Author: Eugene Fischer

An Unnecessary Proof by Contradiction

Until Covid, I was meeting up with a friend once a week for some recreational abstract algebra. This was an early problem in the textbook we used:

Prove that for any positive integer m there exists a sequence of m consecutive positive integers that are all composite.

My gut instinct was to try to prove this by contradiction, using the pigeonhole principle somehow. But before I could work that out, I saw that it’s easily provable by construction.

Proof by construction

Let n = m!. Then, for all integers k such that 0 < k \le m, we know k \mid n. So we can say that n + k = k (\frac{n}{k} + 1). Since k \mid n, the quantity \frac{n}{k} + 1 is an integer. Therefore n+k has k as a factor, making it composite. So, the sequence

n+1, n+2, \dots, n+m

are all composite. \blacksquare

Pretty trivial. But I still wanted to know if my gut instinct was right; could this be proved by contradiction?

It turns out that it can, but it’s much more complicated, and took me a couple of days to figure out.

Proof by contradiction

Assume that every set of m consecutive positive integers contains at least one prime number.

The positive integers can be expressed as a union of sets

\Z^+ = A_0 \cup A_1 \cup A_2 \cup \dots 

where

A_n = \{nm+1, nm+2, \dots, nm+m\}

Since \vert A_n \vert = m, our assumption requires that every set A_n contains at least one prime number.

Let a_i(n) be the i^{th} element of A_n, where 1\le i \le m. Call i the “position” of a_i(n) within A_n. Since there is only one even prime number, the maximum number of positions in A_n at which there can be a prime is \lceil \frac{m}{2}\rceil.


Lemma: If A_j has a prime p in position i then A_{j+q} cannot have a prime in position i when q is a multiple of p.

Proof: Let A_j and A_{j+q} be sets with prime numbers in position i, and say that a_i(j) = p and a_i(j+q) = p^*. Then

p^* = p + (m-i) +m(q-1)+i

In the sum on the right hand side of this equation, the term (m-i) represents all the positions in A_j after p, the term m(q-1) is all the positions in all of the sets between A_j and A_{j+q}, and the term i is how many positions in A_{j+q} we must move through to reach p^*.

This equation simplifies to

p^* = p +qm

If q is a multiple of p, then for some k \in \Z^+

p^* =p+kpm=p(1+km)

which contradicts that p^* is prime.


Consider a set A_j such that a_i(j) is prime. Let a_i(j) = p_1.

By the above lemma, we know that a_i(j+p_1) is composite. By our assumption, the set A_{j+p_1} must contain a prime number. There remain \lceil \frac{m}{2}\rceil - 1 positions in A_{j+p_1} at which there can be a prime. Let a_{i^*}(j+p_1) be a prime, and call it p_2.

Now consider the set A_{j+(p_1p_2)}. This set has two positions, i and i^*, at which there can’t be a prime, leaving only \lceil \frac{m}{2}\rceil - 2 positions at which there can be a prime. By our assumption, one of these positions must have a prime, which we will call p_3.

Continue this pattern until we find the set A_{k+(p_1 p_2 \dots p_{\lceil \frac{m}{2}\rceil})}. This set contains zero positions where there can be a prime. But this contradicts our assumption, which requires every A_n to contain a prime number. Thus our assumption is false, and there exists an A_n (a set of m consecutive positive integers) with only composite elements. \blacksquare

If there’s a simpler way to prove this by contradiction, I’d like to know about it.

Armadillocon 2021 Schedule

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I’ll be in Austin for Armadillocon this weekend, Oct 15-17. Here’s my full schedule.

Reading: Sat. 11:00 am – 11:30 am, Conference Center

Ethics of Magic: Sat. 12:00 pm – 1:00 pm, Southpark A

Making your Novel more Drool-worthy for Filmmakers and Producers: Sat. 5:00 pm – 6:00 pm, Ballroom E

Pro-Level Writing Workshops: Sat. 8:00 pm – 9:00 pm, Ballroom E

Genre Fiction as Activism: Sat 9:00 pm – 10:00 pm, Ballroom E

A Contract Is A Wish Your Heart Makes

The Science Fiction Writers of America had a press conference today, calling out the Walt Disney Company for failure to pay a writer’s royalties. News that a quarter-trillion dollar company is doing something that screws over individual artists isn’t particularly shocking, but the nature of Disney’s justification in this case is wild.

