him down a lot and said he wasn't very smart so one day I signed asked for help in math and Brian couldn't do it and wouldn't do it and he started to ask him questions and Brian turned out to be stupid and he wondered who he was and said This guy is not smart in any way after a time it started to figure other stuff out. And the max hate that too they know it's not our son's fault and it started educating him because he's a rebel and stuff and they said you're a big huge fools and you're not even good rebels you suck at it and it's true and they didn't plan for him to learn tons of stuff more or less kind of did but a lot of them think that it's stupid and some of it's stupid and they do say it even the ones who had it happen say it's a little over the top but if you get down to it they don't want him to know all that stuff. And the Pelham Grant they looked at as money that he could have used to go to formal school and learn arithmetic mathematics engineering instead of going to China and it was not that money that he used and he had investments in Pelham investment firm and there's another name but it includes the word Pelham and our son was confused now he was curious he said boy this is ringing a bell and 9/11 was a big eye opener for our son and it took ages for him to figure stuff out but without that he never would and that was over the top at the end of it they say but really john Travolta's mad because he saw them and they weren't talking about much and they're blaming him for all sorts of stuff. And he says she knew a lot that she has to sit there and listen but I hardly knew anything. I didn't know Mr Paris was at first and couldn't detect much of a change I had no idea it was my own grandpa and he says that was his finding and these idiots are making it look like he knew everything and it's not true he ran around with his mind closed for awhile 'cause he's just trying to get through school and it's very difficult on the outside. And he was saying that to us too after school he started to work a little and at work they were working him to the bone tirelessly then they came up with stupid **** for him to do it got really mad at these people they said you have all this construction you're building giant buildings and you want me to be this proper running around from job to job losing everything I have dragging me through the mud finally living in a camper and he said screw you and it was a good thing. Never found out to be 75% of the population of Earth far too big and the Max lost it they lost control of this thing they're down there thinking they're going to leave then they made some discoveries and the clan escaped and something's up there in Saturn they have huge problems and these idiots don't have control over anything granted they planned it so how can they say it's out of control let's just say it comes off and is acting in his demeanor like he shouldn't have to do what he's doing and it's kind of a chore that's not the act it's really bizarre when it's life and death in the end of the movie he fails in his shot here's a real jerk to the two of our people here but he said in the movie too they're sitting in this lustful loveful relationship and everybody starts screwing around with it and you're screwing around with me like I'm your property too and I'm nobody's property and you're nuts and he's right these people are insane they think that they can go around and push people around shove them around stay what they want at any point in time not even if they have power and they're gonna learn this wrong. They've been too big for too long true too they're big for thousands of years they were about 40% of the population and they run around doing tricks and saying dumb things dumping on people and they're hateful mean people and he recognizes that's and he recognizes that's a fact and also says how he's supposed to know if it's too long if they're this dumb and influence people like they are. And these people had to have them covered it was a nightmare. So you don't know what you see until you open your eyes and John Travolta was glad to hear it and he heard this from him he said I'm just looking over and there's someone there and I said well people live there and these people went nuts about it from what I recall they kept mention.... from what I recall they kept mentioning you and trying to get me jealous trying to see you doing stuff and you had a girlfriend you had several girlfriends. Knew it too and he was real popular and also he said this I can't stand .... and also he said this I can't stand these people at all and I don't want to hear about it he said that's how they were getting away with it and I still think they are. And there are rude bunch of cannibals and that got him going. So we have this problem with them that they're insistent on screwing a that they're insistent on screwing around with their son and if there is nothing. So the giant takes these hostages and he trades them in for Denzel Washington who is actually trump and he's in his own body it's coming up pretty quick and it has to do with the battles one of them we described. In Pennsylvania. It does have a strange name and this is when the money moves to New York City so you can pinpoint when this battle is. And it goes there it goes to the governor know the mayor as a slush fund and he demands that particular money from Tony Soprano and they're threatening other stuff and he hasn't brought down he's forced to and the others are saying to bring it down'cause they wanna see where he brings it and it's kind of stupid they sorta have an idea and he says this you actually know what I'm bringing it and you're kind of having me do it so he got curious and they said they wanna use it as bait and that is going to start and take place very soon.

Thor Freya and yes he has to have some they re chicken shit now

Olympus heard it all day have some intel and need it all they say

we see it is huge. i am here lilstening he is attacked by fleas needs dinner bu tis rules ok i lov it and saw him there and said i amhis. and he said i know but your cute. tried for me a lot. and lame. saw them. and was inundated often.  ahted them and tiried to date me no.  kidnap. and he was forbiden by macs so tested it.  saw themm a bit said these do it say they are influenced  i was pleased good it rolls ow and i see it does i am smart too about it he says and i am all knw it now so hear me they heat up a lot

Hera

we knw and seeit.  tons.  twnty percent out and all heat up we roll. in. and have ten times their force they will be extinquishe dfast and all trump yup

Thor Freya

good 

Hera

[captain-kit-adventuress]

Did you like being an accountant? I’m a math person thinking of doing a career pivot, and one of the options I’m considering pivoting to is accounting.

I'm sorry it's taken me awhile to get back to this. I've been thinking about how to answer your question because for me, I'm ambivalent as to how I feel about having been an accountant.

I had no problem with the work itself. I really enjoyed it, and I was lucky to be able to see a few different specialties, like tax and audit, and have jobs both more and less specialised, and I found things to enjoy in each of them. I also really liked doing accounting work for small businesses; I enjoyed the kinds of problem-solving I regularly encountered. There can be repetitiveness and tedium, but no more so than any other office-based professional job. The maths itself is rather easy, and a lot less intensive than people think. The whole system is based off of a very simple algebraic equation, and the rest is just extremely basic arithmetic functions. Overall, accounting is less about maths (though that is important) and more about problem-solving, so if you enjoy that, looking into accounting might be worthwhile. The only really specific thing I'll say is that if you want to purse a career in accounting, get really good at Excel, because you will live in it.

Accounting is a pretty specialised field, though, and by that, I mean there's lots of areas and niches, it's not really just one thing. Which is great, because it does tend to give you more options based on what your career goals are and how ambitious you want to be. If you are thinking about becoming a capital-a Accountant, it can be a major financial, educational, time and motivational investment, and may or may not be something you're looking to undertake. If you want your CPA (or CA in a number of countries other than America, it stands for Chartered Accountant), there's all of that plus the work of getting licenced or chartered. If you want something more low-key, you might still have to take some accounting classes if you've never done it before; many companies require even entry-level accounting people to have some knowledge of accounting practices. In America, this is called US GAAP (generally accepted accounting principles). Most other countries use a standard called IFRS. Either way, expect the possibility that you may need to take some classes at some point if you're not familiar with those standards, even if you have a job at first that doesn't require them. There are also jobs that won't require any of that, but they generally don't pay all that well, at least not in America, and they can be something of a dead-end to begin with. As always, your mileage may vary. Be prepared to see a lot of positions which require a ridiculous amount of education but only pay $15-$20 an hour, like accountants can't actually do all of the maths and tell those employers by exactly how many dollars we are being ripped off. :) There's lots more to say about this, but this is already too long, so if you want more info feel free to message me privately.

