[part 1 ... ;-)]

> Possibly, but where we are missing each other on definitions would likely be more noticeable in a syllogism.

Well, maybe, but part of the problem is that I don't really know yet what assumptions to build on that you would share. There are tons of ways to demonstrate incompatibility between religion and science, depending on which assumptions (definitions, in particular) you build on.

> Is it possible that a testable claim could imply another claim that is not testable?

Yes. Example: The earth being flat implies that invisible unicorns exist. "the earth is flat" is testable, "invisible unicorns exist" is not (arguably), and the implication is true (ex falso quodlibet).

> In other words, could you have an implicit reason for believing that something is true rather than an explicit one?

For one, testability is not a reason to believe a claim, as can be seen from the claim above that the earth is flat, which is testable, but demonstrably false. Lack of testability is a reason to not believe (because it's indistinguishable from claims that were just made up), testability is not a reason to believe (because the test might very well disprove the claim).

But the bigger problem with that question seems to be that it assumes some sort of essentialist testability. If we turn the implication around, we get a correct statement: If some claim B implies a claim A, and A is testable, then that is reason to not disbelieve B for lack of testability. But it makes no sense under any circumstances to label B as "untestable": Disproving A implies disproving B, and thus if you have some method for testing A, that method is also a test of B, and thus B is testable.

If a test could disprove your claim, then that is what it means for the claim to be testable, the amount of indirection between the claim and the test is irrelevant.

> So, if it is possible that some aspects of reality are not testable, then it is possible that a claim about such an aspect of reality is not compatible with science?

Yes. Making claims that are indistinguishable from stuff that's just made up is not compatible with science, because science is about making claims that are as reliable as possible, and claims that are indistuinguishable from stuff that's just made up is notoriously unreliable.

Or another way to put it: Making claims about the untestable implies the claim that you are able to detect the undetectable, which is self-contradicting. Either you can detect something, then you can use that to construct a test, or you can't, then you have no basis for claiming that you can.

> That is not the case. I think there is great evidence. If it is false, then it is likely that my willingness to accept things is too generous, but I'll make any effort I can to get past that if that is indeed the position I find myself in.

Well, granted, that is a likely alternative!

(Though I would say the problem is not necessarily that the evidence is false, but that it is insufficient ... or rather, that it should be. I mean, unless you really do happen to have the evidence that everyone has been waiting for ... ;-)

Also, good luck with that! :-)

> So, might you be attempting to say that non-testable claims are incompatible with doing science, rather than non-testable claims being incompatible with science itself?

I don't understand what the distinction is that you are making here.

> I am.

Well, let's see ...

http://www.atheist-experience.com/

Seems like a good starting point overall: Various hosts with different perspectives and styles, focus on talking to theists, so chances are they will address (or have addressed) what you are interested in, and obviously you could also try calling yourself if you feel like it, plus they regularly have guests who might have their own sites or youtube channels or whatever, so also a good way to discover more resources. You might also want to have a look at the tons of other programs they (the atheist community of Austin) have started to produce recently ... really, just dig around a bit and see whether anything fits your taste.

https://www.youtube.com/user/SansDeity

One of the hosts of the above, addressing specific topics more in depth.

https://www.youtube.com/user/AntiCitizenX
https://casualentropy.blogspot.com/

Not necessarily very diplomatic, but they do dive into the more philosophical questions, so might be worth a read/listen.

https://www.youtube.com/user/magnabosco210

One guy (not the only one by far, you might also want to look for others) practicing street epistemology, with many talks addressing questions of religious beliefs.

https://www.recoveringfromreligion.org/resources

Not sure whether it's too relevant for you, but then, they have a collection of quite a wide range of different resources, so who knows, digging around there might turn up something interesting ...

If you want to read a book or two, the classics (almost) from Dawkins, Hitchens, Harris, and Dennett probably aren't the worst idea, Dennett in particular if you want to get more into the philosophy side of things. Other than that, I guess I would rather start from the above and look out for recommendations from there.

> So, is there a chain of scientific methods that establish further scientific methods with no beginning, or is there some first self-evident method, that can't be tested for reliability without being circular?

Neither, really, though more the latter than the former. You could say that the basis that everything else builds on is the assumption that our senses are somewhat reliable, i.e., that if an overwhelming majority of people agree about some observation, the observation is probably a reasonably accurate description of something that actually happened. And mind you that not much reliability is necessary in order to bootstrap this, because you can largely test that, too.

But when you think about it, you'll notice that that assumption isn't actually necessary if you are careful with formulating your goal. Namely, if your goal is to have as accurate a model of reality as possible, then that still holds if you have absolutely no way to make any reliable determination about reality: If you have no way to make any reliable determination about reality, then, by definition, no method will give you a better result than pure randomness, and thus every method is optimal (and pessimal at the same time). And as soon as there is even a little bit of structure to reality that you could figure out, building on your senses is going to outperform ignoring your senses.

> So are you saying "you can't make a claim about John be about Jack, because John is not Jack", and to parallel "you can't make a claim about math be about reality because math is not reality"?

More precisely, because a statement about math is not a statement about reality, other than that math is part of (human) reality.

> What if John was Jack's middle finger, how could a claim about John not be about Jack?

That's just a pointless question. And that is not because it is categorically wrong, but because it brings no insight. There is not a single statement that you couldn't object to with "but what if some words had completely different meanings than what the speaker presumably intended, based on common usage?" "'The earth is round' is true!"--"But what if by 'true' they meant false, or by 'earth' they meant a banana, or by 'round' they meant flat?" The answer is simply: They didn't.

> Given the Peano axioms, and their evaluations and derivations by humans, does '1 < 2' have a referent?

Yes, those axioms, evaluations and derivations (well, particular aspects of them). Just the same way that 'Game of Thrones' has a referent, given the series, and characters of that series have a referent, given the series. None of that implies some essence that exists absent those human artefacts.

> You objected to my claim that "everything has an explanation" thinking I had not substantiated it. I get that you're not making the opposing claim, but in order to substantiate my claim I showed that the opposing claim can't be true.

No, you haven't. At best, you have shown that holding that position is not justified (arguably, you haven't done that either, but it doesn't really matter for this). But lack of justification for one position does not imply that the position is false, nor that there is justification for the opposing position. It simply means that there is no good reason to believe that one position. If neither a given position nor the opposing position is justified, then we simply don't know which of the positions is correct.

> Even if your not making the opposing claim, your objection has no case because the opposing claim can't be true. If "not everything has an explanation" can't be true then "everything has an explanation" must be true. So you can choose to reject my claim, but that is a choice to reject a substantiated claim. If you don't think I have substantiated it, let me know why.

Well, see above. You have to show that it is impossible for something to be unexplainable, not just that we don't know whether something is unexplainable.