How do we know the speed of light is constant and spacetime dilates rather than vice versa? Some conspiracy nut was telling me that Einstein was BS and there's a giant conspiracy that he's wrong but scientists would loose all their jobs if they admitted it. Of course this is all baloney, satellite clocks need to be adjusted for time dilation and Mercury's orbit is weird. But that got me thinking about how we know light is constant and space/time changes instead of the other way around? Can we measure space stretching without using light? If light goes faster and we use it to measure a space, we would just say the space contracted. Can we measure time without using the speed of light? Does the math work out if you set space and time constant but have changing light speed?
 A: You might be confusing some issues.
In special relativity, space and time do not stretch or compress. It really comes down to measurements with clocks and rulers made by people that are moving uniformly with respect to each other.
One option that is consistent with observations for SR is that there is one family whose clocks and rulers are right and everyone else is wrong (the rulers the moving people are using are too short and their clocks run too slowly). This is possible, but if that were true then the laws of physics are such as to have a vast conspiracy because there are no measurements anyone could do to say who is the special observer that is right and therefore who is wrong. To each person their stuff looks right and they think they could be the right one.
What is the point of saying one person is right when they all look the same? So SR postulates that all the clocks and rulers are equally accurate and they are all measuring actual geometrical facts about our spacetime and that the spacetime has a Minkowski geometry instead of some other geometry. That is what is consistent with our measurements and still allows all these people to be equally good at their jobs. It's is based on the idea that no inertially moving person should be treated differently. The advantage is everyone can look the same instead of one person being right but everyone else's errors cancelling out so they can't tell they are wrong.
Now in GR there is a different geometry, which is only approximately like Minkowski in a small region of space and time. So each event has a small region where things are approximately a Minkowski geometry.
In GR, spacetime is curved. If you use some coordinates, the rate your position coordinates change compared to your time coordinate changes could vary. But when people say the speed of light is constant what they meant was those two inertially moving people pull out their clocks and rulers and measure a speed and they get the same result.
We should say invariant rather than constant, since it is talking about how it appears to different people. But the word constant is already out there so we will have to hear questions about that word. And GR has that too. Since the laws of physics actually predict that light moves at the numerical value of $c$ if differently moving inertial observers measured it differently then different inertial observers would have to have different laws of physics.
That's annoying to have to write different laws for people that are no different except that they are moving inertially with different starting velocities. It's not even what we want. One person could get the laws we are used to and then what, everyone else has to get different ones? And after all, we'd just ask how you decide to assign laws and someone would try to call that the law and then it will end up not depending on the observer any more or else no one will know how to assign that big law and we'll be back to that alternative to SR where one person is right and everyone else is wrong but no one knows who.
Why bother? We can write laws that treat everyone the same and these agree with observation. They might seem weird to people that are only used to people moving slowly compared to each other and who stay away from strong curvature. But the alternative is so painfully biased or complicated almost nobody even bothers to consider those options.
A: 
How do we know the speed of light is constant and spacetime dilates rather than vice versa?

We know that the speed of light is not constant. I'm afraid it's a popscience myth that the speed of light is constant. See Irwin Shapiro talking about it here:


Some conspiracy nut was telling me that Einstein was BS and there's a giant conspiracy that he's wrong but scientists would loose all their jobs if they admitted it. 

This isn't true. But nor is what you've been told about the speed of light. Einstein said the speed of light was not constant, repeatedly: 

He didn't abandon this idea in 1911 like this article used to say. Search the Einstein digital papers for more examples.  

Of course this is all baloney, satellite clocks need to be adjusted for time dilation and Mercury's orbit is weird. 

The GPS clocks employ microwaves, which are light waves in the wider sense. Such clocks go slower when they're lower because light goes slower when it's lower, just like Einstein said.  

But that got me thinking about how we know light is constant and space/time changes instead of the other way around? 

The locally-measured speed of light is constant because we use the local speed of light to define the second and the metre, which we then use to measure the local speed of light. The constancy is a tautology. See http://arxiv.org/abs/0705.4507. 

Can we measure space stretching without using light? 

Gravity features a radial length contraction, which is the opposite of a stretch. 

If light goes faster and we use it to measure a space, we would just say the space contracted. 

Space is radially contracted closer to the planet, and the "coordinate" speed of light is lower closer to the planet. See Baez for more on this. 

Can we measure time without using the speed of light? 

No. The second is just the duration of 9,192,631,770 microwaves passing you by. If the light goes slower, the second is bigger. 

Does the math work out if you set space and time constant but have changing light speed?

There is a way to do it, but not the way you suggest. The changing light speed means the time isn't constant. However the metre is. The metre is the length of the path travelled by light in vacuum during a time interval of 1/299,792,458th of a second. When the light goes slower your second is bigger, but the slower light and the bigger second cancel each other out, leaving the metre unchanged.   
