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Could you have a ship equipped with fusion reactors and all the rest actually function with just clever engineering, human oversight, and clockwork computers? What issues would arise?

A fission rocket on a civilian mission on a well defined trajectory with lots of time to check for problems and correct them might do ok. Anything more complex than that though? Dubious.

A war ship of the sort of capabilities you're talking about? Probably not. Your meatbags and clockwork simply won't have the reaction times needed. You'll also have a pretty hard time operating your massive coilguns (switching those coils requires fast electronics, everything operating in sync) or your fusion rockets (they'll be difficult enough even with computers, let alone without it). Point defense systems will be utterly impractical, and getting firing solutions on distance targets will take too long to be useful in a fight. You won't be able to use guided missiles. You won't be able to use projectiles with fancy warheads such as bomb-pumped lasers or casaba howitzers.

If your magitech opponents are still able to use computers, you will simply get wasted. Their reaction times will be thousands of times faster than you, they'll see you before you see them, they'll hit you before you know what is happening, they'll have a wider range of weapons for all circumstances and they'll have a better ability to protect themselves. You'll be like knights facing off against modern ground-strike drones, though at least you'll have some idea of what you're up against and can surrender before the slaughter begins.

Obviously you're talking about magic here, but remember there are many ways to make computers, not just electronics or mechanical systems. I've always liked fluidics, but there's also optical computing which is entirely practical and quite powerful. Naturally, you can handwave that your magic somehow affects any kind of computer without scrambling regular brains, somehow, but do bear the alternatives in mind.

As an alternative though, consider biological control systems. Your missile warheads can have a carefully cloned and trained brain from a bird of prey. Your sensors might be wired up to wetware from a more nervous kind of tasty prey animal, one used to keeping an eye on the sky for predators. Neither will compare to the speed or robustness of a computer, but they are an alternative, and one that might well be practical with the level of technology you're thinking about.

Our brain cells are incredibly slow (the speed of nerve signals is significantly slower than the speed of sound in air!). And certainly much larger than modern transistors. Yet we are able to do a lot of things efficiently, including some which need a lot of computing power to do with current computers, and some which we haven't figured out how to do at all. So it is well possible that knowledge has improved until your story's time, so that computers built on the principles of our brain can be built. Such computers would still be pretty powerful even if the hardware is at sizes and speeds of the 1950s. Also note that our brain is essentially analogue (yes, it has discrete voltage spikes, but the information in in the frequency of those spikes which varies continuously; also at the synapses, the information is translated into the concentration of chemicals, which is inherently analogue, too).

And everything that would interfere with brain-like computers would also interfere with brain activity, in which case the computers would be your least problem.

Yes you can do this, but it's not easy and requires both a larger and a more highly skilled crew. It is important to note that computers don't currently do things we cannot; every computer does what it does because it is programmed to; meaning that at least one human understood the process sufficiently to encode it into a computer program. Some times it doesn't seem like that is the case I grant you, but effectively computers only do what we already know how to do, even if they do it faster and more reliably than us.

Many of the things we get computers to do now in lieu of a person is because of the consistency of the computer in the face of a known process. SCADA control systems in electrical grids for instance detect problems and alert humans, or take known reactive measures as a result of what they see happening in a critical utility grid. Even manned passenger aircraft have computers like TCAS, detecting nearby aircraft and alerting the pilots so they do something so as to avoid the collision.

SCADA and TCAS don't get distracted, go off for a minute to fetch a new cup of coffee, etc. and as such they're consistent. But, that's also not the same as better; a human can use experience and insight to deviate from the known script when circumstances dictate it happens. Computers can't. If you have enough eyes on the parts or systems that can fail, you don't actually need computers in space. But, what that means is that your crew has to know what they're doing.

Navigation can literally be done by the stars and planets around you. You can still look out portholes that are magnified or even use telescopes to look out for asteroids or other debris in your path. Fusion reactors can have analogue gauges that are constantly monitored by trained professionals who know what to do if one starts to twitch. Manning the guns can become a thing again.

Knowing whether you're in orbit around a planet might be as simple as checking that the stars in the front are rising at the same rate as the stars behind you are falling, and that the planet in the bottom of your porthole isn't getting bigger. We knew long before computers where the planets were going to be at a specific time and date; instead of putting that into computers we go back to almanacs and plot our courses with charts, compasses and the like. On the high seas on Earth, it's a 2 dimensional course but in space it will be more complex because it's 3 dimensional, but the concepts are the same.

The only reason why computers are 'essential' in space is because it limits the number of specialists needed to work on a given problem. If on the other hand you can get mass and life support into space relatively cheaply, then of course human minds can work around the lack of computers. If you're in doubt of this, then check out just how little computing power the Apollo missions had. If you can put that much computer power into space, you can already do a lot with your ship but when you get right down to it the benefit of computers in space is simply that it limits the number of heavy humans (with all their accompanying life support) you have to get out of Earth's gravity well; first by focusing the attention of the pilots only on what needs their attention, then secondly by reducing the need for some specialists or roles completely.