Background: Lucasfilm and 20th Century Fox used to be big, independent media companies. Now they’re part of Disney. Since Disney bought them, they now own the Star Wars and Alien media franchises, each of which contain a teeny-tiny crumb made up of novels.

Alan Dean Foster wrote the original novelization of Star Wars, and a sequel novel, Splinter of the Minds Eye. He also wrote the novelizations of Alien, Aliens, and Alien 3. Apparently, all five of these books are still in print. But, while the books are still for sale, Alan Dean Foster is no longer receiving royalties. Disney has explained to the Science Fiction Writers of America that this is all fine:

Disney’s argument is that they have purchased the rights but not the obligations of the contract. In other words, they believe they have the right to publish work, but are not obligated to pay the writer no matter what the contract says.

SFWA president Mary robinette kowal

I thought I was already maximally cynical about the habits of megacorporations, but I’m actually astonished at how brazen this is. I mean, I’m sure there’s a deep, rich body of legal theory that I’m about to ignorantly stomp all over, but: isn’t the entire point of a contract that you don’t get to pick and choose after the fact which parts apply?

I’d be shocked if this went to court. I suspect #DisneyMustPay will result in Foster getting what he’s due, if only because I can’t imagine the legal fees being less than the royalties. If this did go to court, though, I would assume it was because Disney has purchased “the rights but not the obligations” of so many contracts that it was worth the expense. And Foster’s experience does make one wonder: has Disney been paying royalties to any of the authors whose books came along with those acquisitions?

Also, is this a fight that Disney really wants to win? They license a lot of IP. Do they really want it to be possible for someone to just buy up a licensee, then carry on selling Mickey Mouse-branded red wine or whatever without paying any more fees? Because, by their own argument, that seems like it should be a-okay. Would that really be better than just paying authors the rounding error on Disney’s books that is their royalties?

Medium’s Big Rights Grab

I’ve had “The New Mother” up on Medium since 2015.

Today, I had to take it down.

In 2015, Medium was a fairly new platform, one that made it very easy to put lengthy things online in an attractive, easy-to-read format. It integrated nicely with Twitter, which I still used a lot back then. It had a comprehensive analytics page for tracking engagement. It seemed like a great place to put “The New Mother” to get it some more attention. And it was; hundreds of people who might never have otherwise read the story scrolled through that entire page over the next few months.

Eventually, attention dwindled, as I knew it would. But I figured I might as well leave the story up until some compelling reason came along to take it down.

Over the years there were changes to Medium’s platform (the establishment of a filtered paywall, subscriptions, the Medium Partner Program), but nothing I couldn’t opt out of. The licensing setting on the page remained “all rights reserved.” That was enough for me, as the Terms of Service always contained language roughly equivalent to:

You own the rights to the content you create and post on Medium.

By posting content to Medium, you give us a nonexclusive license to publish it on Medium Services, including anything reasonably related to publishing it (like storing, displaying, reformatting, and distributing it). In consideration for Medium granting you access to and use of the Services, you agree that Medium may enable advertising on the Services, including in connection with the display of your content or other information. We may also use your content to promote Medium, including its products and content. We will never sell your content to third parties without your explicit permission.

Medium.com Terms of service, march 2016 – August 2020

That language is going away at the end of the month. Effective September 1, 2020, Medium’s new Terms of Service will read:

You retain your rights to any content you submit, post or display on or through the Services.

