The Billy Joel GIF Set On Instagram Is Funny Because You've Got All Your Uptown Girls, Your We Didn't

Around The Galaxy In Eighty Hours

Rey left the Falcon behind, walking up the steps on the Ahch-To island, and she fought the urge to run.

It had taken all this struggle to get here. All this time. The map BB-8 had carried… so many who’d been lost on the way… and now she was here.

She was going to ask Luke Skywalker for help. The legendary Jedi Master, the one who had defeated the Emperor.

As she climbed, though, a niggling little feeling began to gnaw at her.

Where was he, anyway?

She’d been assuming he was somewhere high up, and the Force wasn’t pointing her anywhere else. But she couldn’t see him, and as she reached the very top of the stairs… there was no sign of him.

“Master Skywalker?” she asked, looking around. “Master Luke?”

“Jee-dhai?” one of the locals asked, in a curious voice.

“Huh?” Rey replied, turning. “I… well, I don’t think… I want to be, but I’m not one yet… do you know where Master Skywalker is?”

The hooded alien shrugged, and pointed to one of the rock huts.

Curious, Rey entered.

It was immediately obvious Master Skywalker wasn’t in the hut. There wasn’t room. There was barely room for Rey… but, after a moment, she spotted something odd.

A folded piece of flimsiplast, with a metal-rimmed piece of crystal on it.

Taking the crystal, Rey was surprised to find that it felt… warm, and tingly. It fizzed with an unidentifiable but oddly familiar energy, and she turned it over before opening the flimsiplast.

It held only one sentence.

Use the Force on the crystal.

“...is this going to be a riddle?” Rey asked. “Or a trial of some sort?”

Silence answered her, and she took a deep breath before closing her eyes and focusing.

It was still… difficult, to call on the Force at will, but she could do it.

As she did, the crystal glowed, then filaments of light streamed out of it to form a face.

Master Skywalker’s face. She was sure of it.

“To whoever has found this,” he began. “Firstly, if this is Ben… well done for coming back to the light. And if not… I’m glad there are others besides myself who can use the Force without being tainted by the Dark Side. This crystal has been constructed using the techniques of the ancient Holocrons, which would shatter if they were forced open by the Dark Side.

He paused. “The Caretakers have a few of them, in case they need to replace one. Anyway… if you came here, then either the Force guided you here to Ahch-To or you came following the map. And if you came following the map, you came looking for me.”

Master Skywalker’s expression turned rueful. “So I’m sorry to disappoint you. I’m not here. I left. I grew up on a desert planet, and this place just… unsettles me. It gives me the creeps to see all that water. Hurricanes should be illegal, and this planet has some really nasty ones… anyway, I’ve moved somewhere where I don’t need to worry about that. You’ll find me in the Bespin system, on Cloud City…

Rey’s eye twitched, as the blue illusion of Master Skywalker’s face listed off an address.

The crystal fizzed slightly, and she dropped it before she could break it somehow, then crouched down and picked it up again – not accessing it with the Force, this time.

“Right,” she said, her voice tight, and turned to go right back down the slope again.

“You’re back early,” Chewbacca said, concern in his voice.

“Luke’s not here,” Rey replied, hitting the switch to raise the Falcon’s ramp. “Do you know where Bespin is?”

Chewie blinked.

“What?” he asked. “Yes, I know where Bespin is… you’re saying he’s on Bespin?”

“Apparently,” Rey replied. “Though I suppose the map is a map to where he went, not where he is. It’s not like he was updating it…”

Cloud City was an amazing sight, though it had begun to pall slightly for Rey when it took them half an hour to get a landing permit.

Eventually Chewbacca called in a favour from someone called Lobot, and ten minutes after that Rey rang the door chime on the address Luke had given her.

Then she stood outside, waiting.

It was strange to be in a completely built environment. Even the ground under her feet ultimately had nothing beneath it but air… and yet all this was kept in the air by technology.

If Rey hadn’t known quite so much about how solidly built repulsorlift units were, she might have been unsettled.

The door hissed open, and a woman looked out. “Yes?”

“I’m looking for Luke Skywalker?” Rey asked, awkwardly.

“Oh!” the woman said. “You know, he didn’t leave a forwarding address, but he did ask that something be given to anyone who came looking for him… hold on a moment, please.”

The door hissed closed again, and Rey leaned on her staff and groaned.

“I’m guessing we’re leaving?” Chewbacca asked.

“We’re leaving,” Rey confirmed. “For somewhere called the Dagobah system.”

She held up the crystal she’d been given. “If you’ve never heard of it, this should help, at least. It’s got a planetary map, as well… and a long, long complaint about vertigo.”

“He did once fall out the bottom of Cloud City,” Chewbacca volunteered. “That would give anyone vertigo… here, anyway.”

“So after spending a month here, I realized what training with Master Yoda had let me forget until then,” the pseudo-visible Jedi Master explained, as Rey focused – not without some annoyance – on the crystal she’d found in a hut. “Which is that Dagobah is damp. I can’t walk very far without sinking into the swamp, the only food available is moss soup… Master Yoda stayed here for decades, and I can see the argument that a Jedi should be inured to physical discomfort, but I just can’t take it any more. I’m going to Ajan Kloss.”

“Really?” Rey asked. “Really?”

She focused, drawing out her anger, and expelled it with a sigh.

Where on Ajan Kloss was she supposed to be looking, anyway?

The holocron-alike crystal shimmered, showing an Ajan Kloss planetary map, and Rey committed it to memory before closing her hand around the delicate-seeming crystal.

“All right,” she said. “Ajan Kloss, then! And there had better be a Jedi Master there.”

There was not.

“So it’s been the rainy season…” the next crystal declared. “And it’s not as swampy as Dagobah or as rainy as Ahch-To, but it’s a lot warmer and the combination is absolute hell. I thought it was the rainy season when I was here before, but it turns out that it was actually the dry season. This is the rainy season, and it never gets dry. Nothing gets dry. The humidity is absolutely one hundred percent constantly. The floor’s covered with millipedes and our robes are growing fungus on them.”

