There is a simple and cheap way you can prove (or disprove) that the earth is flat or spherical, using classical physics. All you need is a gyroscope mounted in a set of gymbals and some patience.
A gyro spinning at constant angular velocity will tend to maintain a constant orientation of its axis with regard to a distant point in space, rather than the earth, due to conservation of momentum. This simply means that the gyro doesn't actually rotate about a secondary axis, as long as no external torque is acting on it.
The experiment is simple. First align the gyro axis to point in the east-west direction. If the earth is actually spinning, then the gyro will appear to rotate through one full revolution in 24 hours. However, the way that it rotates will tell you whether the earth is flat or spherical.
Flat earth proponents say that the earth is essentially a flat disc. If this is the case, the gyro will slowly rotate in a horizontal plane. That is, 6 hours after the experiment begins, the gyro axis will point north & south.
However, if the earth is more or less spherical, the gyro will slowly rotate through a vertical plane throughout the day. That means that 6 hours into the experiment, the gyro axis will point up & down.
I haven't done this experiment, but it could be done in less than a day and settle the argument once and for all. Rather than argue about it, why not go and see what kinds of results you get? I'll be waiting.
We had gyroscopes to play with when I was a kid. They had a diameter of about 3 inches. You'd wind a string around the shaft that supported the disk that spins; pull the string as fast as you can and the gyroscope would spin. These came with a little stand that you could set the spinning toy on and watch it spin for a couple of minutes before inertia caused the disk to gradually slow down and eventually stop after the gyroscope fell off the stand.
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Supply me with a self-powered gyroscope that will spin at a constant speed for a 24-hour period and I'll do the experiment for you.
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Regardless of the results obtained, that will leave me with a problem. I won't know "why" gyroscopes behave as they do, and neither do the physicists, philosophers, inventors, scientists, and elementary school teachers who teach us "why" they do. It's a phenomena that does what it does and all of us can only come up with theories to explain why.
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I can't disprove your experiment. You have a list of variables and a line of reasoning that apparently accounts for the behavior of the device. The experiment is repeatable. What alternate theories account for those behaviors?
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Take a different phenomenon, time travel. Prove to me that time travel does, or does not exist. My contention is that it's impossible for me to "know" whether time travel exists or not because I'm not aware of ever having witnessed it. Science and philosophy have "proven" that time travel is impossible; and there are scientists and inventors working to make it a reality.
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I can't prove that I am not a hologram having a form of consciousness attached to it and living in a holographic universe. Whether I am a hologram or not, I'm also like a fish in a fish bowl; I have no way at this time to get out of the fish bowl to prove to myself there is a different, and larger reality that surrounds me and my fish bowl.
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Give any 5-year old child with "normal" intellectual functioning a gyroscope to play with. Eventually they will come up with a reason that explains why the thing spins and moves the way it does. That explanation will most likely not involved a spinning globe and the law of inertia. That child will believe their explanation - until you do something unexpected with the gyroscope that contradicts their line of reasoning.
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Stage magicians do this type of thing all the time. They have a good understanding of how human perception seems to work and they know how to distract their audiences while they manipulate objects in ways that defy the internalized rules we've created to explain how the physical universe behaves.
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I can tie a string around the stem of a goblet and set it on the edge of a table. I can predict that when I pull the string the goblet will fall toward the floor and the string will arrest that movement before it actually hits the floor. I can explain that gravity is the "force" that causes the glass to move downward, but I can't show you gravity. I can only show you a bunch of objects that behave the same way. When I've got you convinced that all objects always fall to the lowest solid surface beneath them, I can show you a plane and you will never believe it will fly through the air until I start the engines and release the brakes, and the wheels leave the runway.
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Believing in things we cannot verify with our physical senses is called faith. This is where physics leaves off and metaphysics begins. It is faith, believing in things that we cannot see or touch, that is the foundation upon which we organize religions and create dogmas. These are the theories that govern the worlds of metaphysics. When people live their lives in accord with these dogmas, we wind up with things like the Spanish inquisition and Jesuit priests persecuting people for publicly contradicting the Bishop of Rome who declared the world to be flat - until the the Jesuit priest Copernicus formulated the concept of the earth being a spherical object rotating around a central sun in heliocentric solar system. After a period of time this theory became the preferred theory and a different Pope codified a new doctrine that declared the earth is, indeed, round. And since it was round, it permitted more Jesuit priests to go out and spread the "gospel" and cast the net of spiritual and political control over indigenous populations in those new worlds that hadn't existed prior to the voyages of Columbus and Balboa and other explorers.
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It was through propaganda that Hitler's minions convinced the German people to accept as an article of faith that Jews and Gypsies were an inferior form of life and not worth worrying about the fate they suffered. It is through propaganda and faith that fundamentalist Muslims and Christians are being convinced that war between the two sides is inevitable. Personally, I find the "keep an open mind" attitude of "flat earthers" to be more humane and preferable to that alternative.