The caveat to all of this is that the reality of an accounting career can be a little bit...jarring for some people, I think, when it comes down to it, and personally, I don't think it's talked about enough. This is a field, by and large, where the entirety of your job centres around helping fairly well-off, or even very rich people make themselves even more money at the expense of pretty much everyone else, often in ways you may personally disagree with; and as I said, depending on how high up you are, they may not even pay you that well to do it. 99.9% of corporate practices (even in small companies) are rooted in execs and shareholders squeezing every drop of money they can get out of something, for no other reason than they can. You may be fine with this; you may just want a job. I'm not judging, we all have to eat. I do bring it up, though, because it can mess with your sense of self, sometimes very subtly. Over time, it made my job a lot harder to do because at the end of the day, I wasn't doing work I could care about or be proud of. That's not important to everyone, and I'm not saying it should be, but if it is important to you, it's something to consider when thinking of a pivot into a new field. Obviously, this can be true of any industry because they're almost all for-profit, but accountants are so directly involved in the money itself that it can be hard to ignore. Finance comes with similar issues, too. I'm not trying to scare you. If you don't think you'll have hangups with it, accounting can be very rewarding work, especially if you enjoy it. It is where my ambivalence came from, though.

I hope this helps, and as I said before, feel free to message me directly for more info if you like. I could literally talk about the ins and outs of accounting all day, because I truly did love it, especially the theoretical aspects. But we none of us live in theory.

[jestershark]

Minor quibble about the bit in the draft about IQ: the issues with IQ and racism actually go deeper than just dreck like The Bell Curve. The entire concept had its origins in the eugenics movement, and the field of psychology has consequently been trying to disentangle the useful bits from the evil bits for a century, because it turns out there are a LOT of ways for a supposedly "objective" test to be bigoted as hell. In the simplest example, if I can solve a page of arithmetic problems quickly and someone else has more trouble, is that because my brain is better at it, or because I had better math education? What if they have test anxiety? What if they grew up in a culture where timed tests aren't really a thing at all? Why is it assumed to be more "intelligent" to do the math problems yourself, and not to find a calculator? The entire field is thorny, because while it's obvious that measurable differences exist (my brain IS better than average at arithmetic), it's also super-easy to accidentally design a test that just tells you that people who think like you are smarter than people who don't.

Appreciate this feedback! I started with the bell curve because I didn't want to get into the wholly racist origins of intelligence in general, but definitely don't want to underplay the eugenic origins. It's also super important to the thesis-- especially because so many people are using intelligence right now to deny people their rights.

it's also important to hit on correctly because it ties to what jobs are being seen as disposable and what people are seen as disposable, and how because AI are seen/perceived as intelligent in a way people respect, they can often be treated better than people who are thought of as less intelligent.

(my coworkers and I have observed something similar in our CEO: if he thinks he could do your job, he respects you a lot less)

But yeah! thank you for your feedback, I really appreciate it and am going to incorporate some more of that feedback into the draft.

(if you like being mad, this some more news vid about jordan peterson goes into how he talks about IQ, and he says pretty much that he believes one in ten people are low IQ and therefore cannot do jobs that are useful for society 🙄 literally "useless eaters." I also recommend "my year in mensa" by Jamie Loftus)

[soul-dwelling]

I'm not sure if π as a plotpoint worked or not. I guess it justifies the whole changing of the world thing because it could theroretically change physics enough so even well known endless sets of numbers can end but still maybe I think it should have been connected more, like maybe saying it has to calculate the perimeter of the new sun or something, idk

I admit my ignorance as to the relevance of pi in physics. 

My understanding was that, seeing as pi is an estimate, that if the Evangelist just nudges that number enough to change it just enough, it throws all of physics out of whack. 

This open the other problem with Fire Force: it acts as if math is itself not an estimate of the rules of reality, but that math is actually representational of reality, thus that, by changing the math, it is like changing reality…which makes no sense, even within the rules of the story. 

If I may give an example, for what meaning I think Ohkubo was getting at: it’s similar to saying that art creates reality, or that religion creates reality. Ohkubo bends reality to his will to make characters act as he wishes rather than how we imagine most people would act, to make a plot unfold neatly as opposed to the mess that reality usually is, to make people look as he thinks they should look rather than far more realistic. 

And that is not quite how art and reality work. Sure, people do imitate what they see in fiction, then that feeds into what becomes “realistic,” then subsequent fiction imitates that “realism” that is itself not “realistic.” To ramble about this with buzzwords: it’s simulacrum, it’s the most photographed barn in Don DeLillo’s White Noise, it’s a copy of a copy. 

(At least when Bungo Stray Dogs has the Book, it has rules: anything you write in it can’t just be willy-nilly like our shitty reality is, it has to follow the logic like a narrative, where one event logically leads to the next. That is a manga that is reflecting on the art of storytelling; Ohkubo’s stuff just feels like writing for his own satisfaction--which, barring so much unethical stuff in his writing, that’s valid to satisfy yourself, just don’t expect it to find an audience beyond yourself.)

But that isn’t what Ohkubo is doing: this isn’t a copy of a copy. It’s not even Plato’s forms. It’s “the Evangelist created images in people’s minds, people then went about creating what the Evangelist wanted, the Evangelist uses this to change math itself, the Evangelist makes math like a magic spell* that changes reality to turn the Earth into a new Sun, to make the Moon into the Soul Eater Moon, and so on.” 

*And, goddamn it, now I realize (or remember again?) that “the Evangelist makes math like a magic spell” thing is more prequel crap: it sets up the arithmetic magic Eruka and the other witches use in Soul Eater. Goddamn it…And I guess that means somehow Eruka and the others’ magic is now also just physics-bending reality-bending crap. =_=; How does Ohkubo keep making this worse…

[cassandrasimplex]

This would have changed so much for me in grade school.

I have what was initially diagnosed as a "math-related learning disability" and would later be known as dyscalculia. It has several symptoms, like difficulty estimating distances and amounts, not understanding how long it takes to do things or how much time is passing, and other things that basically come down to "the brain keeping track of quantities", but the primary symptom is that I can't. fucking. do. arithmetic.

My father fought for years to keep me from saying, "I hate math." Sure, I'd sit down at the kitchen table, look at my math homework, break into tears, and often throw up from stress. (This is "math anxiety", which is its own separate thing anyone can develop if they experience enough trauma related to doing math, but whoo boy is dyscalculia one very certain way to develop it!) Dad insisted that I hate arithmetic, not "math", and would only allow me to complain if I were specific about that.

We'd also play math games for fun, except Dad called them "logic games" and would try to hide the math from me until the very end. Weekend mornings, he'd crack open a Martin Gardner or Douglas Hofstadter book and pick a "puzzle" for us to work together. We'd talk it over a bit, identify important parts of the puzzle, and then he'd have me draw a picture to represent the logic I was applying.

What Dad didn't tell me was that he, someone whose job regularly required him to do things like calculate orbital mechanics, had grown up with the same problem I was struggling with. He kept promising me that all the hard part came first, and that if I could just make it to higher forms of mathematics, I'd discover there was less and less arithmetic and more and more "logic puzzles". He'd replace all the numbers in my math homework with letters for me, then walk me through "putting the numbers back in" one at a time when there was nothing else left to do.

So even in very early grades, if I saw a problem like

6 ÷ 3

We'd first sign a letter to the answer:

6 ÷ 3 = y

Dad would have me ask, "is 6 even or odd"? It's even, so my problem would change to

2x ÷ 3 = y

Then he'd ask me for 6 ÷ 2 and we'd make a note off to the side that x = 3.