Rey shuddered involuntarily.

It did sound bad.

They were fortunately in the dry season again, or at least she assumed so since the rain coming down outside was only moderately heavy and the geography hadn’t been entirely covered by cloud.

“What’s worse, the plants here even grow at night,” Luke complained. “So that’s it. I’m done with this place. We’re moving somewhere where there’s no need to worry about plant life at all…”

“Are you sure this is necessary?” Rey asked, two hours later.

“Yes,” Chewbacca replied, giving her another parka, and Rey put it on somewhat awkwardly. “You’re from a desert world. You know how Dagobah was cold and wet?”

“I’m having trouble forgetting,” Rey replied.

“Well, that’s about fifteen degrees,” Chewbacca explained. “Hoth is minus forty. I was cold there.”

Rey stared.

“...do you have any more warm clothes?” she asked.

Eventually, with some difficulty, Rey struggled into the ruins of the Rebellion’s Echo Base.

It was below freezing cold, and intensely annoying, and what was worse was that there wasn’t even a Jedi Master there. Instead, there was another crystal.

It mostly contained Luke complaining about how kriffing freezing it was, and that he’d spent three days here before electing to move to the Forest Moon of Endor.

“What is this?” Rey asked, after extracting herself from the parkas and as the Falcon sped towards the Endor system. “Is it some kind of sick joke?”

“I’ll give this for Endor, it’s warmer than Hoth,” Chewbacca contributed.

The Endor map led to an Ewok village, where they treated Chewbacca like an old friend and sniffed at Rey with great suspicion before Chewbacca managed to make himself understood enough to explain that she was a friend.

Then an Ewok shaman said… something… and Rey found herself involved in some kind of blessing ceremony. It was surprisingly useful, in that it actually involved the Force, but Rey was struggling to concentrate by the second hour… and it wasn’t until the fifth that she actually managed to convey the question she had.

The Ewoks discussed amongst themselves, then finally realized what she meant, and led her to a large treetop hut.

An empty hut, with nothing but some folded flimsiplast on the table, and a crystal on top of it.

Rey wanted to scream, but she didn’t want her hosts to take it the wrong way.

“If you’ve ever met Ewoks, you’ll know they’re brave warriors and good people,” Luke said, as Rey slumped over the Dejarik table on the Falcon.

Both she and Chewbacca were watching Luke’s latest message, and part of Rey hoped that wherever it was going to be was far away enough that she could get some rest.

The rest of her was wondering if they could just give up looking.

“But they’re also… a bit much,” Luke went on. “It took a month or two, but ultimately it got to be too much for us, so we decided to move on. This time we’re going to somewhere where we should be able to be alone, and as a bonus we can be out of the rain as well… it’s a lot like a homecoming, in some ways. We’re going to the Great Temple on Yavin Four.”

Chewbacca muttered something, and went to set the autopilot.

“I never thought I’d say this, but I have actually got bored of green,” Rey said, as they flew low over the jungles of Yavin Four. “I didn’t think it was possible to get bored of something that quickly.”

Chewbacca shrugged.

“Are we picking anything up?” he asked.

“Not on the long range,” Rey replied, sitting down and checking the scanners. “Nothing on passive… that’s just because Luke wants to hide, right?”

She detected a note of desperation in her voice. “It’s not because he’s moved on again, right?”

Chewbacca didn’t say anything, but he did raise an eyebrow at her.

Searching the Great Temple took about an hour, and they didn’t find a Jedi Master.

They did, however, find one of the now all-too-familiar crystals, and Rey stared balefully at it before clasping her hands and letting out her anger.

Again.

Then she snatched it up, wanting to know where they were going to have to go this time.

“You know…” Rey said, as they broke orbit. “I actually almost sympathize with that one.”

“You do?” Chewbacca asked.

“Yeah,” Rey agreed. “Knowing that the temples here were literally built by slaves who were members of the original Sith species… it’s a Sith Temple. I imagine any Jedi would be uncomfortable with that.”

She looked down at the crystal. “I really wish he’d put one of these on Ahch-To, though.”

“No argument there,” Chewie mumbled. “At least Naboo is an easy one…”

“I don’t know much about the place,” Rey said. “Only that it was involved with the Clone Wars, somehow. Or maybe something before the Clone Wars.”

The crystal pointed them to a very fine town house in Theed, which did not have Master Luke in it.

Instead, it had a droid, who beeped and whistled at them.

“We’re looking for Master Skywalker,” Rey said. “Please tell me you know where he went.”

The droid beeped again.

“...Master Amidala?” Rey repeated. “But Master Skywalker said to come here…”

“Same person, it’s just his mother’s surname instead of his father’s,” Chewbacca provided. “Show the droid one of the crystals?”

“It can’t hurt,” Rey conceded. “Is this some kind of ancestral home, then?”

She activated one of the crystals, and the droid whistled gleefully before opening an internal compartment and depositing another crystal in her palm.

“Right,” Rey said, rubbing her forehead with her free hand. “It’s a good thing the Falcon is so fast. We must have done a lap of the galaxy by now.”

“We’ve mostly been going through the middle, but yes,” Chewie agreed. “Where now?”

“That’s always the question,” Rey conceded, focusing.

If there was one thing this was good for, it was learning to master her anger.

“I know, I know, I said we’d be here for good,” Luke apologized. “But I ran into a Palpatine on the street yesterday, and it freaked me out.”

He shook his head. “I know, they’re from a different branch of the family, not everyone called Palpatine is evil… but it really unsettled me and I can’t feel comfortable here any more. Not after I heard from Binks about how Palpatine exploited both my parents… and him.”

