So I have become very enthusiastic about the concept of magic built out of math, but my brain is currently fried because What Is Sleep so I cannot articulate much of it at the moment. Anyway I want to talk to people about this and I know other people have similar ideas (@jacktrash 's novel The God Eaters was a huge inspiration when I read it nine years ago (and every time I've read it since)) and I want to hear about them! Also, I am bound and determined to write a book set in a world where math and magic are inexorably linked, but I will not use anyone else's ideas without their express permission.
Hm, well... physics and math are pretty closely linked... So, my personal magic system for my comics project has to do with things interacting across different dimensions of the same universe. Basically, some dimensions have discrete intervals (there's spaces between layers, like pages in a book) and patterns on one layer induce patterns in the others, and vice versa. This lets all sorts of Weird Shit exist, depending on what the pattern transferations are. For example, if in plane A the pattern a moves energy left on plane B, and moving energy left on plane B causes plane A to move in pattern a.......................... you get a perpetual motion machine. There's also a concept of being able to gather/remove energy, and it has something to do with distorting the web of spacetime on these other planes. (n.b. 'planes' as they exist in D&D are, with these definitions, a kind of spacetime pocket) The whole idea is derived from Analysis and that understanding of how functions work, basically. As I go through abstract algebra I'm starting to get a clearer idea of how a system like this could work. Oh, and you can add in all sorts of other dimensions, like folding dimensions that let you make two parts of spacetime arbetrarily close so you can hop from one point to another. I have a semihumerous and more detailed writeup of this that constitutes an early page of my WIP webcomic... I'll see if I can track it down later.
My first thought when I read this is that mathematical magic could be based on veridical paradoxes. There are some cool strange things you can prove true with math that don't work in the real world. Usually this is because infinity is involved somehow: the mathematical proof describes how to divide something into infinitely tiny pieces or do a procedure infinitely many times. Since the universe is finite, the "true" results don't actually work here. Which is where magic would come in. The mathematician could - somehow - treat the pieces of the real world as if they followed the purest theoretical laws. She could do things we've been able to describe mathematically but not implement in the real world. For example, the Banach-Tarski paradox describes a way to divide a sphere into two identical spheres the same size as the first. Here's a youtube video describing how it works: In summary, you give every single infinitely small point on a sphere a unique name (naming things to exert power over them is a good classic magic technique) then divide the sphere into six weird jagged shapes by grouping together similar names, then put the six pieces back together into two spheres the same shape and size as the original sphere. Spoiler: tangent with naked people surrounded by piles of oranges It's only marginally related, but there is a wonderfully strange and silly music video made by the math department at the University of Copenhagen about the Banach-Tarksi paradox and a pile of oranges that keeps doubling in size. https://www.youtube.com/watch?v=uFvokQUHh08 Real world, you can't name infinitely small pieces of sphere because matter doesn't come in infinitely small pieces, and there isn't anything sharp enough to cut the jagged shapes. But maybe the math magician can - sometimes - treat the real world like all the pure math rules apply. So maybe she could just ... duplicate things. Some mook gets the jump on the math magician and tells her to drop her weapon. She puts down her weapon but is still holding an identical one. And smirking. The mook tells her to put down that knife. She does, and is still holding one. Three hundred knives and one really good smirk later, the mook surveys the enormous pile of knives on the floor and says something like, "This is why everyone hates math, you know." Or there's Smale's Paradox, which describes a way to turn a sphere inside out without folding or tearing it. Here's a video explaining that one: It doesn't work in the real world because it requires the sphere to be made of something infinitely stretchy and able to intersect itself, and you can't make that out of atoms. But it's fun to imagine a math magician, say, turning a safe inside out without damaging or opening it. The more I think about this paradox-based magic system, though, the less I like it. It feels cheap, like video game glitches. An Infinite Knife Bug seems like a bug a real video game could have, and random objects being inside out because someone made a sign mistake calculating their normal vectors is actually a very common computer graphics bug. I guess it's not too surprising that defining magic as "the ability to enforce a set of consistent mathematical rules that ignore the limitations of the real world" would lead to magic that feels like a collection of bugs, glitches and cheap tricks. There are some stories that would work well in (dystopian stories with themes about breaking rules, perhaps, or stories where the magicians are artificial intelligences) but not very many. Maybe there could be a cost to using your will to briefly make the world more perfectly mathematical so you can do a mathematically-true-but-physically-impossible-thing? Like maybe each time the mathematician does it, she has to to choose to turn away from the beautiful truth of pure math and back to a world where things are made of atoms and truth doesn't work quite right, and it gets harder and harder each time to choose to come back. Maybe most math magicians, at least the really good ones, end their days spray painting unnoticed equations of stunning genius on overpasses and train cars, no longer caring about whatever it was they were trying to achieve with math in the first place. Only the math is real to them now. The math magician seeking out her mentor to convince him to teach her one final proof, but the proof is too lovely for him, and he is too far gone to care about their once-shared goals or show it to her. Maybe he only recognizes her by the arcs of their lives intersecting, and uses a parametric equation instead of her name. Is she able to derive the proof from the diagrams he scrawls on the fogged window with his fingertips? Is it safe to even look at them? That sort of human consequence could make the idea of paradox magic feel a little less cheap, maybe, but I still can't quite get past the glitchy impression. It would take a much more skilled writer than I am to pull this off, I think, balancing the fact that what the magic does is pretty much exploiting video game bugs in the real world against how beautiful the math itself is supposed to be. This is a great thread, thanks! I enjoyed the invitation to brainstorm, even if my idea didn't go anywhere.
You know what also involves math? A lot of fibercrafts. Knitting requires math, so does weaving - and depending on what exactly you're making, it can involve some pretty weird calculations; quilting and crochet involve math too, and so does clothes-making. And I recall that the Norse (I think the Vikings, but not 100% sure) believed that math in general was magic, and women's magic specifically - so things like household accounting were given over for the women to do. So what if there's basically two types of math-magic - one where you're doing weird paradox stuff and risking madness, and one where you're doing seemingly more mundane stuff but stuff that keeps the world going. Discworld-style wizardry and witchcraft. A math magician can glitch walls into not existing or warp herself from one part of the world to another, with the right equations. A numbers witch can knit socks that really are lucky, or a cardigan that is always just the right temperature for the weather; she can weave a set of curtains that let in just the right amount of light for the time of day, or sew a quilt that's perfectly fits the bed you put it on...no matter what size the bed is. Perhaps carelessly done math leaves imprints on the world - fractures where that more perfect world is trying to impose itself on our reality, and it's very hard for a math magician to fix it with the equations they're used to using. Some math magicians (perhaps the really, truly good ones) might not even care, too far gone in their equations to be aware of why this might be undesireable. But a numbers witch is used to fixing dropped stitches and darning holes - the only difference between doing it on mundane yarn and doing it on the fabric of reality is the material you need to use to do it successfully.
A plot: Harried numbers witch is Very Done with fixing this particular math magician's fuck ups, and hunts them down to give them a piece of her mind... Only for them to discover something much worse, and have to team up to fight it. (Both are able to pull in others of their craft to help. At least one nonmagical mathematician receives a frantic phone call about some advanced mathematical theory they wrote their thesis on in the middle of the night. They help, bemused, and are thanked profusely and assured that their contribution will help save the world. They go back to sleep.)