By rejecting the principle of stationarity, it's not carbon dating that you put in doubt but almost everything science has ever said or done. We suppose that when we find a rule, it has always been the same because that's the simplest hypothesis that is coherent with our observations. All findings done via carbon dating are coherent with previous scientific knowledge. The "previous" part is important because this implies that no human tried to skew either theories in order to make it work, it just so happens that everything fits in. That's not a very frequent thing with scientific hypotheses and a very strong indication that something is right with what is being done. For example, Kepler's law linking the mass of planets and their orbit (I don't remember the exact terms but that's not relevant) worked for all known planets at the time. That's not really a feat because anybody can come up with a mathematical expression complicated enough to match the data. The impressive thing about it is that it was extremely simple (an equation of the type x^2 = y^3) and, more importantly, that we later found out that it correctly matched the data of a planet Kepler had no idea existed. The same thing occurs with theories that are validated through carbon dating.
This principle is not based on belief, it is based on pragmatism and also the most basic aspect of scientific work : find the simplest model which fits known data. Up until now, we never needed anything evolving with the age of the universe, so up until now we assumed there was none. By breaking this principle (I'm sure you have heard of the name Occam's Razor) you can basically add any spurious and trivial hypothesis you want and call it science.
Btw, this also means that in the same line of thought I can make the hypothesis of the existence some phenomemon that would have happened at some point in 1044 and which added a negation in an important sentence in all representations of the Bible, including people's memories. You can't prove that it never happened, just like I can't prove there is nothing that changes with the age of the universe, but it is also equally useless to explain the universe as we know it and therefore not accepted by the scientific community.
Suppose, suppose, suppose! Yes that's right! We suppose they work because we haven't been contradicted. If, at anypoint, we find that stationairity changed, what happens then? In terms of Science, nothing, actually. Most science isn't built on past reference data(other than theories of change and evolution for things like the universe or soil mechanics, etc).
Plate tectonics happens. Proven.
Did it happen faster, same, or slower: Science says same, but we can't prove.
Same with other major theories. The basic Application is PROVEN, but anything that relates to length of time outside of 10,000+ years is speculation on that stationary principle.
It doesn't change science. Algebra, Calculus, Newtonian Physics are all unchanged by this principle(Unsure of modern Physics applications as they are outside of my interest/learning). That's the thing about Math, it is a logical, and precise tool. Carbon dating, though is an empirical application of the current mathematical application of Radioactive decay.
All I said was that we take it as a fact that stuff happened "linearly," for lack of better word, through time.
And that's why science is nice.
Theory(based on Observation) > Data > Confirmation or Change > New Theory > Law(End)
But what we have on some things is this:
Theory(based on Observation) > No previous data > Confirmation > New Theory > Law(End)
It's similar to Quantom Mechanics. We have knowledge its there(mathematically, it was shown to be true, if I remember correctly), but we are still at the DATA stage.
Carbon Dating: Is it False? NO, but maybe, we can't say for sure, but it is not logical or mathematical to say it hasn't happened.
We climb to high, oft times, and don't pay careful attention to our floors.
I actually had a serious calculus book(grad level) that mentioned all of this. It wasn't some crazy right wing math nut, but a normal non partisan book.
Always be careful that while it's mathematically possible, doesn't mean it can happen IRL.Here's a Fun question: If we lose the gravity from Sol, how long until we lose the light too?
(That's a classic Newtonian Answer)
@Armpit: Good job selectively quoting. I'm talking about a key theory of science, ignoring religion entirely.