The Jedi Master let out a long sigh. “But being somewhere I inherited… it helped, really. It reminded me of the other place that I inherited. We’re going back home. Beggar’s Canyon and the Lars homestead. Ben, if you’re the one hearing this… I’m sorry that we couldn’t give you the childhood that my aunt and uncle gave me.”

The force hologram disappeared, and Rey closed her eyes.

“That didn’t even give us a planet,” she said.

“No problem,” Chewbacca replied. “I know where we’re going. I know where Luke grew up.”

He nodded to the droid. “Thanks for your help.”

The droid whistled, waving a probe cheerfully.

Naboo to Tatooine. Mos Eisley to the Jundland Wastes to the Lars homestead, and from there on to Beggar’s Canyon.

Rey could feel the tension building in the air. Like the signs of a sandstorm, but more positive.

Signs of… something. Maybe signs of hope.

“Found something,” Chewbacca said. “Zeroing in on it now.”

The Falcon banked, slowing, and Rey went to the ramp as it opened. Around her, the light transport hovered on repulsorlifts, and she held on to a stanchion as she leaned out into the hot, dry air.

“I can see something!” she reported, through her comlink. “Bring us down another four metres… all right… I’m getting out here, land as near as you can.”

“Got it,” Chewbacca replied, and Rey slipped out of the door.

She landed with a roll, and shaded her eyes to look closely at what she’d spotted.

There was no mistaking it. It was a hangar bay. Built into the side of Beggar’s Canyon, concealed from above except at exactly the right angle, and big enough to service plenty of ships at once.

There were ships there, in fact. Two transport shuttles, a light and utilitarian variety, and a heavier and heavily modified yacht. But there was space for several more, and Rey frowned as she approached.

This didn’t feel empty in the way the other places had been, a difference that only made sense now she’d felt both sides of it.

It felt… lived in.

Then three young adults – a strange four-legged two-armed half-equine, a more familiar Bothan, and a human – came out of a doorway, all looking at her warily.

“Who are you?” the bothan asked. “Why are you here?”

“I’m looking for Master Luke Skywalker,” Rey explained.

“...oh, well, you just missed him,” the half-equine replied. “He’ll be back-”

“Lusa!” the Bothan protested. “Operational security!”

“Right, right,” the now-identified Lusa said. “Why do you want to speak to him?”

“Because we need him,” Rey said, simply. “To fight the First Order. I… brought his old lightsaber?”

She held it out.

“Whoa,” all three youngsters said, at once.

Then the Falcon came flying back over, still looking for a landing spot, and the human gasped.

“Is that the Millennium Falcon?” he asked. “Did you come here with Han Solo and Chewbacca? Does that mean Ben-”

“No,” Rey replied. “Han’s dead. He… Ben killed him.”

That put a damper on the mood.

“...so, where is Master Luke?” Rey asked, after a few seconds. “Who are you? What are you doing here? I’ve been following his messages for more than a day!”

“Well…” Lusa began. “We’re… trainees?”

“The old word was padawans,” the Bothan supplied. “Master Luke decided that… uh… he said that he remembered what Master Yoda said, and that the only thing that mattered was the spirit. That you had to learn to avoid the Darkness, and that everything else you could learn at your own pace, however fast or slow that was.”

“And all the teachers left about two hours ago in their X-Wings,” the human contributed. “So we’re the ones defending the Academy!”

“I am going to need some time to process this,” Rey said. “...wait, in X-Wings?”

“We had a fleet,” Poe said. “Now we’re down to one ship, and you’ve told us nothing!”

He waved his hands, for emphasis. “Tell us that we have a plan! That there is hope!”

Admiral Holdo stared back.

“There is a plan,” she said. “But I don’t have to tell you what-”

“Admiral!” someone interrupted. “Hyperspace signatures! It looks like… they’re snub fighters, twelve of them!”

Holdo’s shoulders slumped.

“And there it is,” she declared, as the tension left, and she sat back into her seat. “Turn the ship! Prepare for close engagement!”

The radio crackled.

“All wings report in,” came a voice, Luke Skywalker’s voice, and it was so unexpected that Poe staggered back a pace.

“Katarn standing by,” one of the fighters reported.

“Horn, standing by,” another voice added.

The reports came, one by one. Jade, Dracos, Solusar, Durron, Ikrit, Binks, Desann, Korr, Penin. Then they broke for an attack run, and Poe could only stare.

He knew he was a good pilot. One of the best.

But even he had to admit that he couldn’t outdo that squadron.

Just stumbled upon a great philosophical debate while listening to a playlist on shuffle.

The esteemed philosopher Bill Joel puts forward that "It's still rock and roll to me" confidently declaring that rock and roll remains to some extent constant in how much it rocks.

However, his point is immediately challenged by philosopher Bob Seger who posits that "Todays music ain't got the same soul" and that he "Likes that old time rock and roll". Confidently declaring that rock and roll has declined in its rockness.

This is truly one of the most important debates facing philosophy today.

RAHHH THE ONLY CORRECT WAY TO START 2025!!

A Vision of the Labour Party, 2017

Liz Kendall has returned to a quiet life as a backbencher and constituency politician, but due to border change legislation passed by a gloating Tory Government, her constituency is now Hades.

Yvette Cooper is serving a ten year sentence for the murder of her husband Ed Balls after his 167th utterance of “At least I lost to the Millibands…”

Jeremy Corbyn, in what is considered one of his more unusual leadership moves, proposes legislation to ban the word “Crown” from any and all pub names. The Labour conference that year is held in the “People and Anchor” pub in Islington.

Andy Burnham turned back into a wooden doll at the stroke of midnight after the leadership election, where he came third place, as a result of the particular brand of contractual wish magic that gave him life.