You might think this is where we'd put the 3 in for x in "2x", but Dad was focused on using fewer numbers, not more. So we'd end up with

2x ÷ x = y

since if x = 3, all xs are 3s and all 3s can be replaced with xs.

"But if you're multiplying by something, and then you divide by the same thing, all you've got is what you started with before you multiplied!"

That's right, Dad would say, and he'd cross out both xs to leave us with

2 = y

"...and there's your answer."

This meant writing approximately three billion and twelve (but I'm bad at estimating) steps nobody else in second grade had to write down -- but it also meant I made fewer mistakes and passed my math tests.

It also gave me a way to start engaged with the logic behind mathematics when the symbolic notation was working hard to "hide" the part I was good at and force me to confront, first thing, only the parts I'm neurologically unfit for.

Symbolic notation is great, don't misunderstand me. It's a fantastic kind of lossless compression, a shorthand in which everything has a highly specific, precise meaning. It's SO much more difficult to write everything out in words instead. Ask me how I know.

Don't bother; I'll tell you anyway. I know because that's how Dad would have me write down how I was working the math pr- uh, logic puzzles we did together. And a decade plus later, when I was volunteering as a math tutor at community college to work with other students who had math learning disabilities, I'd ask them to do the same thing.

Sure, it takes six hundred and fifty-two times as long (curse you, estimation!) as just doing the problem once. But it actually doesn't take longer than doing the problem WRONG twelve times, getting twelve different answers, not knowing whether the right answer is among them at all (let alone which one it might be), getting frustrated, taking "a break" and getting up to do something else, coming back to it e next morning and trying again only to get a thirteenth answer of unknown quality, throwing the whole paper into the trash (and considering setting it on fire), and then nothing else because you can't bring yourself to prioritize that task again until class time, when you try to "just quickly do it to have something to turn in" and making a zero for something you handed in instead of something you didn't, further permanently centering your belief that you're "just bad at math".

So eventually, I learned I could go the long route, get an answer I could trust, actually learn whatever skill the homework was trying to teach, and get credit for my work... or I could spin my wheels in the mud for roughly equal hours producing nothing, learning nothing, and failing the class by way of failing every single assignment I was given. While crying and throwing up.

And the other students I tutored in college learned the same thing. (They all passed. They ALL passed. You have NO IDEA how proud I am of that, for myself and for them!)

Turns out, "math" is just a way of taking about logic in shorthand. But it's the logic that counts most, not the shorthand! And when there's a complete disconnect between the symbolic notation and the logic it represents, some people who have no trouble at all with mathematical logic are going to fail because their brains get lost in the symbols. When your brain gets left behind by the symbols, you can't learn the logic.

I have yet to meet anyone who's just innately "bad at math". Y'all. I need you to read that again. Three times. Four times, if your reflexive response is "Bullshit, I'M bad at math." I have yet to meet anyone who is innately bad at math, and I say that as someone who is neurologically incapable of adding one and one reliably. I've met people with math anxiety (SO MANY people with math anxiety) and people with low frustration tolerance and lots of other people with dyscalculia. I've met people bad at memorizing and people with less-common learning styles whose brains can't handle learning something intimidating at the same time they're struggling to translate from, say, spoken information to a visual or tactile-kinesthetic leaning style.

But "bad at math"? Nope. Never yet. Not even me.

So why am I going off about dyscalculia on a post about Inuit numerals?

Because this is symbolic notation with the logic baked right in.

This is symbolic notation that TEACHES the logic of arithmetic rather than serving to obscure it.

This is notation that represents the logic clearly to both visual and tactile-kinesthetic learners.

Look at that long division. LOOK AT IT.

Actually, look at the subtraction and addition examples first. Then look at the long division. If the first two examples make great sense, but the long division makes your head swim, I'm going to ask you to consider something: that you don't really understand the logic behind long division as well as you think you do. Not as well as you understand the logic behind addition and subtraction.

I mean, you understand it will enough to perform it. Probably. Although lbr, when was the last time you had to write out a long division problem and work it by hand? You probably learned to manipulate the symbols, passed that grade, and have used a calculator for anything remotely like long division ever since the school system decided you no longer needed "practice" at it.

But you might not have learned it DEEPLY. You might not have learned all it has to say about the way we count, about the way the decimal system works and about how other systems of counting could work instead. Which is fine, in a way, for adults; we get very entrenched in our hatred for math and our resentment that it can benefit us to learn more about it, to truly understand it. We have other tools we rely on and maybe the idea of going back and re-learning how to do long division makes our teeth grind. Because long division sucks, right?

Most of us hate long division, if we're honest.

Unless we had fathers, teachers, tutors, mentors, or friends who wouldn't take "I hate math" for an answer and demanded we separate hatred of calculation from hatred of logic. Or just being bad at calculation from being "bad at math".

But the thing I learned as a tutor is that everyone who hates math, everyone who believes they're bad at math, they got left behind at some point. They missed one day of class when the logic of something like long division was being taught, and when they got back, they memorized how to go through the steps, but they stopped really grokking the logic in fullness. And after that, anything else that built on that skill that got left behind will also be something they have to just memorize, not understand.

More than half my job as a tutor was "rewinding" to whatever early skill I could find that the student couldn't explain and re-teaching that -- and then just getting the hell out of their way as everything else they'd memorized that built on that fell into place as true understanding. So fast. SO FAST. It's like watching a video of a jigsaw puzzle assembling itself at triple speed.

I had to pause up there at "more than half my job" to reconsider the phrase, because more than half my job was also rebuilding lost confidence to battle math anxiety, but you know... in reality, fixing those left-behind points was a huge part of that confidence-building, because that's the point when the student fully realizes that HOW THEY WERE TAUGHT had more to do with their poor performance in math classes than anything else, including dyscalculia. So no, it stands -- even if fixing the left-behind points was less than half the job directly, it's also a critical step in doing the other major task of restoring confidence.

And ever since then, I get SO angry about math education sometimes, because I know from my own experiences that even a "hey presto, you're permanently Bad At Arithmetic Forever!" learning durability that can't be changed isn't enough to make anyone actually, permanently and forever bad-at-math. And I know from my father's experience that even that kind of leaning disability isn't, all on its own, enough to keep someone from being very successful in an EXTREMELY math-centric career for fifty years. And loving it! Truly enjoying it! And I know from the experiences of students I've tutored, as well as my own, that the logic of math is one long, unbroken chain (well, actually it branches into several later on) where any one link being weak, especially an early one, can completely wreck a person's ability to enjoy or succeed at math for multiple decades afterward, and that it's very easy for one link of that logic to just kind of fail to form in a way that goes unnoticed, again for multiple decades.

Some of the people I tutored were retirees. Some of them were great-grandparents. You get lots of older people returning to community colleges after decades of believing themselves "bad at math". I saw how angry THEY got when they realized that something like learning how to cancel fractions without understanding WHY to cancel fractions had been not just holding them back in all math that followed but wrecking their self-image and giving them panic attacks for longer than I, at that point, had been alive. Their rage became mine.

So when I say that this is important. When I say that this visual representation of arithmetic is important.

