Formulaic Magic

Your math textbooks were right: math is magic!

"Using the power of Mathemagic, you can multiply power, subtract hope, and cube-root the dreams of man himself."

— Mathemagic Skill-tree Description, Dungeons of Dredmor

Modern science loves explaining complex physical phenomena through comparatively simple-looking mathematical abstractions. Motion of an object? F = ma. Energy content of a mass? E = mc². State of a gas? pV = nRT. Exceptions to these rules are themselves often explainable through math — more intimidating calculations, of course, but still math. These equations are only useful, however, because they accurately describe physical interactions in nature; without this connection, they're just meaningless jumbles of symbols.

Authors often neglect to establish this link between math and reality, instead treating equations as if they possess a power all on their own.^note There are power functions, though! And some theorems are said to be more powerful than others. As of today, real life mathematicians still struggle on how to harvest math's magical potential. This often manifests in two ways:

• Reduction of math to a magical artifact: The simple knowledge of an equation is powerful in itself. Committing a mathematical expression to memory might unlock superpowers, for instance.
• Giving math a will of its own: The mere presence of some mathematical statement on paper or in someone's mind has some effect on its surroundings — discoverers immediately go insane, or logical reasoning ceases to work in its presence.

Such "magical math" (or "mathemagic" or "Mathamancy") doesn't have to be connected to any aspect of reality, so it will often be rather nonsensical as a result. In this way, such math is somewhat related to the Numerological Motif, in which numbers get intrinsic properties.

On the flip side, writers will occasionally make systems of magic describable through math, much like their physical science counterparts. This use of math is more readily justified, though usually still not explained in sufficient detail to make mathematical sense.

This may be Functional Magic under rule magic. Arguably Truth in Television if you consider science to be a form of magic, and that its formulas are being applied to create technology, thus hacking the universe and having magic-like powers over things. Also, we don't actually know why reality is describable by maths - this is a surprisingly profound philosophical problem rarely addressed in fiction.

Compare Calculus Is Arcane Knowledge, in which math is not exactly magic.

---

Examples

Anime and Manga

• In The Case Study of Vanitas, 16th Century alchemist Paracelsus created the World Formula, a mathematical equation that is reality. Any changes to the formula also changes reality. It's believed that vampires resulted from the formula for humans being changed, and Astermite resulted from the formula for coal being changed. Vampires are able to see the World Formula and change it on the fly, giving them magical powers like Playing with Fire and so on. In turn, Chasseurs have developed technology to blind a vampire's ability to see the Formula, thus disorientating and weakening them.
• Calcumon of Digimon Universe: App Monsters is able to alter any data, including the power stats of other Appmon, by applying mathematical operations to it. It renders most of the Buddy Appmon powerless by plugging in multiplication and division operations with the buttons on its arms and then firing a Breath Weapon at them. It's one of the few instances where the Appmon's stat screens are actually story-relevant. The only one this doesn't work on is Dokamon, who is Hot-Blooded enough to straight-up ignore it.
• In The Epic of Zektbach, the Ristaccia grants its wielders the ability to see and understand the fundamental building blocks of reality, represented as mathematical equations, as a form of Combat Clairvoyance. Lien Clewing's prodigious math skills even allow him to transcend reality and understand the ultimate truth of the world.
• JoJo's Bizarre Adventure: Steel Ball Run takes geometry very seriously. A short two-page lesson on the golden ratio gives Johnny the power to create miniature wormholes.
• It's revealed in the supplementary manga of Lyrical Nanoha that constructing the Instant Runes needed for magic in that setting requires a very good grasp of math. In fact, one of the reasons why Nanoha and Fate are such powerful mages is because they're both math geniuses. The Devices aren't just conduits for mana, they're also highly advanced calculators.
• Magic in Miss Kobayashi's Dragon Maid is incredibly similar to computer programming, to the point that wizards that migrated to Earth used it as the basis for the programing language at Kobayashi's workplace.
• Witches in Soul Eater use "calculation magic" to make their spells more precise.
• Mages of the Acedia archive in Trinity Seven use Logos Arts, magic that manipulates movements, space, and time through numbers and calculation.
• In Yu-Gi-Oh! GX, Misawa has a monster called Mathematician, a goofy-looking wizard wearing a mortarboard and academic road. Its attack is called "Battle Curriculum" (or "Number Cruncher" in the dub) and consists of a beam of magical energy full of numbers and mathematical symbols. Konami eventually made it into a real card.