Explaining math jokes

So yeah I said I was gonna do it and now is the time I think (if I wait any longer I'm gonna have too much jokes to explain) IMPORTANT: I had a lot of trouble writing this because I don't really know what background to assume the reader has. So at each new explanation, assumed background changes. Difficulty of the concepts explained is in no particular order. So if there is something you don't understand, that's fine, just go read something else. Dually, if there is something you already know well, don't throw away the whole post. Though it is very possible you already know everything I'm gonna rant about lol Anyways let's get to it

geometric group theory talk but on the speaker’s slide instead of the Cayley graph of the free group on two generators there’s just loss

(link) Geometric group theory is a subfield of math that studies groups using geometry. A particular geometric thing that is often interesting to study for a given group is its Cayley graph, which roughly speaking is a graph that reflects in a way the structure of the group. My neurodivergent brain thought the Calyey graph of the free group on two generators lowkey looks like the abstracted loss meme in the original post:

sooo, turns out the #latex tag is not for typography enthusiasts

(link) LaTeX is a typesetting engine and the industry standard for math. It's the thing almost everyone uses to typeset beautiful math equations and stuff on computers. If you're seriously interested in math I'd recommend you learn it. A good place to start would be Overleaf which is a free online LaTeX editor and has some tutorials on how to get started, though eventually you may want to switch to doing LaTeX on your computer directly

so a homological algebraist goes to see their therapist and says “doc, i’ve got complexes”

(link) Homological algebra is a branch of algebra that was born from algebraic topology. It has become widely used in many parts of math because of the computational power it brings. The gist of it is that people define a thing called a chain complex, that is a sequence of abelian groups (or modules, or maybe something even more general, look up abelian categories), with homomorphisms from one abelian group to the next called differentials, such that doing one differential then the next always gives you zero:

If you're more comfortable with linear algebra, you can replace "abelian group" with "vector space" and "homomorphism" by "linear map". The fact that doing one differential then the next gives you zero means that the image of one differential is contained in the kernel of the next. Homological algebra is about finding ways to calculate exactly how far away we are from the image of a differential being exactly the kernel of the next. This is made precise when one defines homology groups, which are the quotient Ker(d_n)/Im(d_{n+1}). What happens in practice when we apply homological algebra is that we try to define an interesting chain complex related to what we are doing, so that the homology groups tell us something interesting about what's going on with whatever math thing interests us, and then we apply methods from homological algebra to calculate them. Any serious example of homological algebra being used is going to require a bit of math background, but two examples I can give are singular homology, from algebraic topology, and de Rham cohomology, from differential geometry (don't worry about the co-, it just means the indices go up instead of down). So yeah a homological algebraist would have complexes

If you’re not careful and you noclip out of reality in the wrong areas, you’ll end up in Hilbert’s Hotel

(link) Hilbert's Hotel is an imaginary hotel with infinitely many rooms, that is one room labeled zero, one room labeled one, one room labeled two, and so on. One room for every natural number. It reminds me of the backrooms, hence the joke. Hilbert's hotel is commonly used as a metaphor to think about how infinity behaves and how bijections work. For instance, if the hotel was full, but one new guest showed up, you could still get them a room: simply tell the person occupying room number n to move to room number n+1. Then room 0 will be empty and the new guest can stay in it. However, quite interestingly, it is possible for too many guests to show up and the hotel to be unable to give a room to all of them.

If you speak French, excellent math youtuber El jj (I heavily recommend you subscribe to his channel!) has a very good video on Hilbert's Hotel.

If you don't speak French but still speak English, here's a Veritasium video on Hilbert's hotel, and a Ted-Ed video on it.

“cats are liquid” factoid actually formalized by mathematicians as saying a cat is only truly defined up to homeomorphism

(link) Topological spaces are mathematical objects that abstract away the concept of nearness. What do I mean? Well, a topological space is a set X, together with a collection of subsets called "a topology", that in way specifies which points are "near" each other. This allows us to generalize a lot of concepts from real analysis, for instance limits: if you have a sequence of points (x_n), and they get "nearer" and "nearer" to some point x, well that point can be called the limit of the sequence. But topology also turns out to be massively useful to geometry: if I only gave you the set of points of a sphere, you wouldn't know they make up a sphere because you wouldn't know how to assemble them. But if I give you the set of points of a sphere and the correct topology on it, then you can actually know it is a sphere and do stuff with it. But as always, in math, we only consider things up to some notion of being "the same". This notion of "the same" for topological spaces is called "homeomorphism", and two things being homeomorphic corresponds to the more intuitive geometric intuition of "I can continuously deform one thing into the other without cutting or gluing stuff". For instance, a cube is homeomorphic to a sphere:

Or more famously, a mug is homeomorphic to a donut:

So cats only truly being defined up to homeomorphism kinda works to say they're liquid. Not math, but physicist Marc-Antoine Fardin did actual physics on cats being liquid and was award the 2017 IgNobel prize in physics for it.