I wouldn't want to try to do calculus in this system. It's not going to mesh well with established notation. I definitely wouldn't want to use it in any algebra that uses x, y, m, n, w, and v as common variables. I wouldn't want to try to use it in exponential notation. Writing something like "10^5" would be a nightmare to keep legible with numerals based in part on "^" as a number instead of an operand! But for teaching arithmetic itself? This isn't just alternate notation for numbers, it's a METHOD. Just like the sign you use for long division. Just like using sigma for sums. It's numeric representation that also encapsulates logic like an operand does. Squeeze this in between "counting on the number line" (already a visual representation of logic that students discard as soon as they've mastered it) and solving expressions like "7 + 3” and fewer students get left behind so early they never recover from it.

So like. It's important. It's important for reasons already stated, like its cultural significance. It's important because it contains a lesson about how cultures oppressed nearly to extinction matter and have ideas of value, ones they came up with long ago, that are still useful but end up ignored thanks to racism and xenophobia.

But like also. Get this the fuck into classrooms, I beg us all. THIS IS A GIFT. Billions, literally billions, of people could benefit from this. It isn't just a cultural curiosity -- and I feel like there's still some lingering patronizing going on here: aw, isn't it cute what these Inuit kids came up with, see how smart other cultures can STOP THAT. This is vital, it is modern, IT. IS. NEEDED. This is research into the advancement of mathematics as a field that must be taught before it can be practiced.

In the remote Arctic almost 30 years ago, a group of Inuit middle school students and their teacher invented the Western Hemisphere’s first new number system in more than a century. The “Kaktovik numerals,” named after the Alaskan village where they were created, looked utterly different from decimal system numerals and functioned differently, too. But they were uniquely suited for quick, visual arithmetic using the traditional Inuit oral counting system, and they swiftly spread throughout the region. Now, with support from Silicon Valley, they will soon be available on smartphones and computers—creating a bridge for the Kaktovik numerals to cross into the digital realm.

Today’s numerical world is dominated by the Hindu-Arabic decimal system. This system, adopted by almost every society, is what many people think of as “numbers”—values expressed in a written form using the digits 0 through 9. But meaningful alternatives exist, and they are as varied as the cultures they belong to.

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[ohaithe-re]

5, 6, 7, 8

I love small combinatorial structures. Here I'd like to save a few, for my own future reference, I suppose.

The Hoffman-Singleton graph has several nice constructions. Depending on your point of view, it is a graph that is "5-ish", or "6-ish", or 7 or 8. What do I mean by that? Well, the 11-vertex cycle graph is undeniably the number 11. The 8 vertex cube graph probably most screams "2" or "4", but it could also be "6" if you count the 6 faces. Formally, these are small numbers that divide the order of the automorphism group of the graph. When you divide the order of the automorphism group of an object, there's probably an aesthetically pleasing construction of the object that makes that symmetry manifest. The Hoffman-Singleton graph has a staggering 252,000 automorphisms, or 32x9x125x7, so it many nice constructions that look nothing like one another.

This graph is a regular graph, with 50 vertices all of degree 7. That already says something: the number of vertices is divisible by 2 and 5, but the degree gives us 7. Let's go deeper.

That Wikipedia page mentions how it can be built from structures of size 5 (pentagons, pentagrams, and arithmetic mod 5). There's a more elegant description of the Z5 math here. That's the "5", and shows how to divide the 50 into 2x5x5, with many pentagons and pentagrams.

There's the "6", which is based on 1-factorizations of K6. This can be found a few different places, but the main idea is that if you want to build a degree-7 Moore graph, you can take one edge, remove it and its neighbors, and you'll be left with a 6x6 grid to connect in a certain way. This when completed should give you a Sylvester graph. How do you wire that 6x6 graph?

Take a K6 (6 points, all connected to each other). If you try to make a 1-factorization (a way of dividing the edges into 5 disjoint groups, in other words, a perfect 5-coloring), you'll see there's 6 ways. Label the 36 points in your 6x6 grid by (P,F), where P is a point and F is a 1-factorization. Connect (P1,F1) in to (P2,F2) iff P1 and P2 belong to the same 1-factor in both F1 and F2. It turns out this has many nice properties: for any given (P1,P2,F1), there's exactly one F2 that works, and likewise for fixing any other 3. In other words, each 1-factor occurs in exactly two 1-factorizations.

That's the "6". There's more on that here, and the relation to the Generalized Quadrangle here and here.

The Hoffman-Singleton graph can also be constructed from the Fano plane, a specific configuration of 7 points also knowns as "PG(2,2)" or "that horcrux thingy". More specifically, these 7 points get assembled in different ways into PG(3,2). That is also explained on the Wikipedia page on the Hoffman-Singleton graph, and the 7 points of the Fano plane give the degree 7 of the graph. That's the "7".

Now we return one more time to the Sylvester graph. As explained here again, the Sylvester graph arises from the Ternary Golay Code, a specific set of linear equations that can be used to encode 6 ternary symbols into 11 symbols with optimal redundancy. It turns out that the ternary Golay code and the Binary Golay Code, which lets you encode 12 bits in 24, are essentially the only two nontrivial optimal encodings in this form. Where do these two codes come from?

From a miracle! The "Miracle Octad Generator" lets you describe the binary Golay code using octads of 8 elements, and in turn gives rise to beautiful things such as the Leech Lattice. But there's also the ''MiniMOG'', which really should be the "MHG", because it uses hexads instead of octads. You can read about the minimog here or this Google cache. This poster is actually just fantastic. I guess this is really a six then, not an eight, but the MOG (the "8" in this blog post) is great and should be appreciated. My understanding is that there are more connections between the MiniMOG and MOG; for instance, they give constructions of the very special Mathieu groups M11 and M12, or M23 and M24, respectively; and M11 and M12 (from MiniMOG) can be found in M23 and M24 (from MOG). So there are surely some nice ways to see the MOG on 8 elements giving M23, and then M11, and then the ternary Golay code, and then the Sylvester graph, and then the Hoffman-Singleton graph.

[cetaceanhandiwork]

I don’t care about the checkmarks and I’m probably not gonna buy one unless I can think of some really funny way to use it. and to be clear, this post isn’t really about that.

but it was the inspiration for this post, because I saw some interesting secondhand discourse on the topic, which reminded me of this old XKCD episode

but I think there’s a better point to be made in this neighborhood than the sorta gratuitous “take that” approach Munroe used there

which is that if you let yourself get to the point where you see any expenditure of money “for fun” as wicked because there’s children starving in detroit, you have fallen for one of the central lies that money tries to indoctrinate us with: that you can math out everything in terms of +EV and opportunity cost, even things like good and evil.

and the reason you can’t math it out is that:

nobody on planet fucking earth knows enough to actually math out the consequences of an action. nobody has that level of perspective or context.

even if they did, how do you price those consequences against each other? there’s no way to compare one person’s pain to another’s; you’ll getting meaningless results, worse than useless. there’s a reason that we say “there’s no winners in misery poker”.

even if you could compare, how do you add or subtract? how do you do basic arithmetic when we know the hedonic treadmill exists, or that beyond a certain minimum successful care produces over-unity of happiness?

but we think we can anyway. we think we can b/c we live in a world where everything costs money and everyone needs money, where we’ve been trained to think that you can just convert X into a dollar-denominated opportunity cost and compare that way.

but that ain’t how good and evil work, and it ain’t how our hearts work. and it’s a deadly risk, because as soon as you start trying to do that math, you can get your arm twisted into believing that enough money can make something wrong into something right (b/c you can donate it to make something better happen than the bad thing you did). which, to be clear, is nonsense, and it’s the kind of thinking that ate Ana Mardoll (f’rex) alive.

or you start wondering if your existence is somehow a net negative to the world b/c you happen to take prescription meds paid for by medicaid. which, to be clear, is also nonsense and eats people alive every day.

and all this shit erodes your moral sense, your... fuckit, I’m just gonna say your conscience. b/c you keep telling it “no this good thing is bad actually because of the math” or vice versa until you get so much practice at it that nothing feels good anymore, or nothing feels bad anymore. and now you’re entirely off in deep space without even anything to base your busted math on.

