Both Magic and Science provide constraints.

I think one problem with mixing magic and science fiction is that science fiction readers prefer some whiff of plausibility in however science is extended for the purpose of the story; so you run the risk of disappointing those readers by just using "magic" when you don't know the science.

And vice-versa! Magic, like science, is expected to be a tool, with constraints on how it is used. Your story cannot just be filled with deus-ex-machinas because they are convenient for you as an author. You can't just hand-wave everything away as "magic", or your heroes aren't heroic at all.

Remember you are writing a story. In order to be a good story, the hero must solve a problem and struggle to do that; they can't just wave their wand and solve the problem on page 1. Or page 100! Readers keep reading to find out what happens, which means the outcome (in the next few pages, by the end of this chapter, by the end of the section [Act], by the end of the book) has to feel up in the air. That means you need constraints on how the hero can solve the problems, and these need to feel (to the reader) near crippling; so they are kept wondering how in the world the hero will get it done.

Your magic system cannot be "anything goes," that just kills the suspense. Your scifi cannot be "anything is possible," either. For the same reason. No suspense.

By combining them, you run the risk of loosening the constraints (just use magic if you can't use scifi), and thus killing the suspense.

That is why fantasy stories have limitations on what magic can do, and why scifi stories have limitations on what science can do. In both cases, they allow non-realist stories to take place, but are actually obstacles that must still be overcome by human ingenuity and spirit. Harry Potter and his crew must still be brave and risk their lives in the face of lethal danger, because magic alone isn't enough.

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Amadeus‭

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I'm of the opinion that "Don't Mix" should be seen more as a caution than a prohibition. A very useful caution, but a caution nonetheless.

Hard sci-fi and pure fantasy work with very different world-views - one embraces technology and the other magic. Technology works within the framework of physical law, where what you want is entirely beside the point. Magic is (traditionally) based on a rejection of physical law, and the desires/will of the magician are what count. Technology is above all impersonal: predictable, although often obscure - Mother Nature is a coy lady whose secrets need effort and intelligence to uncover. Magic is (or was) fundamentally personal, full of irrationality and wonder. Attempting to mix the two runs into the strong temptation to mix the worst of both worlds, producing either technology which is indistinguishable from wish-fulfillment, or magic which is, well, boring.

Over the last several decades, there has been a tendency for fantasy to be infected by technological influences. I personally blame Larry Niven's "The Magic Goes Away" for starting the movement, with his concept of magic being limited by a resource (mana) which can be used up and never replenished, very much like, say, oil reserves. While modern fantasy can produce good stuff, many writers seem to treat magic as just another skill, just another set of rules to be obeyed. There is an impersonal quality to the practice, and magic is not dark, irrational and dangerous. Magicians tend to be more like mechanics than wizards.

In other words, the magic gets sucked out of magic.

While the converse is also to some degree true of technology, the danger has always been there, since the author can simply describe almost any level of effect and invoke "advanced" concepts to justify it. As Arcanist Lupus pointed out, Arthur C. Clarke identified the problem at least 45 years ago, although to be fair, he was referring to real technology, rather than fictional, so the meaning is rather different.

One way to look at the problem is to consider dramatic tension, which is the heart of most stories. Stories are about protagonists overcoming obstacles. If a knight battling a dragon can whip out a heavy machine gun and shoot it out of the sky, it takes a lot to make the story interesting. If a scientist trying to invent a serum to stop a plague can call up a demon to do the job, well, who cares?

The trick in either case is one of tone, plot, and ingenuity. Crossovers can work. It's just that striking the balance gets harder, since crossovers need to address the rules of both schools simultaneously - or break them simultaneously, if you prefer. Either way, it's tricky to do well.

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WhatRoughBeast‭

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A lot of wonderful books combine both science-fiction and fantasy.

And why not?

If magic can exist in books set in the modern age or in the past, why not in books set in an otherwise realistic future?

Why can't mythical creatures go to space?

What reason is there to ban ghosts and purveyors of the supernatural from a world of great technology?

One of the most common crossovers involves time travel. Time travel can be either science-fiction or fantasy, depending on how it's used and what other elements are present. If you're Wesley Chu (Time Salvager), it's 100% science-fiction. If you're Deborah Harkness (A Discovery of Witches), it's absolutely fantasy.

Why do people advise against blending the two?

My guess is because most readers who enjoy one genre eschew the other. Or at least that's the belief. I love both and so do a lot of readers. But if you write both together, you run the risk of turning off a subset of potential readers. Of course, you also have a lot to gain from crossover readers.

Write what you love. With luck, the audience will follow.

Examples: Dune (series), Ender's Game (series), The Bone Season (series), The Golden Compass (series), A Wrinkle in Time, Star Wars (movie series).

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Cyn‭

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The reason this is often recommended against is because by mixing them you find yourself unable to meet certain genre conventions. Fantasy readers want swords and lords, sci-fi readers want spaceships and aliens. Even more importantly fantasy readers want epic tales of good versus evil, while sci-fi often want to grapple with trans humanist ideas. But trying to match both markets, you match neither.

That said, I don't think it is very good advice. It is not the writer's job to market, it is their job to write a good story. If you have a good story that involves fantasy and sci-fi elements, write it.

Plenty of existing works mix genres to varying degrees and been successful. Star Wars is mostly sci-fi with a little fantasy, while the Shannara books are fantasy with some subtle sci-fi.

Mixing genres can be a challenge, but don't limit yourself by blindly following a rule.

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rtpax‭

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Any sufficiently advanced technology is indistinguishable from magic. - Arthur C Clarke

There's a reason that science fiction and fantasy are frequently shelved together - separating the two is usually a fools errand.

The Dragonriders of Pern features a preindustrial society where flying, firebreathing, teleporting, and telepathic dragons defend the skies from horrible creatures that rain down from above. They could easily be considered pure fantasy - up until the book where they discover the spacecraft that the human ancestors flew in on, and the labs where they bioengineered the dragons.

In Star Wars (arguably the most famous science fiction property of all time), the Force is really just space magic called another name. And they do all their fighting with swords.

The lists go on and on. Science fiction and fantasy are united in that they explore the impossible. Fantasy uses elements that will never be possible, while science fiction uses elements that theoretically might be possible. But the impossibility is shared.

Science fiction and fantasy are not a single genre - they are many

Epic fantasy (eg Wheel of Time) and space opera (eg Star Wars) are more similar to each other than they are to urban fantasy or "hard" science fiction.

In some ways, it's inaccurate to call science fiction and fantasy genres at all. They're setting elements. A romance doesn't stop being a romance because it's set on a space station, and a murder mystery isn't any less mysterious because it was an elf who was murdered.

The key to mixing these elements successfully is to understand the expectations of your readers, and meeting those expectations (but in surprising ways). Genre helps define expectations.

If you're writing hard SF, readers expect to see a world that's close enough to our own that they can believe that our world could become the world of the story - magic obviously has no place here.

On the flip side, urban fantasy readers expect the author to have considered the modern world and how it might interact with the impossible. Extrapolations from science to science fiction may be natural depending on the exact nature of the setting.

If the reader has led to expect that the serial killer is a Scooby-Doo villain, then the sudden reveal that the murders were done with actual magic will violate their expectations and frustrate them (in general - there will of course be exceptions). But if the story is framed as a space cop trying to capture a vampire despite not believing that vampires are real, the readers will accept that premise as well as any other.

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Arcanist Lupus‭

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