What came first, the atheist or the theist?

[Tiberius]

Science uses a combination of logical arguementation based on empirical evidence, experimentation, and peer review to make claims. An unproven hypothosis could be formulated (like M-theory) based on evidence and startlingly breathtaking mathmatics to make a case, so a hypothosis might be considered to be 'non-empirical evidence' - but we don't call it that. The term is self-defeating because it is technically an oxy-moron. That's why it's called a hypothosis. It's a logical arguement not based on any direct evidence.
Other logical arguements are called just that - somtimes with their own terminology, but you don't go around telling people you have 'evidence without proof' because most people will tell you that evidence without proof isn't.

So what you really meant to say is that "in science, non-empirical proof is an oxymoron". I'd agree with you. Science depends on empiricism.

That isn't true of any other branches of thinking; philosophy, mathematics, etc. The three laws of logic are absolute truths, yet cannot be proved empirically. Instead, they are proved through logical reasoning which falls apart if any one of them is untrue.

Technically speaking, that is empirical evidence.
The ability to think can and has been tested in humans and many other creatures. It's how we've found out that some creatures who can't actually speak the worlds "I thnk, therefore I am" are thinking creatures, like Chimpanzees, dolphins, and octopi (if that's the plural form of octopus.)

Evidently you haven't done any research on the actual proof "I think, therefore I am". It proves that the thinker exists; it does not prove anyone (or anything) else does. It is non-empirical since it cannot be tested by external observers, and is inherently subjective.

The proof stems from the fact that the action "to think" is an ability associated with existing things. That is, there cannot be non-existent things that can think. Therefore, by simply thinking about whether one exists, you have proved that you do.

This proof cannot address anyone else though. If you were to come up to me and say "I think, therefore I am", I could do any number of tests to verify that you could indeed think, but I would have to put a level of trust on my readings; I would have to assume at some point that they are correct. This is the assumption that science rests on; that the natural world we live in exists, and that we can learn things about it by observing it. It is, nonetheless, an assumption.

At the end of the day, nothing science does proves anything. Science cannot prove anything; nor does it attempt to do so. Science tells us what the most likely truths are, based on observations and reasoning, and the underlying assumption that the observations and reasoning we use is real.