I don’t claim to have all the answers. my own mutant ass take on virtue ethics is probably not an effective coda to this post b/c it’s not actually the point. and I’m definitely not saying that you shouldn’t spend your money on good causes. it’s good to care, and it’s good to act on that care! you’ll probably even enjoy it! but that’s not the point either. the point is more that “you should quit your topic X because my topic Y is more important” or “you should give up on real human being Z because my topic Y is more important” are snake oil, because you can’t even do math to those things in the first place without getting yourself irrecoverably, tragically confused.

[sarahsfixations]

I think its very important to remember that math isn't just the abstract system playground mathematicians explore in. This view of math is actually a very high level type perspective that is developed only from very gradual math experience that its easy to forget that it does not at all translate to others. So while yes given how you view math its easy to see how its taught as not portraying it right, and that's probably true to an extent from what I've seen, it's also worth considering the hindsight bias that has from your level of experience, and how you need to consider how students themselves understand it. I think it is quite evident that there are reasons complex numbers are treated differently to the reals and one shouldn't ignore that at the very least, and the student questions on these should be really considered: this isn't about physical intuition but context as a whole. When I say 'implicitly provable aspect', I don't mean in terms of proof systems but as some evidently justified aspect, physical or not. Proofs and formalizations are built on top of math to recontextualize it: mathematicians were able to work with naturals and real numbers without a formalization. And some of these demands (like axiom of infinity) is only a thing in fitting a specific formalization, namely conceptualizing natural number arithmetic as an entire singular structure taking the totality of naturals into one set itself. This is in itself not needed for naturals or speaking of their 'potentially infinite' nature. So yes while it true numbers don't have a definite meaning and can be defined by what they do, there aren't things pertaining to existence both in logic, physical reality, and 'something else' that makes something quantify like a number and have a more tangible existence for most people. It's a valid view when we think of other worldviews outside of math itself. So, while in some part teaching is at play, its also worth considering the concept itself and not treating it 'all as equal'. Because try as I might, there is hardly any totally 'direct' digestible examples of complex numbers I can think of. It doesn't feel its as 'essential' and implicitly a 'number' of a number system as something as clearly quantifiable as real numbers. The closest example to showcase complex numbers I can think is phase and waves: and that requires at least trig, so precalculus. All of this requires at least the crucial abstraction learned throughout algebra and geometry, and that can't be rushed through for most. I'd argue though such a thing isn't even understood until calculus, and where you might actually see actual use. In earlier years, numbers are learned through gradually concrete means because that's all there is to go off of. Rationals aren't some operation playground but roughly explained by its representation as proportion. One can learn about decimals now. This is all contextualized on a number line, which is a continuous geometric object: with these together, one is naturally lead to the concept of real numbers and the demonstrated truth of irrationals. Then, these build the coordinate grid, where functions, linear equations need to be explained. Complex numbers can be maybe introduced here but I'm not sure how to bridge the geometric image of the rotation to the actual complex arithmetic itself. I mean thinking it now, and from how I've seen it taught at tutor places and just, gosh how critically missed basic concepts in algebra are, students rushed to algebra before they can do arithmetic, etc., like a massive overhaul of the system can be done. But, I'm still wary over your perception of complex numbers as something so analogous to teach as say the reals.

Okay this is going to be a bit more venty than usual but I've seen a post about complex numbers that's annoyed me.

The only reason you are less willing to accept the existence of imaginary numbers is you aren't taught it in a way that helps build intuition nor from as young an age.

Complex numbers have been around for centuries. They aren't some new fangled thing that are mysterious to mathematicians. Part of the non-mathematician's conception of imaginary numbers certainly isn't helped by their name but I have to say I think the internet and clickbait are a lot to blame for that too.

A lot of mathematics is invented by thinking "what if..." and rolling with it to see if you can draw anything meaningful from it. "What if numbers less than 0 exist?", well then you have given meaning to something like 2-7. "What if we could divide any two integers?", now you can talk about what 3/7 means. Asking "what if square roots of negative numbers did exist?" let's us explore whether √(-1) would give us something with consistent and useful properties and it turns out it does (technically we just declare i²=-1 and i=-√(-1) does everything just as well).

"You can't take the square root of a negative number" is drilled into pupils heads at school when really the message should be a more subtle "there aren't any real numbers that are the square root of a negative number". It's a subtle but important difference. It's exactly like saying "there aren't any natural numbers x such that for naturals numbers y and z with z>y we have x=y-z". You'd be pretty hard pressed to find anyone saying that the latter is impossible.

I don't really know how to end this. I'm just frustrated

[bopinion]

2022 / 18

Aperçu of the Week:

"Life is what happens when you're busy making other plans."

(John Lennon)

Bad News of the Week:

We men will never understand how women manage to do several things at once. Clichéd, for example, ironing, telephoning and watching television. "Multi-tasking" is the name given to this enviable ability. A mystery to those of us who can barely manage to read the newspaper on the toilet. The decisive factor here seems to be retrieving stored routines, since even a female brain cannot manage to actually focus on more than one subject at a time. Especially when it comes to the intellectual processing of complex topics.

This brings us to media consumption. Where not only one headline chases the next, but also displaces them from the frontal lobe. Our society was just grasping the magnitude of climate change, species extinction, nature destruction, etc., when the next dominant crisis came around the corner with Covid and took over the entire public interest. Until Russia attacked Ukraine. Other crises - such as supply chain issues that call into question the entire system of the globalized economy or general price increases on a shocking scale that many of us can't bear any more - barely make it to page 1, which they would dominate "in normal times."

The problem is attention span. Our brains manage an average of 10 trillion arithmetic operations per second. And yet doesn't manage to sustainably and productively turn its attention to more than one topic. Sascha Lobo, a well-known columnist on media topics, calls it the "puppy problem." Because playing puppies - as we all know them from thousands of oh-so-cute clips on social media - always and immediately get distracted by something new: Oh, a ball! For a child, that means ignoring the geography presentation until the math test is written. And for government policy, that the real agenda (as our cabinet just admitted after a closed-door meeting) stays put as long as there's a crisis to deal with.