Comic Books

• The DCU:
  • The Anti-Life Equation from New Gods is usable as a method of Mass Hypnosis. It runs Loneliness + Alienation + Fear + Despair + Self-worth ÷ Mockery ÷ Condemnation ÷ Misunderstanding x Guilt x Shame x Failure x Judgment, n=y where y=Hope and n=Folly, Love=Lies, Life=Death, Self=DARKSEID.
  • The pre-Crisis on Infinite Earths Round Robin maxiseries DC Challenge had a series of numbers in the first issue (written by Mark Evanier) that were somehow important — but none of the other writers could ever really figure out how. Several of them later used the numbers in formulas for this, that, and the other. The secret? Add them up on a calculator and turn the calculator upside down; it spelled out the name "ELI ELLIS", a character from the first issue who was key to the whole crisis.
  • The Golden Age superhero Johnny Quick and later his daughter Jesse Quick accessed their superspeed powers by reciting the mathematical formula "3X2(9YZ)4A". More recent materials have retconned this as a mantra that allows him to tap into the mystical Speed Force.
  • One member of the Green Lantern Corps is a sentient mathematical equation, and so is his ring.
  • Dr. Sivana, longtime enemy of Captain Marvel, broke out of Cardboard Prison by memorizing and reciting a mathematical equation that allowed him to walk through the wall.
  • In the New 52 Stormwatch series, Jenny Quantum uses glowing mathematical symbols to activate her Reality Warper powers. In the zero issue, her predecessor a thousand years earlier, Janni Algebra, used something similar.
• In Empire (by Delany and Chaykin), knowledge of catastrophe theory, enshrined in a crystalloid device, holds to key to ruling a galactic empire.
• Marvel Universe:
  • Amadeus Cho (the seventh-smartest person on the planet) sees the world as mathematical formulas. The Incredible Hulk does as well, according to Cho (albeit unconsciously), which explains how he somehow doesn't hurt any innocent bystanders while Hulk-smashing.
  • The small-time villain Mathemanic has an array of math psycho-powers. On his first appearance, he makes Firestar believe that she's the equivalent distance between the Earth and the sun away from everything else, up to and including the smallest speck of matter. She promptly becomes catatonic.
  • One Fantastic Four arc involves a mathematical formula that gains sentience. It becomes a low-grade Reality Warper that deals with reality in mathematical terms and sees Reed, its accidental "creator", as its only equal — to the point that it resembles a Stalker with a Crush. Reed stops it by creating a formula version of himself, balancing the equation.
  • The Ultimate Marvel version of the Scarlet Witch needs to work out the statistical probability of her hexes before she casts them for desired effects.
• In Numbercruncher, working out a particular mathematical theorem allows the mathematician to transcend the wheel of karma and retain his memories on his next reincarnation. Other forumulae let him escape the attention of God's enforcers, and also choose to lose his memories in his final rebirth.

Comic Strips

• Bloom County: Child Prodigy Oliver Wendell Jones claims to have completed the "Great Unification Theory", a set of equations that explains everything in the universe... except flightless waterfowl. As a result, Opus the penguin is erased from reality — that is, until Oliver realizes he "didn't carry the 2". Once he fixes his mistake, Opus pops back into existence.
  Opus: NOW CUT THAT OUT!!

Fan Fiction

• Magic in The New Math works via complex math equations. The most powerful mages are also really, really good at math.