i would describe my body type as only defined up to homotopy equivalence

(link) Homotopy equivalence is a weaker notion of two topological spaces "being the same". I won't go into details but I have seen it being describe as kind of like a homeomorphism, but you are also allowed to inflate/deflate objects. For instance, a filled cube is a "3d object" in a way (when you are inside the cube, you can move in 3 directions). This means it will never be homeomorphic to a 2d square because a property of homeomorphisms is that they preserve dimensions. But, the cube is homotopy equivalent to the square, because you can "deflate" the cube and squish it to make it a square. In fact, it is even homotopy equivalent to a point (you can just deflate it completely). Homotopy equivalence is weaker (more permissive) than homeomorphism, that is if two things are homeomorphic, then they must be homotopy equivalent, but not the other way around. You may ask yourself why we would care about a notion weaker than homeomorphisms that can't even tell apart points and cubes, and that's a fair question. I will provide one answer but there are definitely many more I haven't even learned yet. In algebraic topology, we are concerned with studying spaces by attaching algebraic thingies to them. Why do we do that? Because telling apart spaces is hard. Think about it: how do you prove a donut is not homeomorphic to a sphere? You'd have to consider all possible deformations of a donut and show none of them is a sphere. This is mathematically hopeless. Algebraic topology solves this by attaching algebraic invariants to spaces. What do I mean? Well we have a way of saying that a donut has "one hole" and a sphere has "zero holes", and we have a theorem saying that if two things are homeorphic they must have the same number of holes (the number of holes is an invariant). Therefore we know that a donut cannot be homeomorphic to a sphere. Usually we have more sophisticated invariants (homotopy groups, homology groups, the cohomology ring, and other stuff) that are not just numbers but algebraic structures, but the same principle remains. It turns out a lot of these invariants are actually invariants for homotopy equivalence, that is, they will not be able to tell apart homotopy equivalent spaces. This is useful to know: for instance, a band and a Möbius band are both homotopic to a circle, so you know that if you want to tell them apart, you're going to need more than the classical algebraic invariants (if you know a bit of algebraic topology and you're curious about that, this can be done by thinking of them as vector bundles, but also through more elementary methods, see this stackexchange post). Also, if you want to calculate some invariants for a complicated space, a good place to start can be to try to find a less complicated space that is homotopy equivalent to the original space (and this is often doable since homotopy equivalence is a kind of weak notion).

in ‘Murica land of the free every module is born with a basis

(link) In 1st-year linear algebra, we study vector spaces over fields. But in more advanced linear algebra, we study modules over rings, which are basically vector spaces, but over rings instead of fields. It turns out dropping the condition that every non-zero scalar must be invertible makes the algebra much more complicated (and interesting!). When a module has a basis, we say it is free, hence the joke. If this basis is finite, we say that the module has finite rank, and the length of the basis is the rank of the module (exactly like dimension for vector spaces!), hence the tag "not every module ranks the same though".

testicular torsion? this wouldn’t happen over a field

(link) Continuing on modules, modules can sometimes have what is called torsion. Let's take Z-modules, or as you may know them, abelian groups! Indeed, a "vector space over Z" is actually the same as an abelian group: any module has an underlying abelian group (just forget you know how to scale elements) and conversely, if you take an abelian group, you know that any element a is supposed to be 1a, so 2a must be (1+1)a = 1a + 1a = a + a. More generally, for any positive integer n, n.a = a + ... + a, n times, and if n is negative, n.a = (-a) + ... + (-a), n times. So knowing how addition works actually tells us how Z must scale elements. With that out of the way, take the Z-module formed by the integers mod n, Z/nZ. It is an abelian group, so a Z-module, but something weird happens here that doesn't happen in a vector space: n.1 = 0. You can scale something, by a non-zero scalar (in fact a non-zero-divisor scalar), and still end up with 0. This is known as torsion, and vector spaces (modules over fields) don't have that. So yeah, testicular torsion? that wouldn't happen over a field. Also, watch out: the notion of torsion for a module over a ring is not necessarily the same as the notion of torsion for the underlying abelian group. Z/4Z doesn't have torsion, when seen as a Z/4Z-module.

Mathematical band names

(link) For these posts, I'll be quickly explaining each band name, and i'll be including good additions from other peeps! (with proper credit of course, you can't expect a wannabe-academic to not cite their sources) (also plagiarism is bad) (if no one is credit that means I thought of the band name)

Algebrasmith, The Smathing Pumpkins, System of an Equations, My Mathematical Romance, I Don't Know How But They Found X, Will Wood and the Tape Measures (by @dorothytheexplorothy), DECO*3^3 (by @associativeglassdesert), The Teach Boys, Dire Straight Lines, n Directions, XYZ Top, Mathallica: I have nothing to explain here

Rage against the Module: if you've read the parts of this post about modules, you get it (partly inspired by the commutative algebra class I'm taking right now, I love it, but I've been stuck on a problem for some time)

The pRofinite Stones: a profinite space is a topological space obtained by some process involving finite, discrete spaces. They are usually called Stone spaces, hence the joke

Mariah Cayley: Arthur Cayley was a mathematician. It's the same Cayley from the Cayley graphs! (also Cayley-Hamilton, if you've heard of the theorem)

Billie Eigen: eigenvalues and eigenvectors are linear algebra concepts. For a given operator on a space, its eigenvalues are scalars that tell us a lot about the operator. This is not my field but I have heard in quantum physics physical quantities like mass, speed, etc are replaced by operators, and eigenvalues correspond to states the physical system can be in

Smash Product: in algebraic topology/homotopy theory, the smash product is an operation on pointed topological spaces that is interesting for categorical purposes (it gives a symmetric monoidal category structure to the category of compactly generated pointed topological spaces, if you know what that means) somebody once told me, the world is categories

FOIL out boy (by @mathsbian): FOIL is a way of remembering how to expand products that some people learn. It means First, Outer, Inner, Last. So if you expand (a+b)(c+d) using FOIL, you get ac + ad + bc + bd.

Sheaf in a Birdcage (by @dorothytheexplorothy): a (pre)sheaf is a way of assigning algebraic data to a topological space (or a generalized notion of space). A presheaf is a sheaf if the data respects some locality condition. (pre)sheaves were introduced by Jean Leray but really used by Grothendieck to completely transform algebraic geometry, and are now widely used in modern geometry (they show up to abstract the notion of "a geometric thing"). I can't explain much more as I am still learning about sheaf theory!

The Curry-Howard correspondents (by @dorothytheexplorothy): the Curry-Howard correspondence, in logic/theoretical computer science, essentially says that algorithms (computer programs) correspond to mathematical (constructive) proofs. I'm no computer scientist or logician so I'll avoid saying dumb stuff by not trying to explain more, but I know it can be made more precise using lambda calculus.