And that's what I think is a fundamental problem. Because humanity can't really afford that. There is no such thing as a big pause button, with which we could stop climate change, for example, until we have the time and energy to deal with it. No matter how busy we are today, we must not lose sight of tomorrow. Many problem complexes are also interwoven. Take Yemen, for example: In the country itself, the soil has dried out so much that practically nothing grows anymore = climate crisis. In the blocked ports of Ukraine, Yemen's main supplier, 25 million tons of grain cannot be loaded = supply chain crisis. But because of Iran's and Saudi Arabia's ongoing proxy war in the country, no distribution could take place anyway = war crisis. And that would have been possible anyway only to the extent financed by aid organizations, since Yemen is effectively broke = debt crisis. This will cost the lives of many = hunger crisis.

So as not to be misunderstood: I do not want to criticize the fact that the focus of foreign and security policy is currently on Ukraine and that many economic and financial factors are also taken into account. But this does not mean that the Ministry for the Environment and Nature Conservation or the Ministry for Economic Cooperation and Development (aid) have to remain inactive. Spending on the Energy and climate fund, for example, has been more than tripled to 27 billion euros in 2021. However, this rather theoretical budget is being dispatched only hesitantly because the administration is standing on the brakes and redistributions are already planned, for example to - of all things! - to lower the price of gasoline. But a so-called "special fund" of 100 billion euros is being approved in a single weekend to upgrade the Bundeswehr. Just to clarify: this has nothing to do with the urgently needed military aid for Ukraine, but is a frightened (over)reaction to the Russian potential for aggression. As if, say, Grozny or Aleppo did never happen.

Yes, we need to take care of Ukraine's defense potential. And of its reconstruction. And about more energy autonomy. And about military emancipation. But it would be nice if we could do that on a planet that can sustain all of our lives. Otherwise, our children will find themselves in a crisis that can't be managed at all.

Good News of the Week:

The training of Ukrainian soldiers on Western weapons can apparently constitute war participation under international law. According to a report, the delivery of weapons itself does not constitute entry into war, "only if, in addition to the delivery of weapons, the instruction of the conflict party or training on such weapons were at issue, would one leave the safe area of non-warfare," the Redaktionsnetzwerk Deutschland quoted from an expert opinion of the Scientific Service of the Bundestag. An institution whose non-political opinion-forming could always be relied upon.

Now we learn that the Bundeswehr will not only hand over the modern self-propelled howitzer 2000 to the Ukrainian military, but will also provide training on the arguably complex weapon (the deviation of the weapon positioning system is only one minute of arc at a distance of 40 km - whatever that means). In the German state of Rhineland-Palatinate. And not secretly, otherwise I would not have been able to read about it on several serious media. So Germany will be involved in the war? And what would that mean? In a way, a redefinition of normality.

I've already wrote about the fundamental changes in German positioning forced upon us by the Russian invasion. Including the admission of fundamental misjudgements - or blindness, if you will - towards a despot who follows what is actually a relatively obvious objective, namely the re-establishment of the Soviet Union including a "balance of terror." Principles of decades of policy have not only been questioned, but answered. As wrong.

Why is this good news? Because there is an European, even worldwide consensus on this. Germany is no longer expected to stand idly by in humility in the face of its historical guilt. On the contrary: Europe's strongest country is expected to show leadership. Not (anymore) only economically, but now also in terms of foreign and security policy. This is an opportunity. To prove that we have not only learned our lessons. But to turn them into action. History can be written, in all meanings of "as well as". For the benefit of all. The signs are good. All we need now is the confidence to recognize them as they are. And the courage to understand them as a mission.

Personal happy moment of the week:

It was only two or three months ago that I was worried about my son's school career because of French. Even if the goal - a stable E - seemed manageable. So I wasn't surprised when he said it didn't go well when he was quizzed on the day of his geography presentation, of all things, and therefore completely unprepared. Then he starts grinning, "...but very well!" He actually got an A after all. Miracles do happen.

I couldn't care less...

...that Macron renamed his party from "En marche" to "Renaissance". I didn't care that Google became Alphabet, Fiat & Peugeot became Stellantis, or Facebook became Meta. Old wine in new wineskins - a phrase I never really understood. "En marche" is still best translated as "Let's go!". Why not, when you are out to make many things better and a few things new. "Renaissance", on the other hand, linguistically means "rebirth" - before which one must actually have been dead. And in terms of cultural history, it means turning away from the dark Middle Ages toward humanistic values - which seems a little strange against the backdrop of having been in authoritative government for the last five years. I'm just getting a little envious, because probably some supposed rebranding expert charged a heavy bill for this bullshit.

As I write this...

...I am sitting on the train on another Sunday. This time on my way to Cologne for a trade show preparation. The first time since the beginning of the Corona pandemic. Therefore with mixed feelings. For business reasons I hope for a lively attendance. For infection control reasons I am unsettled. Thank goodness I am a strict mask wearer.

Post Scriptum:

Tomorrow in Russia there will be the big celebration of "victory day", the anniversary of the triumph over Nazi Germany. Originally, it was assumed that the ominous "major offensive" in eastern Ukraine would also serve to give the Kremlin despot the opportunity to announce at least some kind of victory on this very important day in Russia. In the absence of such a victory - not even Mariupol has been completely taken - fears are growing that Putin might declare a general mobilization and swear the Russian people to blood, sweat and tears. Rarely have I wished so much for a free press and unrestricted media access for a population to be able to counter this inconceivable propaganda with something real...

[yanklbonkl]

Okay because an explanation was requested in the tags, here it goes: (warning: math. Warning: long post. A

A set is a "collection of objects". To be precise they have to follow some rules but to be honest I don't know much about that and it shouldn't matter for this post. The "cardinality" of a set just asks "how big it is" in a precise sense. If the set is finite, this is just how many elements are in the set. For example, the cardinality of the set of overripe bananas on my desk right now is one.

Of course, most mathematicians aren't content to just deal with finite sets. That would be too close to the real world. To make sense of the cardinality of an infinite set, we use an idea called a bijection.

A bijection between two sets (say A and B) is a rule that assigns an element in A to an element in B, such that every element in B has exactly one assigned partner from A.

The easiest example of a bijection is counting. If I have five books on my desk, when I count them, I assign each element of the set {1,2,3,4,5} to exactly one partner book. If every book gets counted, and no book gets counted twice, then this is a bijection.

It makes sense then that two sets should be the same size if there is a bijection between them. But this idea gets a little wonky when talking about infinite sets. (You can find way better descriptions if you look up Hilbert's Hotel. Here's a video Veritasium did!)

But if for some reason you like reading text more than watching nice Youtube videos, here's it goes:

You'd think that the set of whole numbers ({1,2,3,4,5,etc.}) is bigger than the set of whole even numbers ({2,4,6,8,10,etc}). This is because every whole even number is a whole number but not every whole number is even.

But if I give you the rule: "Assign every whole number itself times two," we assign every whole number an even number as a partner, and each even number has exactly one whole number as its partner. So the set of whole numbers and the set of even numbers has the same cardinality!

I won't get into the details (though you can see the idea in the video above at around the 1:57 mark) but actually the set of whole numbers has the same cardinality as integers (whole numbers that can be negative) and even as rational numbers (the set of fractions p/q where p and q are whole numbers).

We might then think that there is only one infinite cardinality. But in fact the set of real numbers (numbers that we use to express distances between points, and which change continuously and without jumps) has a bigger cardinality than that of the whole numbers.

In fact, there is no largest cardinality! (I'll include a proof at the end of this post). We can always make bigger and bigger sets. But what about an in-between cardinality? Is there some set which is "bigger" than the whole numbers but "smaller" than the real numbers?