Unless otherwise agreed in writing, by submitting, posting, or displaying content on or through the Services, you grant Medium a nonexclusive, royalty-free, worldwide, fully paid, and sublicensable license to use, reproduce, modify, adapt, publish, translate, create derivative works from, distribute, publicly perform and display your content and any name, username or likeness provided in connection with your content in all media formats and distribution methods now known or later developed without compensation to you.

medium.com terms of service as of september 2020

That is a very different set of things Medium suddenly wants permission to do with my story.

I don’t know many fiction authors interested in giving away the right to publish their work royalty- and compensation-free, anywhere in the world. Not only does Medium want license to do that, they want it “sublicensable”—they want to be able to give other people the right to publish my story, too.

And what about “derivative works?” Does Medium want carte blanche to, say, hire someone to write sequels to my story? Because I don’t see anything in here that stops them from doing so. It’s a nonexclusive right, so they couldn’t sue me if I also wrote and published a sequel, but that would be a very small comfort if a global publishing platform started promoting alternate versions of my characters.

This looks like a huge rights grab. The minute I read the new ToS, “The New Mother” came down.

You can still read it, though. It’s right here on my website now, where the only one who decides what happens to it is me.

Lee Mandelo on the Crucial Importance of Generous Reading

Easily the best thread I’ve seen about the the most recent fiasco of SFF Twitter is this thread on literary criticism and heterogeneity of marginalized experience by Lee Mandelo. It’s so good that, with Lee’s permission, I’m reproducing it here in its entirety, in case my prayers are someday answered and Twitter actually does burn down and fall into the sea.

Art does not exist to be evaluated on a scale of “harm” to “uplift,” and if we want to talk dog-whistles, that right there is a huge one: it’s deeply anti-intellectual, and it centers a form of toxic individualism that evacuates solidarity/difference in favor of moral purity. Also, relevant from other recent intra-community trans Discourse: the fact that something triggered or hurt you, personally, is real— but that doesn’t actually make it bad, or wrong, or Harmful ™ because you *are not the center of the universe.* Other trans folk who have different experiences of gender and the world might be deeply seen by the art that you think is morally bad and harmful personally.

To some extent, we know why this is common: traumatic stress forces your focus to be survival oriented, internal, and evaluative. It’s hyper-vigilance! However, what it is *not* is healthy or productive— especially when turned relentlessly outward to hold others responsible for your bad feelings as opposed to processing them, or saying “ouch, not for me.” (Which is not to say artists shouldn’t be cognizant of other people’s pain and the larger social implications of their work, so please don’t reduce what I’m saying here to “fuck it, who cares.”)

The other huge flaw with “the story harmed me” or flat harm-critique is the lack of acknowledgement that, if we’re using that metric, then your insistence on the story harming you is EQUALLY harming to the other trans folk for whom the piece was a revelatory story, or productive. It’s powerfully self-centered and not feasibly sustainable. This is where the whole “criticism is an art itself and has theory” thing comes in. Because Sedgwick wrote re: queer theory’s internal failings a long ass time ago about “paranoid” vs “reparative” reading practices.

What we saw here was a classic case of destructive/paranoid readings that (1) FORCIBLY OUTED A TRANS WRITER and (2) caused a lot of misery and stress across the board for everyone… but that stress has been processed unevenly. Paranoid readings are also a valid understandable response to a violent world that seeks to harm us! But they close in on themselves and each other like a fucking bear trap. Reparative readings are open to pain as useful and potential, and are by definition attempting generosity. Generosity in critique MATTERS. And furthermore, here’s where I get mad as hell: direct-effects audience theory has been discarded for like 40 years for a reason, but it HAUNTS twitter discourse like a hideous revenant. This framing of art and culture is very conservative, pretty fucked up, & spooky to someone who does this stuff professionally. If your replies are full of people saying “hell yes this is critical theory RUN AMOK” I want you to think hard about that.