That said, if you can't have the computers, you have to have the people and the principal issue is efficiency. It just means larger crews, requiring more water, food, air, and warmth, meaning larger ships and more cost to get them off Earth. In essence, this is going to mean that you're going to be building battleships; not guided cruise missile frigates. But, with the right crew, you can still function in space without a computer.

What is essentially being asked for is a space warship common in 1950 to mid 1960 era science fiction. Real life counterparts such as Project Horizon (a US Army moonbase) or RoBo (Rocket Bomber, the precursor to modern ICBMs) are also largely analogue with human pilots and workers. The Apollo missions had many instances where the astronauts performed their own work, such as Neil Armstrong taking control of the LEM when it seemed about to land n a boulder field, or the Apollo 13 crew making a manual control burn Using the Mark 1 eyeball and a stopwatch because they could not spare any electrical power for the flight control computer.

Robert A Heinlein's descriptions of working spacecraft in the "juvenile" novels have standing watch crews equipped with slide rules to carry out calculations and manning control stations for every conceivable function the spacecraft does. In many cases the watch officers are simply supervising automatic systems with the ability to override and take manual control when needed. Human damage control parties also are part of the ships compliment, in many ways Heinlein's spacecraft are the counterparts of the warships he served on in the 1930's.

Given the extreme distances that can be observed with telescopes, and if you accept limitations like chemical thrusters, ion engines and maybe NERVA style nuclear thermal main engines, then ships can sense and engage (using missiles) at ranges which allow actual evasive manoeuvres - keeping in mind that you are going to start your manoeuvre when you see the enemy launch, and the missile might take 9 hours to cross a distance equal to the distance between the Earth and the Moon, and both the launching spaceship and the target can only move either at a fraction of a g using an ion drive, or a very short burst of high thrust which rapidly depletes the available reaction mass or fuel. Spacecraft are not going to move like fighter planes and break missile lock with a violent "split S" at the last second.

So you will need to adhere to some very realistic constraints in order to make this workable.

You can sure fly around in space without a computer. One can do orbital maths with a pen and paper. Kepler described the motions of orbiting planets in 1619. Mathematics, sextants and telescopes were sufficiently advanced 300 years ago to enable orbital flight and navigation to a useable accuracy. It just took materials technology a long time to catch up.

You can do all of the required computing work with some slide rules, an abacus, pen, paper, a smart mathematician and an accurate sextant and stopwatch. You may benefit and achieve greater fuel efficiency from accurate measurements of fuel tank and engine pressure, and doppler radio measurements, but these are only fine tuning.

The problems become more complex when multiple strong gravity fields are interacting, such as transferring between moons of jupiter, but this too can be iteratively solved using little more than some slide rules, a lot of paper and some quick fingers to some degree of accuracy. Modern astrodynamics use very precise measurements to be very fuel efficient, but if you want to wave some hands over that then performing multiple course corrections and adjustments and refinements as you go then sure, flying around in space can be made work without a computer. It's just accurate observations, and math.

The ability to fight effectively is a very different question. You will need extreme accuracy in your observations, and in the manouvers of projectiles. I don't think it is feasible to hit a target with some kinetic weapons without computer assisted trajectory calculation, radar systems, accurate inertial guidance, and a large degree of autonomy (since you probably won't be able to transmit instructions to a missile when its on the other side of a planet. These systems are not exactly computers per se, but generally would need a very computer-like system to integrate them. If you're ok with purpose-built controller circuits, but not general purpose instruction-executing machines then maybe this too might work. But you're really going to be walking a thin line on what defines a computer and what doesn't.

You can do surprisingly large amounts of things with old technology. The hard part will be fighting with the enemy.

As many of the other answer have pointed out, we've done astonishing things with limited computing power. The Space Shuttle AP-101s were roughly on par with the power of a Nintendo Game Boy! (Gameboy is faster but 8 bit, AP-101 is slower but could do 32 bit arithmetic). It also relied on core memory, which looks something like this:

(Yes, those are wires woven through individual ferrite cores. From what I have read, much of core memory was produced by women because it was hard to find men who could weave such wires with enough precision)

And this was a powerful computer. Gemini ran on a 7,000 instruction per second computer.

So we did accomplish a lot with old technologies. However, there's one common thread to all space exploration so far: there was no space combat.

In space combat, you're not just up against nature, you're up against another individual who is trying to kill you with the utmost ferocity. That changes the game somewhat. The Apollo missions may have time to have ground control calculate a course deviation and then punch it into a computer. If you have a salvo of weapons bearing in on you, you must respond much faster.

And herein lies the rub. Your technological civilization has one brutal handicap. This means that, unless the magical civilization has a similar handicap, no amount of "can it be done" will be sufficient. The magic users will simply out maneuver, out defend, out attack, and overall out-win your technology users. You might as well have asked if Bear Grylls can survive on a modern battlefield unarmed. Sure, his survival skills are famous, and he could probably actually find usable food and water on the battlefield, but the army of individuals with guns and tanks and aircraft is going to win.