Films — Live-Action

• π: Max starts the film with a dogmatic belief that "mathematics is the language of nature", which straddles the line between "math is capable of explaining everything in existence" and "math determines our existence". He later encounters a 216-digit number that seems to have a catastrophic effect on anything it touches: the stock market, his mentor's health, and eventually his (already tenuous) sanity.
• Prince of Darkness has ancient, complex mathematical equations of extraterrestrial origin with seemingly magical powers. They predict the existence of a God and Anti-God and, once understood, they open up whole range of bizarre phenomena, including leaping headlong through mirrors into an abyssal darkness on the other side.

Literature

• Grimoires and parchments in A Certain Magical Index seem to contain complex formulas which need to be deciphered in order for it to work — kind of like math books, except those don't attempt to destroy their user.
• In "Career Day", one of the stories in the Chicks in Chainmail anthology, the protagonist comes from a world where magic is done with mathematical formulas. She brings her daughter's class there on a field trip to observe her in her career as a barbarian swordswoman. When her opponent in a duel cheats by hiring a wizard to help him, the other chaperon on the trip, her daughter's math teacher, counteracts the magic with his knowledge of calculus. It Makes Sense in Context. Later expanded to a full novel, called Mathemagics with this entire premise for the magical system.
• Magic in the Cthulhu Mythos is often related to mathematics, and it makes sense considering H. P. Lovecraft's preference for Alien Geometries. In "The Dreams in the Witch House", for instance, the protagonist is a mathematician who researches geometrical principles that would allow him to travel outside normal space. If you can understand the true nature of the universe, you can use that knowledge to do things that seem physically impossible to us... provided you don't first Go Mad from the Revelation or attract the attention of some Eldritch Abomination, of course.
• The Aons (runes) necessary to make the magic system from Elantris work are very much like a combination of mathematical symbols and a very complex alphabet. Learning magic is incredibly difficult for this reason, and it can take weeks for even an experienced practitioner to write out the more complex spell "equations".
• Fine Structure uses science, not magic. In later stories, scientists fluent in Eka can unlock the "The simple knowledge of an equation is powerful in itself" kind of formulaic magic. It's also revealed that the universe, or more accurately the imprisoning god, doesn't like anyone messing around with Eka.
• In The God Eaters, this appears to be how the very powerful 'Pattern Magic' works.
• The Harold Shea books are heavily based on mathematics causing magic.
• The Harry Potter series has an Arithmancy class. The reader doesn't get to hear much about the subject, though, beyond the fact that Hermione's studying it (and apparently enjoys it).
• The reality warping abilities of human interfaces in Haruhi Suzumiya seem to work like this.
• InCryptid: All Johrlac have an unusual affinity for math, and Imaginary Numbers, the first book with one as the main POV character, reveals that a sufficiently powerful Johrlac can use their mathematical skill combined with their Psychic Powers to rewrite reality and tear a hole between dimensions, which would destroy Earth (and fry their own mind) in the process.
• The Laundry Files shows magic as mathematics, to the point that computers solving certain equations can warp reality as per magical spells. Becomes a bit of a problem when the walls around reality start weakening, to the worst-case scenario of somebody solving equations in their head running the risk of accidentally summoning an Eldritch Abomination.
• Life, the Universe and Everything introduces the field of Bistromathics, math that can be used to warp spacetime for purposes of Faster-Than-Light Travel by taking advantages of the unique properties numbers take on when written down within the confines of an Italian restaurant.
• Miskatonic University - Elder Gods 101: Sheryl Mason discovers that her proclivity with mathematics allows her to wield powerful supernatural forces. Unfortunately, her attempts to combine them with occult forces leads to the resurrection of Herbert West.
• In Monday Begins on Saturday, a book called Equations of Mathmagic is mentioned.
• In A Practical Guide to Evil, most of the major characters use Trimegistan Magic, based on spell formulas, experiments and quantifiable effects. This isn't the only magical theory, but it is the most reproducible and teachable. However, because it was developed by The Dread Empire Of Praes on the back of a foundational theory by the series Big Bad, other nations tend to be wary of it.
• In Pyramids, You Bastard the camel is a genius mathematician, able to use complex equations to step in and out of pocket dimensions.
• Ra is about actual magic, which is a poorly-understood branch of physics. Magic is accessed by reciting specific incantations while performing mental exercises stated to be not dissimilar to solving differential equations. More generally, spells also require a lot of math to design, and resemble computer programs (right down to concepts like modularity, recursion, and writing interpreters to parse gestures as incantations).
• This is the premise behind Simon Bloom.
• The Great Equation from the Well World series. A couple of supercomputers are capable of warping reality retroactively (that is, those who didn't see the change actually happen are incapable of realizing that anything actually changed) by "altering" the Equation, which basically is reality. By moving a few numbers in the equation, the result — that is, our reality — changes to suit.
• In John Langan's "What Is Lost, What Is Given Away", and by extension his larger mythos, space-bending magic has heavy mathematical elements to it, to the point where a math teacher states he was one of the best students a senior sorcerer had ever had. He himself is imprisoned in an extradimensional pocket by that same sorcerer, which requires him to retrieve his books and study more of the occult mathematical equations.
• In the Young Wizards series, all magic is based in math and science, and the kids have quite high-level discussions of these things, because part of magic is being able to completely describe what you want to change.