And regarding some subtweets: it is, in fact, some people’s job—a job for which they have trained extensively!—to do critical work. That does not mean your opinion doesn’t matter, but it does mean (as I teach students every semester!!) that when doing heavy lifting with art, perhaps the metric of “who is allowed to speak about rhetoric and discourse” is not *solely* an identity based category. That’s a dangerous game. All of us can read badly, or be missing the background that a piece is speaking from, and being trans is NOT a guarantee against that. I’m exhausted and upset by the idea that we can’t have things that dig into more than 101 level exploration of gender, or our pain and tropes and violence, because it won’t be perfect for Everyone. And a queer woman who has the background to engage with what rhetoric and discourse and criticism do, weighing in specifically on those things, is not out of line— and neither is a trans person speaking to their identity experiences. Both can coexist and be discussed with an ethical approach to critique that is not infuriating.

I’m extremely tired and frankly feel violated by the level of anti-intellectual rhetoric and vitriol that cropped up in this discussion, and I’m not talking about fair critiques of a story’s functions or failure to fulfill those. Shit got personal quick, in unproductive ways. In short: harm-based critique of art sounds reasonable on the surface but its application & implications are intensely problematic and almost impossible to ethically or properly deploy, particularly when applied not to, like, egregious hate speech, but affectively difficult art.

Lee Mandelo

Things I wondered after viewing Star Wars: The Rise of Skywalker

Emperor Palpatine:

What the heck was his long term goal anyway? I mean, literally what did he want to achieve? If he wanted to rule the galaxy again, he already had the tools necessary: a fleet of mini Death Stars and a loyal military to use them. Why did he bother waiting for Kylo Ren to show up before making a move? Why did he leave the ships hanging out on his planet where they’re most vulnerable? What was he trying to accomplish militarily/politically?

If Palpatine’s Plan A for regaining youthful vigor was for Rey to strike him down in anger, thereby making herself susceptible to the dark side and subsequent bodily possession by Palpatine, why did he tell her about it? She was there to kill him. If he just kept his mouth shut, or goaded her into avenging her parents, he would have won. Why did this arch manipulator galactic puppetmaster turn all Republic serial villain just in time to make Rey hesitate?

After Palpatine discovers that he can use the power of the Force connection between Rey and Kylo to heal his own body, why does Palpatine act like Rey is a threat? Sure, he has a Plan B now that doesn’t involve stealing Rey’s body, but Plan A should still work, right? The only thing that’s changed is Palpatine is stronger and Rey is weaker. So when Palpatine is attacking the rebel fleet with Force lightening and Rey pops up to try to kill him again, why doesn’t he just keep blowing up x-wings and let her kill him if she wants to, as per the original plan?

Having inexplicably decided to fight Rey, why does Palpatine keep shooting Force lightening after it starts ricocheting and tearing his face off? I mean, I get why he does this in Episode III; he is manipulating Anakin while keeping Mace Windu from decapitating him. But here he’s not trying to manipulate anyone by looking helpless, and, until the last five minutes, getting decapitated by Rey is what he has desired the entire movie.

Luke Skywalker:

If Luke knew that Rey was related to Palpatine, why didn’t he tell her? I seem to recall Luke was displeased when his own Jedi Master concealed his relationship with Darth Vader.

Why, during the many years Luke apparently spent looking for the two MacGuffins that point the way to Planet Sith, did it never occur to him to look in Emperor Palpatine’s room on the Death Star? Shouldn’t combing through that wreckage been one of the first things the New Republic did after the Battle of Endor? Isn’t the Emperor’s sock drawer the first place someone searching for Sith artifacts would think to look?

Kylo Ren:

Speaking of the MacGuffins, if Luke couldn’t find one despite years of looking, how does Kylo Ren track one down so easily? He makes it seems effortless. (Apparently the official companion publications say that Kylo finds MacGuffin 1 on planet Mustafar, inside what used to be Darth Vader’s fortress. So Luke spent years hunting for these things, but didn’t look in Emperor Palpatine’s room or Darth Vader’s house.)

Why does Kylo Ren bring MacGuffin 1 to Rey when he’s trying to stop her from finding MacGuffin 2? His whole deal at that point is forcing her to rely on him to find the Emperor. He even destroys MacGuffin 2 so she can’t possibly use it. So why did he bring another one along with him?

