[part 1 ... again!]

> Why define 'incompatible with science' this way over utilizing some form of the top definitions of 'incompatible' resulting from a quick google search (e.g. "so opposed in character as to be incapable of existing along with science", "incapable of association or harmonious coexistence with science", "that which cannot coexist or be conjoined with science", "not able to exist or work with science")?

All of those say pretty much the same, in so far as they are applicable to the situation?

> Also, are you saying that it is the "making" of the claim that is incompatible when using non-demonstrated methodology, or is it the claim itself that is incompatible?

Science is a "method for making/justifying claims", and as such, it is non-scientific methods that are (potentially) incompatible with it? Plus, obviously, you can justify claims that do have scientific justification via non-scientific methods (like, I dunno, "masses attract because I think 5 is a lucky number")--that justification is obviously non-scientific, but that doesn't change the fact that the claim that masses do attract does have scientific justification.

> How can a methodology be demonstrated to be reasonably reliable for determining the truth of a claim?

Using science.

> Why are formal claims excluded from being considered incompatible with science?

Because science is concerned with reality, and determining what is true about reality, and formal claims don't say anything about reality.

> Is it possible that a claim could be really true even though the methodology used to arrive at that claim has not been demonstrated to be reasonably reliable for determining the truth of such a claim?

Yes, obviously?

> What is the definition of science/"the scientific method"?

As far as this discussion is concerned, it's the idea that methods you use to justify claims about reality should be demonstrably reliable.

> Regarding the second part of your description of numbers, to what does "measuring or counting" refer (i.e. what is being measured or counted)?

Distances (in spaces) and cardinality (of sets).

> When we distinguish a rock from a picture of a rock, from the word "rock", and from the concept of a rock, we are noting that the rock exists apart from a picture of it, a word of it, and a concept of it. Now that's not difficult to accept because a rock is physical. So when I ask if we can do the same of mathematics, I am seeking an acknowledgement of the non-physical and a reason why if we can, and for a reason why if we cannot.

If we can what? Distinguish the word 'mathematics' from the concept of mathematics?

> Are you saying "not concerned with physical representation" is equivalent to "not material", as in "a movie is not material because it is not concerned with physical representation but is an instance of an abstraction which we have no evidence for or against being physical"?

No, it's not an equivalence, it is just one of the definitions of "not material". It is generally accepted usage to call concepts that are not concerned with material/physical aspects "not material". But there is also the usage of "something possessing attributes otherwise associated with the material, but also not being material", such as "an immaterial horse" or something along those lines. Those two definitions are not interchangeable.

> So all mathematicians agree that they don't intend to make statements about reality when they declare axioms?

No, just as not all humans agree that the earth is a sphere.

> What your definition of 'axiom'? The most readily available definitions I can find either roughly state that an axiom is "a self-evident truth", "a statement widely accepted as true", or "an assumption without proof for the sake of studying the consequences that follow". None of these indicate that axioms aren't intended to be statements about reality.

Which might be because they are not all talking exclusively about mathematics?

But really, I don't get what the point of all this even is. You can distinguish claims in formal systems and claims about reality, right? And you can distinguish statements declared as assumptions from statements declared as derived or observed truth, right? So, who cares whether all mathematicians agree on the names for these things (no, they don't, if only because mathematicians speak differen languages)? I have pointed out that those categories can be distinguished, and that showing something for one of those categories does not demonstrate truth for the other categories. So, how about we get back to the substance of that, instead of trying to figure out what people call it?

> So would I be correct in stating that you could just be defining 'axiom' however you want, and the statement "axioms are not claims about reality" simply conforms with your definition?

Well, yes and no. No, in that I used the definition that is commonly accepted, and didn't mean to introduce any unusual definition. But also yes, in the sense that in so far as we are not primarily talking about what the generally accepted definitions are or what other people mean, it doesn't make much sense to do anything other than to go with "well, ok, that's what they mean when they use the word", as long as it is being used consistently.

> So am I correct in that, given the Peano axioms, it is not obvious to you that '1 < 2' has a referent, because the Peano axioms may or may not be a purely artificial construct of humans.

No. It is not obvious that there is such a thing as "given the Peano axioms", absent humans (or possibly more generalized evaluators). You constantly want to make this assumption, but that assumption really is just the assumption of the existence of the immaterial, and as such it's not a useful basis for demonstrating the existence of the immaterial, because every false claim could be assumed to be true in order to demonstrate that it is true, and thus assuming a claim in order to demonstrate that it is true is not a reliable method for determining the truth of a claim.

> So you acknowledge that you don't know whether non-physical things exist?

That depends on the definition of "exists". As far as the most commonly used definition is concerned, the non-physical does not exist, because existence by definition implies physicality (when a normal person says "there exists no grocery store in this street", they are not excluding the possibility that a non-physical grocery store exists in the street). For a more general definition that you might find to be used by philosophers, I don't know.

> Because all of reality as we experience it can be intelligibly explained by science

Hu? Do you mean by that that there is nothing that's left unexplained by science? Or are you saying that whatever science doesn't have an explanation for yet, is (by definition) not part of reality? Or what?

> and since changes in state can be observed by science then there are intelligible explanations for those changes.

Hu?! I mean, yeah, we can observe changes in state, I guess, but then we also can observe changes in state that we don't have explanations for, can't we? So ... no?

> If in reality there were some changes of state that could not be intelligibly explained, then reality taken as a whole wouldn't be intelligible.

Well ... maybe? It smells like a setup for equivocation ...

> If reality as a whole was unintelligible, then it couldn't be fully and intelligibly asserted that "reality is not entirely intelligible". Such a statement is self defeating.

Well ... yeah? The point being? Has anyone made the claim that reality is not entirely intelligible?

> The electric field is physical, we can observe its effects and things that indicate it has been affected. The potential(s) for the electron field being in a different state is/are not observable via effects or things that indicate it has been affected.

Oh, so your claim is that the potential for the electric field to change state is something distinct from the electron? And analogously for the potential for the universe to change state?

Could you explain then what there is to this potential to change the state of the electric field beyond what is described by the theory of the electron and other charged particles (I used only "electron" before for simplicity, but I guess some more precision is appropriate here)?

As far as I can tell, "the potential to change" generally is simply a way to describe certain attributes of a thing, and you haven't demonstrated that that isn't all that's going on here.

When someone says "this lamp has the potential to change color", there isn't a thing in addition to the lamp that we call "the potential to change color", neither a physical nor a non-physical one, it's simply a way of expressing the fact that it has colored light filters built in, or uses different-colored LEDs, and that there are controls or something to influence in which proportions those are used to produce light. If you think there is something like "the essence of the potential to change color" that has some sort of existence beyond what is described by the mechanism of the lamp, that would be for you to demonstrate.

> "If you cared to attain maximum wealth, and you believed that maximum wealth could be attained through shooting yourself in the head, then shooting yourself in the head would be the way to go"--was an assertion that a person ought to follow their beliefs.

So, you think they ought to follow their belief even though they demonstrably (I think you would agree?) will achieve the exact opposite of their stated goal by doing so?