> To be incompatible with science means to be making factual (as in real/non-formal) claims based on methodology that is not demonstrated to be reasonably reliable for determining the truth of such a claim.

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")?

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?

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

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

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?

> The definition of what science/"the scientific method" is?

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

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> Well, then it's not. ... I obviously kindof agree that the first part is pretty ambiguous, which presumably is why I added the second part

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

> Well, obviously not the same, as there is no such thing as "a picture of a mathematics"?! But obviously, we can, say, distinguish mathematics from the word "mathematics". So, I guess I am not really sure what the question is?

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.

> I agree that instances of abstractions that are not concerned with physical representation do exist in reality. That does not mean that I agree that instances of those same abstractions exist without physical representation, for lack of evidence of that being a thing.

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"?

> Because that is how "axiom" is defined (by mathematicians)

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

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.

> Well, yes, in the sense that anyone can obviously redefine words however they want.

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?

> But the moment the other party tells you they don't agree with your assumption, the equivalent of "but I think you should agree" is not really going to help anyone.

I haven't only told you "I think you should agree" though. I also asked why you disagree...

> If the Peano axioms are a purely artificial construct of humans, then "'1 < 2' absent humans" has no referent ...
... Because as far as I can see ['1 < 2'] is a statement in a formal system, which has no meaning without that formal system.

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.

> How would I know? I mean, I have only ever experienced material (in the farthest sense) copies and originals, so I see no reason to think that non-material ones are a thing. But who knows ...

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

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> How do you know that? Couldn't there be mechanisms by which something changes state that can not be determined by science? How do you know there aren't?

Because all of reality as we experience it can be intelligibly explained by science, and since changes in state can be observed by science then there are intelligible explanations for those changes.

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. 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.

> And in the particular case: How did you determine that the mechanism by which the universe changes state can not be determined by science?

I don't know what this is asking.

> So, electrons are non-physical after all, then?

What have I said that lead you to this conclusion?

> I assume you agree that the electric field is physical? The potential to change the electric field is not observable, but it exists, therefore, it is non-physical? (I am paraphrasing, obviously, but I think that is roughly your argument applied to electrons?)

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.

> Exactly. So, why would you expect the same reason to be convincing when you use it?

What line of reasoning?! My agreement to the modification you gave to my scenario--"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.

Unless there is some ambiguity somewhere, "would be the way to go" doesn't bear much resemblance to "you will indeed attain".

You modified "then shooting yourself in the head would be the way to go" to say "you then shoot yourself in the head, [then] you will indeed attain maximum wealth", which seemed to become a line of questioning to determine whether I thought a person will get what is in line with their beliefs. It should be fairly obvious that I don't think that is the case, unless one's beliefs are in line with what is true. Have I established what I think is true: no.

I acknowledged that I had not substantiated a convincing argument to care about morality (given my definition) when you highlighted that "'care' and 'listen to' are two different questions as well".