Live-Action TV

• Doctor Who:
  • "Logopolis" features a planet where they've developed a branch of mathematics in which the act of performing the calculations changes the fabric of reality. The entire population takes shifts in calculating a never-ending formula that holds the end of the universe at bay. Bonus points to the Master for destroying an entire quarter of the Universe by killing a couple of the guys doing the math.
  • A more abstract example is seen in the episode "The Shakespeare Code"; the Carrionites use words as magic, like witches, and the Doctor explains this as completely non-magical by comparing it to mathematics on earth; "With the right string of numbers you can split the atom!" Except, of course, that saying the number out loud will not cause an atom to spontaneously divide in two.
• While not stated in Lost itself, the Alternate Reality Game The Lost Experience states that the numbers (4, 8, 15, 16, 23, 42) are parameters in the Valenzetti equation used to derive the time remaining before humanity's extinction.
• The 2005 miniseries Supernova involves a weird secular magic variant. The solar flares predicted to destroy the earth, and which actually start to destroy the Earth, stop suddenly. This happens as soon as the scientist discovers he made an error in the formula which predicted them.

Music

• The Rhapsody of Fire song "The Mystic Prophecy of the Demonknight " has the narrator mention the heroes using "black geometry" to create a bridge over a deep chasm. When being pursued on the way back, they don't have enough time to work it out and are forced to make a Leap of Faith.

Tabletop Games

• In Dungeons & Dragons 3.5's Complete Arcane rulebook, there is a Prestige Class called the Geometer. They learn to describe magic as using abstract geometric designs. This is an advantage, if nothing else, because they can save a ton on spellbooks: they need only a single page to depict any spell, regardless of complexity, whereas with the traditional method one would need more pages with higher-level spells. They can also use scribe glyphs of these designs to cast a spell silently.
• In Magic: The Gathering, several Blue-aligned schools of magic are based around this, but the Strixhaven college of Quandrix is particularly about this trope. Most of the spells depicted have a mathematical perfection about it, and students can even summon Fractals, which are described as math-elementals.
• Pathfinder has the Sacred Geometry feat, which turns spellcasting into a minigame resembling Countdown. A player who wants to add certain metamagic effects to their spell can roll several dice. If they can reach a specific target number by adding, subtracting, multiplying, and dividing the values of the dice, then they get to apply the metamagic "for free" (i.e. without using a higher-level spell slot).
• In Vampire: The Masquerade, one branch of Tremere Thaumaturgy concerns itself with Kabbalah, Jewish mysticism heavily based in numerology derived from the Torah. In the same way other magi would seek out a person or thing's True Name, Kabbalists use calculations to derive someone or something's True Number. Blood magic as derived through math. It's worth noting that, like basically all magic in the Old World of Darkness — blood, hedge, or mortal magick — this method only serves as a focus for the user's will. The equations need to have conscious intent in order to reshape reality. Hence why numbers scrawled on primary school blackboards don't end up melting the instructor's face.