Le(ast com)mon Demoninator (by @dorothytheexplorothy): I don't think I have to explain anything here (let's be honest, if you're reading this, you probably already know what a least common denominator is), but I will say that the band name being spoofed here is Lemon Demon (Neil Cicierega's musical project) and I love his music go listen to it. Also I love the word demoninator thank you for that dottie

Taylor Serieswift (by @associativeglassdesert): a Taylor series is an infinite sum that approximates a nice-enough (analytic) function around a point. This is useful because the Taylor series only depends on the derivatives of the function at one point but can approximate its behavior on more that one point, and also because the Taylor series is a power series, so a more tractable kind of function. In particular if we truncate it, that is stop at some term, we get a polynomial that approximates our function well around a point, and polynomials are very nice to work with (this is where kinda cursed stuff you may have seen in physics like sin(x) = x or tan(x) = x comes from!)

mxmmatrix (by @associativeglassdesert): you may have heard a matrix is a table of numbers. Actually, it's much more than that. Matrices are secretly functions! In fact, very special kind of functions (linear maps) between very special kind of objects (finite-dimensional vector spaces). And if you've seen how to multiply matrices before but have not been told why we do it that way, be not afraid, there is actually an answer. The answer is that when we take some x, do one linear map f to get f(x), then another linear map g to get g(f(x)), we actually end up with a new linear map, gf. And if you take a matrix representing f and multiply it (left) by a matrix representing g, you get a matrix representing gf. This is why the matrix product is done like that: it's actually composition of functions! If this interests you, consider reading more about abstract linear algebra.

Ring Starr (by @associativeglassdesert): a ring is an algebraic structure. Take the integers. What can we do with them? We can add them together, addition is associative (when adding a bunch of stuff we don't need parentheses), commutative (a+b = b+a), we have zero that doesn't do anything when adding (a+0 = a), and we have opposites: for any integer a, we have another integer -a such that (a + (-a) = 0). But we also have multiplication: multiplication is associative (no need for parentheses again), commutative, we have 1 and multiplying by 1 doesn't do anything, and multiplication distributes over addition. Now, re-read what I just said but replace "integer" by "real number". Or "complex number". When seeing such similar behavior by different things (there are in fact many more examples that those I just gave), mathematicians are compelled to abstract away and imagine rings. A ring is a set of stuff, with some way to add the stuff and some way to multiply the stuff that satisfies the properties I talked about above. Sometimes we also drop some properties, for instance we allow multiplication to not be commutative (ab =/= ba). By allowing this, square matrices of a given dimension form a ring! Quaternions, if you know what they are, also form a ring. A lot of things are rings. Rings are cool. Learn about rings.

WLOGic (by @associativeglassdesert): WLOG is mathematician speak for "without loss of generality".

Alice and Bob Cooper: in many math problems, people are called Alice and Bob. Because A and B. Yes there is a Wikipedia page for this

The four Toposes: a topos is a kind of category meant to resemble a topological space. Grothendieck toposes are used in algebraic geometry and elementary topoi are used in logic. I can't explain more since I don't really know anything about topoises besides that they are kinda scary and that people really like to argue about what the plural of "topos" should be

Green-Tao Day: the Green-Tao theorem says that if you have a positive integer n, then you can find prime numbers p1, p2, ..., pn, such that they are evenly spaced (or equivalently, in an arithmetic progression). It's pretty neat. I have no idea how the proof goes, though. It must be pretty complicated, since it was proven in 2004.

Aut(Kast)/Inn(Kast): I'm really proud of that one. So if you have a group G, you can look at bijective group homomorphisms from G to G, or as they are more well-known, automorphisms of G. Together with composition, they form a group, called Aut(G). Now we already know of some automorphisms of G: if g is any element of G, then x ↦ gxg^{-1} is an automorphism of G (proof is left as an exercise to the tumblr). These automorphisms are called inner automorphisms of G, and they form a normal subgroup of Aut(G). The quotient group Aut(G)/Inn(G) is called the outer automorphisms of G and denoted Out(G), which is reason behind the band name.

Depeche modulo: modulo is a math word that means "up to [some notion of being the same]". For instance the integers modulo 7 are the integers but we declare that two integers a and b are the same if 7 divides a-b. From there we get modular arithmetic which you may have heard of. This kind of operation is called a quotient and is insanely useful in all branches of mathematics.

Phew! We're done with the band names. For now.

"oh you like math? what's 1975 times 7869?" well that's a great question Jimmy but to answer it first I need to construct the natural numbers. [...]

(link) So this is a post about a type of response math people get when they say they do math which is that people automatically assume this give us insane mental math power. It does not. The rest of the post is about constructing the natural numbers in the ZFC axiomatic system. I'm kinda lazy and don't want to get into all that but here's a good video by certified good math channel Another Roof about it: what IS a number? The same channel has several other videos on that same topic, go watch em if you're interested

1957 times 7869 (IF IT EVEN EXISTS) is the universal object with morphisms into 1957 and 7869

This is a joke by @dorothytheexplorothy in the notes of the previous post. The joke here comes from interpreting "times" are referring to the categorical notion of product. I'm actually not gonna explain anything here because 1) this post is taking forever to write and 2) I will probably rant about category theory in the future. Here are two videos by Oliver Lugg you can watch:

27 Unhelpful Facts About Category Theory (funni video)

A Sensible Introduction to Category Theory (serius video)

and here are two books you can use to learn more if you're interested:

Seven Sketches in Compositionality (very applied, very nice, I think easy to read)

Basic Category Theory (less applied, is a typical math book)

she overfull on my \hbox till i (5.40884pt too wide)

(link) This she on my till i joke is based on a LaTeX warning you get when it can't figure out how to typeset your document well and that leads to a margin being exceeded.

