How do you make a machine that has perfectly flat/plumb/level metal components without the aide of a machine to begin with?
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This is an awesome question that I hope every person would explore a bit.
You can start with certain natural forces (gravity is perfectly perpendicular to the horizon level of water in a vessel). This gives you straightness and square.
You can also start with geometric principles, like how a string between two points must be straight, since it’s the shortest line. (Some philosophers contend this was the starting point of mathematical and scientific thinking–points and lines and how they intersect into angles.) Or you know that square should be reversible and end on the same line after 4 sides are produced. Or that a square triangle will have units of 3, 4 and 5–known to very ancient people, even before Pythagoras’ theorem. So people could work with certain basic principles like flatness, squareness, and parallel very early on.
Workers still use lines, plumb bobs, and spirit levels, btw. The masons who built European cathedrals used a hanging plumb bob against a fixed wooden square to build 125 and 150 foot spires. (Only the 150 footer fell)
Before you go beyond that, I have to say that the level of precision possible just by eye is fairly impressive. You can see a line diverting from a string and know it’s not parallel better than you might guess, with practice.
Gauges transfer known flatness, squareness, or length. Surface plates are gauges, and gauges bring awesome accuracy with zero math. As others have mentioned, rub 3 plates, get a flat surface, and start looking for light or charcoal dust or blue ink against your straightedges. You can get very flat this way and transfer flatness and straightness and squareness to any machine you build. You can build to parallel thickness by feel to within a couple thousandths. You can make every other part fit together by feel, too.
After gauges, there will be some element of trying to settle on a named gauge that counts as the official standard, like the inch or meter, but this only matters for interchangeable parts. You can build single machines that are totally serviceable on their own: lathes, surface grinders, planers, etc. All based on a straight line, squareness, and parallelism.
A universal standard is hugely important because you can describe parts accurately. But you can get precise fitment without ever knowing how long or thick something is exactly on its own.
Geometry and math give little tricks we have used effectively, too. I have a screw with 20 threads to the inch, so how much does one turn of the screw move the screw lengthwise? 1/20 or .05 inches. Divide the clockface into regular geometric marks (like just 4 marks at 15, 30, 45 and 60 minutes) and you can get down to (1/4 x 1/20) = .0125 in. Or use a vernier scale, which is a scale of 9 set against a scale of 10 (dividing calipers help you do this easily without math), and you will get down to 1/10 resolution, so (1/10 x 1/20) = .005 inches pretty easily actually. If you can set out 60 minutes on the clock dial, that gets you down to .0008, an incredible level of accuracy. Building the thing without slop is the challenge now, but you don’t need any math or special tools to build a single tool to itself without slop.
You can build your own vernier calipers with cardboard and get some of these measurements yourself. Don’t even start with an inch–make your own unit up, and imagine how you could build one machine according to that unit alone….but you and your friends need to decide on units in order to swap parts effectively is all.
The technical achievements are incredible, but the mathematical insights inside common tools and techniques really makes you imagine why people thought of pythagoras as a mystic, and why numbers held special insight into the universe. The geometry that unlocks all this stuff is quite elegant. And it is beautiful, too: check out the rosette art made with dividing calipers.
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