Video Games

• Reinhart Manx, the playable mage character in Dungeon Siege III employs both this and Magitek, using the power of math rather than fireballs.
• In Dungeons of Dredmor, one of the skill sets you can choose at start is Mathemagics. It gives you access to fun spells such as Curse of the Golden Ratio and Beklam's Diminishing Calculus!
• In The Elder Scrolls, the Altmeri (High Elven) Clan Direnni, who once ruled nearly 1/3 of Tamriel's land mass as the Direnni Hegemony in the 1st Era, were known to use "hidden magic" in the form of signs and runes in their financial reports in order to keep them out of the hands of their competitors. "Sorcerous precautions" were required on the part of the reader, lest these magic numbers prove fatal. Per one text on the subject:
  "Crucial pages were covered in spittle of the previous translator, who had babbled idiotically over the text for days before catching fire."
• Final Fantasy:
  • Final Fantasy VII has an interesting example: When Safer Sephiroth casts Super Nova, four equations fly by the screen. They are, roughly, the potential attractive force between the sun and the planet, the Earth's potential attractive force, and... the area of a circle. Still, 75% good scientific formulae being used in a magic spell is better than average.
  • This is echoed in Dissidia: When doing Supernova as his EX Burst, Sephiroth will begin chanting the equations, which briefly appear on-screen.
  • The Final Fantasy Tactics series has a Calculator class that uses various magical effects, such as casting spells that hit every target whose level is a prime number. How their abilities work from an in-universe perspective is not specified. According to the remake for the PSP's tutorial, Calculators are attuned enough to the flow of mist that they can manipulate it in unique and bizarre ways which other classes just can't pull off. In other words: SE knew it made absolutely no sense and Hand Waved it.
  • In Final Fantasy XIV, the background lore about the Arcanists reveals that they study their magic through the art of mathematics and geometry. Apparently in this realm, finding X on the Triangle, understanding Pi, the square root, cubic powers, and combining multiple symbols together, gives you access to magic and the ability to summon gods.
• Rita's non-spell special attacks in Tales of Vesperia take the form of mathematical equations.
• The World Ends with You: Minamimoto summons a nuke by initializing the Level i Flare, which consists of reciting pi to 150 digits. Of course, there are a lot of people capable of reciting pi to 150 digits, but they aren't Minamimoto, and they don't have a graffiti covered dead end alleyway and eldritch tattoos.^{Fun bit of math/Final Fantasy info here} "Level x Flare" is a type of spell that casts Flare (one of the most powerful attack spells) on everything with a level divisible by whatever number x is. i is a mathematical term referring to the square root of -1, the consummate imaginary number. In addition to that, as the root of -1, that also makes it a possible factor in every number, real or imaginary. It would affect everything, across all possible realities. Useful when your target can travel through dimensions,

Visual Novels

• Chaos;Head has fun¹⁰ × int⁴⁰ = Ir2, the equation Takumi discovered in elementary school and the basis of the Noah Project.
• In Umineko: When They Cry, the Power Levels for Leviathan and Kyrie are determined in how many hours they've experienced intense envy. In-game, Leviathan attempts the multiplication for the latter's multiplication formula, along with revealing her own, giving players the idea; in the TIPS, Kyrie is shown actually chanting her formula as she runs to give her husband (flirting with a younger girl) what for.