Time for the math battle reblog chain

(half of the posts are by @dorothytheexplorothy)

fuck you *forgets your group is a group and only remembers it's a set now*

So any group has an underlying set, and any group homomorphism is actually a map between these underlying sets. This means that the operation of "forgetting a group is a group and only remembering it's a set" is a functor. This is less useless than you might think, because of adjunctions.

two can play at that game *constructs a free group over this set, even bigger and better than the one I had*

So basically in lots of cases the functor that forgets some structure is (right) adjoint to some other functor. You do not need to know exactly what this means to read the rest, don't worry. What it means is basically that from the operation of forgetting some structure, we can get another operation, which adds structure, in a "natural" way. In the case of forgetting a group is a group and only remembering it's a set, the adjoint functor is the "free group" functor, that takes a set and constructs the free group on it. This idea of free objects works not just for groups but for a whole lotta stuff. See this part of the Wikipedia page on forgetful functors for some information.

oh don't get me started *abelianizes your free group, now it's just a big direct sum of Z's*

A non-abelian group can be turned into one through abelianization, which is quotienting out by the commutator subgroup. This makes sense: commutativity is asking ab = ba for all a, b, which is asking aba^{-1} b^{-1} = 1 for all a, b, which is precisely what we get when quotienting by the subgroup generated by words of the form aba^{-1} b^{-1}. Abelianization is also a functor, so it fits the theme. The abelianization of a free group is a free abelian group, and a free abelian group is a direct sum of a bunch of copies of Z.

big mistake, friend *moves over to the endomorphism group over that group and treating composition as multiplication, thus replacing it with a unital ring*

The endomorphism group of an abelian group is actually a ring (like in linear algebra, endomorphisms form a unital ring with composition as multiplication). I don't think this construction is functorial, though? (correct me if I'm wrong on that. correct me if I'm wrong on anything, really. if i'm wrong about stuff send me an ask and i'll fix it)

you fool, you fell right into my trap! *takes the field of fractions of your ring* have fun working in the category of fields! now you only have monomorphisms and your eyes to shed tears

So I thought I had the advantage here because fields are, categorically speaking, very bad. This is (I think) mainly because a homormorphism of fields is always injective (so is a monomorphism, that is left-cancellable). In fact, products of fields don't exist, direct sum of fields don't exist, a lot of categorical constructions we usually like don't exist in this category. We basically only have inclusions. I will elaborate on why I was wrong in my post here in a bit

fuckkk idk enough about schemes or whatever to get out of this! you've bested me X(

Schemes are the main objects of study of algebraic geometry. I won't being to try and explain what they are because it is very abstract and I don't even really understand the definition (yet). I just know they're algebraic geometer's analogue of a "geometric object", like how smooth manifolds are to a differential geometer.

wait actually I just realized the ring of endomorphisms of a free abelian group has no business being an integral domain, or even commutative. so I think taking the field of fractions makes no sense, and I actually lost the battle.

So the field of fractions construction only makes sense for integral domains. The name of this construction is really explicit: passing from an integral domain to its field of fractions is the same thing as passing from Z to Q, or from k[X] to k(X) if you know what that is. However I made a mistake, since the ring we were talking about is almost never commutative (much like matrices).

WON ON A TECHNICALITY LET'S GOOOOO

well played, dottie

yeah, uh, we oidified your boyfriend. yeah we took his core concept and horizontally categorified it. yeah he's (or they're?) a many-object version of himself now. sorry about your one-object boyfriendoid

(link) Oidification (also known as horizontal categorification but "oidification" sounds funnier) is a way of categorifying a concept, by turning it in a "many-object" version of itself. For instance, a one-object category is precisely a monoid, so the concept of category is the oidification of the concept of a monoid. A category where every morphism is invertible is called a groupoid, and a one-object groupoid is precisely a group. The name "oidification" probably comes from the fact that after being oidified, the name of the concept gets added the suffix -oid. So a category is a monoidoid. In fact, you can even have monoidal monoidoids. Category theory really is well-suited to shitposting huh

My advisor [...] stared into my soul and noticed I liked categories. It's over for me, i am going to end up a homotopy theorist, or worse, a youtuber

(link) Category theory has the reputation of being abstract nonsense. I don't disagree. I guess I have a slightly-above-average tolerance to category theory and algebra. This has led to a not-insignificant amount of people in my life telling me I'm gonna end up in one of the abstract-nonsense-related fields like homotopy theory, infinity-category theory, etc. The "or worse, a youtuber" part was stolen from the following quote

Research shows that when someone becomes personally invested in an idea, they can become very close-minded. Or worse, a youtuber.

-hbomberguy, Vaccines and Autism: A Measured Response (4:12)

(this video is incredible, if you haven't seen it yet, go watch it)

PHEW.

I'm done. For now. This took multiple hours to write. I hope you enjoyed this post! If you enjoyed it, please let me know! If you have any questions or want to tell me "youre doing good lad" or want to yell at me, my asks are open! Thank you for reading this far! If there is a post I talked about here you found funny, you can click on the (link) to look at the original post. Give me those sweet sweet statistics. I crave them. I NEED that dopamine hit of knowing someone interacted with my blog. ok bye

It’s impossible to give directions in Boston. Nothing makes sense. There are inexplicable one way streets, there are streets that change their names as you move from one area to another. There’s a road in my area where the gps literally tells you to take TWO full right turns to “stay on” the same road which is at a right angle to the original. There’s like four different Massachusetts Avenues. Sometimes you have to be in the left lane to turn right. The gps can’t even get directions to my workplace correct; it tells me to take a left on a road where lefts are not allowed, and the only way to not have to go across the river and take 15 minutes to turn around is to remember this shit one block early and make the left THERE. Recently, they’ve restricted 1 of 2 lanes on each side of major thoroughfares to only allow bikes and buses, and the government officials seemed to genuinely believe that would somehow EASE traffic. Oh and don't try to drive on Memorial Drive on Sundays; they close it for pedestrians. Just because. And when you DO drive on Memorial, there's one exit that will make your gps lose its mind and start chanting random sequences of numbers for four minutes straight. You can't take a Uhaul on Storrow Drive because the bridges that go over it are too short, and every year some doofus college student ignores this rule and proceeds to "get Storrowed" when they shave off the top of the truck on the overpass and get stuck. I-93 turns into I-95 and makes a big circle around the city, so a lot of the time you'll be on I-95 north but driving east or west.