This gets even weirder when you consider that he crashed his first ship earlier in the movie. Are we to understand that Kylo wired the MacGuffin into a ship, crashed that ship, retrieved the MacGuffin from the wreckage, wired it into a new ship, and then flew that ship straight to Rey?

(The two ships do look the same, so I suppose it’s possible that he just has, like, a garage of identical pointy black ships. But then why did he pick the one with a MacGuffin in it to chase Rey? At first I thought maybe he still needed it so he could take them both back to Planet Sith, but later it’s a major plot point that anyone who knows the way can fly there, no MacGuffin required.)

General Pryde:

When General Pryde is watching the rebels ride warthog-horses across the top of his spaceship to break his very important antenna, why does he not just tilt the ship and tip ‘em off?

Lando Calrissian:

Why was Lando on Pasaana, anyway?

How did General Lando recruit an enormous fleet of reinforcements, when all previous rebel distress calls for the past two movies have gone utterly ignored?

Finn:

How did Finn fail to notice an entire extra spaceship? Either the transport he watched Chewbacca get loaded into had already taken off while he was getting Rey and he missed it, or else the one that Rey accidentally blew up was the first to take off and Chewie’s ship was still just sitting there where Finn left it. Either way, how do you miss an extra spaceship in the flat open desert?

Sith Knife Maker:

How did they know the exact shape that the wreckage of the Death Star would make when viewed from a specific cliff with enough accuracy to make a knife with the same outline?

Dead Jedi:

Why wouldn’t they talk to Rey all the earlier times she was trying to get in touch with them? Seems like every Jedi who ever lived was willing to chat at the end of the movie. So what changed?

Rey:

Why symbolically lay Luke and Leia to rest on Tatooine of all places? Luke hated it there, and Leia’s experience of the place primarily consisted of being enslaved to a giant slug monster.

Other:

Why bother pretending to kill Chewbacca when you’re not going to allow the characters to respond to his supposed death before learning that he’s actually still alive?

What narrative purpose is served by taking us to Poe’s planet, introducing us to Poe’s old friends, blowing up Poe’s planet, then quickly revealing that all of Poe’s old friends miraculously survived?

Why tell the audience exactly how C3PO’s sacrifice is going to be reversed before he even makes it? Sure, C3PO thinks R2D2’s memory banks are notoriously unreliable, but the audience knows that’s not true, so it doesn’t even work as a misdirection.

Are there deleted scenes in which Rose gets to actually do anything?

That split-second lesbian kiss at the end lampshading the widely criticized lack of LGBTQ representation in the series, was that a Legend of Korra-type thing where the parent company literally forbade the creatives from doing more, or was that the extent of directorial efforts to address the issue?

Was the entire script reverse-engineered from a checklist of fan service moments?

Is this the most perfunctory event movie I’ve ever seen?

The Joker and John Wick

Todd Phillips’ move Joker isn’t out yet, but critics have seen it. While many have given it glowing reviews, others have said that it glorifies real-world patterns of violent behavior. Plot synopses have led some to worry the film might normalize the beliefs of the incel community, which has produced several murderers already.

Todd Phillips responded to that criticism in an interview, saying,

The movie still takes place in a fictional world. It can have real-world implications, opinions, but it’s a fictional character in a fictional world that’s been around for 80 years. The one that bugs me more is the toxic white male thing when you go, “Oh, I just saw John Wick 3.” He’s a white male who kills 300 people and everybody’s laughing and hooting and hollering. Why does this movie get held to different standards? It honestly doesn’t make sense to me.

This comment has gotten a lot of attention on Twitter. I’ve seen some people argue that it’s a faulty comparison because John Wick’s victims provoked him to kill them, by attacking him and killing his dog. Others are saying that Phillips has a good point, and those critical of violence in media shouldn’t get to pick and choose which violence they abjure.

The people who say it’s a bad comparison are correct, but not for the reasons given.

