Proof by Counter Example

"The proof is trivial." (Oh, cool. Guess I’m just an idiot then.)

"Left as an exercise." (Translation: You’ll never solve this in a million years.)

"It’s obvious, really." (Sure, if you’re a demigod.)

"By inspection." (Stares harder at problem… still nothing.)

"For small values of epsilon." (How small? Subatomic? Microscopic? Vibes?)

"WLOG (Without Loss of Generality)." (Oh, we’re just assuming it doesn’t matter now? Alright.)

"Details omitted." (Because apparently, you don’t need to understand it.)

"By the usual argument." (Which you somehow don’t know because you weren’t born in 1702.)

"Assume the rest holds." (That’s some impressive optimism right there.)

"The usual abuse of notation." (Why does this feel like an emotional wound?)

"Almost surely correct." (But also possibly wrong? Cool, thanks for the clarity.)

"A non-rigorous approach." (I thought math was supposed to be precise?!)

"Assume it’s obvious." (Buddy, NOTHING about this is obvious.)

"The reader may verify." (No, the reader may CRY.)

"To the interested reader." (Guess I’m not interested enough, huh?)

"Well-behaved functions only." (We’re function-shaming now?)

"Obvious to the trained eye." (Guess I’ll never make it out of amateur league.)

"A trivial case analysis." (Trivial to WHO??)

"Integrate by parts, twice." (Bold of you to assume I got it the first time.)

"As you can clearly see." (Oh, I clearly see my FAILURE, alright.)

"It works in practice too." (Unlike me, who barely works at all.)

"Assume a spherical cow." (Are we doing math or abstract sculpture?)

"A standard result." (Not in my standards, pal.)

"We skip the tedious algebra." (No, no, please—I wanted to suffer MORE.)

"Assume non-zero solutions exist." (Okay, now we’re just assuming life works out.)

"The usual topology." (Bro, I don’t even know the unusual topology.)

"Finitely many cases left." (Just kidding, there’s 72.)

"By virtue of symmetry." (Virtue? I have none left.)

"Don’t worry about the constant." (The constant is probably my GPA dropping.)

"Assume continuity." (I’m assuming my brain is breaking.)

"Smooth functions only." (Guess I’ll leave, I’m clearly not smooth enough.)

"The simplest non-trivial case." (Simplest? NON-TRIVIAL? Pick a side!)

"Epsilon goes to zero." (Epsilon isn’t the only one losing it.)

Naturals, integers, rationals: yeah, easy.

Computable irrationals: ok, we can work with that. You want 20000 digits of pi? One million? Coming right up.

bed of hot coals meeting

A spell the wizard uses to light the group's campfire has an error somewhere in its depths, and sometimes it doesn't work at all. The wizard spends a lot of his time trying to track down the exact conditions that cause the failure.

The wizard is attempting to create a new spell that marries two older spells together, but while they were both written within the context of Zephyrus the Starweaver's foundational work, they each used a slightly different version, and untangling the collisions make a short project take months of work.

The wizard has grown too comfortable reusing old spells, and in particular, his teleportation spell keeps finding its components rearranged and remixed, its parts copied into a dozen different places in the spellbook. This is overall not actually a problem per se, but the party's rogue grows a bit concerned when the wizard's "drying spell" seems to just be a special case of teleportation where you teleport five feet to the left and leave the wetness behind.

A wizard is constantly fiddling with his spells, making minor tweaks and changes, getting them easier to cast, with better effects, adding bells and whistles. The "shelter for the night" spell includes a tea kettle that brings itself to a boil at dawn, which the wizard is inordinately pleased with. He reports on efficiency improvements to the indifference of anyone listening.

A different wizard immediately forgets all details of his spells after he's written them. He could not begin to tell you how any of it works, at least not without sitting down for a few hours or days to figure out how he set things up. The point is that it works, and once it does, the wizard can safely stop thinking about it.

Wizards enjoy each other's company, but you must be circumspect about spellwork. Having another wizard look through your spellbook makes you aware of every minor flaw, and you might not be able to answer questions about why a spell was written in a certain way, if you remember at all.

Wizards all have their own preferences as far as which scripts they write in, the formatting of their spellbook, its dimensions and material quality, and of course which famous wizards they've taken the most foundational knowledge from. The enlightened view is that all approaches have their strengths and weaknesses, but this has never stopped anyone from getting into a protracted argument.

"This spell causes the hair to fall off cats." "That's fixed in Xaranthius' latest publication, you just have to rewrite your entire spellbook for compatibility."