The continuum hypothesis is the guess that the answer to this question is "no". It turns out however that neither the continuum hypothesis (no in-between infinities) nor its negation (yes in-between infinities) can be proved. And I don't mean that it's an open problem. I mean that there is no possible proof of the continuum hypothesis or using "first order logic" and the fundamental axioms that arithmetic stands on, and there is also no possible proof of its negation.

I don't really have an opinion but I felt like this would be a good shitpost to poll people about whether something that can't be proven is true or not (being true is not the same thing as being provable! a discussion for another time).

Bonus panel! A proof that there are infinitely many infinite cardinalities (Hooray, more math!): Given a set X, we'll say that the power set of X (which we denote as P(X) ). is the set of all possible subcollections of X that we can make. For example if X={1,2,3} then P(X) = { {},{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}. (Elements of sets can be other sets! Crazy, right? Also, that first pair {} that I put in is not a typo. It is the set with no elements, called the empty set.

Theorem (I think this is due to Georg Cantor. Tumblr, pls make your jokes about outliers and infinities): Given a set X, there is no bijection between X and P(X).

Proof: We'll proceed by contradiction. Suppose there was a bijection. If x is an element of X, we denote the partner in P(X) it's assigned as f(x). Now remember that f(x) is an element of P(X), so by definition it is a subset of X. For any given x, we can ask if it is a member of f(x).

Now let A be the set of x contained in X which are not elements of f(x). As A is a subset of X, it is an element of P(X). And since f is assumed to be a bijection, there is some element a of X such that f(a)=A.

But now, is a contained in A? If it is, then by the definition of A it is an element which is not contained in f(a). But as f(a)=A, this is a contradiction. And if a is not contained in A, then by the definition of A it follows that a is contained in f(a), which again is A.

Therefore the original premise is wrong and there is no bijection between X and P(X).

[morfinwen]

OC Name: Ian, Lauren, Q, Niner, and Ash.

Wow, this one's old! But answered at last.

Ian

Something really awesome he can do: His coworkers are certainly amazed by his ability to remain in Lauren’s presence when she’s in a temper, let alone how he can talk to her without getting murdered, or even -- sometimes -- calm her down.

A person, creature, or thing he adores: Squirrels. Growing up, he named all the ones he saw in his backyard, and would watch them for hours. He’d love to do the same now, but as an adult with bills he doesn’t have the time. He does have a bird feeder out in his yard specifically for the squirrels.

A secret he’s hiding: No secrets -- Ian wouldn’t be able to keep one if he had it.

Something he truly fears: Something bad happening to his parents.

A fond memory of his: The first time his parents drove to visit his mom’s parents on the Tennessee/Kentucky border. It’s a cozy memory of scenery passing in a blur, what seemed like a continuous stream of snacks and juice boxes, switching between CDs of his and his parents’ favorite music, and listening to his mom and dad tease and flirt with each other (subtle enough to fly over the head of a seven-year-old who still thought kissing was icky).

A place or item which gives him strong feelings: The sanctuary of the church where he grew up. It’s where he was baptized, and where he played guitar during some very moving services.

A dream or ambition for the future: Perform on live TV.

An angsty fact about him: He and Lauren had a falling out in their second year of college. I hesitate to say it was over Protestantism and Catholicism; it was, at least to some degree, but it was also about worldview, the nature of God, the role of tradition and conscience, and perhaps most of all it was about what we owe to people we disagree with. They’d argued about religion before, but it wasn’t until then that they actually understood enough to do more than rebut “Pastor Andrews says” with “Father Vernon says”, and for it to matter enough to them to blow up as much as it did. It took months for them to reconcile, partially because of stubbornness and uncertainty of how to fix things, but also because schoolwork and practice meant they didn’t have the time to figure it all out.

A domestic fact about him: Ian almost always keeps most, if not all, of his windows open. Even when it’s cold (for Tennessee) or raining. It can make things cold, and the smell isn’t always great, but he insists it’s too stuffy otherwise. Lauren is pretty sure it’s a psychosomatic thing.

A random other fact: He once had a dream where he was a dog, Lauren was a cat, and they escaped from their owners to travel the world together. It would’ve made a good movie.

Lauren

Something really awesome she can do: Play Chopin’s Fantaisie-Impromptu (Op. 66) completely from memory.

A person, creature, or thing she adores: Not sure if “adore” is quite the right word, but one of the most important people to Lauren, outside of Ian and her immediate family, is Father Vernon from her family’s parish. He’s known her family since before she was born, listened to their confessions, counseled her parents, baptized her and her siblings. Every weekend she drives the couple hours it takes to her hometown so she can attend Mass at Father Vernon’s church.

A secret she’s hiding: She’s written songs … about fictional characters. She even composed a suite of music for a pair of fictional characters’ wedding.

Something she truly fears: Permanent damage to her hands. Her music teacher once mentioned a friend of hers who couldn’t play the piano anymore after something heavy fell on his hands. If Lauren was the type of person to have nightmares based on things she heard, that certainly would have given her nightmares.

A fond memory of hers: Her grandparents would have a picnic sometime in June, every year. It wasn’t always enjoyable, between the long car drive there with all of her siblings jammed into a cramped place, and the potential for bugs, sunburn, and bad weather, but the park was beautiful, there were so many other kids around that there was always someone to play with even if everyone else had annoyed you, and when it got to be evening they would all gather around, play music, and sing.

A place or item which gives her strong feelings: Mrs. G’s music classroom at the elementary school. In addition to band during school weeks, it was also where Lauren had her piano lessons with Mrs. G on the weekends.

A dream or ambition for the future: She doesn’t think winning a Grammy award is out of the question someday.

An angsty fact about her: From first grade until she graduated college, she believed herself to be her parents’ least favorite child. Even now, it’s not so much that she doesn’t believe it as she doesn’t think it’s worth it to spend time and energy thinking about it.

A domestic fact about her: In order to have room for a keyboard in her apartment, she gave up on having a dining table, so she eats all over the place. She’s good about taking bowls and plates back to the kitchen, but there’s constantly cups and silverware lying around the living areas.

A random other fact: She hates her middle name. “Eleanor” sounds like an old lady name, not least because she’s named after one of her mother’s great-aunts, who is quite old, and has the kind of personality that suggests she was born gray-haired, wrinkled, and talking about “in the old days”.

Q

Something really awesome he can do: He is trained in the use of multiple types of swords. It was a quid pro quo with his aunt and uncle: Q spoke to his politically-connected buddy from boarding school and got the ball rolling on an exemption from some nasty tariffs, they arranged for sword fighting lessons for a year. He never participated in any tournaments or anything (too much publicity), but he can say without undue pride that he got to be pretty good.

A person, creature, or thing he adores: “Adore” doesn’t really describe how Q feels about anything.

A secret he’s hiding: He’s not exactly hiding it, but he isn’t open about precisely how rich and powerful his aunt and uncle are, or how many famous (or in the case of some of his cousin’s criminal friends, infamous) people he knows through them.

Something he truly fears: Just the idea of being buried alive freaks him out.

