Note: We seem to be getting side-tracked with tons of irrelevant details. Instead of answering all your questions, I will try try reformulating my position on some things in the hopes of maybe clearing up some of the confusion.
> Do you think they mean the same thing as your definition? If so why, otherwise where did you get your definition of 'incompatible with science' from?
What does any of this matter? It is really simple: I, just as scientists generally, use the word "science" to mean "the collection of methods that are demonstrably reliable for determining facts about reality". Religion uses methods that are not demonstrably reliable for determining (supposed) facts about reality. So, religion will make (real) truth claims and will claim that those are reliable, where science says that those claims are unreliable, which means that the statements are in conflict.
That is the fact of the matter. If you want to argue about whether there are some people who hold positions as scientists who happen to use a different definition, or whether this is true, ultimate incompatibility when it is possible for both to exist in the same universe ... really, I am just not interested, because none of that changes anything about this pretty easy to understand fact.
What I am interested in are substantial arguments that show that in fact the methods used by religions to make (real) truth claims are comparable in reliability to scientific methods.
> How is science demonstrated to be reasonably reliable for determining the truth of a claim?
By testing whether the predictions made by whatever method is suggested as part of scientific methodology do come true with a probability higher than chance, preferably much higher than chance.
> How do you know that formal claims don't say anything about reality?
How do you know that claims about football don't say anything about the car industry?
First and foremost, that is a nonsensical question. When someone makes a claim that they intend to say something about football, then that is what their claim is about, by the definition of what it means to make a claim. You are making it way more complicated than necessary when you try to then argue about whether the statement could somehow also be interpreted as being about the car industry. Maybe it can, but that doesn't change anything about the fact that whoever made the claim was not trying to convey anything about the car industry. The speaker didn't mean what they didn't mean, no matter how much you argue about the labels used to describe the situation and whether the statement could be reinterpreted to mean something else than what was meant by the speaker.
The fact that the same sequence of words can be used to describe distinct facts about reality does not mean that when you show that one of those facts is true, all other facts that could be described using the same words are also true, so it is a completely futile exercise to keep arguing about whether some word or some statement could also mean something else.
> Then, how do you know that mathematical claims are not about reality when it may simply be the case that the rules that derive them have not been demonstrated to be reasonably reliable for determining the real truth of such a claim?
Because there is nothing in the claim that even refers to reality, just as there is nothing in the claim about football that refers to the car industry. This has absolutely nothing to do with whether some method is reliable--there is simply no method at all, it's all just a category error.
You might as well be asking how I know that claims about English grammar are not about reality when it may simply be the case that the rules to derive English sentences have not been demonstrated to be reasonably reliable for determining the real truth of such a claim.
When someone says "'The tree jumps over the moon.' is a grammatically correct English sentence", it is nothing more than a category error to ask what that statement says about reality. It simply doesn't. It is, very obviously, talking about the formal system also known as the English language, and whether the sentence 'The tree jumps over the moon.' can be derived from the "axioms" of the English language.
Also, it is completely irrelevant whether someone somewhere maybe does make statements that are intended to say something about both English grammar and medicine. The fact that that might somehow be possible still does not change that in general, when someone talks about English grammar, they are not talking about medicine. And most importantly for this conversation: When I am telling you that I am explicitly not talking about medicine, but only about English grammar, it is just completely besides the point to keep going on about how someone maybe could also mean to say something about medicine.
> What is distance?
https://en.wikipedia.org/wiki/Metric_(mathematics)#Definitio...
> What are spaces?
https://en.wikipedia.org/wiki/Metric_space#Definition
> What is cardinality?
https://en.wikipedia.org/wiki/Cardinality
> What are sets?
https://en.wikipedia.org/wiki/Axiomatic_set_theory#Axiomatic...
The point is: A bunch of axioms and what results from them.
> Not the word but mathematics itself. Just like you can distinguish a rock from the concept of a rock.
So, what would you point to for mathematics if you wanted to point to mathematics while avoiding pointing to the concept of mathematics? I don't see what could possibly be meant by that, so if you want to make the claim that there is something there, it's up to you to demonstrate that there is.
> Then why state "... (by mathematicians)" rather than "(by some mathematicians)"
For the same reason that you don't say "The earth is considered to be an oblate spheroid (by most physicists)", and the fact that you are asking this makes me wonder how much you are actually trying to understand, rather than just trying to avoid justifying your own position. It is just completely normal and not the slightest bit of a problem that you generalize when talking about groups of people, when it is just completely irrelevant whether some small minority of that group deviates from what otherwise defines that group.
> We have singled out formal truths from real truths to highlight your position that distinguishes them. I have yet to agree that formal truths are not real truths because I don't think that you have established that as fact.
There is no fact to establish here. I am simply telling you about the distinction that I am making (as are many other people). And it doesn't even matter whether formal truths are also real truths. What matter is that you can distinguish formal truths from not-formal truths.
> So, it is not obvious to you that '1 < 2' does or does not have a referent, because the Peano axioms that '1 < 2' is derived from may or may not be a purely artificial construct of humans (or possibly more generalized evaluators)?
I guess so? I mean, I would say it's pretty obvious that the way we know them they are an artificial construct of humans (like, the axioms were formulated by humans). If you think there is more to it than that, that would be for you to demonstrate.
> Not having an explanation now doesn't mean that there can't be one.
You are shifting the burden of proof, again. It is really getting annoying. You implicitly claimed that there would be explanations for all things we have no explanations for yet. I rejected that claim. When I reject your claim that there will be explanations, I am not making any claim, and in particular I am not making the claim that there will be things that will be unexplained forever, so stop turning my asking for your justification around into "but some other claim I have made up is unjustified as well!".
If you make the claim that there will be explanations for everything we can observe, then it is up to you to substantiate that claim. You are not justified in making that claim just because I have not demonstrated that something will stay unexplained forever--a claim that I never made, and that is in no way implied in my rejection of your claim.
If there is a question to which we don't know the answer, your unsubstantiated answer does not win by default until someone disproves it. If we don't know the answer, we don't know the answer, and it's up to you to demonstrate that you do know the answer if you think that you do.
> Science has yet to fail in giving us further explanations for things. So, unless you want to deny the effectiveness of science you have no case.
Science constantly fails to give us answers? Like, exactly everything we don't know, which is, like, a lot? The effectiveness of science is very obviously limited, given how much we would want to know, but don't. That doesn't mean it's ineffective, it just as obviously is not, but there is no contradiction between recognizing that it is limited, and possibly fundamentally so (in the sense that some things might stay unexplained forever), and recognizing that it is effective.
> If a wall was painted completely blue and then someone added a spec of orange somewhere, then wall as a whole would not be blue.
Yeah, sure. The problem is when you then build an argument on that and then use that argument to conclude something about almost completely blue walls.