Sure, "High Tower is a powerful language, but it's such a pain to write. I just use Unity* as it's simple to write and can do nearly everything I need" "cranky because you can't memorize all the conjugations and declensions, aren't you?" "LOOK MAN, I CAN MEMORIZE ANYTHING, INCLUDING THE FACE OF YOUR MOTHER IN ECSTASY. IN FACT, BEHOLD!" *a little time window appears between them, demonstrating exactly that. The first wizard (seen through the window) turns around and winks at the "camera".

"you kids today with your lizardman. How can you get anything done in a language without gendered pronouns? It's like fingerpainting. Sure you can learn on it but once you've got the basics you should switch over to a REAL language"

"the Kalic have been here already. We better get out before the rest of their army marches in." "how can you be sure?" "you see that teleport?" "no" "well, if you COULD see it, you'd see it's written in Adevic Yevi. That's the Kalic magic language." "couldn't it be someone else? We saw those Monon traders, maybe one of them..." "no. No one writes Adevic Yevi unless they're being paid to. It's a language written by committee."

Wizards going on a quest to get the spellbooks for a lost spell, only to find out that it was written in skydove cant. No one can read that shit! The creator must have been one of those weird "functional wizards". (They're obsessed with making sure their spells have no side effects)

There's a small library on the outskirts of Freeport which tries to collect versions of basic spells in every language. The Adevic Yevi version of "fireball" takes up 7 pages, mostly boilerplate setting up the interfaces with fire and explosions and ExplodingMagicalBallFactorySingletons. The Lizardman version is basically "AHAHAHA, YOU GO BOOM!"

There's a bunch of wizard apprentices working on porting an old "Summon Bread and Fishes" spell from the absolutely archaic language it was written in. Once it's in Unity, it'll be easy to modify and teach to more wizards, which'll obviously be good for disaster areas. It's just too expensive to keep paying the ancient guys who can still do magic in TRAN-FOR.

Eccentric wizards keep inventing new languages for spells. You look at them and they're neat, but it'll never catch on. And either you're right, or the next time you're applying to be a court wizard, the advisors want to know if you have at least 5 years experience in Tilted Runic and you're like "it only came out 2 years ago!" "aren't you a chronomancer?" "oh good point. Yeah I've been using it for 20-30 years."

There's wizards who will spend incredible amounts of time doing silly things with spells in strange ways. There's this guy (Vorth) who made his own language where there's only one basic spell: fireball. Everything else is basic magic glue tying multiple fireballs together. So like, he's got a breakfast spell. Stand back (good advice for all his spells), and you'll see a fish get knocked out of the local pond, flung through the air by successive explosions, and eventually it lands on his plate, nicely cooked and deboned, if slightly charred (the glass of milk is harder to explain). His magical door locks involve a quicksilver sphere and molten lead changing shape when heated... It's tricky but it seems to work. He's working on a teleport spell, but so far it's mainly just killed test subjects (primarily sheep from a nearby farm).

It's wands vs staves vs bare hands.

Wanders are like "they're available everywhere and once you learn how to do it it's so powerful!"

Staffguys always talk about how you can do ANYTHING with a staff. Wanders claim it's a pain to carry around an overpowered device that can do ANYTHING when you just need to cast fireball or a simple one man teleport.

Meanwhile the bare wizards are showing off how they don't need any magical tools and can just do hand motions.

Northern coyotes (Canis latrans incolatus) Potter Hill, Alaska, USA

The very first book she asked for was a book on Morse code. I was instantaneously enthralled with this child. Over the years, she took out books on first nations religions, the war of 1812, Ulysses by james joyce, books about tracking animals and identifying pawprints, basket weaving, loom weaving, the battle of Agincourt, honestly way too many things to remember. 

100% of the topics she was into were not typically “age-appropriate” (although I am a staunch believer in there being no age-appropriate subjects, just age-appropriate ways of explaining them) and about half of them required interlibrary loan requests because the topic was so esoteric the only books on it existed in some university library on the other side of the country. 

Some evenings she would come in after school and it would be a big snow day so there was hardly anyone there, and we’d just talk until the library closed, looking things up on google when she had a sudden inspiration to know something. Topics ranged from the physics behind a woodpecker’s tongue wrapping around its brain (she taught me about that one) to “Horrible bear bear” and the “bear circle” and the “anti-bear circle” (latin and greek roots of bear names and the Arctic.) She was just like, the coolest person. 

I really hope to meet up with her again someday because I really think these kinds of things have a circular effect- I’m thinking back to when I first moved to that town at 13 and went into that same library, marching up to the front desk and asking for books on astrophysics (probably looking even younger, I’m 26 now and people peg me for 14-16 consistently), and how the librarian- who later became my boss- must have felt about tiny me when she was putting in interlibrary loan requests for specific textbooks from MIT to go along with the OpenCourseWare courses I was taking online (which I gained much greater respect for when I became the interlibrary loan technician- holy Hannah that’s a lot of steps.)