It’s not a bad comparison because of who gets killed, or the motives of the main character. It’s a bad comparison because the movies operate in two different rhetorical modes. Or, to put it another way, because not all fictional worlds are the same.

The John Wick series are solidly action fantasy movies. The copious violence is choreographed and balletic; combination fist/gun/knife/car/axe fights that would be impossible in reality. The main character belongs to an international secret society of genteel super-assassins, with its own baroque customs, currency, and global infrastructure. This is a fictional world that is blatantly unreal. No one worries about John Wick inspiring copycat crimes because it would be literally impossible. Behaving as John Wick does requires living in a world that doesn’t exist.

Joker, from its marketing and reviews, appears to be working in the rhetorical mode of psychological realism. This is the same mode as a great deal of mainstream literature—fiction which concerns invented but believable people, doing things that didn’t happen but believably could have. Phillips admits this in another interview, saying, “I literally described to Joaquin at one point in those three months as like, ‘Look at this as a way to sneak a real movie in the studio system under the guise of a comic book film.’” So while the world of Joker is, as he says, fictional, this depiction of it is striving to be realistic. That’s why people are criticizing it in a way they don’t criticize John Wick.

John Wick 3 valorizes a kind of violence that doesn’t (and can’t) actually happen. Joker, according to some critics, valorizes a kind of violence that does. Those critics may turn out to be wrong, but they aren’t hypocrites for treating the two movies differently.

Armadillocon 2019 Schedule

If you’ll be in Austin this coming weekend, you can find me at the Omni Southpark hotel for Armadillocon. Here’s my schedule:

Friday, Aug. 2, 5:00 pm — Art of the Short Story — Ballroom F

Saturday, Aug. 3, 8:00 pm — What Sciences Haven’t Been Used in Science Fiction? — Southpark A

Sunday, Aug. 4, 11:30 am — Author Reading — Southpark B

Online Class on Writing and Selling Short Science Fiction

On August 13th, 7:00-9:30pm CST, I’m teaching an online class through The Writing Barn for people getting started with science fiction short stories. We’ll be talking about elements of craft, as well as how to publish your work. If you are an early career writer of SF, consider signing up!

Kawhi Has a Great Idea For a Novel, Just Needs You To Write It and Split the Money Fifty-fifty

I mean, not quite. But it’s close. Kawhi Leonard used to have an endorsement deal with Nike. Now he has an endorsement deal with New Balance. Also, Nike and Kawhi are suing each other, both claiming authorship of the logo that was on Kawhi’s stuff from Nike. The linked article has an image:

corporate needs you to find the differences between this picture and this picture

A Klawmedy of Errors

—Act 1—

Nike: [Gives Kawhi a swimming pool full of money for seven years of endorsement rights.]

Kawhi: Cool, thanks. Maybe my logo could be, like, my hand, and also my initials, and also the number 2 that I wear on my sports shirt when I am being a professional athlete?

Nike: Sure, we employ designers who can do something with that.

Designer: [Spends years in training, maybe gets a degree in graphic design, builds a portfolio, is hired by Nike to create brand marks that will be worth millions of dollars. Uses this expertise to turn Kawhi’s vague notion into an actual professional logo.]

Nike: Cool, thanks. Here’s your paycheck, we own the logo now.

Designer: Yes, that is my job.

Nike: [Spends more money marketing Kawhi and making the logo recognizable, recoups that money by selling products featuring the logo.]

—ACT 2: Seven Years Later—

Kawhi: The contract for selling products featuring my logo has expired. I am arguably the best at my sport in the world now, so a new contract will require thirty-three swimming pools full of money.

Nike: That is too many swimming pools.

Kawhi: I must leave where I am and go to a new place.

New Balance: [Gives Kawhi all of the money pools.]

Kawhi: Now New Balance is allowed to sell products featuring the exact logo that Nike paid a designer to create and spent the past seven years marketing.

Nike: Oh Kawhi, you are such a joker.

Nike & Kawhi: [Both laugh because Kawhi made a very good joke. The laughter sounds entirely natural.]

—End—