Webcomics

• Dresden Codak: In a game of Dungeons & Discourse, Kim Ross plays a "Bayesian Imperimancer". Bayesian probability concerns itself with the likelihood of a given event based upon the outcome of past events; for example, if a coin has landed heads ten times, Bayesian probability dictates it is highly likely to land heads an eleventh time (because it's probably a Two-Headed Coin). Kim's class is a play on "Bayesian Imperative" as in, she can basically tell the world how to work if she makes strong enough judgments about how it should work.
  Kim: I am 87% confident you will burst into flames. (*fwoosh*)
• Erfworld has Mathamancy, which involves "analyzing probabilities", "predicting outcomes", and "the raw calculations thereof". It should be noted, however, that Mathamancy in Erfworld (or at least Parson's Mathamancy Bracer) seems to be more than merely math from the real world: for example, Parson has a magical deal to give Charlie twelve Mathamancy calculations whenever he wants. At one point, when negotiating whether to give up the deal in exchange for information, Charlie asked Parson to calculate whether the information was worth the nine calculations remaining. The answer was that there was a probability of 4% that it was worth the exchange, something that should not be possible to calculate.
• The Khert in Unsounded is a Background Magic Field that oversees the setting's reality and powers its magic system, pymary. The incantations for spells are a very precise shorthand spoken in the Language of Magic describing which materials and physical properties are to be manipulated and how. Logic errors, on the other hand, get messy: when someone tried to liquefy a material that turned out to be saturated with water already, the Khert glitched and melted his casting hand instead.
• xkcd:
  • The last panel of Comic #804 treats the (very counter-intuitive) Banach-Tarski paradox as a result you can get in real life if you take the axiom of choice.
  • Comic #687 abuses dimensional analysis to "prove" that certain absurd constants must be related by some fundamental law. So if they build a better Prius, either England must drift out to sea, or the pressure at the Earth's core must rise slightly.

Web Original

• SCP Foundation:
  • SCP-033 is an equation which adds up to a "missing number", an integer which non-paranormal mathematics has somehow missed. If it's recorded in any sort of computing device or written on machine-made paper, whatever it's recorded on starts to physically degrade.
  • SCP-233 is a 23-sided (impossible) polyhedron that magically accelerates calculations done in base-23 but reacts extremely violently to the number [REDACTED] (nine) and causes rounding errors in every calculation not done in base-23.
  • During this lecture, the speaker explains that almost anybody who does not believe in luck cannot create a luck charm with magic, unless they are an actual statistician, because they know what it would actually take to create luck, as an example of how not everyone can or can't do the same things with magic.
  • SCP-1313 is an equation whose solution is a live grizzly bear, which materializes when the equation is solved (and quite possibly mauls the solver).

Western Animation

• In an episode of Camp Lazlo, being flipped upside down gradually changes the Too Dumb to Live characters Chip and Skip into geniuses (it increases the blood flow to their heads). At the height of their intellect, they use their brains to flip themselves right-side up again: this involves rattling off a lengthy Newtonian formula, and then simply glowing with a pink light that levitates them into position. Unfortunately, this reduces the blood flow to their brains, and they soon turn right back into idiots.
• Looney Tunes has Egghead Junior, who can accomplish anything, even to the point of bending reality, so long as he has a few seconds to write down an appropriate equation for doing so.

Real Life

• From the times of ancient Egypt mathematics were associated with mystical properties and powers, and various theorems and systems passed in and out of cult fashion. The most famous of these were the Pythagoreans, who maintained a hermetic hierarchy back in the days when you could get away with executing your members for revealing their secrets.
  • And don't even ask what they did to people who failed to box their answers.
  • Legend has it that the Pythagoreans were so obsessed with all numbers being rational that, when someone worked out that the square roots of non-square integers were irrational (square roots being very important when dealing with, y'know, the Pythagorean theorem), they grabbed the guy and hurled him off a cliff.
  • Another telling of the story mentions how he was so pleased with the discovery that he sacrificed every sheep in town.
• During the 19th century, 4th dimensional mathematics was seen by occultists as the key to understanding ghosts and the spirit world: if people could just teach themselves how to think and move in four dimensions rather than the normal three, they could become like ghosts themselves, teleporting and becoming intangible at will. While that's technically true (if oversimplified), today it's believed that, if higher spatial dimensions such as hyperspace exist, they're curled up far too small for human beings to move through, and/or necessitate immense amounts of energy in order to access.
• The notion to get as close as possible to Quantum Physics is quite strong in modern Chaos Magick, of which a prominent example is the Octavo: A Roundworld Edition, a spellbook consisting of eight formulae-spells, by Peter Carroll.

---