It’s not limited to driving either. The Arlington train station is not in Arlington, it’s in the middle of downtown. Harvard Square is not a square, it’s more like a pentagon. There are four different green line train routes, and they’re labeled B for Boston College, C for Cleveland Circle, D for… Riverside, and E for… Heath Street. The Silver line is listed on the train map but is entirely run on buses which have to be connected and disconnected from power lines every time you go through the route. The Blue line goes to (and I’m not joking) Wonderland. The two red lines are labeled for their southern points: Braintree line goes to Braintree, and the Ashmont line goes to… Mattapan. To be fair, the train itself stops in Ashmont and you continue to Mattapan on a trolley, but that doesn't make it better. South Station and North Station are 1 mile apart and the easiest way to get from one to the other is just to walk it because otherwise you have to travel through 4 or 5 train stops on two different lines. But make sure you memorize the route because there's a good chance your gps will lose signal in the Financial district because it can't get through the buildings. In Boston Commons there are two train stops within line of sight of each other, on the same street, and one of them screams. To get to the trains at Porter Square, you have to ride down escalators 105 feet below street level, or you could just take the 3 flights of stairs totaling 199 steps (presumably because the engineers had something against nice even numbers). The North End is south of East Boston. Castle Island is part of the mainland.

No matter where you're going or how you're getting there, it takes 45 minutes (no wrong turns) or an hour and a half (one wrong turn). It doesn't matter if you're going one stop on a train; it will take 45 minutes. If it's summer, there's a better than 50% chance you'll be in the train car that lost it's AC; if it's winter, you're guaranteed to be in the car where the heat has it up to 80 degrees and the inside of your winter coat will be a sauna. Check the Red Sox schedule before you go south of the river, or you'll be trapped in the waves of baseball fans flooding the streets and days will go by before you're found again. And just... don't go outside on September 1.

If you're thinking that this sounds eldritch as shit, you're right. The entire city is an arcane lock keeping the ghoulies and ghosties from haunting the rest of the nation. We charge it with every "fuck" we utter while we travel our labyrinthine paths and drink our Dunks. You're welcome.

I have a new post up on my blog, continuing the Fictional History of Numbers series. In part 1 we started with the natural numbers and built up the algebraics, which let us solve equations. In part 2 we started asking geometric questions, and constructed the real numbers.

But the real numbers are weird and hard to define. In part 3 we see one way they're extremely strange, and then talk about why we want them anyway. In the end, we shouldn't worry about the definition of the reals; we should worry about what they allow us to do. And it turns out they're exactly what we need to make calculus function as it should.

Suggested Alternatives to the One China Policy

Currently, the policy of the United States on the Taiwan question is that the US recognizes that polities on both sides of the Taiwan Strait hold that there is only one China and that Taiwan is part of China. In the current tense international climate, it may be useful to considers alternatives to that policy.

Two Chinas Policy: The United States recognizes the independence of Taiwan as a sovereign state, separate from the People's Republic of China.

Three Chinas Policy: The US recognizes Taiwan, Hong Kong, and the mainland as independent states.

Four Chinas Policy: The US recognizes Taiwan, Hong Kong, Macau, and the mainland as independent states.

One China Policy (Retro 1978): The US switches its diplomatic recognition back from the PRC to the ROC.

One China Policy (Retro 1911): The US recognizes the Qing Dynasty as the legitimate government of China and finds some schmuck to play Emperor-in-Exile.

Many Chinas Policy: The US recognizes the sovereign independence of every Chinese province.

Too Many Chinas Policy: Hong Kong makes a perfectly fine city-state, so why not let everyone do that? The US recognizes every Chinese municipality as its own independent state.

1436506450 Chinas Policy: The US recognizes the sovereign independence of every Chinese person.

2^1436506450 Chinas Policy: The US recognizes the sovereign independence of every subset of of the set of all Chinese persons.

2^1436506450-1 Chinas Policy: Same as above, but not including the empty set, because that doesn't even make sense because it's already claimed by Germany.

Infinite Chinas Policy (Countable): The US recognizes that (1) The PRC is a China and (2) for every China c, the successor S(c) is also a China, and (3) for every China c, c != S(c).

Infinite Chinas Policy (Uncountable): The US recognizes that the set C of all Chinas is an ordered field, and that every non-empty subset of C with an upper bound in C has a least upper bound in C.

No Chinas Policy: The United States embraces mereological nihilism and recognizes only atoms and the void.

What issues would a Jewish Werewolf face? I mean with a lunar calendar and so many of the holidays near the full moon, they would have to get pretty inventive, just think about sleeping in the succah, or since Yom Kippor is about 4 days from the full moon, it should make things interesting as in most stories weres start to lose control near the full moon.

HMMM! (and thank you for sending me these anons!)

I suspect it depends on what tradition we’re drawing from. Werewolves as a whole are mostly a European thing, although people changing into or communing with animals is pretty much a worldwide myth. 

Some things to think about: If you’re not fully conscious (or not conscious in the same way) when you’re a wolf, are you accountable for any destruction you cause? Does transformation count as work? (Also, if you can’t stop yourself from doing work, you probably aren’t breaking Shabbat..) Can you attend synagogue as a wolf?

 And we do have recorded cases of nice werewolves! In Latvia in 1692, an eighty-year-old man named Thiess confessed to being a werewolf who, with other werewolves, regularly went to Hell three times a year to fight Satan to ensure a good harvest. This would be a great tradition for Sukkot, Shavuot, and Tu B’Shevat, and I propose we all adopt the custom immediately!

physics is scary what the hell is the Riemann zeta function doing in the equation for the critical temperature of a Bose-Einstein condensate