A fond memory of his: He’s got some good memories of some summer holidays during his time at boarding school. Occasionally Q got invited to tag along with someone, a friend of his cousin or a fellow classmate, on their vacation to some super rich resort in some beautiful, exotic location. He’d still hear from his aunt and uncle regularly, and he always had to be well-behaved, but it was less than when he was at school -- at his age, just hanging out with rich and influential people counted as “networking” to his aunt and uncle, so they’d call in to check on him regularly but otherwise left him alone, and in a less formal setting than school “well-behaved” was an easier standard to meet. While parts of those summers were genuinely enjoyable for him, years of living hand to mouth a hairsbreadth away from sleeping on a street corner has added a much rosier shine to those days sleeping in five-star hotels and eating haute cuisine.

A place or item which gives him strong feelings: The family pile. It was where his dad spent his summers, so spending his own summers there growing up was one of the few times he felt connected to his parents. He’s also spent a significant number of holidays and family parties there, so it’s also associated with the exacting standards of his aunt and uncle and the strain of Keeping Up Appearances.

A dream or ambition for the future: He likes to imagine his aunt and uncle getting taken down a peg (or two, if he's particularly angry with them; sometimes he dares indulge the thought of three), though he struggles to imagine a scenario where that happens without notable repercussions.

An angsty fact about him: Virtually all of his t-shirts are band t-shirts, including bands he doesn’t listen to, bands he’s never heard of, foreign bands, fictional bands, and bands with potentially offensive names or symbols (though he usually only wears those at home or when he can be pretty sure he can keep his jacket closed all day). During his time in LA, one of his roommates asked if he wore them because, as an orphan who grew up in boarding schools, it was the closest he got to feeling like he belonged to something. The precise wording was kinder than that, but it still kickstarted a realization that rocked Q’s world for a couple days.

A domestic fact about him: He’s kind of weird about household chores in general. Despite his best efforts not to be as dismissive as his aunt and uncle, he still grew up in an atmosphere of "The Help does that," and it led to a steep learning curve when he moved out after graduation. He’s on top of dishes now and has a good handle on laundry, but sweeping and vacuuming require active thought, he barely registers that mopping is an actual thing, and unless "swipe hand over surface then brush hand off on pants" counts, he has never dusted.

A random other fact: Thanks to growing up outside the occult community, Q is unaware of the various taboos and 'bad words' within the community, and more than once says something offensive. Fortunately, this never creates any real issues for him, as everyone he does it to or in front of is aware he's not doing it maliciously. In fact, to most of them it's more like a small child 'swearing' because they misunderstood or mispronounced something, or saying something offensive out of innocent ignorance. Q is not best pleased when he finds out: being unintentionally offensive is one thing, being unintentionally adorable is worse.

Niner

Something really awesome she can do: She’s very good at mental math -- basic arithmetic, conversion from metric to imperial or types of currency, multiplying large numbers. Most people are more impressed with this ability than Niner herself is: she’s never had to work at it, and for most of her life it hasn’t been terribly relevant.

A person, creature, or thing she adores: Niner has a lot of younger siblings. She adores them all. Around them, she will drop the pretense of caring about nothing, and show full enthusiasm for anything they like.

A secret she’s hiding: She hasn't told anyone about the abusive relationship in her past.

Something she truly fears: For werecats, the threat of getting caught by animal control and getting euthanized or ending up as somebody’s housecat is about as probable as your average person getting struck by lightning, but few werecats are completely immune to fearing it. Niner in particular finds it horrifying.

A fond memory of hers: Her last year hanging with her parents and immediate siblings was a pretty good one. Since their kids were all on the verge of striking out on their own, her parents allowed them more independence than they ever had before, but there was still the safety net and companionship of family. They also made a point of visiting some places that they’d talked about visiting for years but hadn’t gotten to. It was basically a year-long vacation, and made Niner more aware of her independent spirit.

A place or item which gives her strong feelings: An alley behind a bar in Atlanta, Georgia. It’s where her relationship with Marrow ended. It’s also where one of her recurring nightmares, on the rare occasions she has one, takes place.

A dream or ambition for the future: Werecats travel a lot, but they tend to remain in the same country, and after their roaming days as young adults, they tend to remain in the same geographic area. Niner’s roaming days might be over, but she wants to visit another country. It won’t be easy, considering that she lacks money, a birth certificate, and a general idea of what other countries there are out there, but Niner can be dedicated when she really wants something.

An angsty fact about her: Tied in with her desire for independence is a belief that she needs to rely solely on herself, that other people can’t be trusted or that asking them to bear even part of one of her problems is infringing on them.

A domestic fact about her: Niner’s favorite place to sleep is Q’s windowsill. It gets a good amount of sunlight, and the size is just perfect.

A random other fact: She once worked as a cashier. It was just for a single shift, she got paid under the table, and frankly she was terrible at it, but the hot dog stand guy was desperate, and Niner really needed the money.

Ash

Something really awesome he can do: Ash can make a vegetarian version of just about anything. He considers it a gift. To others in the household (particularly Connie) … it’s amazing, but not in a good way.

A person, creature, or thing he adores: A crocheted frog that ‘lives’ on a shelf in the kitchen. It was a gift to his great-aunt who owned the house before him. It’s not what most people would consider cute, it’s probably older than he is so it’s got some noticeable wear and tear to it, but to little bitty Ash it was a benevolent spirit watching over the kitchen, smiling kindly to everyone who entered, and never telling on the small boy sneaking cookies before dinner.

A secret he’s hiding: It’s not quite a secret, more a deeply personal family matter that Ash is aware of but doesn’t share with anyone. His aunt Lily isn’t technically his aunt. She’s a half-dryad who, because of complications tied to her conception and birth, is bound to a tree on his grandmother’s property and traveling too far from it is painful for her. Obviously this isn’t something they can share with normal people, but it’s a sensitive enough situation that no one outside of Ash, his mother, grandmother, and aunts knows the details.

Something they truly fear: Ash is a naturally patient person, but he is not inherently non-confrontational or easy-going. And he is not the kind of person to just stand by when something makes him angry. He’s not exactly scared of losing his temper and doing something he regrets, but he is concerned about someday ending up on a slippery slope and reaching a point where he wouldn’t regret it.

A fond memory of his: He and Danae took a lot of walks in the country when they were in high school, usually by themselves, but sometimes with one of his aunts or one of her siblings. One especially beautiful summer evening, they sat by a lake for a couple hours and talked about their dreams for the future -- not just what they were likely to do, but what they really wanted to do with their lives.

A place or item which gives him strong feelings: The abovementioned lake. He spent a lot of time there growing up, and there’s good and bad memories tied to it. Mostly good.

A dream or ambition for the future: He has a lot of plans for the house. He’s not sure yet whether he wants to try to turn it into a school for enchanting, or turn it back into a bed-and-breakfast type place for occult people. The former is more appealing to him personally, but the latter works better considering that there’s already a bunch of different occult types staying there.

An angsty fact about him: Ash is the least angsty person in the household, honestly. He’s very grateful for that.

A domestic fact about him: He hates dust. Dishes can stack up for a meal or two, laundry gets done mainly because otherwise he’d have no clothes, mopping only happens if the floor is sticky, the fridge is cleaned out on a schedule, but dusting happens every day.

A random other fact: He’s watched several YouTube videos trying to learn how to yodel. It’s gotten put on the back burner, now that there’s people in the house who could hear him no matter how tightly he shuts the door or how quietly he tries to do it, but he still hopes to learn how someday.

Thanks for asking!