Assuming you properly handicap the mages, this could work. The real secret to many of these devices is that we push them to the limits. We use computers to stabilize them, but that's only because we pushed it to begin with. If we relaxed our standards a bit, we'd find that things are much easier to manufacture. They won't be as good, but they'll work. And anyone knows that in combat, something working is better than something not working.

There will be limits. You wont have the same delta-V that you would with your computers. However, in many cases you can resolve this because you still have computers back home. Build the warships at the edges of your gravity wells, where escape velocity is as low as possible, and you don't need quite as much delta-V. Use your efficient craft where you can, and only send the dumb craft where you have to.

You will have to rely on dumb weapons, which will be more of a challenge. This will mean getting closer to the enemy than you are comfortable with, and wasting more reaction mass than you are comfortable with. If the mages are on their game, this will probably be a death wish, but if you've handicapped the mages properly, you may get away with it.

A big challenge will be sensing. The distances in realistic space combat are enormous. Without modern computers and ICs providing your sensing technologies, it's going to be hard to see your opponent before they whizz past you. Fortunately, the enemy will have the same limitations as you. Maybe you can borrow some of their magic amulets or what-not to sense with!

It's worth mentioning that Spaceflight is a mechanical challenge, not a computation challenge.

What do I mean by this? Well, spaceflight existed here, on Earth, pretty much before computers did:

• Sputnik 1 was launched in 1957, and the electronics on board was a radio transmitter and a power supply. If you look at the schematic for the transmitter, it had 3 active elements (vacuum tubes), so can hardly be considered a computer.
• The Zenit spy sattelites launched by the Soviets in 1960's used film cameras, and returned the film to earth via parachute. Same for the Corona spy satellites launched by the US at the same time.
• The Titan rocket used to launch the gemini in 1964 and the Saturn 1 rocket did have a "guidance computer". Was it really a computer? By modern standards a bunch of analog components with hardcoded logic and a tape player for time-based events isn't a computer at all.
• The gemini capsule itself had a computer that took 140ms to do an addition. It took 420ms to do a single multiplication. This was the first digital computer in space.
• And of course, people are fond of pointing out that men landed on the moon with a computer with about 4kb of ram. This was enough for the lander to be theoretically capable of automated landing. Also, there was a sextant present on the LEM to verify the trajectories.

Even modern space-based have tiny computer systems. The curiosity rover runs with 256 MB of RAM, and 2 GB of "disk", and a CPU that clocks at 366Mhz. The slowest smartphone has more than that!

Why is it that these systems can get away with being so low power? Well, the math involved in spaceflight is essentially ballistics. The model of gravity used to get to the moon was known as "patched conics" where you assume that the spacecraft is influenced only by one body at a time. As a result, the computation is bounded to some form of conic section (ellipse, parabola, hyperbola), which are computable by a high-school student. Actually computing transfers between bodies is harder than solving a single equation, but nothing that can't be done by a human using these approximations and some dedication. While the orbits for the Apollo missions were probably calculated and optimized by computer, I consider it extremely likely that they were checked by hand and slide-rule.

So if computation isn't the challenge of spaceflight, whey didn't it happen before computers? Well, I think in some ways it happened at about the same time because both computers and spaceflight come as a society gets better at engineering and materials science. Computers come as an understanding of semiconductors improve. Rockets powerful and accurate enough can only exist with sufficiently high understanding of metallurgy. You can't make a rocket nozzle out of cast iron. You can't get a blacksmith to build you a turbopump.

It's worth mentioning that the Saturn V rockets were constructed by hand operated machinery. CNC machines just didn't exist at the time. Can you imagine welding rocket nozzles by hand? Well, that's how we got to the moon.

If you read 1960's sci-fi, you'll find all sorts of gems. Anne McCaffery's spaceships use tape players for navigation. Arthur C Clarke wrote about an engineer fashioning an Abacus and using it after the main ship's computer failed.

So how about combat? Several other responses here suggest that combat is more complex than spaceflight itself, but I'm not sure:

• For kinetic projectiles, the ballistics for the projectile is exactly the same as it is for the spacecraft itself. Predicting motion is something that's been done as long as anti-aircraft systems have existed, and there were purely mechanical solutions for that.
• Chemical lasers work just as well without computers, and there's not much difference between a telescope and a laser: aka line up the telescope, if it's in the cross-hairs, fire the laser.
• Missiles tracking infrared sources have existed since the 1950's, and torpedo guidance systems have been done since a long time before then.

Warfare in space without computers would be different, but certainly not impossible.

Stanisław Lem's commercial space travel crews (check out his "Pilot Pirx" stories) included navigators whose basic work was calculating trajectories and locating the ship in space using star positions, not unlike to what a navigator on ocean vessels was tasked to do in commercial sea travel. The long and boring travel times in space, not unlike to ship travel of old, provided ample time for getting stuff right and correcting course timely in order not to waste all that much fuel (Lem mostly employed nuclear reactors about as trustworthy as the worst power plants of the time, mostly skipping over the ongoing question of how to convert energy into propulsion when in space).