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spacecadet2563 said:

How? Why? He sees the Earth's clock moving slowly relative to his. The reverse would be true. Unless something happens when he turns around.

Are you sure you are in the right frame of mind to learn relativity?

spacecadet2563 said:

If someone stationary on Earth watches a vehicle leaving Earth very fast he sees that time is dilating for that vehicle. Any clocks he sees slow down. But since motion is relative the traveler sees the Earth race away from him at a high speed and the same thing happens. He sees the Earth clock slow down. This is a problem. Who is right? Saying both are is not good enough.

You have overlooked the relatively of simultaneity.

Say that observer A is located on earth, B on the ship, and both agree that the ship leaves earth at 12:00 noon. One hour later A looks at their clock and sees that it reads 1:00 PM, as we expect. A uses the Lorentz transformation to calculate what B's clock reads AT THE SAME TIME that their clock reads 1:00, or equivalently they are watching B's clock through a telescope and by allowing for light travel time they can see what B's clock read when light left B AT THE SAME TIME that A's clock reads 1:00 PM. Either way, they find that B's clock reads 12:30 AT THE SAME TIME that A's clocks reads 1:00. In other words, the events "A's clock reads 1:00" and "B's clock reads 12:30" are simultaneous when we use the frame in which A is at rest.
Clearly B's clock is running slow, by a factor of two.

But this is where the relativity of simultaneity comes in. Using the frame in which B is at rest, these two events do not happen at the same time. Instead, the event "A's clock reads 12:15" is simultaneous with the event "B's clock reads 12:30" so just as clearly A's clock is running slow by a factor of two.

And that’s how they’re both right.

As an aside, this setup is different from the twin paradox because the two clocks are not at the same place when we read them. Confusingly, people often use the term “time dilation” for both this case and the twin paradox even though they are completely different things.

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I think it helps to suspend your disbelief for as long as it takes you to learn how it works. Putting your mental energy into arguments showing why it can't be true is a waste of your time. Trust me, it will all make sense in the end.

PS, quantum mechanics is even worse.

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spacecadet2563 said:

If someone stationary on Earth watches a vehicle leaving Earth very fast he sees that time is dilating for that vehicle. Any clocks he sees slow down. But since motion is relative the traveler sees the Earth race away from him at a high speed and the same thing happens. He sees the Earth clock slow down.

Seen in terms of geometry, it is closely analogous to two cars driving at equal speeds ##v## down straight roads that make an angle ##\theta## to one another. Both cars will observe that the the other car is progressing at ##v\cos\theta##, so both will see the other falling further behind. This is not a paradox, it's just that they have different ideas about what "forward" means.

Similarly, the two relativistic observers have different ideas about what "not moving in space, only advancing in time" means, and both say that the other is moving at an "angle" (called the rapidity) to them and hence their clocks are ticking slow.

Start from scratch. It's about how people in an inertial reference frame synchronize their clocks at different locations:
1) Something about the clocks is surprising
Suppose a person is traveling from left to right at high speed. You shine a flash to the right. Experiments show that both of you measure the speed of the light flash at c relative to them, even though the traveling man is going in the direction of the light flash. Something about clock settings is weird.

2) It takes two of your clocks to measure how fast the moving man's clock is ticking. Suppose when he passes you, both of your clocks indicate 12:00. Call those clocks "start clocks" When he passes your clock to the right, after one of your hour, his clock indicates 12:55 instead of your far clock's 1:00 (call that clock your "stop clock"). How you synchronized your two, widely separated, clocks is critical. We will assume that you did it so carefully that all standard physics works out and the speed of light between your two clocks is exactly c.

3) The traveling man did exactly the same thing. He measures your clock rate using two of his clocks. Now, here is the important part: His "stop clock" is in the opposite direction from your "stop clock". When the two "start clocks" were together at 12:00, your "stop clock" was to your right and his "stop clock" was to your left. You are looking at your two clocks from left to right and he is looking at his two clocks from your right to left. All your clocks to the right are ahead of his (his appear to run slower), and all your clocks to the left are behind his (he thinks yours run slower).

4) Conclusion: Your clocks to the right of you are set farther and farther ahead of his. Your clocks to the left of you are set farther and farther behind his. When he travels to the right, you use your "stop clocks" to the right to track his one, moving "start clock" rate. His "start clock" looks slow to you. When he tracks your "start clock" rate, he is using his "stop clocks" to the left. Your "start clock" looks slow to him.