[part 1, of course!]

> Could you please state your argument, regarding things being testable as they relate to science, in the form of a syllogism that ends with the conclusion [...]

I might try to, but I suspect the problem is more the nailing down of definitions than the logical structure of the argument, and I suspect in particular that there might be some misunderstanding about ...

> A preemptive question to the syllogism that you may give (to possibly help its formulation): Are all aspects of reality testable by science?

... what "testability" means in the context of the philosophy of science, as this question doesn't really make all that much sense in that context.

First of all, there is no such thing as "testable by science". Science is not a method of testing things. At most, you could maybe say that science is the collection of all of the reliable methods of testing things known so far.

I'll go on somewhat of a tangent, but I think it might be helpful: You do maybe know the saying "There is no such thing as alternative medicine, just medicine that works, and medicine that doesn't"? What is meant by that is that medicine is not a fixed set of methods, but rather the collection of all of the methods that have been demonstrated to be effective so far. But the important bit is the "so far": It's very much possible that some other methods are effective as well, even some of those that are suggested as "alternative medicine". But the point is that those methods are not excluded from medicine on the grounds of some dogma or anything like that, but rather simply because they have not been demonstrated to be effective--and as soon as their efficacy is demonstrated, they will simply become part of medicine. So, "alternative medicine" only consists of methods of which the efficacy is unknown, or which are known to be ineffective, because anything that previously was in that category, and for which efficacy now has been demonstrated, becomes "normal" medicine.

Science is the same way. So, in particular, science does not have any dogma against religious ideas or methods just because they are religious ideas or methods. There is no doctrine that says that science is somehow not allowed to come to the conclusion that god exists, say. Nor does science have any doctrine against conclusions about the supernatural or the non-physical ... or anything, really. The demand is just that you can show that whatever methods you use to come to a conclusion are (reasonably) reliable.

The point is: When science demands testability, it doesn't mean "testable using the methods prescribed by science", it just means testable in any way at all--if you need to use a new method for the test, that is perfectly fine, if you can demonstrate that it works. Really, it's not just that any effective method of testing is fine, but rather, any new method demonstrated to be effective would simply start being considered to be part of science. The defining goal of science is to produce reliable predictions, and any method that helps with that is acceptable. If you could demonstrate that prayer can be used to make reliable predictions (including to reliably influence outcomes, in which case the prediction would be that the prayer produces that particular outcome), then prayer would be accepted as a scientific method.

Also, I think it is important to note what specificially testability means in the context of the philosophy of science: Testability is the property of a claim that you know experiments that could be done, and if those experiments gave a result different than predicted by your claim, you would conclude that your claim is false. The point of demanding testability is not to establish that something is true (apparently something that many people think), nor to establish that it is false. The point of testability is to weed out those claims that are indistinguishable from stuff that someone just made up. That is not because what someone just made up is necessarily false, but because almost all claims that someone could potentially make up are false, and thus, if you have no way to distinguish a given claim from one that was just made up, the probability that it is false is extremely high. Importantly, demanding testability is not a dogmatic demand: There is a reason for it, namely that without it you are likely to accept lots of false claims.

So, are all aspects of reality testable by science? Well, for one, any aspects that are testable, by whichever method, are, by definition, testable by science, for any demonstrably reliable test you were to use would become part of science. Which leaves the question: Are all aspects of reality testable? I don't know, possibly not. But the time to believe any claims about those aspects that are not testable is when they become testable, and not before, for the reason I explained above: They are indistinguishable from stuff someone just made up.

> I am making my best attempt to understand your position (you have no idea how nice it would be for me to ditch my Catholic faith if it could be shown that it wasn't true),

Haha, that statement actually indeed makes it very believable that you do, for a reason that you most likely didn't even notice. Let me explain:

What you are doing here is shifting the burden of proof, again--except: You are shifting it to yourself, where it doesn't belong. You are believing something, it seems, because it hasn't been disproven to you. As a general rule, that's a bad idea, and with religions in particular, it will essentially guarantee that you won't find the way out. Mind you that I am not saying that you necessarily should find the way out, because your particular religion could in principle be true, after all. But what should give you pause is that this is true of all religions--and they can impossibly all be true, because they make contradicting claims. All religions have at their core untestable claims, that is, claims which are impossible to contradict with any test. The claims themselves are constructed in such a way that it is impossible to disprove them, even if they are in fact false. So, no matter whether they are true or false, you won't ever find a way to disprove them.

So, if you are trying to disprove the claims of your religion, you might as well give up, you won't, ever, at least for the central claims, and no matter what your religion is. What I would suggest instead is shifting the burden of proof to those making the claim, and to disbelieve claims until they have been demonstrated to be most likely true, not believe claims until they have been demonstrated to be most likely false--which most likely is something that you do practice for most (all?) other claims anyway (like, if someone is trying to sell you something, you won't just take their word for it that theirs is the best product until you have disproved that claim, but rather you will first investigate alternatives before you come to a conclusion about which product is likely to be the best). There should be reasons to believe, not no reasons to not believe.

Now, given the effects of religious indoctrination, especially when coupled with (also untestable) threats for disbelief, that's potentially easier said than done, and probably not something that you'll just switch frome one day to the next, but given what you wrote, I guess it's a perspective worth thinking about. If you are interested, I guess I could also dump a bunch of resources on you that might be helpful in further exploring your reasons for your religious beliefs, just let me know and I'll dig up some URLs ...

> Testing by a method other than science? How do you know that method is reliable?

No, by testing with an established scientific methodology. You simply use methods that you do already know are reliable to test other methods whether they are reliable. (And mind you that "reliable" here does not mean "give results with 100% certainty".)

> Could you please provide a syllogism that ends with the conclusion "Therefore formal claims are not about reality" to establish that mathematical/grammatical claims are not about reality?

Could you provide a syllogism that ends with the conclusion "Therefore claims about John are not about Jack"?

> That's what I'm asking, I'm not claiming one way or the other. If you know what it points to please let me know, and how you know. If you don't know, then you can't rule out the non-physical without knowing whether it points to something physical or not.

I am not ruling out the non-physical, if only because the term is ill-defined. You made claims about the existence of the non-physical, and you have yet to demonstrate that that's a thing, without constantly shifting the burden of proof to me. "Mathematics" clearly is used to refer to stuff that has at least physical representation, so I'll agree that mathematics has at least those physical aspects (like, books, brains, lectures, what have you). If you think there is more to it, then that's for you to demonstrate, not for me to disprove.

> But now you've stated that it depends upon the intent of the majority.

No, I have done that before, because that is how language is used. There is no need to state so explicitly. "Physicists believe earth is an oblate spheroid" does not mean "there are no physicists who don't believe earth is an oblate spheroid". That sentence simply is not the English translation of "forall <physicists>: <physicist> believes that earth is an oblate spheroid". It is common usage in English to use generalized statements with the implied understanding that they do not apply to all individuals, so you are simply deviating from standard English if that is how you choose to interpret the sentence.